FlexEDU Question Bank Explorer

Grade 9

Identify number patterns with a common difference.

Q2: What is the common difference of the arithmetic progression 3, 7, 11, 15, ...?

Explanation: The common difference is found by subtracting any term from its succeeding term. For example, 7 - 3 = 4, or 11 - 7 = 4.

Q3: Calculate the common difference of the arithmetic progression 5, 12, 19, 26, ...

Explanation: The common difference is found by taking any term and subtracting the preceding term. For example, 12 - 5 = 7, or 19 - 12 = 7.

Q4: Find the common difference of the arithmetic sequence 10, 18, 26, 34, ...

Explanation: To find the common difference, subtract any term from the term that follows it. For instance, 18 - 10 = 8, or 26 - 18 = 8.

Q5: What is the common difference of the arithmetic progression 2, 13, 24, 35, ...?

Explanation: The common difference is found by subtracting the preceding term from any given term. For example, 13 - 2 = 11, or 24 - 13 = 11.

Q6: Determine the common difference for the arithmetic progression 20, 23, 26, 29, ...

Explanation: The common difference is the constant difference between consecutive terms. In this case, 23 - 20 = 3, and 26 - 23 = 3.

Q7: What is the common difference of the sequence 1, 6, 11, 16, ...?

Explanation: The common difference is obtained by subtracting any term from the term immediately following it. Here, 6 - 1 = 5, and 11 - 6 = 5.

Q8: Consider the arithmetic progression: 5, 12, 19, ... What are the next three terms of this sequence?

25, 31, 37

26, 33, 40

20, 21, 22

24, 30, 36

Explanation: To find the next terms, first determine the common difference (d). Here, d = 12 - 5 = 7. So, each subsequent term is found by adding 7 to the previous term. The terms are 19 + 7 = 26, 26 + 7 = 33, and 33 + 7 = 40. Thus, the next three terms are 26, 33, 40.

Q9: What is the common difference of the arithmetic progression: 30, 24, 18, 12, ...?

-6

Explanation: The common difference (d) of an arithmetic progression is found by subtracting any term from its succeeding term. For example, d = 24 - 30 = -6. We can verify this with other terms: 18 - 24 = -6, and 12 - 18 = -6.

Q10: Is the sequence 4, 8, 12, 16, ... an arithmetic progression? If so, what is its common difference?

No, it is not an arithmetic progression.

Yes, common difference is 8.

Yes, common difference is 4.

Yes, common difference is 2.

Explanation: An arithmetic progression has a constant difference between consecutive terms. In this sequence, 8 - 4 = 4, 12 - 8 = 4, and 16 - 12 = 4. Since the difference is constant (4), it is an arithmetic progression with a common difference of 4.

Q11: Given the arithmetic progression: 2.5, 3.0, 3.5, ... What are the next three terms of this sequence?

3.6, 3.7, 3.8

4.0, 4.5, 5.0

3.75, 4.0, 4.25

4.0, 5.0, 6.0

Explanation: First, find the common difference (d). Here, d = 3.0 - 2.5 = 0.5. To find the next terms, add 0.5 to the last given term repeatedly. So, 3.5 + 0.5 = 4.0, 4.0 + 0.5 = 4.5, and 4.5 + 0.5 = 5.0. The next three terms are 4.0, 4.5, 5.0.

Q12: Which of the following sequences is NOT an arithmetic progression?

10, 15, 20, 25

2, 4, 6, 8

1, 4, 9, 16

50, 40, 30, 20

Explanation: An arithmetic progression has a constant common difference between consecutive terms. Let's check each option: A) 10, 15, 20, 25 (Common difference = 5, so it's an AP) B) 2, 4, 6, 8 (Common difference = 2, so it's an AP) C) 1, 4, 9, 16 (Differences are 4-1=3, 9-4=5, 16-9=7. The difference is not constant, so it's not an AP. This is a sequence of squares: 1², 2², 3², 4²) D) 50, 40, 30, 20 (Common difference = -10, so it's an AP) Therefore, 1, 4, 9, 16 is not an arithmetic progression.

Q13: Consider the arithmetic progression: 13, __, 23, 28. What is the missing term?

Explanation: In an arithmetic progression, the common difference (d) is constant. We can find 'd' using the known consecutive terms: d = 28 - 23 = 5. Now, to find the missing term, add the common difference to the first term: 13 + 5 = 18. Alternatively, subtract the common difference from 23: 23 - 5 = 18. So the missing term is 18.

Q14: In an arithmetic progression, the 4th term is 15 and the 9th term is 35. What is the common difference?

Explanation: Let the 4th term be a_4 and the 9th term be a_9. We know that a_n = a + (n-1)d. So, a_4 = a + 3d = 15 and a_9 = a + 8d = 35. Subtracting the 4th term from the 9th term: (a + 8d) - (a + 3d) = 35 - 15. This simplifies to 5d = 20. Therefore, the common difference d = 20 / 5 = 4.

Q15: The 3rd term of an arithmetic progression is 25 and its 8th term is 10. Find the common difference.

-5

-3

Explanation: Let the 3rd term be a_3 and the 8th term be a_8. We have a_3 = a + 2d = 25 and a_8 = a + 7d = 10. Subtracting the 3rd term from the 8th term: (a + 7d) - (a + 2d) = 10 - 25. This simplifies to 5d = -15. Therefore, the common difference d = -15 / 5 = -3.

Q16: The annual maintenance cost of a machine increases by a fixed amount each year. In the 2nd year, the cost was Rs. 12,000, and in the 6th year, it was Rs. 20,000. What is the annual increase in maintenance cost?

Rs. 4,000

Rs. 8,000

Rs. 2,000

Rs. 1,600

Explanation: This is an arithmetic progression where the annual increase is the common difference (d). The cost in the 2nd year is a_2 = Rs. 12,000, and in the 6th year is a_6 = Rs. 20,000. The difference in terms is a_6 - a_2 = (a + 5d) - (a + d) = 4d. So, 4d = 20,000 - 12,000 = 8,000. Therefore, d = 8,000 / 4 = Rs. 2,000.

Q17: A plant's height follows an arithmetic progression. At the end of the 3rd week, its height was 15 cm, and at the end of the 7th week, it was 27 cm. What is the weekly growth rate of the plant?

4 cm

12 cm

3 cm

6 cm

Explanation: The weekly growth rate is the common difference (d). The height at the end of the 3rd week is a_3 = 15 cm, and at the end of the 7th week is a_7 = 27 cm. The difference in heights is a_7 - a_3 = (a + 6d) - (a + 2d) = 4d. So, 4d = 27 - 15 = 12 cm. Therefore, d = 12 / 4 = 3 cm.

Q18: In an arithmetic progression, the 5th term is 22.5 and the 10th term is 40. What is the common difference?

17.5

2.5

3.5

Explanation: Let the 5th term be a_5 and the 10th term be a_10. We have a_5 = a + 4d = 22.5 and a_10 = a + 9d = 40. Subtracting the 5th term from the 10th term: (a + 9d) - (a + 4d) = 40 - 22.5. This simplifies to 5d = 17.5. Therefore, the common difference d = 17.5 / 5 = 3.5.

Q19: A person saves money in an arithmetic progression. On the 4th day, they saved Rs. 300, and on the 9th day, they saved Rs. 550. What is the daily increase in their savings?

Rs. 250

Rs. 62.50

Rs. 50

Rs. 100

Explanation: The daily increase in savings is the common difference (d). The savings on the 4th day is a_4 = Rs. 300, and on the 9th day is a_9 = Rs. 550. The difference in savings is a_9 - a_4 = (a + 8d) - (a + 3d) = 5d. So, 5d = 550 - 300 = 250. Therefore, d = 250 / 5 = Rs. 50.

Generate a sequence from a given general term (Tn).

Q20: What are the first three terms of the sequence given by the general term Tn = 2n + 3?

3, 5, 7

5, 7, 9

2, 4, 6

5, 6, 7

Explanation: To find the first three terms, substitute n=1, n=2, and n=3 into the general term Tn = 2n + 3. For n=1: T1 = 2(1) + 3 = 2 + 3 = 5 For n=2: T2 = 2(2) + 3 = 4 + 3 = 7 For n=3: T3 = 2(3) + 3 = 6 + 3 = 9 Thus, the first three terms are 5, 7, 9.

Q21: Calculate the first three terms of the sequence given by Tn = 3n - 1.

-1, 2, 5

3, 6, 9

2, 5, 8

4, 7, 10

Explanation: To find the first three terms, substitute n=1, n=2, and n=3 into the general term Tn = 3n - 1. For n=1: T1 = 3(1) - 1 = 3 - 1 = 2 For n=2: T2 = 3(2) - 1 = 6 - 1 = 5 For n=3: T3 = 3(3) - 1 = 9 - 1 = 8 Thus, the first three terms are 2, 5, 8.

Q22: What are the first three terms of the sequence whose general term is Tn = n + 5?

1, 2, 3

5, 6, 7

6, 7, 8

6, 11, 16

Explanation: To find the first three terms, substitute n=1, n=2, and n=3 into the general term Tn = n + 5. For n=1: T1 = 1 + 5 = 6 For n=2: T2 = 2 + 5 = 7 For n=3: T3 = 3 + 5 = 8 Thus, the first three terms are 6, 7, 8.

Q23: List the first three terms of the sequence if the general term is Tn = 4n.

0, 4, 8

1, 2, 3

1, 5, 9

4, 8, 12

Explanation: To find the first three terms, substitute n=1, n=2, and n=3 into the general term Tn = 4n. For n=1: T1 = 4(1) = 4 For n=2: T2 = 4(2) = 8 For n=3: T3 = 4(3) = 12 Thus, the first three terms are 4, 8, 12.

Q24: What are the first three terms of the sequence defined by Tn = 5n + 1?

1, 6, 11

5, 10, 15

6, 11, 16

6, 7, 8

Explanation: To find the first three terms, substitute n=1, n=2, and n=3 into the general term Tn = 5n + 1. For n=1: T1 = 5(1) + 1 = 5 + 1 = 6 For n=2: T2 = 5(2) + 1 = 10 + 1 = 11 For n=3: T3 = 5(3) + 1 = 15 + 1 = 16 Thus, the first three terms are 6, 11, 16.

Q25: Find the first three terms of the sequence generated by the general term Tn = 2n - 5.

-5, -3, -1

2, 4, 6

-3, -1, 1

7, 9, 11

Explanation: To find the first three terms, substitute n=1, n=2, and n=3 into the general term Tn = 2n - 5. For n=1: T1 = 2(1) - 5 = 2 - 5 = -3 For n=2: T2 = 2(2) - 5 = 4 - 5 = -1 For n=3: T3 = 2(3) - 5 = 6 - 5 = 1 Thus, the first three terms are -3, -1, 1.

Q26: If the general term of a sequence is T_n = 3n + 2, what are the first four terms and the 8th term of the sequence?

5, 7, 9, 11 and 20

3, 6, 9, 12 and 24

5, 8, 11, 14 and 26

5, 8, 11, 14 and 28

Explanation: To find the terms, substitute n=1, 2, 3, 4 for the first four terms, and n=8 for the 8th term. T1 = 3(1) + 2 = 5, T2 = 3(2) + 2 = 8, T3 = 3(3) + 2 = 11, T4 = 3(4) + 2 = 14. T8 = 3(8) + 2 = 24 + 2 = 26. So, the first four terms are 5, 8, 11, 14 and the 8th term is 26.

Q27: For a sequence with the general term T_n = 5 - 2n, what are the first four terms and the 10th term?

3, 7, 11, 15 and 25

3, 1, -1, -3 and -13

3, 1, -1, -3 and -15

3, 1, -1, -3 and 15

Explanation: Substitute n=1, 2, 3, 4 for the first four terms and n=10 for the 10th term. T1 = 5 - 2(1) = 3, T2 = 5 - 2(2) = 1, T3 = 5 - 2(3) = -1, T4 = 5 - 2(4) = -3. T10 = 5 - 2(10) = 5 - 20 = -15. The first four terms are 3, 1, -1, -3 and the 10th term is -15.

Q28: Given the general term T_n = n² + 1 for a sequence, find its first four terms and the 7th term.

1, 4, 9, 16 and 49

2, 3, 4, 5 and 8

2, 5, 10, 17 and 48

2, 5, 10, 17 and 50

Explanation: Substitute n=1, 2, 3, 4 for the first four terms and n=7 for the 7th term. T1 = 1² + 1 = 2, T2 = 2² + 1 = 5, T3 = 3² + 1 = 10, T4 = 4² + 1 = 17. T7 = 7² + 1 = 49 + 1 = 50. The first four terms are 2, 5, 10, 17 and the 7th term is 50.

Q29: What are the first four terms and the 8th term of a sequence whose general term is T_n = 2n² - 3?

1, 8, 18, 32 and 128

-2, 1, 6, 13 and 61

-1, 5, 15, 29 and 125

-1, 5, 15, 29 and 123

Explanation: Substitute n=1, 2, 3, 4 for the first four terms and n=8 for the 8th term. T1 = 2(1)² - 3 = -1, T2 = 2(2)² - 3 = 5, T3 = 2(3)² - 3 = 15, T4 = 2(4)² - 3 = 29. T8 = 2(8)² - 3 = 2(64) - 3 = 128 - 3 = 125. The first four terms are -1, 5, 15, 29 and the 8th term is 125.

Q30: For a sequence with the general term T_n = n² - n + 2, what are the first four terms and the 6th term?

2, 6, 12, 20 and 44

0, 2, 6, 12 and 30

2, 4, 8, 14 and 30

2, 4, 8, 14 and 32

Explanation: Substitute n=1, 2, 3, 4 for the first four terms and n=6 for the 6th term. T1 = 1² - 1 + 2 = 2, T2 = 2² - 2 + 2 = 4, T3 = 3² - 3 + 2 = 8, T4 = 4² - 4 + 2 = 14. T6 = 6² - 6 + 2 = 36 - 6 + 2 = 32. The first four terms are 2, 4, 8, 14 and the 6th term is 32.

Q31: If the general term of a sequence is T_n = -n² + 4n - 1, what are the first four terms and the 5th term?

4, 11, 20, 31 and 44

0, 4, 8, 12 and 15

2, 3, 2, -1 and -4

2, 3, 2, -1 and -6

Explanation: Substitute n=1, 2, 3, 4 for the first four terms and n=5 for the 5th term. T1 = -(1)² + 4(1) - 1 = -1 + 4 - 1 = 2. T2 = -(2)² + 4(2) - 1 = -4 + 8 - 1 = 3. T3 = -(3)² + 4(3) - 1 = -9 + 12 - 1 = 2. T4 = -(4)² + 4(4) - 1 = -16 + 16 - 1 = -1. T5 = -(5)² + 4(5) - 1 = -25 + 20 - 1 = -6. The first four terms are 2, 3, 2, -1 and the 5th term is -6.

Q32: The general term of a number sequence is given by Tn = 3n + 1. Is 40 a term in this sequence? Justify your answer.

No, because n = 13.33...

Yes, because n = 13, which is a positive integer.

No, because n is not an even number.

Yes, because it's close to a multiple of 3.

Explanation: To check if 40 is a term, we set Tn = 40. So, 3n + 1 = 40. Subtracting 1 from both sides gives 3n = 39. Dividing by 3 gives n = 13. Since n = 13 is a positive integer, 40 is the 13th term in the sequence.

Q33: The general term of a sequence is Tn = 2n - 5. Is 10 a term in this sequence? Justify your answer.

Yes, because 10 is an even number.

No, because n = 7.5, which is not a positive integer.

Yes, because n = 7.

No, because the first term (n=1) is -3.

Explanation: To check if 10 is a term, we set Tn = 10. So, 2n - 5 = 10. Adding 5 to both sides gives 2n = 15. Dividing by 2 gives n = 7.5. Since n = 7.5 is not a positive integer, 10 is not a term in the sequence.

Q34: The general term of a sequence is Tn = n² + 2. Is 51 a term in this sequence? Justify your answer.

No, because 51 is an odd number.

Yes, because n = 7, which is a positive integer.

No, because n² = 53.

Yes, because the terms are increasing.

Explanation: To check if 51 is a term, we set Tn = 51. So, n² + 2 = 51. Subtracting 2 from both sides gives n² = 49. Taking the square root of both sides gives n = 7 (since n must be positive for a term number). Since n = 7 is a positive integer, 51 is the 7th term in the sequence.

Q35: Consider a sequence with the general term Tn = 4n - 3. Is 25 a term in this sequence? Justify your answer.

No, because 25 is not a multiple of 4.

Yes, because n = 7, which is a positive integer.

No, because n = 7.25.

Yes, because it ends in 5.

Explanation: To check if 25 is a term, we set Tn = 25. So, 4n - 3 = 25. Adding 3 to both sides gives 4n = 28. Dividing by 4 gives n = 7. Since n = 7 is a positive integer, 25 is the 7th term in the sequence.

Q36: The general term of a sequence is Tn = 5n + 2. Is 33 a term in this sequence? Justify your answer.

Yes, because n = 6.

No, because n = 6.2, which is not a positive integer.

No, because 33 is an odd number.

Yes, because it is close to 32.

Explanation: To check if 33 is a term, we set Tn = 33. So, 5n + 2 = 33. Subtracting 2 from both sides gives 5n = 31. Dividing by 5 gives n = 6.2. Since n = 6.2 is not a positive integer, 33 is not a term in the sequence.

Q37: For a sequence with the general term Tn = (n/2) + 1, is 6 a term in this sequence? Justify your answer.

No, because n = 5.

Yes, because n = 10, which is a positive integer.

No, because n is not an odd number.

Yes, because the sequence terms are always integers.

Explanation: To check if 6 is a term, we set Tn = 6. So, (n/2) + 1 = 6. Subtracting 1 from both sides gives n/2 = 5. Multiplying by 2 gives n = 10. Since n = 10 is a positive integer, 6 is the 10th term in the sequence.

Find the general term (Tn) for a given arithmetic progression.

Q38: What is the general term (Tn) of the arithmetic progression 3, 7, 11, 15, ...?

4n + 3

4n - 1

7 - 4n

Explanation: The first term (a) is 3. The common difference (d) is 7 - 3 = 4. Using the formula Tn = a + (n-1)d, we substitute a=3 and d=4: Tn = 3 + (n-1)4 = 3 + 4n - 4 = 4n - 1.

Q39: Find the general term (Tn) for the arithmetic progression 10, 7, 4, 1, ...

10 - 3n

13 - 3n

7 - 3n

10 + 3n

Explanation: The first term (a) is 10. The common difference (d) is 7 - 10 = -3. Using the formula Tn = a + (n-1)d, we substitute a=10 and d=-3: Tn = 10 + (n-1)(-3) = 10 - 3n + 3 = 13 - 3n.

Q40: What is the general term (Tn) of the arithmetic progression -5, -2, 1, 4, ...?

-5 + 3n

3n - 2

3n - 8

5 + 3n

Explanation: The first term (a) is -5. The common difference (d) is -2 - (-5) = -2 + 5 = 3. Using the formula Tn = a + (n-1)d, we substitute a=-5 and d=3: Tn = -5 + (n-1)3 = -5 + 3n - 3 = 3n - 8.

Q41: Find the general term (Tn) for the arithmetic progression 2, 8, 14, 20, ...

6n + 2

6n - 4

8n - 6

6n - 2

Explanation: The first term (a) is 2. The common difference (d) is 8 - 2 = 6. Using the formula Tn = a + (n-1)d, we substitute a=2 and d=6: Tn = 2 + (n-1)6 = 2 + 6n - 6 = 6n - 4.

Q42: What is the general term (Tn) of the arithmetic progression 15, 19, 23, 27, ...?

4n + 15

4n + 11

4n + 19

11 - 4n

Explanation: The first term (a) is 15. The common difference (d) is 19 - 15 = 4. Using the formula Tn = a + (n-1)d, we substitute a=15 and d=4: Tn = 15 + (n-1)4 = 15 + 4n - 4 = 4n + 11.

Q43: Find the general term (Tn) for the arithmetic progression -10, -7, -4, -1, ...

3n - 10

3n - 13

3n + 13

3n - 7

Explanation: The first term (a) is -10. The common difference (d) is -7 - (-10) = -7 + 10 = 3. Using the formula Tn = a + (n-1)d, we substitute a=-10 and d=3: Tn = -10 + (n-1)3 = -10 + 3n - 3 = 3n - 13.

Q44: What is the general term (Tn) of the arithmetic progression 1, 5, 9, 13, ...?

4n + 1

4n - 1

4n - 3

3n + 1

Explanation: The first term (a) is 1. The common difference (d) is 5 - 1 = 4. Using the formula Tn = a + (n-1)d, we substitute a=1 and d=4: Tn = 1 + (n-1)4 = 1 + 4n - 4 = 4n - 3.

Q45: What is the general term (Tn) of the arithmetic progression 3, 7, 11, 15, ...?

4n + 1

4n - 1

3n + 4

n + 2

Explanation: For the given arithmetic progression, the first term (a) is 3. The common difference (d) is 7 - 3 = 4. Substituting these values into the formula Tn = a + (n-1)d, we get Tn = 3 + (n-1)4. Expanding this expression gives Tn = 3 + 4n - 4, which simplifies to Tn = 4n - 1.

Q46: Find the general term (Tn) for the arithmetic progression 15, 12, 9, 6, ...

15 - 3n

18 - 3n

12 - 3n

3n + 12

Explanation: The first term (a) is 15. The common difference (d) is 12 - 15 = -3. Substituting these into Tn = a + (n-1)d, we get Tn = 15 + (n-1)(-3). Expanding this gives Tn = 15 - 3n + 3, which simplifies to Tn = 18 - 3n.

Q47: Determine the general term (Tn) for the arithmetic progression -5, -2, 1, 4, ...

3n - 2

3n - 8

3n + 5

-5n + 3

Explanation: The first term (a) is -5. The common difference (d) is -2 - (-5) = -2 + 5 = 3. Using Tn = a + (n-1)d, we have Tn = -5 + (n-1)3. Expanding gives Tn = -5 + 3n - 3, which simplifies to Tn = 3n - 8.

Q48: What is the general term (Tn) for the arithmetic progression -2, -6, -10, -14, ...?

-4n - 2

2 - 4n

-4n - 6

4n - 2

Explanation: The first term (a) is -2. The common difference (d) is -6 - (-2) = -6 + 2 = -4. Substituting these into Tn = a + (n-1)d, we get Tn = -2 + (n-1)(-4). Expanding this gives Tn = -2 - 4n + 4, which simplifies to Tn = 2 - 4n.

Q49: Find the general term (Tn) for the arithmetic progression 100, 95, 90, 85, ...

100 - 5n

105 - 5n

95 - 5n

5n + 100

Explanation: The first term (a) is 100. The common difference (d) is 95 - 100 = -5. Using Tn = a + (n-1)d, we get Tn = 100 + (n-1)(-5). Expanding this gives Tn = 100 - 5n + 5, which simplifies to Tn = 105 - 5n.

Q50: What is the general term (Tn) of the arithmetic progression 20, 30, 40, 50, ...?

10n + 20

10n + 10

10n - 10

20n + 10

Explanation: The first term (a) is 20. The common difference (d) is 30 - 20 = 10. Substituting these into the formula Tn = a + (n-1)d, we get Tn = 20 + (n-1)10. Expanding this expression gives Tn = 20 + 10n - 10, which simplifies to Tn = 10n + 10.

Q51: The 3rd term of an arithmetic progression is 10 and the 7th term is 26. What is the general term (Tn) of this progression?

4n + 2

4n - 2

2n + 4

Explanation: Let the first term be 'a' and the common difference be 'd'. T3 = a + 2d = 10 --- (1) T7 = a + 6d = 26 --- (2) Subtract (1) from (2): (a + 6d) - (a + 2d) = 26 - 10 => 4d = 16 => d = 4. Substitute d = 4 into (1): a + 2(4) = 10 => a + 8 = 10 => a = 2. The general term Tn = a + (n-1)d. Tn = 2 + (n-1)4 = 2 + 4n - 4 = 4n - 2.

Q52: The 4th term of an arithmetic progression is 15 and the 9th term is 35. What is the position (n) of the term 67 in this progression?

Explanation: Let the first term be 'a' and the common difference be 'd'. T4 = a + 3d = 15 --- (1) T9 = a + 8d = 35 --- (2) Subtract (1) from (2): (a + 8d) - (a + 3d) = 35 - 15 => 5d = 20 => d = 4. Substitute d = 4 into (1): a + 3(4) = 15 => a + 12 = 15 => a = 3. The general term Tn = a + (n-1)d. Tn = 3 + (n-1)4 = 3 + 4n - 4 = 4n - 1. To find the position of the term 67, set Tn = 67: 4n - 1 = 67 => 4n = 68 => n = 17.

Q53: An arithmetic progression begins with the term 5, and the common difference between consecutive terms is 3. What is the general term (Tn) for this progression?

3n + 5

3n + 2

5n + 3

3n - 2

Explanation: The first term (a) = 5. The common difference (d) = 3. The general term Tn = a + (n-1)d. Tn = 5 + (n-1)3 = 5 + 3n - 3 = 3n + 2.

Q54: In an auditorium, the 3rd row has 12 seats and the 7th row has 24 seats. If the number of seats in each row forms an arithmetic progression, what is the general term (Tn) for the number of seats in the nth row?

3n + 6

3n + 3

3n - 3

Explanation: Let the number of seats in the nth row be Tn. T3 = a + 2d = 12 --- (1) T7 = a + 6d = 24 --- (2) Subtract (1) from (2): (a + 6d) - (a + 2d) = 24 - 12 => 4d = 12 => d = 3. Substitute d = 3 into (1): a + 2(3) = 12 => a + 6 = 12 => a = 6. The general term Tn = a + (n-1)d. Tn = 6 + (n-1)3 = 6 + 3n - 3 = 3n + 3.

Q55: The 5th term of an arithmetic progression is 23 and the 10th term is 48. What is the 15th term of this progression?

Explanation: Let the first term be 'a' and the common difference be 'd'. T5 = a + 4d = 23 --- (1) T10 = a + 9d = 48 --- (2) Subtract (1) from (2): (a + 9d) - (a + 4d) = 48 - 23 => 5d = 25 => d = 5. Substitute d = 5 into (1): a + 4(5) = 23 => a + 20 = 23 => a = 3. The general term Tn = a + (n-1)d. Tn = 3 + (n-1)5 = 3 + 5n - 5 = 5n - 2. To find the 15th term, substitute n = 15 into Tn: T15 = 5(15) - 2 = 75 - 2 = 73.

Q56: An arithmetic progression has its 2nd term as 8 and its 6th term as 20. Which term of this progression is 35?

10th term

11th term

12th term

9th term

Explanation: Let the first term be 'a' and the common difference be 'd'. T2 = a + d = 8 --- (1) T6 = a + 5d = 20 --- (2) Subtract (1) from (2): (a + 5d) - (a + d) = 20 - 8 => 4d = 12 => d = 3. Substitute d = 3 into (1): a + 3 = 8 => a = 5. The general term Tn = a + (n-1)d. Tn = 5 + (n-1)3 = 5 + 3n - 3 = 3n + 2. To find which term is 35, set Tn = 35: 3n + 2 = 35 => 3n = 33 => n = 11.

Solve word problems involving number patterns.

Q57: A plant's height was measured at the end of the first two weeks as 10 cm and 13 cm respectively. If it continues to grow by the same amount each week, what will its height be at the end of the third and fourth weeks?

13 cm, 16 cm

16 cm, 19 cm

19 cm, 22 cm

10 cm, 13 cm

Explanation: The sequence of heights is 10 cm (Week 1), 13 cm (Week 2). The common difference is 13 - 10 = 3 cm. So, the height at the end of the third week will be 13 + 3 = 16 cm, and at the end of the fourth week will be 16 + 3 = 19 cm.

Q58: A car's fuel tank had 45 litres of fuel. If it consumes 5 litres of fuel every hour, how many litres will be left after 2 hours and 3 hours of driving?

40 litres, 35 litres

35 litres, 30 litres

30 litres, 25 litres

45 litres, 40 litres

Explanation: The initial fuel is 45 litres. After 1 hour, 45 - 5 = 40 litres. After 2 hours, 40 - 5 = 35 litres. After 3 hours, 35 - 5 = 30 litres. The common difference is -5 litres per hour. So, after 2 hours, 35 litres will be left, and after 3 hours, 30 litres will be left.

Q59: A stack of chairs has 8 chairs in the first row, 11 in the second, and 14 in the third. If this pattern continues, how many chairs will be in the fourth and fifth rows?

16, 19

17, 20

15, 18

20, 23

Explanation: The sequence of chairs is 8, 11, 14. The common difference is 11 - 8 = 3. So, the fourth row will have 14 + 3 = 17 chairs, and the fifth row will have 17 + 3 = 20 chairs.

Q60: A student saves money. In January, she saved Rs. 150. In February, Rs. 175. In March, Rs. 200. Following this pattern, how much will she save in April and May?

Rs. 200, Rs. 225

Rs. 225, Rs. 250

Rs. 250, Rs. 275

Rs. 175, Rs. 200

Explanation: The sequence of savings is Rs. 150, Rs. 175, Rs. 200. The common difference is 175 - 150 = Rs. 25. So, in April, she will save 200 + 25 = Rs. 225, and in May, she will save 225 + 25 = Rs. 250.

Q61: A library adds new books to its collection. In the first month, they added 20 books. In the second month, 24 books. In the third month, 28 books. How many books will be added in the fourth and fifth months if the pattern continues?

30, 34

32, 36

28, 32

36, 40

Explanation: The sequence of books added is 20, 24, 28. The common difference is 24 - 20 = 4 books. So, in the fourth month, 28 + 4 = 32 books will be added, and in the fifth month, 32 + 4 = 36 books will be added.

Q62: A painter is mixing colors. He starts with 50ml of blue paint. For each subsequent batch, he adds 10ml less blue paint than the previous batch. If the second batch had 40ml and the third had 30ml, how much blue paint will be in the fourth and fifth batches?

30ml, 20ml

20ml, 10ml

10ml, 0ml

40ml, 30ml

Explanation: The sequence of blue paint amounts is 50ml, 40ml, 30ml. The common difference is 40 - 50 = -10ml. So, the fourth batch will have 30 - 10 = 20ml, and the fifth batch will have 20 - 10 = 10ml.

Q63: A runner increases her training distance by 0.5 km each day. If she ran 3 km on Monday and 3.5 km on Tuesday, how far will she run on Wednesday and Thursday?

3.5 km, 4 km

4 km, 4.5 km

4.5 km, 5 km

3 km, 3.5 km

Explanation: The sequence of distances is 3 km (Monday), 3.5 km (Tuesday). The common difference is 3.5 - 3 = 0.5 km. So, on Wednesday, she will run 3.5 + 0.5 = 4 km, and on Thursday, she will run 4 + 0.5 = 4.5 km.

Q64: A baker uses sugar for cakes. For the first cake, she used 250g. For the second, 220g. For the third, 190g. If she continues this pattern, how much sugar will she use for the fourth and fifth cakes?

190g, 160g

160g, 130g

130g, 100g

220g, 190g

Explanation: The sequence of sugar used is 250g, 220g, 190g. The common difference is 220 - 250 = -30g. So, for the fourth cake, she will use 190 - 30 = 160g, and for the fifth cake, she will use 160 - 30 = 130g.

Q65: A plant is 15 cm tall on the first day. It grows 2 cm taller each subsequent day. How tall will the plant be on the 10th day?

35 cm

33 cm

31 cm

17 cm

Explanation: This is an arithmetic progression with the first term (a) = 15 and common difference (d) = 2. To find the 10th term (T10), use the formula Tn = a + (n-1)d. So, T10 = 15 + (10-1) * 2 = 15 + 9 * 2 = 15 + 18 = 33 cm.

Q66: Nimal saves Rs. 500 in the first month. Each subsequent month, he saves Rs. 50 more than the previous month. How much will he save in the 12th month?

Rs. 1100

Rs. 1050

Rs. 1000

Rs. 550

Explanation: This is an arithmetic progression with the first term (a) = 500 and common difference (d) = 50. To find the 12th term (T12), use the formula Tn = a + (n-1)d. So, T12 = 500 + (12-1) * 50 = 500 + 11 * 50 = 500 + 550 = Rs. 1050.

Q67: An auditorium has 20 seats in the first row. Each subsequent row has 3 more seats than the row in front of it. How many seats are there in the 15th row?

65 seats

62 seats

59 seats

23 seats

Explanation: This is an arithmetic progression with the first term (a) = 20 and common difference (d) = 3. To find the 15th term (T15), use the formula Tn = a + (n-1)d. So, T15 = 20 + (15-1) * 3 = 20 + 14 * 3 = 20 + 42 = 62 seats.

Q68: A stack of bricks has 35 bricks in the bottom layer. Each layer above it has 2 fewer bricks than the layer below. How many bricks are there in the 10th layer from the bottom?

15 bricks

17 bricks

53 bricks

33 bricks

Explanation: This is an arithmetic progression with the first term (a) = 35 and common difference (d) = -2 (since it decreases). To find the 10th term (T10), use the formula Tn = a + (n-1)d. So, T10 = 35 + (10-1) * (-2) = 35 + 9 * (-2) = 35 - 18 = 17 bricks.

Q69: A factory produces 120 units on the first day. Each subsequent day, the production increases by 8 units. What will be the production on the 8th day?

184 units

176 units

168 units

128 units

Explanation: This is an arithmetic progression with the first term (a) = 120 and common difference (d) = 8. To find the 8th term (T8), use the formula Tn = a + (n-1)d. So, T8 = 120 + (8-1) * 8 = 120 + 7 * 8 = 120 + 56 = 176 units.

Q70: A cyclist covers 10 km on the first day. On each subsequent day, he covers 3 km more than the previous day. What is the distance he covers on the 10th day?

40 km

37 km

34 km

13 km

Explanation: This is an arithmetic progression with the first term (a) = 10 and common difference (d) = 3. To find the 10th term (T10), use the formula Tn = a + (n-1)d. So, T10 = 10 + (10-1) * 3 = 10 + 9 * 3 = 10 + 27 = 37 km.

Q71: A gardener plants 8 trees on the first day. On each subsequent day, he plants 4 more trees than the previous day. How many trees will he plant on the 12th day?

56 trees

52 trees

48 trees

12 trees

Explanation: This is an arithmetic progression with the first term (a) = 8 and common difference (d) = 4. To find the 12th term (T12), use the formula Tn = a + (n-1)d. So, T12 = 8 + (12-1) * 4 = 8 + 11 * 4 = 8 + 44 = 52 trees.

Q72: A gardener plants a sapling that is 15 cm tall. It grows 3 cm every week. After how many weeks will the sapling be 60 cm tall?

10 weeks

15 weeks

20 weeks

25 weeks

Explanation: The initial height of the sapling is 15 cm. It grows by 3 cm each week. Let 'n' be the number of weeks. The height after 'n' weeks can be represented by the arithmetic progression: H_n = 15 + 3n. We want to find 'n' when H_n = 60 cm. So, 15 + 3n = 60. Subtract 15 from both sides: 3n = 45. Divide by 3: n = 15. Thus, it will take 15 weeks for the sapling to reach 60 cm.

Q73: A student starts saving with Rs. 200 and adds Rs. 50 to their savings at the end of each subsequent month. In which month will their total savings first exceed Rs. 1000?

15th month

16th month

17th month

18th month

Explanation: The student starts with Rs. 200. They add Rs. 50 each month. Let 'n' be the number of months. The total savings after 'n' months will be 200 + 50n. We want to find 'n' when the savings first exceed Rs. 1000. So, 200 + 50n > 1000. Subtract 200 from both sides: 50n > 800. Divide by 50: n > 16. Since 'n' must be an integer, the smallest integer greater than 16 is 17. Therefore, in the 17th month, their savings will first exceed Rs. 1000.

Q74: Shop A sells a certain type of fabric for Rs. 500 per meter, with an initial one-time discount of Rs. 100 for any purchase. Shop B sells the same fabric for Rs. 450 per meter, with no initial discount. At what length of fabric (in meters) will the total cost be the same at both shops?

1 meter

2 meters

3 meters

4 meters

Explanation: Let 'x' be the length of the fabric in meters. For Shop A, the total cost is (500 * x) - 100. For Shop B, the total cost is (450 * x). To find when the total costs are the same, we set the two expressions equal: 500x - 100 = 450x. Subtract 450x from both sides: 50x - 100 = 0. Add 100 to both sides: 50x = 100. Divide by 50: x = 2. So, at 2 meters, the total cost will be the same at both shops.

Q75: Two typists, Anne and Ben, are typing a book. Anne has already typed 150 pages and can type 20 pages per day. Ben starts from scratch but can type 30 pages per day. After how many days will Ben have typed more pages than Anne?

14 days

15 days

16 days

17 days

Explanation: Let 'd' be the number of days. Anne's total pages after 'd' days = 150 + 20d. Ben's total pages after 'd' days = 30d. We want to find when Ben has typed more pages than Anne: 30d > 150 + 20d. Subtract 20d from both sides: 10d > 150. Divide by 10: d > 15. Since 'd' must be an integer, the smallest integer greater than 15 is 16. Therefore, after 16 days, Ben will have typed more pages than Anne.

Q76: Consider a pattern of dots where the first figure has 5 dots, the second figure has 8 dots, and the third figure has 11 dots. If this pattern continues, which figure number will have 35 dots?

9th figure

10th figure

11th figure

12th figure

Explanation: The number of dots forms an arithmetic progression: 5, 8, 11, ... The first term (a) is 5. The common difference (d) is 8 - 5 = 3. The formula for the nth term of an arithmetic progression is T_n = a + (n-1)d. Substitute the values: T_n = 5 + (n-1)3. We want to find 'n' when T_n = 35. So, 35 = 5 + (n-1)3. This simplifies to 30 = (n-1)3, then 10 = n-1, which gives n = 11. Therefore, the 11th figure will have 35 dots.

Q77: A car rental company offers two plans. Plan A charges a flat fee of Rs. 1000 plus Rs. 50 per kilometer. Plan B charges a flat fee of Rs. 500 plus Rs. 75 per kilometer. For what distance (in kilometers) will the cost of both plans be identical?

10 km

15 km

20 km

25 km

Explanation: Let 'k' be the distance traveled in kilometers. For Plan A, the total cost = 1000 + 50k. For Plan B, the total cost = 500 + 75k. To find the distance where the costs are identical, we set the two expressions equal: 1000 + 50k = 500 + 75k. Subtract 50k from both sides: 1000 = 500 + 25k. Subtract 500 from both sides: 500 = 25k. Divide by 25: k = 20. Therefore, for a distance of 20 kilometers, the cost of both plans will be identical.

Q78: A company produces a certain item. On the first day, they produce 100 items. On each subsequent day, they increase production by 20 items. On which day will their daily production reach exactly 400 items?

14th day

15th day

16th day

17th day

Explanation: The production forms an arithmetic progression. First term (a) = 100. Common difference (d) = 20. Let 'n' be the number of days. The production on the nth day (T_n) is given by T_n = a + (n-1)d. We need T_n = 400. So, 400 = 100 + (n-1)20. Subtract 100 from both sides: 300 = (n-1)20. Divide by 20: 15 = n-1. Add 1 to both sides: n = 16. Therefore, on the 16th day, their daily production will reach exactly 400 items.

Identify binary numbers.

Q79: Which of the following is a correctly represented binary number?

`12_2`

`20_10`

`101_2`

`10_3`

Explanation: A binary number (base 2) can only contain the digits 0 and 1. `101_2` uses only these digits and is correctly notated. `12_2` is invalid because it contains the digit 2. `20_10` is a decimal number (base 10). `10_3` is a number in base 3.

Q80: Identify all correctly represented binary numbers from the list: `110_2`, `202_2`, `1001_2`, `15_10`.

`110_2`

`110_2`, `202_2`

`110_2`, `1001_2`

`1001_2`, `15_10`

Explanation: Binary numbers must only contain digits 0 and 1. `110_2` and `1001_2` satisfy this condition. `202_2` is incorrect due to the digit 2. `15_10` is a decimal number.

Q81: From the given options, which list contains only valid binary numbers?

`10_2`, `12_2`

`101_2`, `11_2`

`100_10`, `111_2`

`21_2`, `101_2`

Explanation: Valid binary numbers (base 2) only use the digits 0 and 1. In option B, both `101_2` and `11_2` adhere to this rule. Other options contain numbers with digits other than 0 or 1, or are in a different base.

Q82: Select the option that correctly identifies all binary numbers (base 2) from the following: `101`, `11_2`, `10_10`, `110_2`.

`101`, `11_2`

`11_2`, `110_2`

`10_10`, `110_2`

`101`, `10_10`

Explanation: Binary numbers are explicitly denoted by the subscript '2'. `101` and `10_10` are decimal numbers (base 10). `11_2` and `110_2` are correctly represented binary numbers.

Q83: Which group consists exclusively of correctly represented binary numbers?

`1010_2`, `100_10`, `11_2`

`1101_2`, `102_2`, `01_2`

`1010_2`, `11_2`, `01_2`

`100_2`, `113_2`, `10_2`

Explanation: A binary number must only contain the digits 0 and 1. In option C, all numbers `1010_2`, `11_2`, and `01_2` use only these digits and are correctly notated in base 2. Other options contain decimal numbers or invalid binary numbers (with digits like 2 or 3).

Q84: Among the following numbers, which are valid binary representations? `1_2`, `10`, `111_2`, `20_2`.

`1_2`, `10`

`10`, `111_2`

`1_2`, `111_2`

`111_2`, `20_2`

Explanation: Binary numbers (base 2) must only contain the digits 0 and 1. `1_2` and `111_2` satisfy this condition. `10` is a decimal number. `20_2` is invalid because it contains the digit 2.

Q85: Which of the following is a valid binary number?

$101_{10}$

$210_2$

$110_2$

$10_B$

Explanation: A valid binary number must only contain digits 0 and 1, and be explicitly denoted with base 2. $110_2$ satisfies these conditions.

Q86: What type of number is $101_{10}$?

Valid Binary Number

Decimal Number (using binary digits)

Invalid Number Representation

Octal Number

Explanation: Although $101_{10}$ uses only digits 0 and 1, the subscript $10$ indicates it is a decimal number.

Q87: Which of the following is an invalid binary number representation?

$111_2$

$101_2$

$102_2$

$10_2$

Explanation: A binary number can only contain digits 0 and 1. $102_2$ contains the digit '2', making it an invalid binary representation.

Q88: What is the classification of $110_B$?

Valid Binary Number

Decimal Number

Invalid Number Representation

Hexadecimal Number

Explanation: The subscript 'B' is not a standard or recognized base notation in number systems, making $110_B$ an invalid number representation.

Q89: Select the option that correctly identifies a valid binary number.

$12_2$

$100_{10}$

$1_2$

$110_8$

Explanation: A valid binary number uses only digits 0 and 1, and is explicitly denoted with base 2. $1_2$ meets these criteria.

Q90: Which statement accurately describes the number $10_{10}$?

It is a valid binary number.

It is an invalid number representation.

It is a decimal number (using binary digits).

It is an octal number.

Explanation: The number $10_{10}$ is explicitly shown with base 10, meaning it is a decimal number. It happens to use digits 0 and 1, which are also binary digits.

Q91: A classmate states: "Any number that uses only the digits 0 and 1 is a binary number." Which of the following sequences of digits 0 and 1 is NOT necessarily a binary number?

101_two

101_ten

011_two

1_two

Explanation: The statement is flawed because a sequence of 0s and 1s could represent a number in a different base, such as decimal. For example, 101 in base 10 (one hundred and one) uses only digits 0 and 1 but is not a binary (base 2) number. It is only binary if its base is explicitly stated or understood to be 2 (e.g., 101_two).

Q92: Why is the sequence of digits "10" not always considered a binary number, despite using only 0 and 1?

Binary numbers cannot have a trailing zero.

It could represent the decimal number ten.

It is too short to be a binary number.

It is only valid as a binary number if written as "010".

Explanation: The sequence "10" can be interpreted as the decimal number ten (10_ten), which is not a binary number. To be definitively a binary number, its base must be specified, for example, 10_two (which is 2 in base 10).

Q93: Which of the following examples most clearly shows that a sequence of 0s and 1s is NOT always a binary number?

The binary number 101_two.

The decimal number 101_ten.

The digit 0.

The sequence "0101" without a specified base.

Explanation: The decimal number 101 (one hundred and one) uses only the digits 0 and 1, but it is a base-10 number, not a base-2 (binary) number. This serves as a direct counter-example to the flawed statement.

Q94: What essential information is missing when someone claims "11" is a binary number, based solely on it containing 0s and 1s?

The number of digits.

Whether it's an even or odd number.

The base of the number.

Its value in base 10.

Explanation: For a sequence of 0s and 1s to be definitively identified as a binary number, its base must be explicitly stated or understood to be base 2. Without this, "11" could be the decimal number eleven, or the binary number three (11_two).

Q95: Consider the statement: "The sequence '110' is a binary number because it only uses digits 0 and 1." Why is this statement flawed?

Binary numbers cannot contain the digit 0.

The sequence '110' could represent the decimal number one hundred and ten.

Binary numbers must always start with '1'.

The number of digits is too large for a binary number.

Explanation: While '110_two' is a valid binary number, the sequence '110' without a specified base could also represent the decimal number one hundred and ten (110_ten), which is not a binary number. The flaw lies in assuming any sequence of 0s and 1s *must* be binary.

Q96: Which of the following pairs of examples best illustrates that using only digits 0 and 1 does NOT automatically make a number binary?

(0_two, 1_two)

(10_ten, 11_ten)

(101_two, 110_two)

(2_ten, 3_ten)

Explanation: The numbers 10 (ten) and 11 (eleven) in the decimal system (base 10) use only the digits 0 and 1. However, they are decimal numbers, not binary numbers. This demonstrates that merely using the digits 0 and 1 is not sufficient to classify a number as binary; the base must be base 2.

Convert decimal numbers to binary numbers.

Q97: Convert the decimal number 13 to its binary equivalent.

1011₂

1101₂

1110₂

1001₂

Explanation: To convert the decimal number 13 to binary, we repeatedly divide by 2 and record the remainders. 13 ÷ 2 = 6 remainder 1; 6 ÷ 2 = 3 remainder 0; 3 ÷ 2 = 1 remainder 1; 1 ÷ 2 = 0 remainder 1. Reading the remainders from bottom to top gives 1101₂.

Q98: What is the binary representation of the decimal number 13?

1110₂

1011₂

1101₂

1111₂

Q99: The decimal number 13 can be expressed in binary as:

1001₂

1100₂

1101₂

0111₂

Q100: Which of the following is the correct binary equivalent of 13₁₀?

1010₂

1101₂

1011₂

1110₂

Q101: If a number is 13 in base 10, what is its value in base 2?

1110₂

1001₂

1101₂

1100₂

Q102: Convert the decimal number 'thirteen' into its binary form.

1011₂

1111₂

1101₂

0111₂

Q103: What is the binary equivalent of the decimal number 75?

1101010₂

1001011₂

1010101₂

1001101₂

Explanation: To convert a decimal number to binary, repeatedly divide the decimal number by 2 and record the remainders. The binary equivalent is obtained by reading the remainders from bottom to top. For 75: 75 ÷ 2 = 37 remainder 1 37 ÷ 2 = 18 remainder 1 18 ÷ 2 = 9 remainder 0 9 ÷ 2 = 4 remainder 1 4 ÷ 2 = 2 remainder 0 2 ÷ 2 = 1 remainder 0 1 ÷ 2 = 0 remainder 1 Reading the remainders from bottom to top gives 1001011₂.

Q104: Which of the following binary numbers correctly represents the decimal number 75?

110101₂

1001011₂

1001010₂

1011011₂

Q105: A computer system stores the decimal value 75. How is this value represented in binary?

1101100₂

1001011₂

1001001₂

1000111₂

Q106: When converting the decimal number 75 to binary, what is the final binary representation?

1110011₂

1001011₂

1011010₂

1000101₂

Q107: Select the correct binary representation for the decimal number 75.

1010110₂

1001011₂

1101011₂

1001111₂

Q108: Among the given options, which one accurately shows the binary equivalent of 75?

1101010₂

1001011₂

1001101₂

1010011₂

Q109: What is the largest decimal number that can be represented using exactly 8 binary digits (bits), and what is its binary equivalent?

256, 100000000_2

255, 11111111_2

127, 01111111_2

254, 11111110_2

Explanation: The largest number representable with 'n' bits is 2^n - 1. For 8 bits, it's 2^8 - 1 = 256 - 1 = 255. Its binary representation is 11111111_2, where all 8 bits are set to 1.

Q110: Convert the largest decimal number representable by 8 bits into its binary equivalent.

11111110_2

11111111_2

100000000_2

01111111_2

Explanation: The largest decimal number for 8 bits is 2^8 - 1 = 255. To convert 255 to binary, we represent it as a sum of powers of 2 (128 + 64 + 32 + 16 + 8 + 4 + 2 + 1), which means all 8 bits are 1s, resulting in 11111111_2.

Q111: If an 8-bit binary number is 11111111_2, what is its decimal equivalent? What does this value represent in the context of 8-bit numbers?

256, which is one more than the largest

255, which is the largest possible 8-bit decimal number

127, which is the largest 7-bit decimal number

254, which is one less than the largest

Explanation: The binary number 11111111_2 can be converted to decimal as 1*2^7 + 1*2^6 + 1*2^5 + 1*2^4 + 1*2^3 + 1*2^2 + 1*2^1 + 1*2^0 = 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = 255. This is the largest decimal number that can be represented using exactly 8 binary digits.

Q112: An 8-bit system can represent a range of decimal numbers. What is the highest decimal number in this range, and how is it written in binary?

128, 10000000_2

255, 11111111_2

256, 100000000_2

254, 11111110_2

Explanation: The largest decimal number representable by 'n' bits is 2^n - 1. For an 8-bit system, n=8, so the highest decimal number is 2^8 - 1 = 256 - 1 = 255. Its binary representation, with all 8 bits set to 1, is 11111111_2.

Q113: To find the largest decimal number representable by 8 bits, you calculate 2^8 - 1. What is the result of this calculation, and how would you write it in 8-bit binary?

256, 100000000_2

255, 11111111_2

127, 01111111_2

255, 10000000_2

Explanation: The formula for the largest decimal number representable by 'n' bits is 2^n - 1. For 8 bits, 2^8 - 1 = 256 - 1 = 255. The binary equivalent of 255 is obtained by setting all 8 bits to 1, which is 11111111_2.

Q114: You are given an empty 8-bit register. If you fill all its bits with '1's, what is the decimal value stored in it, and how does this relate to the maximum capacity of the register?

127, it's the largest 7-bit number

256, it's one more than the maximum capacity

255, it represents the maximum decimal capacity for an 8-bit register

254, it's the maximum capacity minus one

Explanation: Filling all 8 bits of a register with '1's means the binary number is 11111111_2. Converting this to decimal gives 1*2^7 + 1*2^6 + ... + 1*2^0 = 255. This value, 255, is indeed the largest decimal number that can be stored or represented using exactly 8 binary digits, thus representing the maximum capacity of an 8-bit register.

Convert binary numbers to decimal numbers.

Q115: Convert the binary number 1011₂ to its decimal equivalent.

Explanation: To convert a binary number to a decimal number, multiply each digit by its corresponding power of 2, starting from the rightmost digit with 2^0, and sum the results. For 1011₂: (1 * 2³) + (0 * 2²) + (1 * 2¹) + (1 * 2⁰) = (1 * 8) + (0 * 4) + (1 * 2) + (1 * 1) = 8 + 0 + 2 + 1 = 11 Therefore, 1011₂ is 11 in decimal.

Q116: What is the decimal representation of the binary number 1011₂?

Q117: The binary number 1011₂ can be expressed in base 10 as:

Q118: Find the decimal value of 1011₂.

Q119: In the decimal number system, what is the value of 1011₂?

Q120: Which of the following is the decimal equivalent of the binary number 1011₂?

Q121: Convert the binary number 1011.01₂ to its decimal equivalent.

10.25₁₀

11.25₁₀

11.5₁₀

1011.01₁₀

Explanation: To convert 1011.01₂ to decimal, we expand it using powers of 2: Integer part: 1*2³ + 0*2² + 1*2¹ + 1*2⁰ = 8 + 0 + 2 + 1 = 11 Fractional part: 0*2⁻¹ + 1*2⁻² = 0*0.5 + 1*0.25 = 0 + 0.25 = 0.25 So, 1011.01₂ = 11 + 0.25 = 11.25₁₀.

Q122: What is the decimal equivalent of the binary number 110.11₂?

7.75₁₀

6.5₁₀

6.75₁₀

6.11₁₀

Explanation: To convert 110.11₂ to decimal: Integer part: 1*2² + 1*2¹ + 0*2⁰ = 4 + 2 + 0 = 6 Fractional part: 1*2⁻¹ + 1*2⁻² = 1*0.5 + 1*0.25 = 0.5 + 0.25 = 0.75 So, 110.11₂ = 6 + 0.75 = 6.75₁₀.

Q123: Convert 1001.1₂ to its decimal equivalent.

9.1₁₀

10.5₁₀

9.5₁₀

8.5₁₀

Explanation: To convert 1001.1₂ to decimal: Integer part: 1*2³ + 0*2² + 0*2¹ + 1*2⁰ = 8 + 0 + 0 + 1 = 9 Fractional part: 1*2⁻¹ = 1*0.5 = 0.5 So, 1001.1₂ = 9 + 0.5 = 9.5₁₀.

Q124: Find the decimal equivalent of the binary number 111.001₂.

7.001₁₀

7.25₁₀

7.125₁₀

6.125₁₀

Explanation: To convert 111.001₂ to decimal: Integer part: 1*2² + 1*2¹ + 1*2⁰ = 4 + 2 + 1 = 7 Fractional part: 0*2⁻¹ + 0*2⁻² + 1*2⁻³ = 0*0.5 + 0*0.25 + 1*0.125 = 0 + 0 + 0.125 = 0.125 So, 111.001₂ = 7 + 0.125 = 7.125₁₀.

Q125: Convert the binary number 10.101₂ to its decimal form.

2.101₁₀

3.625₁₀

2.5₁₀

2.625₁₀

Explanation: To convert 10.101₂ to decimal: Integer part: 1*2¹ + 0*2⁰ = 2 + 0 = 2 Fractional part: 1*2⁻¹ + 0*2⁻² + 1*2⁻³ = 1*0.5 + 0*0.25 + 1*0.125 = 0.5 + 0 + 0.125 = 0.625 So, 10.101₂ = 2 + 0.625 = 2.625₁₀.

Q126: What is the decimal value of 101.011₂?

5.011₁₀

6.375₁₀

5.25₁₀

5.375₁₀

Explanation: To convert 101.011₂ to decimal: Integer part: 1*2² + 0*2¹ + 1*2⁰ = 4 + 0 + 1 = 5 Fractional part: 0*2⁻¹ + 1*2⁻² + 1*2⁻³ = 0*0.5 + 1*0.25 + 1*0.125 = 0 + 0.25 + 0.125 = 0.375 So, 101.011₂ = 5 + 0.375 = 5.375₁₀.

Q127: Convert the binary number 1101.101₂ to its decimal equivalent.

13.101₁₀

13.5₁₀

13.625₁₀

14.625₁₀

Explanation: To convert 1101.101₂ to decimal: Integer part: 1*2³ + 1*2² + 0*2¹ + 1*2⁰ = 8 + 4 + 0 + 1 = 13 Fractional part: 1*2⁻¹ + 0*2⁻² + 1*2⁻³ = 1*0.5 + 0*0.25 + 1*0.125 = 0.5 + 0 + 0.125 = 0.625 So, 1101.101₂ = 13 + 0.625 = 13.625₁₀.

Q128: If (11x1)₂ is a binary number where 'x' represents a binary digit, and its decimal equivalent is greater than 10 but less than 14, determine the possible value(s) for 'x'.

x = 1

x = 0

x = 0 or 1

No possible value for x

Explanation: To convert (11x1)₂ to decimal: (1 × 2³) + (1 × 2²) + (x × 2¹) + (1 × 2⁰) = 8 + 4 + 2x + 1 = 13 + 2x. Given that its decimal equivalent is greater than 10 but less than 14, we have 10 < 13 + 2x < 14. Subtracting 13 from all parts gives -3 < 2x < 1. Dividing by 2 gives -1.5 < x < 0.5. Since 'x' must be a binary digit (either 0 or 1), the only possible value for 'x' in this range is 0.

Q129: If (1x01)₂ is a binary number where 'x' represents a binary digit, and its decimal equivalent is greater than 8 but less than 12, determine the possible value(s) for 'x'.

x = 1

x = 0

x = 0 or 1

No possible value for x

Explanation: To convert (1x01)₂ to decimal: (1 × 2³) + (x × 2²) + (0 × 2¹) + (1 × 2⁰) = 8 + 4x + 0 + 1 = 9 + 4x. Given that its decimal equivalent is greater than 8 but less than 12, we have 8 < 9 + 4x < 12. Subtracting 9 from all parts gives -1 < 4x < 3. Dividing by 4 gives -0.25 < x < 0.75. Since 'x' must be a binary digit (either 0 or 1), the only possible value for 'x' in this range is 0.

Q130: If (10x1)₂ is a binary number where 'x' represents a binary digit, and its decimal equivalent is greater than 8 but less than 10, determine the possible value(s) for 'x'.

x = 1

x = 0

x = 0 or 1

No possible value for x

Explanation: To convert (10x1)₂ to decimal: (1 × 2³) + (0 × 2²) + (x × 2¹) + (1 × 2⁰) = 8 + 0 + 2x + 1 = 9 + 2x. Given that its decimal equivalent is greater than 8 but less than 10, we have 8 < 9 + 2x < 10. Subtracting 9 from all parts gives -1 < 2x < 1. Dividing by 2 gives -0.5 < x < 0.5. Since 'x' must be a binary digit (either 0 or 1), the only possible value for 'x' in this range is 0.

Q131: If (x11)₂ is a binary number where 'x' represents a binary digit, and its decimal equivalent is greater than 5 but less than 8, determine the possible value(s) for 'x'.

x = 0

x = 1

x = 0 or 1

No possible value for x

Explanation: To convert (x11)₂ to decimal: (x × 2²) + (1 × 2¹) + (1 × 2⁰) = 4x + 2 + 1 = 4x + 3. Given that its decimal equivalent is greater than 5 but less than 8, we have 5 < 4x + 3 < 8. Subtracting 3 from all parts gives 2 < 4x < 5. Dividing by 4 gives 0.5 < x < 1.25. Since 'x' must be a binary digit (either 0 or 1), the only possible value for 'x' in this range is 1.

Q132: If (101x)₂ is a binary number where 'x' represents a binary digit, and its decimal equivalent is greater than 10 but less than 12, determine the possible value(s) for 'x'.

x = 0

x = 1

x = 0 or 1

No possible value for x

Explanation: To convert (101x)₂ to decimal: (1 × 2³) + (0 × 2²) + (1 × 2¹) + (x × 2⁰) = 8 + 0 + 2 + x = 10 + x. Given that its decimal equivalent is greater than 10 but less than 12, we have 10 < 10 + x < 12. Subtracting 10 from all parts gives 0 < x < 2. Since 'x' must be a binary digit (either 0 or 1), the only possible value for 'x' in this range is 1.

Q133: If (x010)₂ is a binary number where 'x' represents a binary digit, and its decimal equivalent is greater than 9 but less than 13, determine the possible value(s) for 'x'.

x = 0

x = 1

x = 0 or 1

No possible value for x

Explanation: To convert (x010)₂ to decimal: (x × 2³) + (0 × 2²) + (1 × 2¹) + (0 × 2⁰) = 8x + 0 + 2 + 0 = 8x + 2. Given that its decimal equivalent is greater than 9 but less than 13, we have 9 < 8x + 2 < 13. Subtracting 2 from all parts gives 7 < 8x < 11. Dividing by 8 gives 0.875 < x < 1.375. Since 'x' must be a binary digit (either 0 or 1), the only possible value for 'x' in this range is 1.

Q134: If (1x101)₂ is a binary number where 'x' represents a binary digit, and its decimal equivalent is greater than 20 but less than 26, determine the possible value(s) for 'x'.

x = 1

x = 0

x = 0 or 1

No possible value for x

Explanation: To convert (1x101)₂ to decimal: (1 × 2⁴) + (x × 2³) + (1 × 2²) + (0 × 2¹) + (1 × 2⁰) = 16 + 8x + 4 + 0 + 1 = 21 + 8x. Given that its decimal equivalent is greater than 20 but less than 26, we have 20 < 21 + 8x < 26. Subtracting 21 from all parts gives -1 < 8x < 5. Dividing by 8 gives -0.125 < x < 0.625. Since 'x' must be a binary digit (either 0 or 1), the only possible value for 'x' in this range is 0.

Add and subtract binary numbers.

Q135: What is the sum of 101₂ and 010₂?

110₂

111₂

100₂

011₂

Explanation: To add 101₂ and 010₂: Starting from the rightmost bit: 1 + 0 = 1 Middle bit: 0 + 1 = 1 Leftmost bit: 1 + 0 = 1 Therefore, 101₂ + 010₂ = 111₂.

Q136: Add the binary numbers 101₂ and 001₂.

100₂

110₂

111₂

000₂

Explanation: To add 101₂ and 001₂: Rightmost bit: 1 + 1 = 10₂ (write 0, carry 1) Middle bit: 0 + 0 + (carry)1 = 1 Leftmost bit: 1 + 0 = 1 Therefore, 101₂ + 001₂ = 110₂.

Q137: Find the sum: 011₂ + 010₂.

001₂

101₂

111₂

110₂

Explanation: To add 011₂ and 010₂: Rightmost bit: 1 + 0 = 1 Middle bit: 1 + 1 = 10₂ (write 0, carry 1) Leftmost bit: 0 + 0 + (carry)1 = 1 Therefore, 011₂ + 010₂ = 101₂.

Q138: What is the difference when 010₂ is subtracted from 110₂?

000₂

100₂

111₂

010₂

Explanation: To subtract 010₂ from 110₂: Rightmost bit: 0 - 0 = 0 Middle bit: 1 - 1 = 0 Leftmost bit: 1 - 0 = 1 Therefore, 110₂ - 010₂ = 100₂.

Q139: Subtract 001₂ from 101₂.

110₂

100₂

000₂

010₂

Explanation: To subtract 001₂ from 101₂: Rightmost bit: 1 - 1 = 0 Middle bit: 0 - 0 = 0 Leftmost bit: 1 - 0 = 1 Therefore, 101₂ - 001₂ = 100₂.

Q140: Evaluate: 110₂ - 001₂.

111₂

101₂

100₂

010₂

Explanation: To subtract 001₂ from 110₂: Rightmost bit: 0 - 1. We need to borrow from the middle bit. The middle bit (1) becomes 0, and the rightmost bit (0) becomes 10₂ (which is 2 in decimal). So, 10₂ - 1 = 1. Middle bit: Now 0 - 0 = 0. Leftmost bit: 1 - 0 = 1. Therefore, 110₂ - 001₂ = 101₂.

Q141: What is the result of adding the binary numbers 1011₂ and 0111₂?

10100₂

10010₂

11000₂

10001₂

Explanation: To add binary numbers, we follow the rules: 0+0=0, 0+1=1, 1+0=1, 1+1=10 (0 with a carry of 1), and 1+1+1=11 (1 with a carry of 1). Performing the addition: 1011₂ + 0111₂ ------- 10010₂

Q142: Calculate the sum of 11011₂ and 01101₂.

100100₂

110010₂

101000₂

100010₂

Explanation: Using binary addition rules (0+0=0, 0+1=1, 1+1=10, 1+1+1=11): 11011₂ + 01101₂ ------- 101000₂

Q143: What is 10101₂ + 11011₂?

101100₂

110000₂

100000₂

110100₂

Explanation: Adding the binary numbers: 10101₂ + 11011₂ ------- 110000₂ Each column is added, and any carry is moved to the next column.

Q144: Subtract 0111₂ from 1101₂.

0010₂

1010₂

0100₂

0110₂

Explanation: To subtract binary numbers, we follow the rules: 0-0=0, 1-0=1, 1-1=0. When we need to subtract 0-1, we must borrow from the next higher position. A borrowed 1 becomes 10₂ (decimal 2) in the current position. 1101₂ - 0111₂ ------- 0110₂

Q145: What is the result of 10110₂ - 01101₂?

01011₂

00101₂

01001₂

11001₂

Explanation: Performing binary subtraction with borrowing: 10110₂ - 01101₂ ------- 01001₂ When subtracting 0-1, you borrow 1 from the next position, making the current position 10₂ (decimal 2).

Q146: Calculate 11001₂ - 10110₂.

01111₂

00111₂

00011₂

00001₂

Explanation: Binary subtraction involves careful handling of borrowing, especially when borrowing across zeros: 11001₂ - 10110₂ ------- 00011₂ When a '0' needs to borrow, it borrows from the next '1', which then reduces by 1, and the '0' becomes '10₂'. If there are multiple '0's, the borrowing propagates.

Q147: What is the value of X in the binary equation: X + 1011₂ = 11100₂?

10101₂

10001₂

10111₂

11001₂

Explanation: To find X, we need to subtract 1011₂ from 11100₂. 11100₂ - 1011₂ = 10001₂.

Q148: Determine the value of X in the binary equation: 11010₂ - X = 1011₂.

10001₂

1111₂

10111₂

11101₂

Explanation: To find X, we need to subtract 1011₂ from 11010₂. 11010₂ - 1011₂ = 01111₂ (or 1111₂).

Q149: Find the binary number X such that 10110₂ + X = 100001₂.

1101₂

1001₂

1011₂

1111₂

Explanation: To find X, we need to subtract 10110₂ from 100001₂. 100001₂ - 10110₂ = 001011₂ (or 1011₂).

Q150: If X - 1101₂ = 10101₂, what is the value of X?

11010₂

100100₂

100010₂

101100₂

Explanation: To find X, we need to add 1101₂ to 10101₂. 10101₂ + 1101₂ = 100010₂.

Q151: What is the value of X in the binary equation: X + 111₂ = 10110₂?

1011₂

1001₂

1101₂

1111₂

Explanation: To find X, we need to subtract 111₂ from 10110₂. 10110₂ - 111₂ = 01111₂ (or 1111₂).

Q152: Determine the value of X in the binary equation: 10011₂ - X = 110₂.

1011₂

1101₂

1111₂

1001₂

Explanation: To find X, we need to subtract 110₂ from 10011₂. 10011₂ - 110₂ = 01101₂ (or 1101₂).

Recognize situations where binary numbers are used.

Q153: Which of the following scenarios fundamentally relies on binary numbers for its operation?

Measuring the temperature of a room using a mercury thermometer.

A computer performing calculations and storing data.

Calculating the distance between two cities using an odometer.

Weighing vegetables on a traditional balance scale.

Explanation: Computers process and store all information using binary code (0s and 1s). This is fundamental to how they operate, unlike measuring temperature, distance, or weight, which typically use decimal numbers or analog systems.

Q154: In which of these applications are binary numbers essential for storing information?

Measuring rainfall with a rain gauge.

Storing files on a USB flash drive.

Checking time on an analog wall clock.

Reading the temperature from a liquid-in-glass thermometer.

Explanation: USB flash drives are digital storage devices that store all data (files, photos, documents) as binary code (0s and 1s). Analog clocks, rain gauges, and thermometers do not use binary for their core function of displaying or measuring.

Q155: Which activity relies on binary numbers for successful data transmission?

Writing a letter by hand and mailing it.

Transmitting an email over the internet.

Communicating verbally over a short distance.

Sending smoke signals to convey a message.

Explanation: Transmitting data over the internet involves converting all information into binary signals (packets of 0s and 1s) that can be sent digitally. The other options involve manual or analog processes.

Q156: Which of these devices fundamentally uses binary numbers for its internal processing?

An abacus for basic arithmetic.

A digital electronic calculator.

A slide rule for complex calculations.

A sundial to tell the time.

Explanation: A digital calculator processes all numbers and operations internally using binary logic circuits before converting the result to decimal for display. An abacus, slide rule, and sundial are all non-digital tools.

Q157: Which of the following processes converts analog information into a binary format for storage or transmission?

Listening to a live orchestra performance.

Recording a song on a digital audio player.

Painting a landscape using oil paints.

Sculpting a figure from clay.

Explanation: Recording a song digitally converts the analog sound waves into binary data (0s and 1s) so it can be stored on a computer or played back. Listening to live music, painting, or sculpting are analog activities that do not involve binary conversion.

Q158: Which of the following is an example of a system where binary numbers are fundamental for its control and automation?

A gardener watering plants manually with a hose.

An automatic traffic light system at a road junction.

A person writing notes in a physical diary.

A chef preparing a meal using traditional cooking methods.

Explanation: Automated traffic light systems use microcontrollers and digital logic, which operate on binary numbers (0s and 1s) to control the timing and sequencing of lights. The other options describe manual or mechanical processes.

Q159: Why do computers primarily use binary numbers (0s and 1s) for their internal operations?

Binary numbers are easier for humans to read and understand.

Binary numbers allow computers to perform calculations much faster than decimal numbers.

Binary's two states (on/off or high/low voltage) are simple for electronic circuits to represent and process.

Binary numbers can represent a wider range of values with fewer digits compared to decimal.

Explanation: Computers use binary because their electronic circuits operate on two distinct states: 'on' (represented by 1, high voltage) and 'off' (represented by 0, low voltage). This simplicity makes it highly efficient for hardware to process and store information.

Q160: Which of the following best describes why binary numbers are crucial for computer data storage (e.g., RAM, hard drives)?

Binary allows data to be stored in a compact format, reducing physical storage space.

Data is stored as magnetic polarities or electrical charges, which naturally have two distinct states (0 or 1).

Binary makes it easier to convert stored data into human-readable text.

Binary numbers enable faster data retrieval speeds from storage devices.

Explanation: In computer storage, information is physically represented by two distinct states. For instance, on a hard drive, it's magnetic polarity (north or south); in RAM, it's the presence or absence of an electrical charge. These two states perfectly correspond to the 0 and 1 of binary code.

Q161: How are binary numbers essential for a computer's Central Processing Unit (CPU) to perform calculations and logic operations?

Binary allows the CPU to directly understand and execute instructions written in high-level programming languages.

The CPU uses binary because its fundamental building blocks (logic gates) are designed to process only two states (true/false or 0/1).

Binary helps the CPU to display complex graphical user interfaces on the screen.

Binary numbers enable the CPU to communicate with external peripherals more efficiently.

Explanation: The CPU is built from millions of tiny transistors that act as switches, forming logic gates (AND, OR, NOT, etc.). These logic gates inherently operate on binary inputs (0 or 1) to produce binary outputs, performing all arithmetic and logical operations in a computer.

Q162: When a computer displays an image or plays a sound, why is binary representation critical for this process?

Binary allows for the compression of image and sound files, making them smaller.

Each pixel's color intensity or sound wave's amplitude is converted into distinct binary values that the computer can process and output.

Binary ensures that images and sounds are displayed in high definition.

Binary makes it easier to share multimedia files across different devices.

Explanation: Digital images are made of pixels, and digital sound is a series of samples. Each pixel's color and brightness, or each sound sample's amplitude, are assigned a specific numerical value. These values are then converted into binary code for the computer to store, process, and render them on a display or through speakers.

Q163: How are binary numbers critical in digital communication and networking (e.g., sending emails, browsing the internet)?

Binary helps to encrypt all data transmitted over networks, ensuring security.

All data (text, images, audio) is converted into a stream of binary bits (electrical pulses or light signals) for transmission and reconstruction.

Binary allows for faster connection speeds and lower latency when browsing the internet.

Binary makes it possible for computers to communicate using different human languages.

Explanation: When data is sent digitally, whether through cables, fiber optics, or wirelessly, it is encoded as a series of binary signals. These signals are electrical pulses (on/off), light pulses (light/no light), or radio waves (high/low frequency) that represent the 0s and 1s of the binary data, allowing information to travel between devices.

Q164: What is the primary reason that machine code, the lowest-level programming language, is written in binary?

Binary machine code is easier for human programmers to write and debug than other formats.

Binary allows machine code to be translated quickly into high-level programming languages.

Machine code instructions are directly understood by the computer's processor, which operates solely on binary electrical signals (0s and 1s).

Binary machine code requires less memory to store compared to other forms of programming instructions.

Explanation: Machine code is the native language of a computer's processor. Since the processor's circuitry (transistors and logic gates) operates purely on electrical signals representing binary 0s and 1s, machine code must be in binary form for direct execution without further translation.

Q165: In a digital traffic light system, why are binary numbers fundamental to its operation?

Binary ensures that the traffic lights can display a wide range of colors beyond red, yellow, and green.

Each light's 'on' or 'off' state is a simple two-state condition that can be directly controlled and processed by binary logic circuits.

Binary allows the traffic light system to communicate with emergency services more quickly.

Binary numbers are used to calculate the optimal timing for traffic flow based on real-time data.

Explanation: Traffic lights operate on discrete 'on' or 'off' states for each color (red, yellow, green). Binary numbers (0 for off, 1 for on) are perfectly suited to represent and control these simple two-state conditions within the electronic circuits of the traffic light controller, managing the sequence and timing of signals.

Q166: In a smart home automation system, various devices like lights, thermostats, and security cameras communicate using binary numbers. What is the primary reason binary numbers are chosen for these internal control signals?

They are easier for humans to read and understand.

They can be easily represented by two distinct electrical states (on/off, high/low voltage).

They allow for a wider range of colors in displays.

They require less storage space than other number systems.

Explanation: Binary numbers are fundamental to electronic systems because their two states (0 and 1) directly correspond to the two physical states of electrical signals, such as high or low voltage, or the presence or absence of current. This simplicity and direct mapping make them ideal for control signals within complex digital systems like smart home automation.

Q167: A factory control system uses binary numbers for the internal logic that manages robotic arms, conveyor belts, and sensors. Why is binary particularly effective for this purpose compared to decimal or other number systems?

Binary numbers can represent larger values with fewer digits.

Binary operations are simpler and more reliable to implement using electronic circuits with two stable states.

It simplifies mathematical calculations for human operators.

Binary numbers can store more complex data types directly.

Explanation: Electronic circuits inherently operate on two states (on/off, high/low voltage). Binary numbers, with their two digits (0 and 1), perfectly align with these physical states. This makes the design and implementation of logic gates and control mechanisms much simpler, more reliable, and more efficient in terms of hardware for complex systems like factory control, compared to trying to represent ten distinct states for a decimal system.

Q168: Consider a multi-functional medical device that monitors vital signs, administers medication, and alerts staff. How do binary numbers serve as the foundation for the internal logic and control signals in such a complex system?

They make the device's display more visually appealing.

They allow for direct processing of human language commands.

They enable precise and reliable communication between components through simple on/off states.

They reduce the overall weight and size of the device.

Explanation: In complex systems like medical devices, binary numbers are crucial because their simple two-state nature (0 or 1) can be directly translated into electrical signals (on/off, high/low voltage). This allows for highly reliable and unambiguous communication and control between different components, ensuring that commands are executed precisely and consistently, which is critical for safety and accuracy in medical applications.

Q169: Why would a complex flight control system for an airplane, which manages thousands of interacting components, prefer to use binary numbers for its internal operations rather than the decimal system?

Binary numbers are inherently faster for displaying numerical data to pilots.

Implementing ten distinct electrical states (for decimal) is significantly more complex and less reliable than two (for binary).

Binary numbers take up less physical space on circuit boards.

Decimal numbers are only used for human-readable output, not internal logic.

Explanation: The primary reason for using binary over decimal in complex digital systems like flight control is the practical difficulty and reduced reliability of implementing circuits that can consistently distinguish between ten different electrical states (for decimal). Binary, with its two clear states (on/off), is much simpler, more robust, and more efficient to build into electronic hardware, ensuring the necessary reliability for critical systems.

Q170: A modern robotics system, capable of performing diverse tasks like object recognition, movement, and manipulation, relies heavily on binary numbers for its internal processing. Which of the following best describes how binary numbers contribute to its multi-functional capabilities?

They allow for the storage of vast amounts of raw image and video data.

They provide the fundamental 'yes/no' or 'on/off' decisions that form the basis of all digital logic and control.

They make the robot's external casing more durable and resistant to damage.

They simplify the complex mathematical equations required for advanced AI algorithms.

Explanation: Binary numbers are the foundational language of all digital computing and control. Every complex task, from object recognition to motor control, is broken down into a series of elementary 'yes/no' or 'on/off' decisions. Binary numbers (0 and 1) perfectly represent these fundamental logical states, allowing for the precise execution of instructions and the interaction of various components within the robotic system.

Q171: In a large-scale industrial automation system, numerous sensors, actuators, and processors interact to optimize production. Why are binary numbers the most practical choice for the underlying control signals and logic within this comprehensive system?

They require less power consumption for electronic components.

They simplify the process of converting human voice commands into machine instructions.

Their two-state nature (0 or 1) directly corresponds to the fundamental on/off operations of electronic switches and gates, ensuring efficiency and reliability.

They enable the system to automatically correct errors in data transmission.

Explanation: The fundamental advantage of binary numbers in industrial automation and similar complex systems lies in their direct correspondence to the physical reality of electronic components. Electronic switches and logic gates operate in two distinct states (open/closed, high/low voltage). Binary 0 and 1 map perfectly to these states, making the design, implementation, and operation of digital control circuits highly efficient, reliable, and cost-effective.

Q172: Which of the following best explains why binary numbers are crucial for the internal logic of a complex traffic light control system that manages multiple intersections and adapts to real-time traffic flow?

They enable the system to calculate optimal light timings in advanced mathematical formats.

The two distinct states (on/off) of binary numbers perfectly match the electrical nature of digital circuits, ensuring robust control.

Binary numbers are easier for traffic engineers to program directly into the system.

They allow for the visual representation of complex traffic patterns on a monitor.

Explanation: Binary numbers are ideal for complex systems because they allow for the clear, unambiguous representation of 'on' (1) or 'off' (0) states, which directly correspond to the electrical signals used in digital circuits. This simplicity makes it highly efficient and reliable for the internal logic and control signals of a traffic light system, where precise and rapid decisions are necessary to manage traffic flow effectively.

Simplify expressions involving fractions with the 'of' operator.

Q173: What is 1/3 of 12?

Explanation: To find '1/3 of 12', we multiply (1/3) by 12. So, (1/3) * 12 = 12/3 = 4.

Q174: Calculate 2/5 of 20.

Explanation: To find '2/5 of 20', we multiply (2/5) by 20. So, (2/5) * 20 = (2 * 20) / 5 = 40 / 5 = 8.

Q175: Find the value of 1/2 of 3/4.

4/6

3/8

3/2

2/3

Explanation: To find '1/2 of 3/4', we multiply the two fractions. Multiply the numerators and the denominators: (1/2) * (3/4) = (1 * 3) / (2 * 4) = 3/8.

Q176: What is 2/3 of 1/5?

3/8

2/15

10/3

2/8

Explanation: To find '2/3 of 1/5', we multiply the two fractions. Multiply the numerators and the denominators: (2/3) * (1/5) = (2 * 1) / (3 * 5) = 2/15.

Q177: Determine 3/4 of 24.

Explanation: To find '3/4 of 24', we multiply (3/4) by 24. So, (3/4) * 24 = (3 * 24) / 4 = 72 / 4 = 18.

Q178: Calculate 2/3 of 5/6.

7/9

10/6

5/9

4/5

Explanation: To find '2/3 of 5/6', we multiply the two fractions. Multiply the numerators and the denominators: (2/3) * (5/6) = (2 * 5) / (3 * 6) = 10/18. This fraction can be simplified by dividing both numerator and denominator by 2, resulting in 5/9.

Q179: Simplify 1 1/2 of 2/3.

1/2

3/4

Explanation: First, convert the mixed number 1 1/2 to an improper fraction: 3/2. Then, multiply 3/2 by 2/3. (3/2) * (2/3) = (3*2)/(2*3) = 6/6 = 1.

Q180: Simplify 7/4 of 1 1/3.

7/12

7/3

28/12

2 1/4

Explanation: First, convert the mixed number 1 1/3 to an improper fraction: 4/3. Then, multiply 7/4 by 4/3. (7/4) * (4/3) = (7*4)/(4*3). Cross-cancelling the 4s gives 7/3.

Q181: Simplify 3/5 of 10/9.

30/45

13/14

2/3

6/5

Explanation: To simplify 3/5 of 10/9, we multiply the two fractions: (3/5) * (10/9). We can cross-cancel: 3 into 9 (gives 3 in denominator) and 5 into 10 (gives 2 in numerator). This results in (1/1) * (2/3) = 2/3.

Q182: Simplify 1/2 of 3/4 of 4/5.

3/20

3/10

1/5

12/40

Explanation: The word "of" indicates multiplication. So, we multiply the three fractions: (1/2) * (3/4) * (4/5). We can cross-cancel the 4 in the numerator of 4/5 with the 4 in the denominator of 3/4. This gives (1/2) * (3/1) * (1/5) = 3/10.

Q183: Simplify 1/3 of 1 1/2 of 2/5.

1/10

3/10

1/5

6/30

Explanation: First, convert the mixed number 1 1/2 to an improper fraction: 3/2. The expression becomes (1/3) * (3/2) * (2/5). We can cross-cancel the 3 in 1/3 with the 3 in 3/2, and the 2 in 3/2 with the 2 in 2/5. This leaves (1/1) * (1/1) * (1/5) = 1/5.

Q184: Simplify 2 1/4 of 2/3.

1 1/4

1/2

3/2

18/12

Explanation: First, convert the mixed number 2 1/4 to an improper fraction: 9/4. Then, multiply 9/4 by 2/3. (9/4) * (2/3). We can cross-cancel: 3 into 9 (gives 3 in numerator) and 2 into 4 (gives 2 in denominator). This results in (3/2) * (1/1) = 3/2.

Q185: What is the value of 3/4 of 120 minus 1/5 of 100?

110

Explanation: First, calculate '3/4 of 120' which is (3/4) * 120 = 90. Next, calculate '1/5 of 100' which is (1/5) * 100 = 20. Finally, subtract the second value from the first: 90 - 20 = 70.

Q186: A farmer has 240 mango trees. 1/3 of them are ripe, and 1/4 of the remaining trees are semi-ripe. How many trees are ripe or semi-ripe in total?

120

140

160

Explanation: Number of ripe trees = (1/3) * 240 = 80. Remaining trees = 240 - 80 = 160. Number of semi-ripe trees = (1/4) * 160 = 40. Total ripe or semi-ripe trees = 80 + 40 = 120.

Q187: Simplify: (2/5 of 75) ÷ (3/4 of 20)

1/2

Explanation: First, calculate '2/5 of 75' which is (2/5) * 75 = 2 * 15 = 30. Next, calculate '3/4 of 20' which is (3/4) * 20 = 3 * 5 = 15. Finally, divide the first result by the second: 30 ÷ 15 = 2.

Q188: After spending 2/5 of his money on books and 1/3 of the remaining money on food, John had Rs. 600 left. How much money did John have initially?

Rs. 1000

Rs. 1200

Rs. 1500

Rs. 900

Explanation: Let the initial money be X. After spending 2/5 on books, the remaining money is (1 - 2/5)X = (3/5)X. He then spent 1/3 of this remaining money on food, so the money left after food is (1 - 1/3) of (3/5)X = (2/3) * (3/5)X = (2/5)X. We are given that (2/5)X = Rs. 600. So, X = 600 * (5/2) = Rs. 1500.

Q189: Evaluate: (1/2 + 1/3) of 24 - (1/4 ÷ 1/8)

Explanation: First, evaluate the parentheses: (1/2 + 1/3) = (3/6 + 2/6) = 5/6. Then, calculate '5/6 of 24' which is (5/6) * 24 = 20. Next, evaluate the second parentheses: (1/4 ÷ 1/8) = (1/4) * (8/1) = 8/4 = 2. Finally, subtract: 20 - 2 = 18.

Q190: A tank is 1/3 full of water. If 1/4 of the water in the tank is removed, and then 120 liters are added, the tank becomes 1/2 full. What is the total capacity of the tank?

360 liters

480 liters

288 liters

720 liters

Explanation: Let the total capacity of the tank be C. Initially, the tank has (1/3)C water. If 1/4 of the water is removed, the amount removed is (1/4) * (1/3)C = (1/12)C. The remaining water is (1/3)C - (1/12)C = (4/12)C - (1/12)C = (3/12)C = (1/4)C. After adding 120 liters, the tank becomes 1/2 full, so (1/4)C + 120 = (1/2)C. Subtract (1/4)C from both sides: 120 = (1/2)C - (1/4)C = (2/4)C - (1/4)C = (1/4)C. Therefore, C = 120 * 4 = 480 liters.

Simplify expressions involving fractions with brackets.

Q191: Simplify: (1/2 + 1/3) × 6

6/5

Explanation: First, simplify the expression inside the brackets: 1/2 + 1/3 = 3/6 + 2/6 = 5/6. Then, multiply this result by 6: 5/6 × 6 = 5.

Q192: Simplify: (3/4 - 1/2) × 8

1/32

Explanation: First, simplify the expression inside the brackets: 3/4 - 1/2 = 3/4 - 2/4 = 1/4. Then, multiply this result by 8: 1/4 × 8 = 2.

Q193: Simplify: (2/3 + 1/6) × 1/2

5/6

1/6

5/12

10/6

Explanation: First, simplify the expression inside the brackets: 2/3 + 1/6 = 4/6 + 1/6 = 5/6. Then, multiply this result by 1/2: 5/6 × 1/2 = 5/12.

Q194: Simplify: (5/8 - 1/4) × 2/3

3/8

1/3

1/4

9/16

Explanation: First, simplify the expression inside the brackets: 5/8 - 1/4 = 5/8 - 2/8 = 3/8. Then, multiply this result by 2/3: 3/8 × 2/3 = 6/24 = 1/4.

Q195: Simplify: (1/5 + 3/10) × 10

1/2

8/3

Explanation: First, simplify the expression inside the brackets: 1/5 + 3/10 = 2/10 + 3/10 = 5/10 = 1/2. Then, multiply this result by 10: 1/2 × 10 = 5.

Q196: Simplify: (7/9 - 1/3) × 3/4

4/9

1/2

1/3

3/4

Explanation: First, simplify the expression inside the brackets: 7/9 - 1/3 = 7/9 - 3/9 = 4/9. Then, multiply this result by 3/4: 4/9 × 3/4 = 12/36 = 1/3.

Q197: Simplify the following expression: $ \frac{1}{2}(4x + 6) + \frac{1}{3}(3x - 9) $

$ 3x + 6 $

$ 3x $

$ x $

$ 3x - 6 $

Explanation: First, distribute the fractional coefficients into the brackets: $ \frac{1}{2}(4x) + \frac{1}{2}(6) + \frac{1}{3}(3x) - \frac{1}{3}(9) $. This simplifies to $ 2x + 3 + x - 3 $. Next, collect the like terms: $ (2x + x) + (3 - 3) $. This results in $ 3x + 0 = 3x $.

Q198: Simplify the following expression: $ \frac{1}{4}(8x - 12) - \frac{1}{2}(2x + 4) $

$ x - 1 $

$ 3x - 5 $

$ x - 5 $

$ 3x - 1 $

Explanation: First, distribute the fractional coefficients: $ \frac{1}{4}(8x) - \frac{1}{4}(12) - (\frac{1}{2}(2x) + \frac{1}{2}(4)) $. This becomes $ 2x - 3 - (x + 2) $. Be careful with the negative sign: $ 2x - 3 - x - 2 $. Next, collect like terms: $ (2x - x) + (-3 - 2) $. This results in $ x - 5 $.

Q199: Simplify the following expression: $ \frac{2}{3}(6x - 9y) + \frac{1}{5}(10x + 15y) $

$ 6x + 3y $

$ 2x - 3y $

$ 6x - 3y $

$ 6x - 9y $

Explanation: First, distribute the fractional coefficients: $ \frac{2}{3}(6x) - \frac{2}{3}(9y) + \frac{1}{5}(10x) + \frac{1}{5}(15y) $. This simplifies to $ 4x - 6y + 2x + 3y $. Next, collect the like terms: $ (4x + 2x) + (-6y + 3y) $. This results in $ 6x - 3y $.

Q200: Simplify the following expression: $ \frac{1}{2}(x + 4) + \frac{1}{3}(2x - 6) $

$ \frac{3x}{5} $

$ \frac{7x}{6} + 4 $

$ \frac{7x}{6} $

$ \frac{5x}{6} $

Explanation: First, distribute the fractional coefficients: $ \frac{1}{2}(x) + \frac{1}{2}(4) + \frac{1}{3}(2x) - \frac{1}{3}(6) $. This simplifies to $ \frac{x}{2} + 2 + \frac{2x}{3} - 2 $. Next, collect the like terms: $ (\frac{x}{2} + \frac{2x}{3}) + (2 - 2) $. To combine the x-terms, find a common denominator (6): $ \frac{3x}{6} + \frac{4x}{6} = \frac{7x}{6} $. The constant terms cancel out. The result is $ \frac{7x}{6} $.

Q201: Simplify the following expression: $ \frac{3}{4}(8x - 16) - \frac{1}{2}(6x - 10) $

$ 3x - 17 $

$ 9x - 7 $

$ 3x - 7 $

$ 3x + 17 $

Explanation: First, distribute the fractional coefficients: $ \frac{3}{4}(8x) - \frac{3}{4}(16) - (\frac{1}{2}(6x) - \frac{1}{2}(10)) $. This simplifies to $ 6x - 12 - (3x - 5) $. Be careful with the negative sign: $ 6x - 12 - 3x + 5 $. Next, collect like terms: $ (6x - 3x) + (-12 + 5) $. This results in $ 3x - 7 $.

Q202: Simplify the following expression: $ \frac{1}{3}(9a + 6b) - \frac{1}{4}(8a - 12b) $

$ a - b $

$ 5a + 5b $

$ a + 5b $

$ a + b $

Explanation: First, distribute the fractional coefficients: $ \frac{1}{3}(9a) + \frac{1}{3}(6b) - (\frac{1}{4}(8a) - \frac{1}{4}(12b)) $. This simplifies to $ 3a + 2b - (2a - 3b) $. Be careful with the negative sign: $ 3a + 2b - 2a + 3b $. Next, collect like terms: $ (3a - 2a) + (2b + 3b) $. This results in $ a + 5b $.

Q203: Simplify the following expression: `(x/3 + x/6) ÷ (x/2)`

2/9

x²/4

Explanation: First, simplify the terms within the first bracket: `x/3 + x/6 = 2x/6 + x/6 = 3x/6 = x/2`. Then, perform the division: `(x/2) ÷ (x/2) = 1`. The correct application of finding a common denominator within the bracket and then performing division leads to the answer.

Q204: Simplify the expression: `(1/a - 1/2a) ÷ (3/4a)`

-4/3

4/3

2/3

3/(8a²)

Explanation: First, simplify the term in the bracket: `1/a - 1/2a = 2/2a - 1/2a = 1/2a`. Then, perform the division: `(1/2a) ÷ (3/4a) = (1/2a) × (4a/3)`. Cancel out 'a' and simplify the numbers: `(1 × 4) / (2 × 3) = 4/6 = 2/3`.

Q205: Simplify: `( (2/3) + (1/6) ) ÷ ( (5/12) - (1/4) )`

1/3

5/36

2/3

Explanation: First, simplify the left bracket: `2/3 + 1/6 = 4/6 + 1/6 = 5/6`. Next, simplify the right bracket: `5/12 - 1/4 = 5/12 - 3/12 = 2/12 = 1/6`. Finally, divide the two results: `(5/6) ÷ (1/6) = (5/6) × (6/1) = 5`.

Q206: Simplify the expression: `[ (1/x) - (1/2x) ] ÷ ( (3/4x) + (1/8x) )`

-8/7

7/(16x²)

4/7

11/(8x)

Explanation: First, simplify the left bracket: `1/x - 1/2x = 2/2x - 1/2x = 1/2x`. Next, simplify the right bracket: `3/4x + 1/8x = 6/8x + 1/8x = 7/8x`. Finally, divide the two results: `(1/2x) ÷ (7/8x) = (1/2x) × (8x/7)`. Cancel out 'x' and simplify: `(1 × 8) / (2 × 7) = 8/14 = 4/7`.

Q207: Simplify the expression: `( (2m/5) + (m/10) ) ÷ ( (3m/4) - (m/2) )`

m/5

m²/8

3m/4

Explanation: First, simplify the left bracket: `2m/5 + m/10 = 4m/10 + m/10 = 5m/10 = m/2`. Next, simplify the right bracket: `3m/4 - m/2 = 3m/4 - 2m/4 = m/4`. Finally, divide the two results: `(m/2) ÷ (m/4) = (m/2) × (4/m)`. Cancel out 'm' and simplify: `4/2 = 2`.

Q208: Simplify: `[ (y/4) + (y/8) ] ÷ [ (5/6) - (1/3) ]`

y/6

3y/16

(3y+4)/8

3y/4

Explanation: First, simplify the left bracket: `y/4 + y/8 = 2y/8 + y/8 = 3y/8`. Next, simplify the right bracket: `5/6 - 1/3 = 5/6 - 2/6 = 3/6 = 1/2`. Finally, divide the two results: `(3y/8) ÷ (1/2) = (3y/8) × (2/1)`. Simplify: `(3y × 2) / 8 = 6y/8 = 3y/4`.

Identify and apply the BODMAS rule.

Q209: What is the value of 1/2 + 1/3 × 1/4?

5/24

7/12

1/7

2/3

Explanation: According to the BODMAS rule, multiplication should be performed before addition. First, calculate 1/3 × 1/4 = 1/12. Then, add 1/2 + 1/12 = 6/12 + 1/12 = 7/12.

Q210: Calculate the value of 3/4 - 1/2 ÷ 1/4.

-5/4

1/4

5/8

Explanation: According to the BODMAS rule, division should be performed before subtraction. First, calculate 1/2 ÷ 1/4 = 1/2 × 4/1 = 2. Then, subtract 3/4 - 2 = 3/4 - 8/4 = -5/4.

Q211: Find the value of 2/3 × 1/5 - 1/15.

4/45

1/15

7/30

Explanation: According to the BODMAS rule, multiplication should be performed before subtraction. First, calculate 2/3 × 1/5 = 2/15. Then, subtract 2/15 - 1/15 = 1/15.

Q212: Evaluate 5/6 ÷ 5/3 + 1/2.

5/13

1/4

17/9

Explanation: According to the BODMAS rule, division should be performed before addition. First, calculate 5/6 ÷ 5/3 = 5/6 × 3/5 = 1/2. Then, add 1/2 + 1/2 = 1.

Q213: What is the result of 1/2 - 1/3 × 1/4?

1/24

5/12

7/12

Explanation: According to the BODMAS rule, multiplication should be performed before subtraction. First, calculate 1/3 × 1/4 = 1/12. Then, subtract 1/2 - 1/12 = 6/12 - 1/12 = 5/12.

Q214: Calculate 7/10 - 1/5 ÷ 2/5.

5/4

1/5

3/4

31/50

Explanation: According to the BODMAS rule, division should be performed before subtraction. First, calculate 1/5 ÷ 2/5 = 1/5 × 5/2 = 1/2. Then, subtract 7/10 - 1/2 = 7/10 - 5/10 = 2/10 = 1/5.

Q215: Evaluate: (10 + 2) × 3 - 5

-24

Explanation: According to the BODMAS rule, operations inside brackets are performed first. So, (10 + 2) = 12. Then, multiplication (12 × 3 = 36) is performed before subtraction (36 - 5 = 31).

Q216: Evaluate: 20 ÷ 4 + 3 × 5 - 2

Explanation: According to the BODMAS rule, division and multiplication are performed from left to right before addition and subtraction. So, 20 ÷ 4 = 5 and 3 × 5 = 15. Then, addition and subtraction are performed from left to right: 5 + 15 = 20, and 20 - 2 = 18.

Q217: Evaluate: 8 × (12 - 6) ÷ 3 + 1

Explanation: According to the BODMAS rule, operations inside brackets are performed first: (12 - 6) = 6. Then, multiplication and division are performed from left to right: 8 × 6 = 48, and 48 ÷ 3 = 16. Finally, addition is performed: 16 + 1 = 17.

Q218: Evaluate: 40 - 20 + 10 ÷ 5 × 2

Explanation: According to the BODMAS rule, division and multiplication are performed from left to right before addition and subtraction. So, 10 ÷ 5 = 2, and 2 × 2 = 4. Then, addition and subtraction are performed from left to right: 40 - 20 = 20, and 20 + 4 = 24.

Q219: Evaluate: 12 ÷ 3 + 1/2 × (8 - 2)

Explanation: According to the BODMAS rule, operations inside brackets are performed first: (8 - 2) = 6. Then, division and multiplication are performed from left to right: 12 ÷ 3 = 4, and 1/2 × 6 = 3. Finally, addition is performed: 4 + 3 = 7.

Q220: Evaluate: 25 - [ 8 + (10 - 4) ÷ 2 ]

Explanation: According to the BODMAS rule, innermost brackets are solved first: (10 - 4) = 6. Then, operations inside the outer bracket are performed: 6 ÷ 2 = 3, and 8 + 3 = 11. Finally, subtraction is performed: 25 - 11 = 14.

Q221: What is the value of 1/2 + 3/4 ÷ 1/8?

13/2

19/32

7/8

Explanation: According to the BODMAS rule, division must be performed before addition. First, calculate 3/4 ÷ 1/8 = 3/4 × 8/1 = 24/4 = 6. Then, add 1/2 to the result: 1/2 + 6 = 1/2 + 12/2 = 13/2.

Q222: Simplify (2/3 - 1/6) × 4/5.

8/15

2/5

1/2

1/5

Explanation: According to the BODMAS rule, operations inside brackets must be performed first. First, calculate (2/3 - 1/6) = (4/6 - 1/6) = 3/6 = 1/2. Then, multiply the result by 4/5: 1/2 × 4/5 = 4/10 = 2/5.

Q223: Evaluate 5/6 × 3/10 + 2/3 ÷ 4/9.

33/16

7/4

2/3

1/4

Explanation: According to the BODMAS rule, multiplication and division must be performed before addition. First, calculate 5/6 × 3/10 = 15/60 = 1/4. Next, calculate 2/3 ÷ 4/9 = 2/3 × 9/4 = 18/12 = 3/2. Finally, add the results: 1/4 + 3/2 = 1/4 + 6/4 = 7/4.

Q224: Calculate 1/2 - [1/3 × (3/4 - 1/2)].

5/12

1/24

1/4

3/4

Explanation: Follow the BODMAS rule. Start with the innermost brackets: (3/4 - 1/2) = (3/4 - 2/4) = 1/4. Then, perform multiplication within the square brackets: 1/3 × 1/4 = 1/12. Finally, subtract from 1/2: 1/2 - 1/12 = 6/12 - 1/12 = 5/12.

Q225: Simplify 1 1/4 ÷ (2/3 + 1/6) - 1/2.

37/24

13/4

5/12

Explanation: First, convert the mixed number: 1 1/4 = 5/4. Next, apply BODMAS: perform the operation inside brackets: (2/3 + 1/6) = (4/6 + 1/6) = 5/6. Then, perform division: 5/4 ÷ 5/6 = 5/4 × 6/5 = 30/20 = 3/2. Finally, perform subtraction: 3/2 - 1/2 = 2/2 = 1.

Q226: What is the value of [2/5 + 1/2 × (3/4 - 1/5)] ÷ 1/10?

27/4

23/4

99/20

27/400

Explanation: Apply BODMAS: start with the innermost brackets: (3/4 - 1/5) = (15/20 - 4/20) = 11/20. Next, perform multiplication within the square brackets: 1/2 × 11/20 = 11/40. Then, perform addition within the square brackets: 2/5 + 11/40 = 16/40 + 11/40 = 27/40. Finally, perform division: 27/40 ÷ 1/10 = 27/40 × 10/1 = 270/40 = 27/4.

Solve problems involving fractions.

Q227: There are 12 pencils in a box. If 1/3 of the pencils are red, how many red pencils are there?

Explanation: To find 1/3 of 12, we multiply 12 by 1/3. (1/3) × 12 = 12/3 = 4. So, there are 4 red pencils.

Q228: A class has 30 students. If 2/5 of the students are girls, how many girls are there in the class?

Explanation: To find 2/5 of 30, we multiply 30 by 2/5. (2/5) × 30 = (2 × 30) / 5 = 60 / 5 = 12. So, there are 12 girls.

Q229: A baker made 48 cakes. He sold 3/4 of them. How many cakes did he sell?

Explanation: To find 3/4 of 48, we multiply 48 by 3/4. (3/4) × 48 = (3 × 48) / 4 = 144 / 4 = 36. So, he sold 36 cakes.

Q230: A book has 80 pages. Sam read 1/4 of the book. How many pages did Sam read?

Explanation: To find 1/4 of 80, we multiply 80 by 1/4. (1/4) × 80 = 80/4 = 20. So, Sam read 20 pages.

Q231: In a garden, there are 60 flowers. If 2/3 of them are roses, how many roses are there?

Explanation: To find 2/3 of 60, we multiply 60 by 2/3. (2/3) × 60 = (2 × 60) / 3 = 120 / 3 = 40. So, there are 40 roses.

Q232: A farmer has 75 chickens. He sold 2/5 of them. How many chickens did he sell?

Explanation: To find 2/5 of 75, we multiply 75 by 2/5. (2/5) × 75 = (2 × 75) / 5 = 150 / 5 = 30. So, he sold 30 chickens.

Q233: Kasun spent 1/4 of his salary on food and 1/3 on rent. What fraction of his salary is left?

2/7

7/12

5/12

5/7

Explanation: First, find the total fraction of salary spent: 1/4 + 1/3 = 3/12 + 4/12 = 7/12. Then, subtract this from the whole (1) to find the remaining fraction: 1 - 7/12 = 5/12.

Q234: A tank is 3/5 full of water. If 1/10 of the tank's capacity is removed, and then 1/4 of the tank's capacity is added, what fraction of the tank is now full?

1/2

13/20

3/4

7/10

Explanation: Start with 3/5. Subtract the removed amount: 3/5 - 1/10 = 6/10 - 1/10 = 5/10 = 1/2. Then add the amount added: 1/2 + 1/4 = 2/4 + 1/4 = 3/4.

Q235: In a class, 2/3 of the students like Mathematics, and 1/5 like Science. What fraction more students like Mathematics than Science?

1/2

13/15

7/15

1/15

Explanation: To find the difference, subtract the fraction of students who like Science from those who like Mathematics: 2/3 - 1/5. Find a common denominator, which is 15. So, 10/15 - 3/15 = 7/15.

Q236: A farmer planted 3/8 of his land with corn and 1/4 with beans. The rest of the land is used for paddy. What fraction of the land is used for paddy?

5/8

1/2

3/8

2/3

Explanation: First, find the total fraction of land used for corn and beans: 3/8 + 1/4 = 3/8 + 2/8 = 5/8. Then, subtract this from the whole (1) to find the fraction used for paddy: 1 - 5/8 = 3/8.

Q237: A baker used 1/6 of a sack of flour for cakes and 2/9 for bread. What fraction of the flour is left in the sack?

3/15

7/18

11/18

4/5

Explanation: First, find the total fraction of flour used: 1/6 + 2/9 = 3/18 + 4/18 = 7/18. Then, subtract this from the whole (1) to find the remaining fraction: 1 - 7/18 = 11/18.

Q238: In a library, 1/5 of the books are fiction, 3/10 are non-fiction, and the rest are reference books. What fraction more are reference books than fiction books?

1/2

1/5

3/10

1/10

Explanation: First, find the total fraction of fiction and non-fiction books: 1/5 + 3/10 = 2/10 + 3/10 = 5/10 = 1/2. The fraction of reference books is 1 - 1/2 = 1/2. Now, find the difference between reference books and fiction books: 1/2 - 1/5 = 5/10 - 2/10 = 3/10.

Q239: A person spent 1/3 of their salary on rent. From the remaining, they spent 1/4 on food. If they have Rs. 3000 left, what was their total salary?

Rs. 4500

Rs. 5000

Rs. 6000

Rs. 7200

Explanation: If 1/3 of the salary was spent on rent, then 1 - 1/3 = 2/3 of the salary remained. From this remaining 2/3, 1/4 was spent on food. This means 1 - 1/4 = 3/4 of the remaining amount was left. So, the fraction of the total salary left is (3/4) * (2/3) = 1/2. Since 1/2 of the total salary is Rs. 3000, the total salary is Rs. 3000 * 2 = Rs. 6000.

Q240: A water tank was 3/5 full. After 60 liters of water were removed, it was 1/2 full. What is the total capacity of the tank?

300 liters

450 liters

600 liters

900 liters

Explanation: The fraction of water removed from the tank is the difference between the initial and final fractions: 3/5 - 1/2. To subtract these fractions, find a common denominator, which is 10. So, 6/10 - 5/10 = 1/10. This means 1/10 of the tank's capacity is 60 liters. Therefore, the total capacity of the tank is 60 * 10 = 600 liters.

Q241: A farmer cultivated 2/5 of his land with rice and 1/3 of the remaining land with vegetables. If the uncultivated land is 20 acres, what is the total area of his land?

30 acres

40 acres

50 acres

60 acres

Explanation: If 2/5 of the land was cultivated with rice, then 1 - 2/5 = 3/5 of the land remained. From this remaining 3/5, 1/3 was cultivated with vegetables, which is (1/3) * (3/5) = 1/5 of the total land. So, the total cultivated land is 2/5 (rice) + 1/5 (vegetables) = 3/5 of the total land. The uncultivated land is 1 - 3/5 = 2/5 of the total land. Since 2/5 of the total land is 20 acres, 1/5 of the land is 10 acres. Therefore, the total area of his land is 10 * 5 = 50 acres.

Q242: On Monday, a student read 1/4 of a book. On Tuesday, she read 2/3 of the remaining pages. If 30 pages are still unread, how many pages are in the book?

90 pages

100 pages

120 pages

150 pages

Explanation: After reading 1/4 of the book on Monday, 1 - 1/4 = 3/4 of the pages remained. On Tuesday, she read 2/3 of these remaining pages, which means (2/3) * (3/4) = 1/2 of the total book. In total, she read 1/4 (Monday) + 1/2 (Tuesday) = 3/4 of the book. Therefore, the unread fraction is 1 - 3/4 = 1/4. Since 1/4 of the book is 30 pages, the total number of pages in the book is 30 * 4 = 120 pages.

Q243: A father distributed money among his three children. The eldest received 1/2 of the total. The second child received 1/3 of the remaining amount. The youngest received Rs. 4000. What was the total amount of money distributed?

Rs. 8000

Rs. 10000

Rs. 12000

Rs. 15000

Explanation: The eldest child received 1/2 of the total money, so 1 - 1/2 = 1/2 of the money remained. The second child received 1/3 of this remaining amount, which is (1/3) * (1/2) = 1/6 of the total money. The fraction of money remaining for the youngest child is 1/2 (after eldest) - 1/6 (second child) = 3/6 - 1/6 = 2/6 = 1/3 of the total money. Since 1/3 of the total money is Rs. 4000, the total amount distributed was Rs. 4000 * 3 = Rs. 12000.

Q244: A bus covered 2/5 of its journey on the first day. On the second day, it covered 1/3 of the remaining distance. If the remaining distance to be covered is 120 km, what is the total length of the journey?

200 km

250 km

300 km

360 km

Explanation: After covering 2/5 of the journey on the first day, 1 - 2/5 = 3/5 of the journey remained. On the second day, the bus covered 1/3 of this remaining distance, which is (1/3) * (3/5) = 1/5 of the total journey. So, the total distance covered in two days is 2/5 (Day 1) + 1/5 (Day 2) = 3/5 of the journey. The remaining distance to be covered is 1 - 3/5 = 2/5 of the total journey. Since 2/5 of the total journey is 120 km, then 1/5 of the journey is 60 km. Therefore, the total length of the journey is 60 * 5 = 300 km.

Q245: A school had a certain number of students. 1/5 of the students were absent on Monday. On Tuesday, 1/2 of the *remaining* students were absent. If 160 students were present on Tuesday, what is the total number of students in the school?

200

320

400

480

Explanation: If 1/5 of the students were absent on Monday, then 1 - 1/5 = 4/5 of the students were present/remaining. On Tuesday, 1/2 of these remaining students were absent, meaning 1/2 of 4/5 = 2/5 of the total students were absent on Tuesday. This implies that 1 - 1/2 = 1/2 of the remaining 4/5 were present on Tuesday. So, (1/2) * (4/5) = 2/5 of the total students were present on Tuesday. If 2/5 of the total students is 160, then 1/5 of the students is 80. Therefore, the total number of students in the school is 80 * 5 = 400.

Q246: A vendor sold 2/7 of his oranges in the morning. In the afternoon, he sold 3/5 of the *remaining* oranges. If he was left with 40 oranges, how many oranges did he have initially?

100

120

140

Explanation: After selling 2/7 of the oranges in the morning, 1 - 2/7 = 5/7 of the oranges remained. In the afternoon, he sold 3/5 of these remaining oranges. This means 1 - 3/5 = 2/5 of the remaining oranges were left. So, the fraction of total oranges left is (2/5) * (5/7) = 2/7. Since 2/7 of the total oranges is 40, then 1/7 of the oranges is 20. Therefore, the total number of oranges he had initially was 20 * 7 = 140 oranges.

Quantitatively find the profit or loss in a transaction.

Q247: An item was bought for Rs. 500 and sold for Rs. 650. What is the profit or loss amount?

Loss Rs. 150

Profit Rs. 150

Profit Rs. 1150

Loss Rs. 50

Explanation: The selling price (Rs. 650) is greater than the cost price (Rs. 500), so there is a profit. Profit = Selling Price - Cost Price = Rs. 650 - Rs. 500 = Rs. 150.

Q248: A book was bought for Rs. 1200 and sold for Rs. 1000. What is the profit or loss amount?

Profit Rs. 200

Loss Rs. 200

Profit Rs. 2200

Loss Rs. 1000

Explanation: The cost price (Rs. 1200) is greater than the selling price (Rs. 1000), so there is a loss. Loss = Cost Price - Selling Price = Rs. 1200 - Rs. 1000 = Rs. 200.

Q249: A pen was purchased for Rs. 75.50 and sold for Rs. 90.00. What is the profit or loss amount?

Loss Rs. 14.50

Profit Rs. 15.50

Profit Rs. 14.50

Loss Rs. 75.50

Explanation: The selling price (Rs. 90.00) is greater than the cost price (Rs. 75.50), so there is a profit. Profit = Selling Price - Cost Price = Rs. 90.00 - Rs. 75.50 = Rs. 14.50.

Q250: A toy car was bought for Rs. 350.75 and sold for Rs. 320.00. What is the profit or loss amount?

Profit Rs. 30.75

Loss Rs. 30.25

Loss Rs. 30.75

Profit Rs. 670.75

Explanation: The cost price (Rs. 350.75) is greater than the selling price (Rs. 320.00), so there is a loss. Loss = Cost Price - Selling Price = Rs. 350.75 - Rs. 320.00 = Rs. 30.75.

Q251: An article was purchased for Rs. 2000 and sold for Rs. 2350. What is the profit or loss, and its amount?

Loss Rs. 350

Profit Rs. 350

Profit Rs. 450

Loss Rs. 450

Explanation: Since the selling price (Rs. 2350) is greater than the cost price (Rs. 2000), there is a profit. The profit amount is Rs. 2350 - Rs. 2000 = Rs. 350.

Q252: A mobile phone was bought for Rs. 850 and sold for Rs. 780. What is the profit or loss, and its amount?

Profit Rs. 70

Loss Rs. 70

Profit Rs. 1630

Loss Rs. 1630

Explanation: Since the cost price (Rs. 850) is greater than the selling price (Rs. 780), there is a loss. The loss amount is Rs. 850 - Rs. 780 = Rs. 70.

Q253: A chair was purchased for Rs. 1500 and sold for Rs. 1850. What is the profit or loss amount?

Loss Rs. 350

Profit Rs. 350

Profit Rs. 300

Loss Rs. 150

Explanation: The selling price (Rs. 1850) is greater than the cost price (Rs. 1500), so there is a profit. Profit = Selling Price - Cost Price = Rs. 1850 - Rs. 1500 = Rs. 350.

Q254: A shopkeeper bought 15 identical storybooks for Rs. 80 each. He sold all of them for Rs. 100 each. What is his total profit?

Rs. 20

Rs. 300

Rs. 1200

Rs. 1500

Explanation: Total cost price = 15 books × Rs. 80/book = Rs. 1200. Total selling price = 15 books × Rs. 100/book = Rs. 1500. Total profit = Total selling price - Total cost price = Rs. 1500 - Rs. 1200 = Rs. 300.

Q255: A vendor bought 20 kg of mangoes at Rs. 150 per kg. Due to spoilage, 5 kg of mangoes became unusable. He sold the remaining mangoes at Rs. 220 per kg. What was his total profit or loss?

Rs. 300 loss

Rs. 300 profit

Rs. 700 profit

Rs. 750 loss

Explanation: Total cost of mangoes = 20 kg × Rs. 150/kg = Rs. 3000. Quantity of mangoes sold = 20 kg - 5 kg = 15 kg. Total selling price of mangoes = 15 kg × Rs. 220/kg = Rs. 3300. Total profit = Total selling price - Total cost = Rs. 3300 - Rs. 3000 = Rs. 300.

Q256: A stationery shop purchased 25 geometry boxes at Rs. 350 each. Due to a design change, they had to sell all of them at a reduced price of Rs. 320 each. What was the total loss incurred?

Rs. 30

Rs. 750

Rs. 8000

Rs. 8750

Explanation: Cost price per geometry box = Rs. 350. Selling price per geometry box = Rs. 320. Loss per geometry box = Rs. 350 - Rs. 320 = Rs. 30. Total number of geometry boxes = 25. Total loss = Loss per geometry box × Number of geometry boxes = Rs. 30 × 25 = Rs. 750.

Q257: A dealer bought 12 mobile phones for Rs. 25,000 each. He spent Rs. 6,000 on transportation and Rs. 2,000 on minor repairs for all phones. He then sold each phone for Rs. 26,500. What was his total profit?

Rs. 18,000

Rs. 10,000

Rs. 26,000

Rs. 30,000

Explanation: Total purchase cost of 12 mobile phones = 12 × Rs. 25,000 = Rs. 300,000. Additional expenses (transport + repairs) = Rs. 6,000 + Rs. 2,000 = Rs. 8,000. Total cost = Rs. 300,000 + Rs. 8,000 = Rs. 308,000. Total selling price of 12 mobile phones = 12 × Rs. 26,500 = Rs. 318,000. Total profit = Total selling price - Total cost = Rs. 318,000 - Rs. 308,000 = Rs. 10,000.

Q258: A fruit seller bought 50 pineapples at Rs. 120 each. He sold 30 of them at Rs. 150 each. The remaining 20 pineapples were slightly damaged, so he sold them at Rs. 100 each. Calculate his total profit or loss.

Rs. 1000 profit

Rs. 900 profit

Rs. 500 profit

Rs. 200 loss

Explanation: Total cost of 50 pineapples = 50 × Rs. 120 = Rs. 6000. Selling price of the first 30 pineapples = 30 × Rs. 150 = Rs. 4500. Selling price of the remaining 20 pineapples = 20 × Rs. 100 = Rs. 2000. Total selling price = Rs. 4500 + Rs. 2000 = Rs. 6500. Total profit = Total selling price - Total cost = Rs. 6500 - Rs. 6000 = Rs. 500.

Q259: A wholesaler bought 10 dozen eggs at Rs. 200 per dozen. He found that 15 eggs were broken. He sold the remaining eggs at Rs. 25 each. What was his total profit or loss? (1 dozen = 12 eggs)

Rs. 375 profit

Rs. 625 profit

Rs. 200 loss

Rs. 500 profit

Explanation: Total number of eggs bought = 10 dozen × 12 eggs/dozen = 120 eggs. Total cost of eggs = 10 dozen × Rs. 200/dozen = Rs. 2000. Number of broken eggs = 15. Number of eggs sold = 120 - 15 = 105 eggs. Total selling price = 105 eggs × Rs. 25/egg = Rs. 2625. Total profit = Total selling price - Total cost = Rs. 2625 - Rs. 2000 = Rs. 625.

Q260: A farmer bought 40 kg of potatoes for Rs. 4800. He spent Rs. 200 on transport. He then sold all the potatoes at Rs. 150 per kg. What was his total profit or loss?

Rs. 1200 profit

Rs. 1000 profit

Rs. 800 profit

Rs. 200 loss

Explanation: Total purchase cost of potatoes = Rs. 4800. Additional expense (transport) = Rs. 200. Total cost = Rs. 4800 + Rs. 200 = Rs. 5000. Total selling price = 40 kg × Rs. 150/kg = Rs. 6000. Total profit = Total selling price - Total cost = Rs. 6000 - Rs. 5000 = Rs. 1000.

Q261: An item was sold for Rs. 500, making a profit of Rs. 100. What was the original cost price of the item?

Rs. 600

Rs. 400

Rs. 100

Rs. 450

Explanation: When there is a profit, the cost price is found by subtracting the profit from the selling price. Cost Price = Selling Price - Profit = Rs. 500 - Rs. 100 = Rs. 400.

Q262: A book was sold for Rs. 800, incurring a loss of Rs. 150. What was the original cost price of the book?

Rs. 650

Rs. 950

Rs. 150

Rs. 900

Explanation: When there is a loss, the cost price is found by adding the loss to the selling price. Cost Price = Selling Price + Loss = Rs. 800 + Rs. 150 = Rs. 950.

Q263: A chair was sold for Rs. 1200, making a profit of Rs. 250. What was the original cost price of the chair?

Rs. 1450

Rs. 950

Rs. 250

Rs. 1000

Explanation: To find the cost price when there is a profit, subtract the profit from the selling price. Cost Price = Selling Price - Profit = Rs. 1200 - Rs. 250 = Rs. 950.

Q264: A toy was sold for Rs. 650, incurring a loss of Rs. 80. What was the original cost price of the toy?

Rs. 570

Rs. 730

Rs. 80

Rs. 700

Explanation: To find the cost price when there is a loss, add the loss to the selling price. Cost Price = Selling Price + Loss = Rs. 650 + Rs. 80 = Rs. 730.

Q265: A bicycle was sold for Rs. 2500, making a profit of Rs. 300. What was the original cost price of the bicycle?

Rs. 2800

Rs. 2200

Rs. 300

Rs. 2300

Explanation: When an item is sold at a profit, the cost price is calculated by subtracting the profit from the selling price. Cost Price = Selling Price - Profit = Rs. 2500 - Rs. 300 = Rs. 2200.

Q266: A mobile phone was sold for Rs. 3200, incurring a loss of Rs. 400. What was the original cost price of the mobile phone?

Rs. 2800

Rs. 3600

Rs. 400

Rs. 3000

Explanation: When an item is sold at a loss, the cost price is calculated by adding the loss to the selling price. Cost Price = Selling Price + Loss = Rs. 3200 + Rs. 400 = Rs. 3600.

Calculate the percentage of profit or loss and solve related problems.

Q267: A shopkeeper bought an item for Rs. 200 and sold it making a profit of Rs. 40. What is the percentage profit?

40%

20%

25%

10%

Explanation: Percentage profit is calculated as (Profit / Cost Price) × 100%. Here, Profit = Rs. 40 and Cost Price = Rs. 200. So, (40 / 200) × 100% = (1/5) × 100% = 20%.

Q268: An item was bought for Rs. 500 and sold at a loss of Rs. 100. What is the percentage loss?

10%

20%

50%

25%

Explanation: Percentage loss is calculated as (Loss / Cost Price) × 100%. Here, Loss = Rs. 100 and Cost Price = Rs. 500. So, (100 / 500) × 100% = (1/5) × 100% = 20%.

Q269: A dealer purchased a bicycle for Rs. 250 and made a profit of Rs. 75 upon selling it. What is the percentage profit?

25%

75%

30%

Explanation: Percentage profit = (Profit / Cost Price) × 100%. Here, Profit = Rs. 75 and Cost Price = Rs. 250. So, (75 / 250) × 100% = (3/10) × 100% = 30%.

Q270: A mobile phone was bought for Rs. 800 and sold for a loss of Rs. 160. What is the percentage loss?

16%

25%

80%

20%

Explanation: Percentage loss = (Loss / Cost Price) × 100%. Here, Loss = Rs. 160 and Cost Price = Rs. 800. So, (160 / 800) × 100% = (1/5) × 100% = 20%.

Q271: A wholesaler bought a batch of goods for Rs. 1200 and sold them for a profit of Rs. 300. What is the percentage profit?

30%

20%

25%

12%

Explanation: Percentage profit = (Profit / Cost Price) × 100%. Here, Profit = Rs. 300 and Cost Price = Rs. 1200. So, (300 / 1200) × 100% = (1/4) × 100% = 25%.

Q272: A refrigerator was purchased for Rs. 1500 and sold at a loss of Rs. 450. What is the percentage loss?

45%

25%

30%

15%

Explanation: Percentage loss = (Loss / Cost Price) × 100%. Here, Loss = Rs. 450 and Cost Price = Rs. 1500. So, (450 / 1500) × 100% = (3/10) × 100% = 30%.

Q273: An item was bought for Rs. 200 and sold for Rs. 250. What is the percentage profit?

20%

25%

30%

50%

Explanation: Cost Price (CP) = Rs. 200, Selling Price (SP) = Rs. 250. Since SP > CP, there is a profit. Profit = SP - CP = 250 - 200 = Rs. 50. Percentage Profit = (Profit / CP) × 100% = (50 / 200) × 100% = (1/4) × 100% = 25%.

Q274: A book was bought for Rs. 500 and sold for Rs. 400. What is the percentage loss?

10%

20%

25%

100%

Explanation: Cost Price (CP) = Rs. 500, Selling Price (SP) = Rs. 400. Since CP > SP, there is a loss. Loss = CP - SP = 500 - 400 = Rs. 100. Percentage Loss = (Loss / CP) × 100% = (100 / 500) × 100% = (1/5) × 100% = 20%.

Q275: A dealer bought a bicycle for Rs. 800 and sold it for Rs. 960. What is the percentage profit?

15%

16%

20%

25%

Explanation: Cost Price (CP) = Rs. 800, Selling Price (SP) = Rs. 960. Since SP > CP, there is a profit. Profit = SP - CP = 960 - 800 = Rs. 160. Percentage Profit = (Profit / CP) × 100% = (160 / 800) × 100% = (16 / 80) × 100% = (1/5) × 100% = 20%.

Q276: A mobile phone was purchased for Rs. 1200 and sold for Rs. 1080. What is the percentage loss?

10%

11.1%

12%

Explanation: Cost Price (CP) = Rs. 1200, Selling Price (SP) = Rs. 1080. Since CP > SP, there is a loss. Loss = CP - SP = 1200 - 1080 = Rs. 120. Percentage Loss = (Loss / CP) × 100% = (120 / 1200) × 100% = (12 / 120) × 100% = (1/10) × 100% = 10%.

Q277: A toy car was bought for Rs. 400 and sold for Rs. 460. What is the percentage profit?

10%

13.04%

15%

20%

Explanation: Cost Price (CP) = Rs. 400, Selling Price (SP) = Rs. 460. Since SP > CP, there is a profit. Profit = SP - CP = 460 - 400 = Rs. 60. Percentage Profit = (Profit / CP) × 100% = (60 / 400) × 100% = (6 / 40) × 100% = (3 / 20) × 100% = 15%.

Q278: A watch was purchased for Rs. 750 and sold for Rs. 600. What is the percentage loss?

15%

20%

25%

30%

Explanation: Cost Price (CP) = Rs. 750, Selling Price (SP) = Rs. 600. Since CP > SP, there is a loss. Loss = CP - SP = 750 - 600 = Rs. 150. Percentage Loss = (Loss / CP) × 100% = (150 / 750) × 100% = (15 / 75) × 100% = (1/5) × 100% = 20%.

Q279: An item was sold for Rs. 1200, making a profit of 20%. What was the original cost price of the item?

Rs. 960

Rs. 1000

Rs. 1440

Rs. 1500

Explanation: Let the Cost Price (CP) be x. If there is a 20% profit, the Selling Price (SP) is 120% of the CP. So, SP = CP × (100 + 20)/100 = CP × 1.20. Given SP = Rs. 1200. Therefore, 1200 = CP × 1.20. CP = 1200 / 1.20 = Rs. 1000.

Q280: A book was sold for Rs. 750, incurring a loss of 25%. What was the original cost price of the book?

Rs. 562.50

Rs. 937.50

Rs. 1000

Rs. 600

Explanation: Let the Cost Price (CP) be x. If there is a 25% loss, the Selling Price (SP) is 75% of the CP. So, SP = CP × (100 - 25)/100 = CP × 0.75. Given SP = Rs. 750. Therefore, 750 = CP × 0.75. CP = 750 / 0.75 = Rs. 1000.

Q281: A mobile phone was sold for Rs. 2600, making a profit of 30%. What was its original cost price?

Rs. 1820

Rs. 2000

Rs. 3380

Rs. 3714.28

Explanation: Let the Cost Price (CP) be x. If there is a 30% profit, the Selling Price (SP) is 130% of the CP. So, SP = CP × (100 + 30)/100 = CP × 1.30. Given SP = Rs. 2600. Therefore, 2600 = CP × 1.30. CP = 2600 / 1.30 = Rs. 2000.

Q282: A refrigerator was sold for Rs. 1350, incurring a loss of 10%. What was its original cost price?

Rs. 1215

Rs. 1485

Rs. 1500

Rs. 1227.27

Explanation: Let the Cost Price (CP) be x. If there is a 10% loss, the Selling Price (SP) is 90% of the CP. So, SP = CP × (100 - 10)/100 = CP × 0.90. Given SP = Rs. 1350. Therefore, 1350 = CP × 0.90. CP = 1350 / 0.90 = Rs. 1500.

Q283: A television set was sold for Rs. 4500, making a profit of 12.5%. What was its original cost price?

Rs. 3937.50

Rs. 4000

Rs. 5062.50

Rs. 5142.86

Explanation: Let the Cost Price (CP) be x. If there is a 12.5% profit, the Selling Price (SP) is 112.5% of the CP. So, SP = CP × (100 + 12.5)/100 = CP × 1.125. Given SP = Rs. 4500. Therefore, 4500 = CP × 1.125. CP = 4500 / 1.125 = Rs. 4000.

Q284: A bicycle was sold for Rs. 6800, incurring a loss of 15%. What was its original cost price?

Rs. 5780

Rs. 7820

Rs. 8000

Rs. 5913.04

Explanation: Let the Cost Price (CP) be x. If there is a 15% loss, the Selling Price (SP) is 85% of the CP. So, SP = CP × (100 - 15)/100 = CP × 0.85. Given SP = Rs. 6800. Therefore, 6800 = CP × 0.85. CP = 6800 / 0.85 = Rs. 8000.

Identify what discounts and commissions are.

Q285: An item is priced at Rs. 2500. If there is a 10% discount, what is the discount amount?

Rs. 25

Rs. 250

Rs. 2250

Rs. 2750

Explanation: The discount amount is calculated by finding 10% of the original price. 10% of Rs. 2500 = (10/100) * 2500 = Rs. 250.

Q286: A salesperson earns a 5% commission on their total sales. If they make sales worth Rs. 15,000, what is their commission amount?

Rs. 75

Rs. 750

Rs. 14,250

Rs. 15,750

Explanation: The commission amount is calculated by finding 5% of the total sales. 5% of Rs. 15,000 = (5/100) * 15,000 = Rs. 750.

Q287: A book originally costs Rs. 800. If a 15% discount is offered, how much is the discount?

Rs. 15

Rs. 120

Rs. 680

Rs. 920

Explanation: The discount amount is calculated by finding 15% of the original price. 15% of Rs. 800 = (15/100) * 800 = Rs. 120.

Q288: A real estate agent receives a 2% commission on a property sale. If they sell a property for Rs. 5,000,000, what is their commission?

Rs. 1000

Rs. 10,000

Rs. 100,000

Rs. 4,900,000

Explanation: The commission amount is calculated by finding 2% of the total sales. 2% of Rs. 5,000,000 = (2/100) * 5,000,000 = Rs. 100,000.

Q289: A bicycle is sold for Rs. 12,000 with a 8% discount. What is the amount of the discount?

Rs. 800

Rs. 960

Rs. 11,040

Rs. 12,960

Explanation: The discount amount is calculated by finding 8% of the original price. 8% of Rs. 12,000 = (8/100) * 12,000 = Rs. 960.

Q290: A sales representative earns a 3.5% commission on sales. If their sales total Rs. 40,000, what is their commission?

Rs. 140

Rs. 1400

Rs. 38,600

Rs. 41,400

Explanation: The commission amount is calculated by finding 3.5% of the total sales. 3.5% of Rs. 40,000 = (3.5/100) * 40,000 = Rs. 1400.

Q291: A book originally priced at Rs. 2000 is sold with a 10% discount. What is the final price of the book?

Rs. 200

Rs. 1800

Rs. 2200

Rs. 2000

Explanation: The discount amount is 10% of Rs. 2000, which is (10/100) * 2000 = Rs. 200. The final price is the original price minus the discount: 2000 - 200 = Rs. 1800.

Q292: A salesperson earns a 5% commission on all sales. If they made sales worth Rs. 50,000 in a month, what is their commission amount?

Rs. 250

Rs. 5000

Rs. 2500

Rs. 47500

Explanation: The commission amount is 5% of the total sales. Commission = (5/100) * 50,000 = Rs. 2500.

Q293: A television is priced at Rs. 15,000. During a promotion, a 20% discount is offered. What is the price of the television after the discount?

Rs. 3000

Rs. 12000

Rs. 15000

Rs. 18000

Explanation: The discount amount is 20% of Rs. 15,000, which is (20/100) * 15,000 = Rs. 3000. The final price is 15,000 - 3000 = Rs. 12,000.

Q294: A sales executive receives a basic monthly salary of Rs. 25,000 and a 3% commission on sales. If their sales for the month totaled Rs. 100,000, what are their total earnings for the month?

Rs. 25,000

Rs. 3,000

Rs. 28,000

Rs. 280,000

Explanation: The commission on sales is 3% of Rs. 100,000, which is (3/100) * 100,000 = Rs. 3000. Total earnings = Basic salary + Commission = 25,000 + 3000 = Rs. 28,000.

Q295: A jacket is priced at Rs. 1200. If a 15% discount is applied, what is the final selling price of the jacket?

Rs. 180

Rs. 1020

Rs. 1200

Rs. 1380

Explanation: The discount amount is 15% of Rs. 1200, which is (15/100) * 1200 = Rs. 180. The final selling price is 1200 - 180 = Rs. 1020.

Q296: A property agent receives a monthly base salary of Rs. 30,000. In addition, they get a 2% commission on the value of properties sold. If they sold properties worth Rs. 250,000 in a month, what is their total income for that month?

Rs. 5,000

Rs. 30,000

Rs. 35,000

Rs. 280,000

Explanation: The commission on sales is 2% of Rs. 250,000, which is (2/100) * 250,000 = Rs. 5000. Total income = Basic salary + Commission = 30,000 + 5000 = Rs. 35,000.

Q297: An item was sold for Rs. 800 after a 20% discount. What was the original price of the item?

Rs. 960

Rs. 1000

Rs. 640

Rs. 4000

Explanation: If there is a 20% discount, the selling price represents 100% - 20% = 80% of the original price. So, 80% of the original price = Rs. 800. To find the original price, divide 800 by 0.8 (or multiply by 100/80): 800 / 0.8 = Rs. 1000.

Q298: A salesperson received a commission of Rs. 450, which is 5% of their total sales. What was the total value of sales made?

Rs. 22.50

Rs. 9000

Rs. 90

Rs. 45000

Explanation: The commission of Rs. 450 represents 5% of the total sales. To find the total sales, divide the commission amount by the commission percentage (as a decimal): 450 / 0.05 = Rs. 9000.

Q299: After a 10% discount, a book was sold for Rs. 1350. What was the original price of the book?

Rs. 1485

Rs. 1500

Rs. 1215

Rs. 13500

Explanation: A 10% discount means the selling price is 100% - 10% = 90% of the original price. Therefore, 90% of the original price = Rs. 1350. To find the original price, divide 1350 by 0.9: 1350 / 0.9 = Rs. 1500.

Q300: An agent earned a commission of Rs. 720 for a sale, which was 8% of the total sales value. What was the total sales value?

Rs. 57.60

Rs. 9000

Rs. 90

Rs. 72000

Explanation: The commission of Rs. 720 represents 8% of the total sales value. To calculate the total sales value, divide the commission amount by the commission percentage (as a decimal): 720 / 0.08 = Rs. 9000.

Q301: A piece of furniture was sold for Rs. 1800 after a 25% discount. What was its original price?

Rs. 2250

Rs. 2400

Rs. 1350

Rs. 7200

Explanation: With a 25% discount, the selling price is 100% - 25% = 75% of the original price. So, 75% of the original price = Rs. 1800. To find the original price, divide 1800 by 0.75: 1800 / 0.75 = Rs. 2400.

Q302: A real estate agent received Rs. 1200 as commission, which is 6% of the property's selling price. What was the selling price of the property?

Rs. 72

Rs. 20000

Rs. 200

Rs. 120000

Explanation: The commission of Rs. 1200 represents 6% of the property's selling price. To find the total selling price, divide the commission amount by the commission percentage (as a decimal): 1200 / 0.06 = Rs. 20000.

Perform calculations related to discounts and commissions.

Q303: A laptop is priced at Rs. 20000. If a 10% discount is offered, what is the final selling price of the laptop?

Rs. 2000

Rs. 19000

Rs. 18000

Rs. 22000

Explanation: To find the final selling price, first calculate the discount amount. Discount = 10% of Rs. 20000 = (10/100) * 20000 = Rs. 2000. Then, subtract the discount from the original price: Final Selling Price = Rs. 20000 - Rs. 2000 = Rs. 18000.

Q304: A washing machine originally costs Rs. 45000. If there is a 15% discount, what is the discount amount?

Rs. 6000

Rs. 6750

Rs. 38250

Rs. 51750

Explanation: To find the discount amount, calculate 15% of the original price. Discount = 15% of Rs. 45000 = (15/100) * 45000 = 15 * 450 = Rs. 6750.

Q305: A book has a marked price of Rs. 800. If a customer receives a 25% discount, how much does the customer pay for the book?

Rs. 200

Rs. 775

Rs. 1000

Rs. 600

Explanation: First, calculate the discount amount. Discount = 25% of Rs. 800 = (25/100) * 800 = Rs. 200. Then, subtract the discount from the marked price to find the amount paid: Amount Paid = Rs. 800 - Rs. 200 = Rs. 600.

Q306: A bicycle is sold for Rs. 15000 with a 8% discount. What is the discount amount on the bicycle?

Rs. 1200

Rs. 13800

Rs. 14992

Rs. 16200

Explanation: To find the discount amount, calculate 8% of the original price. Discount = 8% of Rs. 15000 = (8/100) * 15000 = 8 * 150 = Rs. 1200.

Q307: A pair of shoes is originally priced at Rs. 3500. If a store offers a 20% discount, what is the final price of the shoes?

Rs. 700

Rs. 2800

Rs. 3300

Rs. 4200

Explanation: First, calculate the discount amount. Discount = 20% of Rs. 3500 = (20/100) * 3500 = Rs. 700. Then, subtract the discount from the original price: Final Price = Rs. 3500 - Rs. 700 = Rs. 2800.

Q308: A television is marked at Rs. 60000. If it is sold with a 12% discount, what is its selling price after the discount?

Rs. 7200

Rs. 58800

Rs. 52800

Rs. 67200

Explanation: First, calculate the discount amount. Discount = 12% of Rs. 60000 = (12/100) * 60000 = 12 * 600 = Rs. 7200. Then, subtract the discount from the marked price: Selling Price = Rs. 60000 - Rs. 7200 = Rs. 52800.

Q309: A jacket is sold for Rs. 4500 after a 10% discount. What was the original price of the jacket?

Rs. 4050

Rs. 4950

Rs. 5000

Rs. 4600

Explanation: If there is a 10% discount, the selling price represents 100% - 10% = 90% of the original price. So, 90% of the original price is Rs. 4500. To find the original price, divide the discounted price by 90% (or 0.90): Original Price = 4500 / 0.90 = Rs. 5000.

Q310: An item was bought for Rs. 720 after a 20% discount. What was its original price?

Rs. 864

Rs. 900

Rs. 576

Rs. 740

Explanation: A 20% discount means the item was sold at 100% - 20% = 80% of its original price. So, 80% of the original price is Rs. 720. To find the original price, divide the discounted price by 80% (or 0.80): Original Price = 720 / 0.80 = Rs. 900.

Q311: A book originally priced at Rs. 800 is sold for Rs. 680. What is the percentage discount?

12%

15%

17.65%

20%

Explanation: First, calculate the discount amount: Rs. 800 - Rs. 680 = Rs. 120. Then, calculate the percentage discount by dividing the discount amount by the original price and multiplying by 100: (120 / 800) * 100% = (12 / 80) * 100% = 15%.

Q312: A smartphone with an original price of Rs. 60,000 is sold for Rs. 54,000 during a promotion. What is the percentage discount offered?

10%

11.11%

12%

Explanation: The discount amount is Rs. 60,000 - Rs. 54,000 = Rs. 6,000. To find the percentage discount, divide the discount amount by the original price and multiply by 100: (6,000 / 60,000) * 100% = (1 / 10) * 100% = 10%.

Q313: A real estate agent earned a commission of Rs. 25,000 at a commission rate of 5%. What was the total value of the property sold?

Rs. 1,250

Rs. 475,000

Rs. 500,000

Rs. 525,000

Explanation: The commission earned (Rs. 25,000) represents 5% of the total sales value. To find the total sales value, divide the commission earned by the commission rate (as a decimal): Total Sales = 25,000 / 0.05 = Rs. 500,000.

Q314: A salesperson receives a 4% commission on all sales. If they earned Rs. 12,000 in commission last month, what was their total sales value?

Rs. 480

Rs. 288,000

Rs. 300,000

Rs. 312,000

Explanation: The commission of Rs. 12,000 represents 4% of the total sales. To find the total sales value, divide the commission by the commission rate (as a decimal): Total Sales = 12,000 / 0.04 = Rs. 300,000.

Q315: An item is priced at Rs. 5000. A store offers a 10% discount on the item. A salesperson earns a 5% commission on the *discounted selling price*. What is the salesperson's commission?

Rs. 250

Rs. 450

Rs. 225

Rs. 500

Explanation: First, calculate the discounted price: Rs. 5000 - (10% of Rs. 5000) = Rs. 5000 - Rs. 500 = Rs. 4500. Then, calculate the commission on the discounted price: 5% of Rs. 4500 = 0.05 * 4500 = Rs. 225.

Q316: A washing machine is originally priced at Rs. 10,000. It is sold with a 20% discount. An agent who sells the machine receives a 10% commission on the *discounted selling price*. What is the net amount the shop receives from the sale after paying the agent's commission?

Rs. 8,000

Rs. 7,000

Rs. 7,200

Rs. 9,200

Explanation: First, calculate the discounted price: Rs. 10,000 - (20% of Rs. 10,000) = Rs. 10,000 - Rs. 2,000 = Rs. 8,000. Next, calculate the commission: 10% of Rs. 8,000 = Rs. 800. Finally, the net amount the shop receives is the discounted price minus the commission: Rs. 8,000 - Rs. 800 = Rs. 7,200.

Q317: An item is priced at Rs. 12,000. Salesperson A sells it with a 15% discount and earns an 8% commission on the discounted price. Salesperson B sells the same item with a 10% discount and earns a 10% commission on the discounted price. Who earns more commission and by how much?

Salesperson A by Rs. 264

Salesperson B by Rs. 264

Salesperson B by Rs. 180

They earn the same commission.

Explanation: For Salesperson A: Discounted price = Rs. 12,000 * (1 - 0.15) = Rs. 12,000 * 0.85 = Rs. 10,200. Commission A = Rs. 10,200 * 0.08 = Rs. 816. For Salesperson B: Discounted price = Rs. 12,000 * (1 - 0.10) = Rs. 12,000 * 0.90 = Rs. 10,800. Commission B = Rs. 10,800 * 0.10 = Rs. 1,080. Salesperson B earns more commission by Rs. 1,080 - Rs. 816 = Rs. 264.

Q318: An agent earns a commission of Rs. 360 from selling an item. A 20% discount was given on the item, and the agent's commission rate is 10% on the *discounted selling price*. What was the original price of the item?

Rs. 3,600

Rs. 4,000

Rs. 4,500

Rs. 5,400

Explanation: Let the discounted price be 'D'. Since the commission is 10% of the discounted price and equals Rs. 360, we have 0.10 * D = 360. So, D = 360 / 0.10 = Rs. 3,600. The discounted price is 80% (100% - 20% discount) of the original price. Let the original price be 'O'. Then 0.80 * O = 3,600. Therefore, O = 3,600 / 0.80 = Rs. 4,500.

Q319: A salesperson earns a basic monthly salary of Rs. 15,000. Additionally, they receive a 12% commission on the *discounted selling price* of items sold. If they sell an item originally priced at Rs. 8,000 after a 5% discount, what are their total earnings for that month?

Rs. 15,960

Rs. 15,912

Rs. 912

Rs. 15,000

Explanation: First, calculate the discounted price: Rs. 8,000 - (5% of Rs. 8,000) = Rs. 8,000 - Rs. 400 = Rs. 7,600. Next, calculate the commission: 12% of Rs. 7,600 = 0.12 * 7,600 = Rs. 912. Finally, add the commission to the basic salary to find the total earnings: Rs. 15,000 + Rs. 912 = Rs. 15,912.

Q320: An item is priced at Rs. 7,500. A 20% discount is offered. A salesperson receives an 8% commission on the *discounted price*. If an alternative commission structure was applied where the salesperson received 6% commission on the *original price* (before discount), what would be the *difference* in the commission earned under these two structures?

Rs. 480

Rs. 450

Rs. 30

Rs. 150

Explanation: Scenario 1 (Commission on discounted price): Discounted price = Rs. 7,500 * (1 - 0.20) = Rs. 7,500 * 0.80 = Rs. 6,000. Commission 1 = Rs. 6,000 * 0.08 = Rs. 480. Scenario 2 (Commission on original price): Commission 2 = Rs. 7,500 * 0.06 = Rs. 450. The difference in commission is Rs. 480 - Rs. 450 = Rs. 30.

Find the value of simple algebraic expressions by substituting directed numbers.

Q321: If x = -3, what is the value of x + 7?

-10

-4

Explanation: Substitute x = -3 into the expression: -3 + 7 = 4.

Q322: Find the value of 5 - y when y = -2.

-7

-3

Explanation: Substitute y = -2 into the expression: 5 - (-2) = 5 + 2 = 7.

Q323: What is the value of a - 4 if a = -6?

-2

-10

Explanation: Substitute a = -6 into the expression: -6 - 4 = -10.

Q324: Evaluate 3k when k = -5.

-2

-15

Explanation: Substitute k = -5 into the expression: 3 × (-5) = -15.

Q325: If m = -8, calculate the value of m / -2.

-4

-16

Explanation: Substitute m = -8 into the expression: -8 ÷ (-2) = 4.

Q326: Find the value of -4p when p = 3.

-1

-12

Explanation: Substitute p = 3 into the expression: -4 × 3 = -12.

Q327: If x = 2 and y = -3, what is the value of 3x - 2y?

Explanation: Substitute x = 2 and y = -3 into the expression: 3(2) - 2(-3). This simplifies to 6 - (-6), which is 6 + 6 = 12.

Q328: Find the value of a^2 + 4b when a = -3 and b = 5.

-1

Explanation: Substitute a = -3 and b = 5 into the expression: (-3)^2 + 4(5). First, calculate the exponent: (-3)^2 = 9. Then, the multiplication: 4(5) = 20. Finally, add: 9 + 20 = 29.

Q329: What is the value of (p - q) / r when p = 8, q = -4, and r = 3?

4/3

-4/3

Explanation: Substitute p = 8, q = -4, and r = 3 into the expression: (8 - (-4)) / 3. First, simplify inside the parenthesis: 8 - (-4) = 8 + 4 = 12. Then, perform the division: 12 / 3 = 4.

Q330: Evaluate xy - z^2 for x = -2, y = 5, and z = -3.

-1

-19

Explanation: Substitute x = -2, y = 5, and z = -3 into the expression: (-2)(5) - (-3)^2. According to the order of operations, first calculate the exponent: (-3)^2 = 9. Then, the multiplication: (-2)(5) = -10. Finally, subtract: -10 - 9 = -19.

Q331: If m = -4 and n = -2, what is the value of 2m - 3(n + 1)?

-11

-5

-2

Explanation: Substitute m = -4 and n = -2 into the expression: 2(-4) - 3(-2 + 1). First, simplify inside the parenthesis: -2 + 1 = -1. So, the expression becomes 2(-4) - 3(-1). Then, perform multiplications: 2(-4) = -8 and 3(-1) = -3. Finally, subtract: -8 - (-3) = -8 + 3 = -5.

Q332: Calculate the value of ab + c - 2 when a = 3, b = -2, and c = 5.

-1

-3

Explanation: Substitute a = 3, b = -2, and c = 5 into the expression: (3)(-2) + 5 - 2. First, perform the multiplication: (3)(-2) = -6. Then, perform addition and subtraction from left to right: -6 + 5 - 2 = -1 - 2 = -3.

Q333: If 3x - 4 = 11, what is the value of x + 7?

Explanation: First, solve the linear equation 3x - 4 = 11. Add 4 to both sides: 3x = 15. Divide by 3: x = 5. Then, substitute x = 5 into the expression x + 7: 5 + 7 = 12.

Q334: If 2y + 5 = -3, what is the value of 4y - 1?

-4

-17

-15

Explanation: First, solve the linear equation 2y + 5 = -3. Subtract 5 from both sides: 2y = -8. Divide by 2: y = -4. Then, substitute y = -4 into the expression 4y - 1: 4(-4) - 1 = -16 - 1 = -17.

Q335: If 5 - 2a = 15, what is the value of 3a + 10?

-5

-25

Explanation: First, solve the linear equation 5 - 2a = 15. Subtract 5 from both sides: -2a = 10. Divide by -2: a = -5. Then, substitute a = -5 into the expression 3a + 10: 3(-5) + 10 = -15 + 10 = -5.

Q336: If x/2 - 3 = 1, what is the value of 2x - 5?

-13

Explanation: First, solve the linear equation x/2 - 3 = 1. Add 3 to both sides: x/2 = 4. Multiply by 2: x = 8. Then, substitute x = 8 into the expression 2x - 5: 2(8) - 5 = 16 - 5 = 11.

Q337: If 7 - b = 12, what is the value of 5b + 3?

-5

-22

-28

Explanation: First, solve the linear equation 7 - b = 12. Subtract 7 from both sides: -b = 5. Multiply by -1: b = -5. Then, substitute b = -5 into the expression 5b + 3: 5(-5) + 3 = -25 + 3 = -22.

Q338: If (m + 1) / 3 = 4, what is the value of 2m - 10?

-4

Explanation: First, solve the linear equation (m + 1) / 3 = 4. Multiply by 3: m + 1 = 12. Subtract 1: m = 11. Then, substitute m = 11 into the expression 2m - 10: 2(11) - 10 = 22 - 10 = 12.

Expand the product of two binomials of the form (x ± a)(x ± b).

Q339: Expand (x+2)(x+3).

x² + 6

x² + 5x + 6

x² + 5x + 5

x² + 6x + 5

Explanation: To expand (x+2)(x+3), multiply each term in the first binomial by each term in the second binomial: x*x + x*3 + 2*x + 2*3. This simplifies to x² + 3x + 2x + 6, which further combines to x² + 5x + 6.

Q340: What is the expansion of (x+4)(x+5)?

x² + 20

x² + 9x + 9

x² + 9x + 20

x² + 20x + 9

Explanation: Using the FOIL method: First (x*x = x²), Outer (x*5 = 5x), Inner (4*x = 4x), Last (4*5 = 20). Adding them together gives x² + 5x + 4x + 20, which simplifies to x² + 9x + 20.

Q341: Expand (y+6)(y+1).

y² + 6

y² + 7y + 7

y² + 6y + 7

y² + 7y + 6

Explanation: Applying the distributive property, (y+6)(y+1) = y*y + y*1 + 6*y + 6*1. This results in y² + y + 6y + 6, which simplifies to y² + 7y + 6.

Q342: Which of the following is the correct expansion of (x+7)(x+8)?

x² + 56

x² + 15x + 15

x² + 15x + 56

x² + 56x + 15

Explanation: The general form for (x+a)(x+b) is x² + (a+b)x + ab. Here, a=7 and b=8. So, x² + (7+8)x + (7*8) = x² + 15x + 56.

Q343: Expand (p+3)(p+9).

p² + 27

p² + 12p + 12

p² + 12p + 27

p² + 27p + 12

Explanation: Using the distributive property: p*p + p*9 + 3*p + 3*9 = p² + 9p + 3p + 27. Combining the middle terms, we get p² + 12p + 27.

Q344: What is the result of expanding (m+10)(m+2)?

m² + 20

m² + 12m + 10

m² + 20m + 12

m² + 12m + 20

Explanation: Using the pattern (x+a)(x+b) = x² + (a+b)x + ab, substitute x=m, a=10, and b=2. So, m² + (10+2)m + (10*2) = m² + 12m + 20.

Q345: Expand (x - 3)(x - 5).

x² + 8x + 15

x² - 8x + 15

x² - 8x - 15

x² + 2x + 15

Explanation: To expand (x - 3)(x - 5), we use the formula (x + a)(x + b) = x² + (a + b)x + ab. Here, a = -3 and b = -5. So, x² + ((-3) + (-5))x + ((-3) * (-5)) = x² + (-8)x + 15 = x² - 8x + 15.

Q346: What is the expansion of (x + 4)(x - 2)?

x² - 2x - 8

x² + 2x + 8

x² + 2x - 8

x² + 6x - 8

Explanation: Using the formula (x + a)(x + b) = x² + (a + b)x + ab, with a = 4 and b = -2. So, x² + (4 + (-2))x + (4 * (-2)) = x² + 2x - 8.

Q347: Expand (x - 7)(x + 3).

x² + 4x - 21

x² - 4x + 21

x² - 10x - 21

x² - 4x - 21

Explanation: Using the identity (x + a)(x + b) = x² + (a + b)x + ab. Here, a = -7 and b = 3. So, x² + ((-7) + 3)x + ((-7) * 3) = x² + (-4)x - 21 = x² - 4x - 21.

Q348: Simplify (x - 6)(x - 1).

x² + 7x + 6

x² - 7x - 6

x² - 5x + 6

x² - 7x + 6

Explanation: Applying the expansion (x + a)(x + b) = x² + (a + b)x + ab with a = -6 and b = -1. So, x² + ((-6) + (-1))x + ((-6) * (-1)) = x² + (-7)x + 6 = x² - 7x + 6.

Q349: Expand (x + 9)(x - 5).

x² - 4x - 45

x² + 14x - 45

x² + 4x - 45

x² + 4x + 45

Explanation: Using the formula (x + a)(x + b) = x² + (a + b)x + ab, with a = 9 and b = -5. So, x² + (9 + (-5))x + (9 * (-5)) = x² + 4x - 45.

Q350: What is the expanded form of (x - 8)(x + 2)?

x² + 6x - 16

x² - 10x - 16

x² - 6x - 16

x² - 6x + 16

Explanation: Using the expansion (x + a)(x + b) = x² + (a + b)x + ab, with a = -8 and b = 2. So, x² + ((-8) + 2)x + ((-8) * 2) = x² + (-6)x - 16 = x² - 6x - 16.

Q351: If (x + 3)(x + 5) = x² + px + 15, what is the value of p?

Explanation: Expand the product of the two binomials: (x + 3)(x + 5) = x(x+5) + 3(x+5) = x² + 5x + 3x + 15 = x² + 8x + 15. Comparing this with the given identity x² + px + 15, we can see that the coefficient of x, p, must be 8.

Q352: If (x - 4)(x + 7) = x² + kx - 28, what is the value of k?

-3

-28

Explanation: Expand the product of the two binomials: (x - 4)(x + 7) = x(x+7) - 4(x+7) = x² + 7x - 4x - 28 = x² + 3x - 28. Comparing this with the given identity x² + kx - 28, we can see that the coefficient of x, k, must be 3.

Q353: If (x - 2)(x - 6) = x² + ax + 12, what is the value of a?

-8

-12

Explanation: Expand the product of the two binomials: (x - 2)(x - 6) = x(x-6) - 2(x-6) = x² - 6x - 2x + 12 = x² - 8x + 12. Comparing this with the given identity x² + ax + 12, we can see that the coefficient of x, a, must be -8.

Q354: If (x + 9)(x - 3) = x² + 6x + m, what is the value of m?

-27

Explanation: Expand the product of the two binomials: (x + 9)(x - 3) = x(x-3) + 9(x-3) = x² - 3x + 9x - 27 = x² + 6x - 27. Comparing this with the given identity x² + 6x + m, we can see that the constant term, m, must be -27.

Q355: Given that (x + 5)(x + 2) = x² + bx + c, find the value of c.

-10

Explanation: Expand the product of the two binomials: (x + 5)(x + 2) = x(x+2) + 5(x+2) = x² + 2x + 5x + 10 = x² + 7x + 10. Comparing this with the given identity x² + bx + c, we can see that the constant term, c, must be 10.

Q356: If (x - 8)(x + 1) = x² + yx - 8, what is the value of y?

-7

-8

Explanation: Expand the product of the two binomials: (x - 8)(x + 1) = x(x+1) - 8(x+1) = x² + x - 8x - 8 = x² - 7x - 8. Comparing this with the given identity x² + yx - 8, we can see that the coefficient of x, y, must be -7.

Verify the expansion of the product of two binomials using area.

Q357: An area model represents the product $(x+3)(y+4)$. What is the algebraic expression for the area of the rectangle formed by sides 'x' and 'y'?

$x+y$

$xy$

$3y$

$4x$

Explanation: The area of a rectangle is calculated by multiplying its length and width. For the rectangle with sides 'x' and 'y', the area is $x \times y = xy$.

Q358: Consider an area model for $(a+2)(b+5)$. What is the algebraic expression for the area of the rectangle with sides 'a' and '5'?

$a+5$

$5a$

$2b$

$ab$

Explanation: The area of a rectangle is found by multiplying its length and width. For the rectangle with sides 'a' and '5', the area is $a \times 5 = 5a$.

Q359: In an area model illustrating the product $(m+6)(n+1)$, what is the area of the rectangle formed by the sides '6' and 'n'?

$mn$

$6n$

$6+n$

$m$

Explanation: The area of a rectangle is the product of its length and width. For the rectangle with sides '6' and 'n', the area is $6 \times n = 6n$.

Q360: For the expansion of $(p+7)(q+3)$ using an area model, what is the area of the rectangle with sides '7' and '3'?

$p+q$

$21$

$7p$

$3q$

Explanation: The area of a rectangle is calculated by multiplying its length and width. For the rectangle with sides '7' and '3', the area is $7 \times 3 = 21$.

Q361: An area model represents the product $(2x+1)(y+4)$. What is the algebraic expression for the area of the rectangle formed by sides '2x' and 'y'?

$2x+y$

$2xy$

$8x$

$y$

Explanation: The area of a rectangle is found by multiplying its length and width. For the rectangle with sides '2x' and 'y', the area is $2x \times y = 2xy$.

Q362: Consider an area model for $(x+3)(2y+5)$. What is the algebraic expression for the area of the rectangle with sides 'x' and '5'?

$x+5$

$10y$

$5x$

$2xy$

Explanation: The area of a rectangle is calculated by multiplying its length and width. For the rectangle with sides 'x' and '5', the area is $x \times 5 = 5x$.

Q363: Which of the following diagrams correctly represents the area model for the expansion of $(x+2)(x+3)$?

A 2x2 grid with sides (x, 2) and (x, 3) and internal areas: x², 2x, 3x, 5.

A 2x2 grid with sides (x, 2) and (x, 3) and internal areas: x², 3x, 2x, 6.

A 2x2 grid with sides (x, 2) and (x, 5) and internal areas: x², 5x, 2x, 10.

A 2x2 grid with sides (x, x) and (2, 3) and internal areas: x², 3x, 2x, 6.

Explanation: The area model for $(x+2)(x+3)$ involves a rectangle with side lengths $(x+2)$ and $(x+3)$. When divided, the internal areas are $x \times x = x^2$, $x \times 3 = 3x$, $2 \times x = 2x$, and $2 \times 3 = 6$. The diagram must correctly show these side lengths and resulting internal areas.

Q364: Consider an area model used to expand $(2x+1)(x+4)$. If the top-left section has an area of $2x^2$, the top-right section has an area of $8x$, and the bottom-left section has an area of $x$, what is the area of the bottom-right section?

$2x^2$

$8x$

$4$

$x$

Explanation: For the product $(2x+1)(x+4)$, the side lengths are $(2x+1)$ and $(x+4)$. The bottom-right section is formed by multiplying the constant terms of each binomial. So, its area is $1 \times 4 = 4$.

Q365: An area model for a binomial product has internal sections with areas $3x^2$, $6x$, $2x$, and $4$. What is the expanded form of the product represented by this area model?

$3x^2 + 12x + 4$

$3x^2 + 8x + 4$

$3x^2 + 8x + 8$

$3x^2 + 4x + 4$

Explanation: To find the expanded form, sum all the internal areas: $3x^2 + 6x + 2x + 4$. Combine the like terms ($6x$ and $2x$) to get $8x$. So the expanded form is $3x^2 + 8x + 4$.

Q366: Which set of internal areas correctly represents the expansion of $(x+5)(x+1)$ using an area model?

{ $x^2$, $5x$, $5$ }

{ $x^2$, $6x$, $5$ }

{ $x^2$, $x$, $5x$, $5$ }

{ $x^2$, $x$, $5x$, $6$ }

Explanation: For $(x+5)(x+1)$, the four internal areas are found by multiplying each term of the first binomial by each term of the second binomial: $x \times x = x^2$, $x \times 1 = x$, $5 \times x = 5x$, and $5 \times 1 = 5$.

Q367: A student is verifying the expansion of $(x+3)(x+4)$ using an area model. They perform the following steps: Step 1: Draw a 2x2 grid and label the sides as $(x, 3)$ and $(x, 4)$. Step 2: Calculate the internal areas: $x \times x = x^2$, $x \times 4 = 4x$, $3 \times x = 3x$. Step 3: Calculate the fourth internal area: $3 \times 4 = 7$. Step 4: Sum the areas: $x^2 + 4x + 3x + 7 = x^2 + 7x + 7$. Which step contains an error?

Step 1

Step 2

Step 3

Step 4

Explanation: Step 3 calculates the product of the constant terms 3 and 4. The correct product is $3 \times 4 = 12$, not 7. This error affects the final constant term in the expanded form.

Q368: Which product of two binomials would result in an area model with internal areas $6x^2$, $9x$, $4x$, and $6$?

$(6x+1)(x+6)$

$(3x+2)(2x+3)$

$(2x+2)(3x+3)$

$(3x+1)(2x+6)$

Explanation: To find the binomials, look for factors that give these areas. The $6x^2$ term suggests factors like $2x$ and $3x$. The constant term 6 suggests factors like $2$ and $3$. If the sides are $(2x+3)$ and $(3x+2)$, the areas are: $2x \times 3x = 6x^2$, $2x \times 2 = 4x$, $3 \times 3x = 9x$, and $3 \times 2 = 6$. These match the given internal areas.

Q369: When expanding $(x+y)(a+b)$ using an area model, which expression correctly represents the sum of the areas of the four internal sections?

$x+y+a+b$

$(x+a)(y+b)$

$xa + xb + ya + yb$

$xy + ab$

Explanation: For the product $(x+y)(a+b)$, the four internal areas are obtained by multiplying each term of the first binomial by each term of the second binomial. These are $x \times a = xa$, $x \times b = xb$, $y \times a = ya$, and $y \times b = yb$. The sum is $xa + xb + ya + yb$.

Q370: Consider an area model for the product of two binomials which results in the total area $x^2 + 8x + 12$. If one side of the model is $(x+6)$, what is the other binomial side length and what are the expressions for the two rectangular internal areas involving $x$?

$(x+12)$; $x$ and $12x$

$(x+2)$; $6x$ and $2x$

$(x+4)$; $6x$ and $4x$

$(x+2)$; $6x$ and $12$

Explanation: To find the missing binomial side, we need to factorize $x^2 + 8x + 12$. We are given one factor $(x+6)$. We need to find two numbers that multiply to 12 and add up to 8. These numbers are 6 and 2. So, the factorization is $(x+6)(x+2)$. Therefore, the other binomial side length is $(x+2)$. The internal areas are found by multiplying the terms. The four internal areas are $x \times x = x^2$, $x \times 2 = 2x$, $6 \times x = 6x$, and $6 \times 2 = 12$. The two rectangular internal areas involving $x$ are $6x$ and $2x$.

Q371: An area model represents the expansion of two binomials, with a total area of $x^2 + 9x + 18$. If one dimension of the model is $(x+3)$, what is the missing binomial dimension and what are the two missing internal areas involving $x$?

$(x+9)$; $3x$ and $9x$

$(x+6)$; $3x$ and $6x$

$(x+6)$; $3x$ and $18$

$(x+15)$; $x$ and $15x$

Explanation: To factorize $x^2 + 9x + 18$, we need two numbers that multiply to 18 and add up to 9. These numbers are 3 and 6. So, the factorization is $(x+3)(x+6)$. Given one dimension is $(x+3)$, the other is $(x+6)$. The internal areas are $x \times x = x^2$, $x \times 6 = 6x$, $3 \times x = 3x$, and $3 \times 6 = 18$. The two missing internal areas involving $x$ are $3x$ and $6x$.

Q372: An incomplete area model shows that the product of two binomials is $x^2 + 10x + 21$. If one side length is $(x+7)$, what is the missing side length and what are the two missing internal areas that sum up to $10x$?

$(x+10)$; $7x$ and $10x$

$(x+3)$; $7x$ and $3x$

$(x+3)$; $7x$ and $21$

$(x+14)$; $x$ and $14x$

Explanation: To factorize $x^2 + 10x + 21$, we need two numbers that multiply to 21 and add up to 10. These numbers are 7 and 3. So, the factorization is $(x+7)(x+3)$. Given one side is $(x+7)$, the missing side length is $(x+3)$. The four internal areas are $x \times x = x^2$, $x \times 3 = 3x$, $7 \times x = 7x$, and $7 \times 3 = 21$. The two internal areas that sum up to $10x$ are $7x$ and $3x$.

Q373: The total area of an expanded binomial product shown in an area model is $x^2 + 11x + 28$. If one of the binomial side lengths is $(x+4)$, what is the other binomial side length, and what are the two internal areas involving $x$?

$(x+28)$; $4x$ and $28$

$(x+7)$; $4x$ and $7x$

$(x+11)$; $4x$ and $11x$

$(x+7)$; $4x$ and $28$

Explanation: To factorize $x^2 + 11x + 28$, we need two numbers that multiply to 28 and add up to 11. These numbers are 4 and 7. So, the factorization is $(x+4)(x+7)$. Given one side is $(x+4)$, the other binomial side length is $(x+7)$. The four internal areas are $x \times x = x^2$, $x \times 7 = 7x$, $4 \times x = 4x$, and $4 \times 7 = 28$. The two internal areas involving $x$ are $4x$ and $7x$.

Q374: For an area model showing the product of two binomials, the total area is $x^2 + 12x + 35$. If one side is $(x+5)$, determine the other binomial side length and the two internal rectangular areas that sum to $12x$.

$(x+35)$; $5x$ and $35$

$(x+7)$; $5x$ and $7x$

$(x+6)$; $5x$ and $6x$

$(x+17)$; $x$ and $17x$

Explanation: To factorize $x^2 + 12x + 35$, we need two numbers that multiply to 35 and add up to 12. These numbers are 5 and 7. So, the factorization is $(x+5)(x+7)$. Given one side is $(x+5)$, the other binomial side length is $(x+7)$. The four internal areas are $x \times x = x^2$, $x \times 7 = 7x$, $5 \times x = 5x$, and $5 \times 7 = 35$. The two internal rectangular areas that sum to $12x$ are $5x$ and $7x$.

Q375: Given an area model for the expansion of two binomials, the total area is $x^2 + 13x + 40$. If one binomial side length is $(x+8)$, what is the other side length, and what are the expressions for the two internal areas that combine to form $13x$?

$(x+40)$; $8x$ and $40$

$(x+5)$; $8x$ and $5x$

$(x+13)$; $8x$ and $13x$

$(x+5)$; $8x$ and $40$

Explanation: To factorize $x^2 + 13x + 40$, we need two numbers that multiply to 40 and add up to 13. These numbers are 8 and 5. So, the factorization is $(x+8)(x+5)$. Given one side is $(x+8)$, the other side length is $(x+5)$. The four internal areas are $x \times x = x^2$, $x \times 5 = 5x$, $8 \times x = 8x$, and $8 \times 5 = 40$. The two internal areas that combine to form $13x$ are $8x$ and $5x$.

Q376: In an area model illustrating the product of two binomials, the total expanded area is $x^2 + 14x + 48$. If one binomial side is $(x+6)$, what is the other binomial side, and what are the two internal areas (excluding $x^2$ and the constant) that constitute $14x$?

$(x+24)$; $6x$ and $24$

$(x+8)$; $6x$ and $8x$

$(x+10)$; $6x$ and $10x$

$(x+8)$; $6x$ and $48$

Explanation: To factorize $x^2 + 14x + 48$, we need two numbers that multiply to 48 and add up to 14. These numbers are 6 and 8. So, the factorization is $(x+6)(x+8)$. Given one side is $(x+6)$, the other binomial side is $(x+8)$. The four internal areas are $x \times x = x^2$, $x \times 8 = 8x$, $6 \times x = 6x$, and $6 \times 8 = 48$. The two internal areas (excluding $x^2$ and the constant) that constitute $14x$ are $6x$ and $8x$.

Factorize algebraic expressions with four terms where the common factor is a binomial.

Q377: Factorize the algebraic expression: `ax + ay + bx + by`

`a(x + y) + b(x + y)`

` (x + y)(a + b)`

` (x - y)(a + b)`

` (x + y)(a - b)`

Explanation: To factorize `ax + ay + bx + by`, group the terms: `(ax + ay) + (bx + by)`. Factor out the common monomial factor from each group: `a(x + y) + b(x + y)`. Now, `(x + y)` is a common binomial factor, so factor it out: `(x + y)(a + b)`.

Q378: Factorize: `2px + 3py + 4qx + 6qy`

`p(2x + 3y) + 2q(2x + 3y)`

` (2x + 3y)(p + q)`

` (2x + 3y)(p + 2q)`

` (2x - 3y)(p + 2q)`

Explanation: Group the terms: `(2px + 3py) + (4qx + 6qy)`. Factor out common monomial factors: `p(2x + 3y) + 2q(2x + 3y)`. Then factor out the common binomial factor: `(2x + 3y)(p + 2q)`.

Q379: What is the factorization of `mn + mp + 2n + 2p`?

` (n + p)(m - 2)`

`m(n + p) + 2(n + p)`

` (n + 2)(m + p)`

` (n + p)(m + 2)`

Explanation: Group the terms: `(mn + mp) + (2n + 2p)`. Factor out common factors: `m(n + p) + 2(n + p)`. Then, factor out the common binomial: `(n + p)(m + 2)`.

Q380: Factorize the expression: `xy - 3x + 2y - 6`

` (y + 3)(x + 2)`

`x(y - 3) + 2(y - 3)`

` (y - 3)(x - 2)`

` (y - 3)(x + 2)`

Explanation: Group the terms: `(xy - 3x) + (2y - 6)`. Factor out common monomial factors: `x(y - 3) + 2(y - 3)`. Then factor out the common binomial factor: `(y - 3)(x + 2)`.

Q381: Which of the following is the correct factorization of `3ab + 6ac + 5b + 10c`?

` (b + 2c)(3a - 5)`

`3a(b + 2c) + 5(b + 2c)`

` (b - 2c)(3a + 5)`

` (b + 2c)(3a + 5)`

Explanation: Group the terms: `(3ab + 6ac) + (5b + 10c)`. Factor out common monomial factors: `3a(b + 2c) + 5(b + 2c)`. Then factor out the common binomial factor: `(b + 2c)(3a + 5)`.

Q382: Factorize `x² + xy + 5x + 5y`.

`x(x + y) + 5(x + y)`

` (x - y)(x + 5)`

` (x + y)(x + 5)`

` (x + y)(x - 5)`

Explanation: Group the terms: `(x² + xy) + (5x + 5y)`. Factor out common monomial factors: `x(x + y) + 5(x + y)`. Then factor out the common binomial factor: `(x + y)(x + 5)`.

Q383: Factorize the algebraic expression: 2x² + 6x - 5x - 15

(x + 3)(2x + 5)

(x + 3)(2x - 5)

(x - 3)(2x - 5)

(2x + 5)(x - 3)

Explanation: Group the terms: (2x² + 6x) - (5x + 15). Factor out common terms from each group: 2x(x + 3) - 5(x + 3). Now, factor out the common binomial (x + 3): (x + 3)(2x - 5).

Q384: Factorize the algebraic expression: 3ab - 6a - 4b + 8

(b + 2)(3a - 4)

(b - 2)(3a - 4)

(b - 2)(3a + 4)

(3a - 2)(b - 4)

Explanation: Group the terms: (3ab - 6a) - (4b - 8). Factor out common terms from each group: 3a(b - 2) - 4(b - 2). Now, factor out the common binomial (b - 2): (b - 2)(3a - 4).

Q385: Factorize the algebraic expression: 12xy + 18x - 10y - 15

(2y + 3)(6x + 5)

(2y + 3)(6x - 5)

(2y - 3)(6x - 5)

(6x - 3)(2y - 5)

Explanation: Group the terms: (12xy + 18x) - (10y + 15). Factor out common terms from each group: 6x(2y + 3) - 5(2y + 3). Now, factor out the common binomial (2y + 3): (2y + 3)(6x - 5).

Q386: Factorize the algebraic expression: x³ - 5x² - 3x + 15

(x + 5)(x² - 3)

(x - 5)(x² - 3)

(x - 5)(x² + 3)

(x² - 5)(x - 3)

Explanation: Group the terms: (x³ - 5x²) - (3x - 15). Factor out common terms from each group: x²(x - 5) - 3(x - 5). Now, factor out the common binomial (x - 5): (x - 5)(x² - 3).

Q387: Factorize the algebraic expression: 14pq - 21p - 4q + 6

(2q + 3)(7p - 2)

(2q - 3)(7p - 2)

(2q - 3)(7p + 2)

(7p - 3)(2q - 2)

Explanation: Group the terms: (14pq - 21p) - (4q - 6). Factor out common terms from each group: 7p(2q - 3) - 2(2q - 3). Now, factor out the common binomial (2q - 3): (2q - 3)(7p - 2).

Q388: Factorize the algebraic expression: 6m + 9n - 2mk - 3nk

(2m + 3n)(3 + k)

(2m + 3n)(3 - k)

(2m - 3n)(3 - k)

(3m + 2n)(3 - k)

Explanation: Group the terms: (6m + 9n) - (2mk + 3nk). Factor out common terms from each group: 3(2m + 3n) - k(2m + 3n). Now, factor out the common binomial (2m + 3n): (2m + 3n)(3 - k).

Q389: Factorize the expression: `ax - ay + bx - by`

`a(x - y) + b(x - y)`

` (x + y)(a - b)`

` (x - y)(a + b)`

` (x - y)(a - b)`

Explanation: Group the terms as `(ax - ay) + (bx - by)`. Factor out the common term from each group: `a(x - y) + b(x - y)`. Now, `(x - y)` is a common binomial factor. Factor it out: `(x - y)(a + b)`.

Q390: Factorize the expression: `3x - 6y - ax + 2ay`

` (x - 2y)(3 + a)`

` (x + 2y)(3 - a)`

` (x - 2y)(3 - a)`

` (2y - x)(3 - a)`

Explanation: Group the terms strategically: `(3x - 6y) - (ax - 2ay)`. Factor out common terms from each group: `3(x - 2y) - a(x - 2y)`. Now, `(x - 2y)` is a common binomial factor. Factor it out: `(x - 2y)(3 - a)`.

Q391: Factorize the expression: `2ab + 6b + ac + 3c`

` (2b - c)(a + 3)`

` (2b + c)(a - 3)`

` (2b + c)(a + 3)`

` (a + b)(2c + 3)`

Explanation: Group the terms as `(2ab + ac) + (6b + 3c)`. Factor out common terms from each group: `a(2b + c) + 3(2b + c)`. Now, `(2b + c)` is a common binomial factor. Factor it out: `(2b + c)(a + 3)`.

Q392: Factorize the expression: `4p² - 6pq + 2pr - 3qr`

` (2p + 3q)(2p + r)`

` (2p - 3q)(2p - r)`

` (2p - 3q)(2p + r)`

` 2p(2p - 3q) + r(2p - 3q)`

Explanation: Group the terms: `(4p² - 6pq) + (2pr - 3qr)`. Factor out common terms from each group: `2p(2p - 3q) + r(2p - 3q)`. Now, `(2p - 3q)` is a common binomial factor. Factor it out: `(2p - 3q)(2p + r)`.

Q393: Factorize the expression: `x²y + 3x - 4xy - 12`

` (x + 4)(xy + 3)`

` (x - 4)(xy - 3)`

` (x - 4)(xy + 3)`

` xy(x - 4) - 3(x - 4)`

Explanation: Rearrange and group the terms: `(x²y - 4xy) + (3x - 12)`. Factor out common terms from each group: `xy(x - 4) + 3(x - 4)`. Now, `(x - 4)` is a common binomial factor. Factor it out: `(x - 4)(xy + 3)`.

Q394: Factorize the expression: `15mn - 20m - 3n + 4`

` (3n + 4)(5m - 1)`

` (3n - 4)(5m + 1)`

` (3n - 4)(5m - 1)`

` (4 - 3n)(1 - 5m)`

Explanation: Group the terms: `(15mn - 20m) - (3n - 4)`. Factor out common terms from each group: `5m(3n - 4) - 1(3n - 4)`. Now, `(3n - 4)` is a common binomial factor. Factor it out: `(3n - 4)(5m - 1)`.

Factorize trinomial quadratic expressions of the form x² + bx + c.

Q395: Which of the following is the correct factorization of x² + 5x + 6?

(x+1)(x+6)

(x+2)(x+3)

(x-2)(x-3)

x(x+5)+6

Explanation: To factorize x² + 5x + 6, we need to find two numbers that multiply to 6 and add up to 5. These numbers are 2 and 3. Therefore, the factorization is (x+2)(x+3).

Q396: Factorize the expression: x² + 7x + 10.

(x+1)(x+10)

(x+3)(x+4)

(x+2)(x+5)

(x-2)(x-5)

Explanation: We need to find two numbers that multiply to 10 and add up to 7. The numbers are 2 and 5 (since 2 × 5 = 10 and 2 + 5 = 7). So, the factorization is (x+2)(x+5).

Q397: What is the factorization of a² + 8a + 15?

(a+1)(a+15)

(a+2)(a+6)

(a+3)(a+5)

(a-3)(a-5)

Explanation: For a² + 8a + 15, we look for two numbers that multiply to 15 and add up to 8. The numbers are 3 and 5 (since 3 × 5 = 15 and 3 + 5 = 8). Thus, the factorization is (a+3)(a+5).

Q398: Factorize y² + 9y + 18.

(y+2)(y+9)

(y+3)(y+6)

(y+4)(y+5)

(y-3)(y-6)

Explanation: To factorize y² + 9y + 18, we need two numbers that multiply to 18 and add up to 9. The numbers are 3 and 6 (since 3 × 6 = 18 and 3 + 6 = 9). So, the factorization is (y+3)(y+6).

Q399: Which of the following is the correct factorization of p² + 11p + 24?

(p+2)(p+9)

(p+4)(p+6)

(p+3)(p+8)

(p+1)(p+24)

Explanation: To factorize p² + 11p + 24, we need two numbers that multiply to 24 and add up to 11. The numbers are 3 and 8 (since 3 × 8 = 24 and 3 + 8 = 11). Thus, the factorization is (p+3)(p+8).

Q400: Factorize the expression: m² + 10m + 21.

(m+1)(m+21)

(m+2)(m+8)

(m+3)(m+7)

(m-3)(m-7)

Explanation: To factorize m² + 10m + 21, we need to find two numbers that multiply to 21 and add up to 10. These numbers are 3 and 7 (since 3 × 7 = 21 and 3 + 7 = 10). Therefore, the factorization is (m+3)(m+7).

Q401: Which of the following is the correct factorization of x² - 7x + 12?

(x+3)(x+4)

(x-3)(x-4)

(x-2)(x-6)

(x+1)(x+12)

Explanation: To factorize x² - 7x + 12, we need to find two numbers that multiply to +12 and add up to -7. These numbers are -3 and -4. Therefore, the factorization is (x-3)(x-4).

Q402: Factorize the expression x² - 10x + 24.

(x+4)(x+6)

(x-3)(x-8)

(x-4)(x-6)

(x+2)(x+12)

Explanation: We need two numbers that multiply to +24 and add up to -10. These numbers are -4 and -6. So, the factorization is (x-4)(x-6).

Q403: What is the factorization of x² + 2x - 15?

(x-5)(x+3)

(x+5)(x-3)

(x+15)(x-1)

(x-15)(x+1)

Explanation: We look for two numbers that multiply to -15 and add up to +2. These numbers are +5 and -3. Thus, the factorization is (x+5)(x-3).

Q404: Factorize x² + 5x - 6.

(x-6)(x+1)

(x+3)(x-2)

(x+6)(x-1)

(x-3)(x+2)

Explanation: We need two numbers that multiply to -6 and sum to +5. The numbers are +6 and -1. So, the factorization is (x+6)(x-1).

Q405: Which of the following is the correct factorization of x² - 5x - 14?

(x+7)(x-2)

(x-14)(x+1)

(x-7)(x+2)

(x+14)(x-1)

Explanation: We are looking for two numbers that multiply to -14 and add up to -5. The numbers are -7 and +2. Therefore, the factorization is (x-7)(x+2).

Q406: Factorize the expression x² - 3x - 10.

(x+5)(x-2)

(x-10)(x+1)

(x-5)(x+2)

(x+10)(x-1)

Explanation: We need to find two numbers that multiply to -10 and add up to -3. These numbers are -5 and +2. So, the factorization is (x-5)(x+2).

Q407: Factorize `2x² + 10x + 12`.

`2(x+1)(x+6)`

`2(x+2)(x+3)`

`(x+2)(2x+6)`

`(x+5)(x+6)`

Explanation: First, take out the common factor `2` from `2x² + 10x + 12` to get `2(x² + 5x + 6)`. Then, factorize the trinomial `x² + 5x + 6`. We need two numbers that multiply to `6` and add up to `5`, which are `2` and `3`. So, `x² + 5x + 6 = (x+2)(x+3)`. Therefore, the fully factorized expression is `2(x+2)(x+3)`.

Q408: Factorize `3x² + 15x + 18`.

`3(x+1)(x+6)`

`(3x+6)(x+3)`

`3(x+2)(x+3)`

`3(x+5)(x+6)`

Explanation: First, take out the common factor `3` from `3x² + 15x + 18` to get `3(x² + 5x + 6)`. Then, factorize the trinomial `x² + 5x + 6`. We need two numbers that multiply to `6` and add up to `5`, which are `2` and `3`. So, `x² + 5x + 6 = (x+2)(x+3)`. Therefore, the fully factorized expression is `3(x+2)(x+3)`.

Q409: Factorize `4x² + 24x + 32`.

`4(x+1)(x+8)`

`(4x+8)(x+4)`

`4(x+2)(x+4)`

`4(x+3)(x+5)`

Explanation: First, take out the common factor `4` from `4x² + 24x + 32` to get `4(x² + 6x + 8)`. Then, factorize the trinomial `x² + 6x + 8`. We need two numbers that multiply to `8` and add up to `6`, which are `2` and `4`. So, `x² + 6x + 8 = (x+2)(x+4)`. Therefore, the fully factorized expression is `4(x+2)(x+4)`.

Q410: Factorize `5x² - 25x + 30`.

`5(x+2)(x+3)`

`5(x-1)(x-6)`

`5(x-2)(x-3)`

`(5x-10)(x-3)`

Explanation: First, take out the common factor `5` from `5x² - 25x + 30` to get `5(x² - 5x + 6)`. Then, factorize the trinomial `x² - 5x + 6`. We need two numbers that multiply to `6` (positive) and add up to `-5` (negative), which are `-2` and `-3`. So, `x² - 5x + 6 = (x-2)(x-3)`. Therefore, the fully factorized expression is `5(x-2)(x-3)`.

Q411: Factorize `2x² + 14x + 20`.

`2(x+1)(x+10)`

`2(x+2)(x+5)`

`(2x+4)(x+5)`

`2(x+3)(x+4)`

Explanation: First, take out the common factor `2` from `2x² + 14x + 20` to get `2(x² + 7x + 10)`. Then, factorize the trinomial `x² + 7x + 10`. We need two numbers that multiply to `10` and add up to `7`, which are `2` and `5`. So, `x² + 7x + 10 = (x+2)(x+5)`. Therefore, the fully factorized expression is `2(x+2)(x+5)`.

Q412: Factorize `3x² - 21x + 36`.

`3(x+3)(x+4)`

`3(x-2)(x-6)`

`3(x-3)(x-4)`

`(3x-9)(x-4)`

Explanation: First, take out the common factor `3` from `3x² - 21x + 36` to get `3(x² - 7x + 12)`. Then, factorize the trinomial `x² - 7x + 12`. We need two numbers that multiply to `12` (positive) and add up to `-7` (negative), which are `-3` and `-4`. So, `x² - 7x + 12 = (x-3)(x-4)`. Therefore, the fully factorized expression is `3(x-3)(x-4)`.

Factorize algebraic expressions which are a difference of two squares.

Q413: Which of the following is the correct factorization of $x^2 - 16$?

$(x-4)^2$

$(x+4)^2$

$(x-4)(x+4)$

$(x-16)(x+16)$

Explanation: The expression $x^2 - 16$ is a difference of two squares. We can write $x^2$ as $(x)^2$ and $16$ as $(4)^2$. Using the formula $a^2 - b^2 = (a-b)(a+b)$, we get $x^2 - 4^2 = (x-4)(x+4)$.

Q414: Factorize $4y^2 - 9$.

$(4y-9)(4y+9)$

$(2y-3)(2y+3)$

$(2y-9)(2y+9)$

$(4y-3)(4y+3)$

Explanation: The expression $4y^2 - 9$ is a difference of two squares. We can write $4y^2$ as $(2y)^2$ and $9$ as $(3)^2$. Using the formula $a^2 - b^2 = (a-b)(a+b)$, we get $(2y)^2 - 3^2 = (2y-3)(2y+3)$.

Q415: What is the factorization of $25 - a^2$?

$(a-5)(a+5)$

$(25-a)(25+a)$

$(5-a)^2$

$(5-a)(5+a)$

Explanation: The expression $25 - a^2$ is a difference of two squares. We can write $25$ as $(5)^2$ and $a^2$ as $(a)^2$. Using the formula $a^2 - b^2 = (a-b)(a+b)$, we get $5^2 - a^2 = (5-a)(5+a)$.

Q416: Choose the correct factorization for $9p^2 - 4q^2$.

$(9p-4q)(9p+4q)$

$(3p-4q)(3p+4q)$

$(3p-2q)(3p+2q)$

$(9p-2q)(9p+2q)$

Explanation: The expression $9p^2 - 4q^2$ is a difference of two squares. We can write $9p^2$ as $(3p)^2$ and $4q^2$ as $(2q)^2$. Using the formula $a^2 - b^2 = (a-b)(a+b)$, we get $(3p)^2 - (2q)^2 = (3p-2q)(3p+2q)$.

Q417: Factorize $49m^2 - 100$.

$(7m-50)(7m+50)$

$(49m-100)(49m+100)$

$(7m-10)^2$

$(7m-10)(7m+10)$

Explanation: The expression $49m^2 - 100$ is a difference of two squares. We can write $49m^2$ as $(7m)^2$ and $100$ as $(10)^2$. Using the formula $a^2 - b^2 = (a-b)(a+b)$, we get $(7m)^2 - 10^2 = (7m-10)(7m+10)$.

Q418: Find the factorization of $1 - 36x^2$.

$(1-36x)(1+36x)$

$(1-6x)^2$

$(1-6x)(1+6x)$

$(6x-1)(6x+1)$

Explanation: The expression $1 - 36x^2$ is a difference of two squares. We can write $1$ as $(1)^2$ and $36x^2$ as $(6x)^2$. Using the formula $a^2 - b^2 = (a-b)(a+b)$, we get $1^2 - (6x)^2 = (1-6x)(1+6x)$.

Q419: Which of the following is the correct factorization of $5x^2 - 45$?

$5(x-3)^2$

$5(x-3)(x+3)$

$(5x-15)(x+3)$

$5(x^2-3)$

Explanation: First, take out the common factor 5: $5(x^2 - 9)$. Then, recognize that $x^2 - 9$ is a difference of two squares, $(x)^2 - (3)^2$. Applying the formula $a^2 - b^2 = (a-b)(a+b)$, we get $5(x-3)(x+3)$.

Q420: Factorize $4y^3 - 16y$ completely.

$4y(y^2 - 2)$

$4y(y-2)^2$

$4y(y-2)(y+2)$

$(2y^2 - 4)(2y)$

Explanation: First, take out the common factor $4y$: $4y(y^2 - 4)$. Then, recognize that $y^2 - 4$ is a difference of two squares, $(y)^2 - (2)^2$. Applying the formula $a^2 - b^2 = (a-b)(a+b)$, we get $4y(y-2)(y+2)$.

Q421: Factorize $(a-b)^2 - 25$.

$(a-b-5)^2$

$(a-b-5)(a-b+5)$

$(a-b-25)(a-b+25)$

$a^2-b^2-25$

Explanation: Recognize that $(a-b)^2 - 25$ is a difference of two squares, where $A = (a-b)$ and $B = 5$. Applying the formula $A^2 - B^2 = (A-B)(A+B)$, we get $((a-b)-5)((a-b)+5)$, which simplifies to $(a-b-5)(a-b+5)$.

Q422: What is the factorization of $(2x+y)^2 - (x-y)^2$?

$(3x)(2y)$

$(x+2y)(3x)$

$(x+2y)(3x-2y)$

$(x-2y)(3x)$

Explanation: This is a difference of two squares, where $A = (2x+y)$ and $B = (x-y)$. Applying the formula $A^2 - B^2 = (A-B)(A+B)$, we get $((2x+y)-(x-y))((2x+y)+(x-y))$. Simplifying the terms inside the brackets: $(2x+y-x+y)(2x+y+x-y) = (x+2y)(3x)$.

Q423: Factorize $81x^4 - 1$ completely.

$(9x^2-1)^2$

$(3x-1)(3x+1)(9x^2-1)$

$(3x-1)(3x+1)(9x^2+1)$

$(9x^2+1)(9x^2-1)$

Explanation: First, recognize that $81x^4 - 1$ is a difference of two squares: $(9x^2)^2 - (1)^2$. This factors into $(9x^2-1)(9x^2+1)$. Then, recognize that $9x^2-1$ is also a difference of two squares: $(3x)^2 - (1)^2$. This factors into $(3x-1)(3x+1)$. The full factorization is $(3x-1)(3x+1)(9x^2+1)$.

Q424: Factorize $7(p-q)^2 - 63$ completely.

$7(p-q-3)^2$

$(7p-7q-21)(7p-7q+21)$

$7(p-q-9)(p-q+9)$

$7(p-q-3)(p-q+3)$

Explanation: First, take out the common numerical factor 7: $7((p-q)^2 - 9)$. Then, recognize that $(p-q)^2 - 9$ is a difference of two squares, where $A = (p-q)$ and $B = 3$. Applying the formula $A^2 - B^2 = (A-B)(A+B)$, we get $7((p-q)-3)((p-q)+3)$, which simplifies to $7(p-q-3)(p-q+3)$.

Q425: Factorize x⁴ - y⁴.

(x² - y²)(x² + y²)

(x - y)(x + y)(x² + y²)

(x - y)(x + y)(x - y)(x + y)

(x² - y²)²

Explanation: The expression x⁴ - y⁴ can be written as (x²)² - (y²)². Applying the difference of two squares formula, a² - b² = (a-b)(a+b), we get (x² - y²)(x² + y²). The term (x² - y²) can be factorized further as (x-y)(x+y). Thus, the complete factorization is (x-y)(x+y)(x² + y²).

Q426: Factorize (a + 3b)² - 16c².

(a + 3b - 4c)(a + 3b + 4c)

(a + 3b - 16c)(a + 3b + 16c)

(a - 3b - 4c)(a + 3b + 4c)

(a + 3b - 4c)²

Explanation: The expression (a + 3b)² - 16c² is in the form of a difference of two squares, A² - B², where A = (a + 3b) and B = 4c (since 16c² = (4c)²). Applying the formula A² - B² = (A - B)(A + B), we get ((a + 3b) - 4c)((a + 3b) + 4c), which simplifies to (a + 3b - 4c)(a + 3b + 4c).

Q427: Factorize 9x² - (y - 2z)².

(3x - y - 2z)(3x + y - 2z)

(3x - y + 2z)(3x + y - 2z)

(9x - y + 2z)(9x + y - 2z)

(3x - (y - 2z))²

Explanation: The expression 9x² - (y - 2z)² can be written as (3x)² - (y - 2z)². This is in the form of a difference of two squares, A² - B², where A = 3x and B = (y - 2z). Applying the formula A² - B² = (A - B)(A + B), we get (3x - (y - 2z))(3x + (y - 2z)). Simplifying the terms inside the brackets gives (3x - y + 2z)(3x + y - 2z).

Q428: Factorize x² + 10x + 25 - y².

(x + 5 - y)(x + 5 + y)

(x + 5 - y)²

(x + 5 - y)(x - 5 + y)

(x² - y)(x + 10x + 25 + y)

Explanation: First, group the terms x² + 10x + 25. This is a perfect square trinomial, (x + 5)². So the expression becomes (x + 5)² - y². This is now in the form of a difference of two squares, A² - B², where A = (x + 5) and B = y. Applying the formula A² - B² = (A - B)(A + B), we get ((x + 5) - y)((x + 5) + y), which simplifies to (x + 5 - y)(x + 5 + y).

Q429: Factorize a² - b² + 6bc - 9c².

(a - b + 3c)(a + b - 3c)

(a - b - 3c)(a + b - 3c)

(a - b + 3c)(a + b + 3c)

(a - (b - 3c))²

Explanation: Rearrange the terms and factor out -1 from the last three terms: a² - (b² - 6bc + 9c²). The expression inside the parenthesis (b² - 6bc + 9c²) is a perfect square trinomial, (b - 3c)². So the expression becomes a² - (b - 3c)². This is in the form of a difference of two squares, A² - B², where A = a and B = (b - 3c). Applying the formula A² - B² = (A - B)(A + B), we get (a - (b - 3c))(a + (b - 3c)), which simplifies to (a - b + 3c)(a + b - 3c).

Q430: Factorize 100 - x² + 6xy - 9y².

(10 - x - 3y)(10 + x - 3y)

(10 - x + 3y)(10 + x - 3y)

(10 - x + 3y)²

(10 - x - 3y)(10 - x + 3y)

Explanation: Rearrange the terms and factor out -1 from the last three terms: 100 - (x² - 6xy + 9y²). The expression inside the parenthesis (x² - 6xy + 9y²) is a perfect square trinomial, (x - 3y)². So the expression becomes 10² - (x - 3y)². This is in the form of a difference of two squares, A² - B², where A = 10 and B = (x - 3y). Applying the formula A² - B² = (A - B)(A + B), we get (10 - (x - 3y))(10 + (x - 3y)), which simplifies to (10 - x + 3y)(10 + x - 3y).

Identify the five basic axioms in mathematics.

Q431: Which axiom is demonstrated by the expression `5 + x = x + 5`?

Associative Property of Addition

Commutative Property of Addition

Distributive Property

Commutative Property of Multiplication

Explanation: The Commutative Property of Addition states that changing the order of the numbers in an addition operation does not change the sum. `a + b = b + a`. The given expression `5 + x = x + 5` directly illustrates this.

Q432: Identify the axiom shown in the equation `3 * y = y * 3`.

Associative Property of Multiplication

Commutative Property of Multiplication

Distributive Property

Commutative Property of Addition

Explanation: The Commutative Property of Multiplication states that changing the order of the numbers in a multiplication operation does not change the product. `a * b = b * a`. The expression `3 * y = y * 3` demonstrates this property.

Q433: What property is illustrated by `(a + 2) + b = a + (2 + b)`?

Commutative Property of Addition

Associative Property of Addition

Distributive Property

Associative Property of Multiplication

Explanation: The Associative Property of Addition states that the way numbers are grouped in an addition operation does not change the sum. `(a + b) + c = a + (b + c)`. The expression `(a + 2) + b = a + (2 + b)` demonstrates this.

Q434: Which axiom is represented by the expression `(x * 4) * y = x * (4 * y)`?

Commutative Property of Multiplication

Associative Property of Multiplication

Distributive Property

Associative Property of Addition

Explanation: The Associative Property of Multiplication states that the way numbers are grouped in a multiplication operation does not change the product. `(a * b) * c = a * (b * c)`. The expression `(x * 4) * y = x * (4 * y)` demonstrates this property.

Q435: The expression `2 * (a + 3) = 2a + 2 * 3` illustrates which property?

Associative Property of Multiplication

Distributive Property

Commutative Property of Addition

Associative Property of Addition

Explanation: The Distributive Property states that multiplying a number by a sum is the same as multiplying each addend by the number and then adding the products. `a * (b + c) = a * b + a * c`. The expression `2 * (a + 3) = 2a + 2 * 3` directly shows this.

Q436: Which axiom is demonstrated by `p * q + p * r = p * (q + r)`?

Associative Property of Addition

Distributive Property

Commutative Property of Multiplication

Associative Property of Multiplication

Explanation: The Distributive Property allows us to factor out a common multiplier from a sum, which is the reverse of distributing. `a * b + a * c = a * (b + c)`. The expression `p * q + p * r = p * (q + r)` is an application of the Distributive Property in reverse (factoring).

Q437: If 3x = 15, then 3x + 5 = _____. Which axiom is used here?

20; Subtraction Axiom

20; Addition Axiom

15; Addition Axiom

5; Part-Whole Axiom

Explanation: The equation 3x = 15 is given. By adding the same value (5) to both sides, we get 3x + 5 = 15 + 5 = 20. This demonstrates the Addition Property of Equality (If equals be added to equals, the wholes are equal).

Q438: If 2a + 7 = 11, then 2a = _____. Which axiom is used here?

4; Subtraction Axiom

4; Addition Axiom

18; Addition Axiom

7; Transitive Axiom

Explanation: The equation 2a + 7 = 11 is given. By subtracting the same value (7) from both sides, we get 2a = 11 - 7 = 4. This demonstrates the Subtraction Property of Equality (If equals be subtracted from equals, the remainders are equal).

Q439: If x = y and y = 7, then x = _____. Which axiom is used here?

7; Addition Axiom

y; Identity Axiom

7; Transitive Axiom

x + y; Part-Whole Axiom

Explanation: Given that x is equal to y, and y is equal to 7. According to the Transitive Property of Equality (Things which are equal to the same thing are equal to one another), if x = y and y = 7, then x must be equal to 7.

Q440: If segment AB is a part of segment AC, then AC _____ AB. Which axiom is used here?

=; Identity Axiom

>; Part-Whole Axiom

<; Subtraction Axiom

+; Addition Axiom

Explanation: The Part-Whole Axiom states that the whole is greater than the part. Since AC is the whole and AB is its part, AC must be greater than AB (AC > AB).

Q441: An object is always equal to _____. Which axiom states this?

its parts; Part-Whole Axiom

itself; Identity Axiom

another object; Transitive Axiom

its sum; Addition Axiom

Explanation: The Identity Axiom (or 'Things which coincide with one another are equal to one another') states that an object is equal to itself.

Q442: If p - 4 = q, then p = _____. Which axiom is used to get this?

q - 4; Subtraction Axiom

q + 4; Addition Axiom

q; Transitive Axiom

4; Part-Whole Axiom

Explanation: The equation p - 4 = q is given. By adding the same value (4) to both sides, we isolate p, getting p = q + 4. This demonstrates the Addition Property of Equality (If equals be added to equals, the wholes are equal).

Q443: Consider the following statement: If a = b and b = c, then it can be concluded that a = c. Which of the five basic axioms directly supports this conclusion?

If equals be added to equals, the wholes are equal.

Things which are equal to the same thing are also equal to one another.

If equals be subtracted from equals, the remainders are equal.

The whole is greater than the part.

Explanation: The axiom 'Things which are equal to the same thing are also equal to one another' states that if two quantities are equal to a third quantity, then they are equal to each other. In this case, 'a' and 'c' are both equal to 'b', so 'a' must be equal to 'c'. This is the Transitive Property of Equality.

Q444: A student is solving the equation 3x - 4 = 11. In the first step, they write 3x - 4 + 4 = 11 + 4. The student claims this step is an application of the axiom 'If equals be subtracted from equals, the remainders are equal.' Is this identification correct?

Yes, the identification is correct.

No, the correct axiom is 'If equals be added to equals, the wholes are equal.'

No, the correct axiom is 'Things which are equal to the same thing are also equal to one another.'

No, the correct axiom is 'The whole is greater than the part.'

Explanation: The student added 4 to both sides of the equation. This action is supported by the axiom 'If equals be added to equals, the wholes are equal,' not the subtraction axiom. Therefore, the identification is incorrect.

Q445: In solving the equation 5y + 7 = 22, a student performs the step: 5y + 7 - 7 = 22 - 7, which simplifies to 5y = 15. Which of the five basic axioms correctly justifies the step 5y + 7 - 7 = 22 - 7?

If equals be added to equals, the wholes are equal.

Things which are equal to the same thing are also equal to one another.

If equals be subtracted from equals, the remainders are equal.

Things which coincide with one another are equal to one another.

Explanation: The step 5y + 7 - 7 = 22 - 7 involves subtracting the same quantity (7) from both sides of an equality. This is directly supported by the axiom 'If equals be subtracted from equals, the remainders are equal.'

Q446: Consider the statement: If x = y + z, where y and z are positive numbers, then x > y. A student claims that this conclusion is supported by the axiom 'If equals be added to equals, the wholes are equal.' Is the student's reasoning correct?

Yes, the reasoning is correct.

No, the correct axiom is 'The whole is greater than the part.'

No, the correct axiom is 'Things which are equal to the same thing are also equal to one another.'

No, the correct axiom is 'If equals be subtracted from equals, the remainders are equal.'

Explanation: The statement 'If x = y + z (where y, z > 0), then x > y' illustrates that the whole (x) is greater than its part (y). This is directly supported by the axiom 'The whole is greater than the part,' not the addition axiom. Therefore, the student's reasoning is incorrect.

Q447: Given that p = q and q = r, a student concludes that p = r. The student identifies the axiom used as 'If equals be added to equals, the wholes are equal.' Is the student's identification correct?

Yes, the identification is correct.

No, the correct axiom is 'Things which are equal to the same thing are also equal to one another.'

No, the correct axiom is 'If equals be subtracted from equals, the remainders are equal.'

No, the correct axiom is 'The whole is greater than the part.'

Explanation: The conclusion 'p = r' from 'p = q' and 'q = r' is an application of the Transitive Property of Equality, which is stated as 'Things which are equal to the same thing are also equal to one another.' The student incorrectly identified it as the Addition Property of Equality.

Q448: Examine the following algebraic simplification steps for the equation 2x + 5 = 15: Step 1: 2x + 5 - 5 = 15 - 5 (Axiom identified: If equals be subtracted from equals, the remainders are equal.) Step 2: 2x = 10 Step 3: 2x / 2 = 10 / 2 (Axiom identified: If equals be added to equals, the wholes are equal.) Step 4: x = 5 Which step contains an incorrect identification of an axiom?

Step 1

Step 2

Step 3

Step 4

Explanation: In Step 3, the operation is dividing both sides by 2. This is an application of the division property of equality, which is derived from axioms but is not itself one of the five basic Euclidean axioms. The student incorrectly identified it as the 'If equals be added to equals, the wholes are equal' axiom.

Establish geometric relationships based on the five axioms.

Q449: Lines AB and CD intersect at point O. If ∠AOC = 70°, what is the measure of ∠BOD?

110°

70°

180°

90°

Explanation: Vertically opposite angles are equal. Since ∠AOC and ∠BOD are vertically opposite angles, ∠BOD = ∠AOC = 70°.

Q450: Two straight lines PQ and RS intersect at O. If ∠POR = 120°, what is the measure of ∠POS?

120°

60°

180°

90°

Explanation: Angles on a straight line sum to 180°. ∠POR and ∠POS are adjacent angles on the straight line RS. Therefore, ∠POS = 180° - ∠POR = 180° - 120° = 60°.

Q451: Straight lines KL and MN intersect at point P. If ∠KPM = 55°, what is the measure of ∠LPN?

125°

55°

180°

90°

Explanation: Vertically opposite angles are equal. Since ∠KPM and ∠LPN are vertically opposite angles, ∠LPN = ∠KPM = 55°.

Q452: Intersecting lines EF and GH meet at point X. If ∠EXG = 40°, what is the measure of ∠GXF?

40°

140°

180°

90°

Explanation: Angles on a straight line sum to 180°. ∠EXG and ∠GXF are adjacent angles on the straight line EF. Therefore, ∠GXF = 180° - ∠EXG = 180° - 40° = 140°.

Q453: Lines ST and UV intersect at W. If ∠SWU = 105°, what is the measure of ∠TWV?

75°

105°

180°

90°

Explanation: Vertically opposite angles are equal. Since ∠SWU and ∠TWV are vertically opposite angles, ∠TWV = ∠SWU = 105°.

Q454: When two straight lines AB and CD intersect at O, ∠AOD = 85°. What is the measure of ∠AOC?

85°

95°

180°

100°

Explanation: Angles on a straight line sum to 180°. ∠AOD and ∠AOC are adjacent angles on the straight line CD. Therefore, ∠AOC = 180° - ∠AOD = 180° - 85° = 95°.

Q455: In the given diagram, two parallel lines are intersected by a transversal. If one of the alternate interior angles is 65°, what is the value of the angle that forms a linear pair (angles on a straight line) with the other alternate interior angle?

65°

115°

25°

130°

Explanation: Let the given alternate interior angle be A = 65°. The other alternate interior angle, B, will also be 65° (alternate interior angles are equal). The unknown angle 'x' forms a linear pair with B, so x + B = 180° (angles on a straight line). Therefore, x = 180° - 65° = 115°.

Q456: In the given diagram, two parallel lines are intersected by a transversal. If one of the corresponding angles is 100°, what is the value of the angle vertically opposite to the other corresponding angle?

80°

100°

180°

20°

Explanation: Let the given corresponding angle be A = 100°. The other corresponding angle, B, would also be 100° (corresponding angles are equal). The unknown angle 'x' is vertically opposite to B, so x = B (vertically opposite angles are equal). Therefore, x = 100°.

Q457: In the given diagram, two parallel lines are intersected by a transversal. If one of the interior angles on the same side of the transversal is 120°, what is the value of the angle vertically opposite to the other interior angle on the same side?

120°

60°

180°

30°

Explanation: Let the given co-interior angle be A = 120°. The other co-interior angle, B, is supplementary to A, so A + B = 180° (co-interior angles sum to 180°). Thus, B = 180° - 120° = 60°. The unknown angle 'x' is vertically opposite to B, so x = B (vertically opposite angles are equal). Therefore, x = 60°.

Q458: In the given diagram, two parallel lines are intersected by a transversal. If one of the alternate exterior angles is 110°, what is the value of the angle that forms a linear pair (angles on a straight line) with the other alternate exterior angle?

110°

70°

50°

20°

Explanation: Let the given alternate exterior angle be A = 110°. The other alternate exterior angle, B, will also be 110° (alternate exterior angles are equal). The unknown angle 'x' forms a linear pair with B, so x + B = 180° (angles on a straight line). Therefore, x = 180° - 110° = 70°.

Q459: In the given diagram, two parallel lines are intersected by a transversal. If one of the corresponding angles is 85°, what is the value of the angle that forms a linear pair (angles on a straight line) with the same corresponding angle?

85°

95°

105°

170°

Explanation: Let the given corresponding angle be A = 85°. The unknown angle 'x' forms a linear pair with A, so x + A = 180° (angles on a straight line). Therefore, x = 180° - 85° = 95°.

Q460: In the given diagram, two parallel lines are intersected by a transversal. If one of the interior angles on the same side of the transversal is 70°, what is the value of the angle that forms a linear pair (angles on a straight line) with the same interior angle?

70°

110°

90°

140°

Explanation: Let the given co-interior angle be A = 70°. The unknown angle 'x' forms a linear pair with A, so x + A = 180° (angles on a straight line). Therefore, x = 180° - 70° = 110°.

Q461: What auxiliary line is typically drawn to prove that the sum of the interior angles of a triangle is 180°?

A median to one of the sides.

A line parallel to one side, passing through the opposite vertex.

An altitude from one vertex to the opposite side.

An angle bisector of one of the interior angles.

Explanation: To prove the sum of interior angles of a triangle is 180°, an auxiliary line is drawn through one vertex, parallel to the opposite side. This construction allows us to use properties of parallel lines.

Q462: After drawing a line parallel to one side of a triangle through the opposite vertex, which axiom is primarily used to relate the interior angles of the triangle to the angles formed on the straight line?

Corresponding angles formed by parallel lines are equal.

Vertically opposite angles are equal.

Alternate interior angles formed by parallel lines are equal.

The sum of angles on a straight line is 180°.

Explanation: The axiom of parallel lines states that when a transversal intersects two parallel lines, alternate interior angles are equal. This is crucial for relating the base angles of the triangle to the angles formed by the auxiliary parallel line.

Q463: In the proof for the sum of interior angles of a triangle, after relating the triangle's angles to angles on a straight line using parallel lines, which axiom is applied to conclude that their sum is 180°?

The sum of angles around a point is 360°.

Complementary angles sum to 90°.

Angles on a straight line sum to 180°.

Supplementary angles sum to 180°.

Explanation: Once the interior angles of the triangle are shown to be equivalent to the angles formed on the straight auxiliary line, the axiom that states 'angles on a straight line sum to 180°' is used to conclude the proof.

Q464: Consider a triangle ABC. If a line DE is drawn through A parallel to BC, and D-A-E is a straight line, which of the following statements is a valid step in proving that angle A + angle B + angle C = 180°?

Angle DAB + Angle ABC = 180° (co-interior angles)

Angle DAB = Angle ABC (alternate interior angles) AND Angle EAC = Angle ACB (alternate interior angles).

Angle DAB = Angle ACB (incorrect pairing)

Angle BAC + Angle ABC = 90° (incorrect assumption)

Explanation: When DE is parallel to BC, and AB and AC are transversals, the alternate interior angles are equal. So, ∠DAB = ∠ABC and ∠EAC = ∠ACB. These equalities are crucial for substituting into the sum of angles on the straight line DAE.

Q465: In proving the sum of angles in a triangle is 180°, if we draw a line XY parallel to BC through vertex A of triangle ABC, then which pair of angles are equal due to the property of parallel lines?

Angle XAB = Angle ACB

Angle XAB + Angle ABC = 180°

Angle XAB = Angle ABC and Angle YAC = Angle ACB.

Angle XAY = Angle ABC

Explanation: When XY is parallel to BC, and AB and AC are transversals, the alternate interior angles are equal. Specifically, ∠XAB is alternate to ∠ABC, and ∠YAC is alternate to ∠ACB.

Q466: The proof that the sum of interior angles of a triangle is 180° fundamentally relies on transforming the interior angles of the triangle into angles that lie on a straight line. Which combination of axioms enables this transformation?

Vertically opposite angles and angles at a point.

Axiom of parallel lines (alternate interior angles) and Axiom of angles on a straight line.

Corresponding angles and complementary angles.

Sum of angles in a quadrilateral and exterior angles.

Explanation: The proof uses an auxiliary line parallel to one side, through the opposite vertex. The 'axiom of parallel lines' (specifically alternate interior angles) allows us to equate two of the triangle's angles to angles on this new straight line. Then, the 'axiom of angles on a straight line' sums these angles to 180°, thus proving the triangle's angle sum.

Solve problems involving calculations using the axioms.

Q467: Which axiom is demonstrated by the statement 5 + 8 = 8 + 5?

Commutative property of addition

Associative property of addition

Distributive property

Identity property of addition

Explanation: The commutative property of addition states that changing the order of addends does not change the sum.

Q468: Identify the axiom illustrated by (3 × 4) × 2 = 3 × (4 × 2).

Associative property of multiplication

Commutative property of multiplication

Distributive property

Identity property of multiplication

Explanation: The associative property of multiplication states that changing the grouping of factors does not change the product.

Q469: Which axiom is shown in the statement 6 × (2 + 7) = 6 × 2 + 6 × 7?

Distributive property

Commutative property

Associative property

Identity property

Explanation: The distributive property states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products.

Q470: The statement 15 + 0 = 15 illustrates which property?

Identity property of addition

Identity property of multiplication

Commutative property of addition

Associative property of addition

Explanation: The identity property of addition states that the sum of any number and zero is that number. Zero is the additive identity.

Q471: Complete the statement: 7 × ____ = 7 to illustrate the identity property of multiplication.

-7

Explanation: The identity property of multiplication states that the product of any number and one is that number. One is the multiplicative identity.

Q472: To illustrate the commutative property of multiplication, complete the statement: 9 × 4 = 4 × ____.

Explanation: The commutative property of multiplication states that changing the order of factors does not change the product.

Q473: What is the value of 18 + 7 + 2 when calculated most efficiently using axioms?

Explanation: To calculate 18 + 7 + 2 efficiently, we can use the Commutative and Associative Properties of Addition. Rearrange the terms to group compatible numbers: (18 + 2) + 7 = 20 + 7 = 27.

Q474: Calculate 5 × 13 × 2 efficiently using axioms.

150

130

260

Explanation: To calculate 5 × 13 × 2 efficiently, use the Commutative and Associative Properties of Multiplication. Rearrange the terms to group compatible numbers: (5 × 2) × 13 = 10 × 13 = 130.

Q475: What is the value of 6 × (10 + 4) when calculated using the distributive property?

Explanation: The Distributive Property states that a × (b + c) = a × b + a × c. So, 6 × (10 + 4) = 6 × 10 + 6 × 4 = 60 + 24 = 84.

Q476: Using the associative property, calculate (25 + 38) + 12 efficiently.

Explanation: The Associative Property of Addition states that the grouping of terms in an addition does not affect the sum: (a + b) + c = a + (b + c). To calculate (25 + 38) + 12 efficiently, we regroup it as 25 + (38 + 12) = 25 + 50 = 75.

Q477: What is the value of 99 × 7 when calculated efficiently using the distributive property?

693

700 - 1

630 + 7

700

Explanation: To calculate 99 × 7 efficiently using the distributive property, we can write 99 as (100 - 1). Then, (100 - 1) × 7 = 100 × 7 - 1 × 7 = 700 - 7 = 693.

Q478: What is the most efficient way to calculate 4 × 17 × 25 using axioms?

4 × (17 + 25)

(4 × 17) + (4 × 25)

(4 × 25) × 17

17 × (4 + 25)

Explanation: To calculate 4 × 17 × 25 most efficiently, we should use the Commutative and Associative Properties of Multiplication to group 4 and 25 together, as their product is 100. So, 4 × 17 × 25 = (4 × 25) × 17 = 100 × 17 = 1700. Option (c) represents this efficient grouping.

Q479: If x - 5 = 10, then the step to get x = 15 is justified by which axiom?

Subtraction Axiom

Addition Axiom

Multiplication Axiom

Division Axiom

Explanation: To get from x - 5 = 10 to x = 15, we add 5 to both sides of the equation. This is the Addition Axiom, which states that if a = b, then a + c = b + c.

Q480: Consider the equation y + 3 = 7. Which axiom justifies the step to obtain y = 4?

Addition Axiom

Subtraction Axiom

Transitive Axiom

Substitution Axiom

Explanation: To get from y + 3 = 7 to y = 4, we subtract 3 from both sides of the equation. This is the Subtraction Axiom, which states that if a = b, then a - c = b - c.

Q481: Which axiom is applied when transforming the equation x/2 = 6 into x = 12?

Division Axiom

Addition Axiom

Multiplication Axiom

Symmetric Axiom

Explanation: To get from x/2 = 6 to x = 12, we multiply both sides of the equation by 2. This is the Multiplication Axiom, which states that if a = b, then ac = bc.

Q482: If 3m = 18, which axiom justifies the step that leads to m = 6?

Multiplication Axiom

Addition Axiom

Division Axiom

Transitive Axiom

Explanation: To get from 3m = 18 to m = 6, we divide both sides of the equation by 3. This is the Division Axiom, which states that if a = b and c ≠ 0, then a/c = b/c.

Q483: Given that a = 2b and c = a + 5, which axiom allows us to conclude that c = 2b + 5?

Addition Axiom

Transitive Axiom

Symmetric Axiom

Substitution Axiom

Explanation: In the expression c = a + 5, 'a' is replaced by '2b' because we are given a = 2b. This is the Substitution Axiom, which states that if two quantities are equal, one can be substituted for the other in any expression or equation.

Q484: If p = q and q = 7, then p = 7. What axiom justifies this conclusion?

Substitution Axiom

Symmetric Axiom

Reflexive Axiom

Transitive Axiom

Explanation: The Transitive Axiom states that if two quantities are equal to the same third quantity, then they are equal to each other. Here, p is equal to q, and q is equal to 7, so p must be equal to 7.

Q485: If x + y = 10 and y = 3, then x + 3 = 10. Which axiom justifies this step?

Addition Axiom

Transitive Axiom

Substitution Axiom

Symmetric Axiom

Explanation: In the equation x + y = 10, 'y' is replaced by '3' because we are given y = 3. This is the Substitution Axiom, which states that if two quantities are equal, one can be substituted for the other in any expression or equation.

Identify and verify theorems related to adjacent and vertically opposite angles formed by intersecting lines.

Q486: Two straight lines intersect at a point. If one of the angles formed is 45°, what is the measure of its vertically opposite angle?

135°

45°

90°

180°

Explanation: Vertically opposite angles are equal. Therefore, if one angle is 45°, its vertically opposite angle is also 45°.

Q487: When two straight lines intersect, an angle of 60° is formed. What is the measure of an angle adjacent to it?

60°

120°

30°

90°

Explanation: Adjacent angles on a straight line are supplementary, meaning they add up to 180°. So, the adjacent angle is 180° - 60° = 120°.

Q488: Two lines PQ and RS intersect at O. If ∠POR = 110°, what is the measure of ∠QOS?

70°

180°

90°

110°

Explanation: ∠POR and ∠QOS are vertically opposite angles. Vertically opposite angles are equal. Therefore, ∠QOS = ∠POR = 110°.

Q489: If two lines intersect and one of the angles formed is 135°, what is the measure of an angle adjacent to it?

135°

90°

45°

180°

Explanation: Adjacent angles on a straight line are supplementary, meaning they add up to 180°. So, the adjacent angle is 180° - 135° = 45°.

Q490: Lines AB and CD intersect at point O. If ∠AOC = 75°, what is the measure of ∠AOD?

75°

15°

105°

90°

Explanation: ∠AOC and ∠AOD are adjacent angles on a straight line (line AB or line CD). Adjacent angles on a straight line are supplementary, so their sum is 180°. Therefore, ∠AOD = 180° - ∠AOC = 180° - 75° = 105°.

Q491: Two straight lines intersect. If an angle adjacent to a certain angle is 140°, what is the measure of the vertically opposite angle to that certain angle?

140°

70°

40°

90°

Explanation: Let the 'certain angle' be X. Its adjacent angle is 140°. Since adjacent angles on a straight line are supplementary, X = 180° - 140° = 40°. The vertically opposite angle to X is equal to X. Therefore, the vertically opposite angle is 40°.

Q492: Two straight lines intersect. If two vertically opposite angles are (3x + 10)° and (5x - 30)°, what is the value of x?

-20

Explanation: Vertically opposite angles are equal. So, 3x + 10 = 5x - 30. Rearranging the terms, 10 + 30 = 5x - 3x, which simplifies to 40 = 2x. Dividing by 2, we get x = 20.

Q493: Two adjacent angles on a straight line are (2x + 40)° and (x - 10)°. What is the value of x?

Explanation: Adjacent angles on a straight line sum to 180°. So, (2x + 40) + (x - 10) = 180. Combining like terms, 3x + 30 = 180. Subtracting 30 from both sides, 3x = 150. Dividing by 3, we get x = 50.

Q494: When two straight lines intersect, one angle is (4x + 20)° and its adjacent angle on a straight line is (2x - 8)°. Find the value of x.

Explanation: Adjacent angles on a straight line sum to 180°. So, (4x + 20) + (2x - 8) = 180. Combining like terms, 6x + 12 = 180. Subtracting 12 from both sides, 6x = 168. Dividing by 6, we get x = 28.

Q495: Two intersecting lines form vertically opposite angles where one angle is (2x + 50)° and the other is (5x - 10)°. Determine the value of x.

-20

Explanation: Vertically opposite angles are equal. So, 2x + 50 = 5x - 10. Rearranging the terms, 50 + 10 = 5x - 2x, which simplifies to 60 = 3x. Dividing by 3, we get x = 20.

Q496: Consider two intersecting lines. If an angle is (7x)° and its vertically opposite angle is (3x + 80)°, find the value of x.

Explanation: Vertically opposite angles are equal. So, 7x = 3x + 80. Subtracting 3x from both sides, 7x - 3x = 80, which simplifies to 4x = 80. Dividing by 4, we get x = 20.

Q497: An angle on a straight line is (x + 15)°. Its adjacent angle on the same straight line is (2x + 30)°. What is the value of x?

Explanation: Adjacent angles on a straight line sum to 180°. So, (x + 15) + (2x + 30) = 180. Combining like terms, 3x + 45 = 180. Subtracting 45 from both sides, 3x = 135. Dividing by 3, we get x = 45.

Q498: Which geometric principle is fundamentally used to prove that vertically opposite angles are equal when two straight lines intersect?

Adjacent angles are complementary.

Angles on a straight line sum to 180°.

Angles around a point sum to 360°.

Vertically opposite angles are equal.

Explanation: The proof for vertically opposite angles being equal relies on the fact that angles on a straight line sum to 180°. By showing that two vertically opposite angles are both supplementary to the same adjacent angle, their equality can be deduced.

Q499: If lines AB and CD intersect at point O, which of the following equations is a correct initial step in proving that ∠AOC = ∠BOD using the concept of angles on a straight line?

∠AOC = ∠BOD

∠AOC + ∠COB = 180°

∠AOC + ∠BOD = 180°

∠AOC + ∠AOD = 90°

Explanation: To prove ∠AOC = ∠BOD, we first use the property of angles on a straight line. ∠AOC and ∠COB are adjacent angles on the straight line AB, so their sum is 180°. This is a fundamental first step.

Q500: Given that lines PQ and RS intersect at O. We know that ∠POR + ∠ROQ = 180° and ∠ROQ + ∠QOS = 180°. What logical conclusion can be drawn from these two statements to prove vertically opposite angles are equal?

∠POR + ∠QOS = 180°

∠POR = ∠QOS

∠POR = ∠ROQ

∠ROQ = ∠QOS

Explanation: Since both (∠POR + ∠ROQ) and (∠ROQ + ∠QOS) are equal to 180°, we can equate them: ∠POR + ∠ROQ = ∠ROQ + ∠QOS. Subtracting ∠ROQ from both sides gives ∠POR = ∠QOS, thus proving the equality of vertically opposite angles.

Q501: Vertically opposite angles are proven to be equal primarily because they are:

Always acute angles.

Formed by two intersecting lines.

Supplementary to the same adjacent angle.

Adjacent to each other.

Explanation: The proof relies on showing that two vertically opposite angles (e.g., A and C) are both supplementary to the same adjacent angle (e.g., B). If A + B = 180° and C + B = 180°, then it logically follows that A = C.

Q502: In the diagram where lines KL and MN intersect at P, if we state that ∠KPM + ∠MPL = 180° (angles on straight line KL) and ∠MPL + ∠LPN = 180° (angles on straight line MN), which pair of angles are proven to be equal by these statements?

∠KPM and ∠MPL

∠KPM and ∠LPN

∠MPL and ∠LPN

∠KPN and ∠MPL

Explanation: From the given statements, we have ∠KPM + ∠MPL = 180° and ∠MPL + ∠LPN = 180°. Since both sums are equal to 180°, we can write ∠KPM + ∠MPL = ∠MPL + ∠LPN. By subtracting ∠MPL from both sides, we get ∠KPM = ∠LPN. These are vertically opposite angles.

Q503: Complete the proof: Let lines XOY and ZOW intersect at O. 1. ∠XOY is a straight line, so ∠XOZ + ∠ZOY = 180°. 2. ∠ZOW is a straight line, so ∠ZOY + ∠YOW = 180°. 3. From (1) and (2), we can conclude that ∠XOZ + ∠ZOY = ∠ZOY + ∠YOW. 4. Therefore, __________.

∠XOZ + ∠YOW = 180°

∠XOZ = ∠YOW

∠XOZ = ∠ZOY

∠XOZ + ∠ZOY = ∠YOW

Explanation: Given ∠XOZ + ∠ZOY = ∠ZOY + ∠YOW. By subtracting the common angle ∠ZOY from both sides of the equation, we are left with ∠XOZ = ∠YOW. This demonstrates that the vertically opposite angles are equal.

Identify angles formed when two straight lines are intersected by a transversal.

Q504: In a diagram where two straight lines intersect, which of the following pairs of angles are vertically opposite?

Angles adjacent to each other on a straight line.

Angles that share a vertex and are directly opposite to each other.

Angles that are on the same side of the transversal and in corresponding positions.

Angles that are on opposite sides of the transversal and between the two intersected lines.

Explanation: Vertically opposite angles are formed when two straight lines intersect, and they are opposite to each other at the point of intersection.

Q505: If two adjacent angles form a straight line, what is their sum?

90°

180°

270°

360°

Explanation: Angles on a straight line (or a linear pair) always sum up to 180 degrees.

Q506: In a diagram with two lines intersected by a transversal, if angle X is the upper-left angle at the first intersection and angle Y is the upper-left angle at the second intersection, what type of angles are X and Y?

Alternate interior angles

Corresponding angles

Consecutive interior angles

Vertically opposite angles

Explanation: Corresponding angles are in the same relative position at each intersection where a transversal crosses two lines (e.g., both upper-left).

Q507: Consider two lines intersected by a transversal. If angle P is the lower-left angle at the first intersection (between the two lines) and angle Q is the upper-right angle at the second intersection (between the two lines), what type of angles are P and Q?

Corresponding angles

Vertically opposite angles

Alternate interior angles

Consecutive interior angles

Explanation: Alternate interior angles are on opposite sides of the transversal and are located between the two lines being intersected.

Q508: When two lines are cut by a transversal, if angle M is the upper-right angle at the first intersection (between the two lines) and angle N is the lower-right angle at the second intersection (between the two lines), what type of angles are M and N?

Alternate exterior angles

Corresponding angles

Vertically opposite angles

Consecutive interior angles

Explanation: Consecutive interior angles (also known as co-interior or same-side interior angles) are on the same side of the transversal and are located between the two lines.

Q509: In a diagram where two straight lines intersect, if angle A and angle B are adjacent angles on a straight line, and angle B and angle C are vertically opposite, which statement is true?

Angle A and angle C are equal.

Angle A and angle C sum to 90°.

Angle A and angle C sum to 180°.

Angle A and angle C are complementary.

Explanation: Since A and B are adjacent angles on a straight line, A + B = 180°. Since B and C are vertically opposite angles, B = C. Therefore, substituting B with C in the first equation gives A + C = 180°.

Q510: In the diagram, lines L1 and L2 are parallel and intersected by a transversal T. If angle ∠1 (top-left at the first intersection) is 60°, what is the measure of angle ∠5 (top-left at the second intersection)?

120°

60°

30°

90°

Explanation: Angles ∠1 and ∠5 are corresponding angles. When two parallel lines are intersected by a transversal, corresponding angles are equal. Therefore, if ∠1 = 60°, then ∠5 = 60°.

Q511: In the diagram, lines L1 and L2 are parallel and intersected by a transversal T. If angle ∠3 (bottom-right at the first intersection) is 120°, what is the measure of angle ∠5 (top-left at the second intersection)?

60°

180°

120°

90°

Explanation: Angles ∠3 and ∠5 are alternate interior angles. When two parallel lines are intersected by a transversal, alternate interior angles are equal. Therefore, if ∠3 = 120°, then ∠5 = 120°.

Q512: In the diagram, lines L1 and L2 are parallel and intersected by a transversal T. If angle ∠4 (bottom-left at the first intersection) is 75°, what is the measure of angle ∠5 (top-left at the second intersection)?

75°

90°

180°

105°

Explanation: Angles ∠4 and ∠5 are interior angles on the same side of the transversal (co-interior angles). When two parallel lines are intersected by a transversal, co-interior angles are supplementary, meaning their sum is 180°. Therefore, if ∠4 = 75°, then ∠5 = 180° - 75° = 105°.

Q513: In the diagram, lines L1 and L2 are parallel and intersected by a transversal T. If angle ∠2 (top-right at the first intersection) is 130°, what is the measure of angle ∠6 (top-right at the second intersection)?

50°

90°

130°

180°

Explanation: Angles ∠2 and ∠6 are corresponding angles. When two parallel lines are intersected by a transversal, corresponding angles are equal. Therefore, if ∠2 = 130°, then ∠6 = 130°.

Q514: In the diagram, lines L1 and L2 are parallel and intersected by a transversal T. If angle ∠4 (bottom-left at the first intersection) is 45°, what is the measure of angle ∠6 (top-right at the second intersection)?

135°

90°

45°

180°

Explanation: Angles ∠4 and ∠6 are alternate interior angles. When two parallel lines are intersected by a transversal, alternate interior angles are equal. Therefore, if ∠4 = 45°, then ∠6 = 45°.

Q515: In the diagram, lines L1 and L2 are parallel and intersected by a transversal T. If angle ∠3 (bottom-right at the first intersection) is 110°, what is the measure of angle ∠6 (top-right at the second intersection)?

110°

90°

70°

180°

Explanation: Angles ∠3 and ∠6 are interior angles on the same side of the transversal (co-interior angles). When two parallel lines are intersected by a transversal, co-interior angles are supplementary, meaning their sum is 180°. Therefore, if ∠3 = 110°, then ∠6 = 180° - 110° = 70°.

Q516: In the given figure, lines AB and CD are parallel. If ∠AEF = (3x + 10)° and ∠CFG = (x + 50)°, find the value of x.

Explanation: Since AB || CD, ∠AEF and ∠CFG are corresponding angles. Therefore, they are equal. (3x + 10)° = (x + 50)°. Subtract x from both sides: 2x + 10 = 50. Subtract 10 from both sides: 2x = 40. Divide by 2: x = 20.

Q517: In the given figure, AB is parallel to CD. If ∠BEF = (2x + 20)° and ∠DFE = (3x - 10)°, find the value of x.

Explanation: Since AB || CD, ∠BEF and ∠DFE are interior angles on the same side of the transversal. Therefore, they are supplementary, meaning their sum is 180°. (2x + 20) + (3x - 10) = 180. Simplify: 5x + 10 = 180. Subtract 10 from both sides: 5x = 170. Divide by 5: x = 34.

Q518: In the figure, PQ is parallel to RS. If ∠PQT = (4x)° and ∠RTU = (2x + 60)°, find the value of x.

Explanation: Since PQ || RS, ∠PQT and ∠QTS are alternate interior angles, so ∠QTS = ∠PQT = 4x. Also, ∠QTS and ∠RTU are vertically opposite angles. Therefore, ∠RTU = ∠QTS. So, (2x + 60) = 4x. Subtract 2x from both sides: 60 = 2x. Divide by 2: x = 30.

Q519: In the given figure, AB is parallel to CD. If ∠BEF = (5x - 20)° and ∠DFE = (3x + 40)°, find the value of x.

42.5

Explanation: Since AB || CD, ∠BEF and ∠DFE are interior angles on the same side of the transversal. Therefore, their sum is 180°. (5x - 20) + (3x + 40) = 180. Combine like terms: 8x + 20 = 180. Subtract 20 from both sides: 8x = 160. Divide by 8: x = 20.

Q520: In the figure, lines AB and CD are parallel. If ∠AEG = (2x + 10)° and ∠DFH = (4x - 50)°, find the value of x.

36.67

Explanation: Since AB || CD, ∠AEG and ∠DFH are alternate exterior angles. Therefore, they are equal. (2x + 10) = (4x - 50). Subtract 2x from both sides: 10 = 2x - 50. Add 50 to both sides: 60 = 2x. Divide by 2: x = 30.

Q521: In the given figure, AB is parallel to CD. If ∠AEF = (3x)° and ∠BEF = (2x + 25)°, and EF is a transversal intersecting parallel lines AB and CD, what is the measure of ∠CFE?

31°

93°

87°

78°

Explanation: ∠AEF and ∠BEF are angles on a straight line, so their sum is 180°. (3x) + (2x + 25) = 180. Simplify: 5x + 25 = 180. Subtract 25: 5x = 155. Divide by 5: x = 31. Since AB || CD, ∠BEF and ∠CFE are alternate interior angles. Therefore, ∠CFE = ∠BEF = (2x + 25) = (2 * 31 + 25) = 62 + 25 = 87°.

Q522: In the given figure, AB is parallel to CD. If ∠AEF = (7x - 5)° and ∠EFD = (4x + 40)°, what is the measure of ∠CFE?

100°

15°

80°

90°

Explanation: Since AB || CD, ∠AEF and ∠EFD are alternate interior angles. Therefore, they are equal. (7x - 5) = (4x + 40). This simplifies to 3x = 45, so x = 15. Now, ∠EFD = (4 * 15 + 40) = 60 + 40 = 100°. Angles ∠CFE and ∠EFD form a linear pair (angles on a straight line CD). Therefore, ∠CFE + ∠EFD = 180°. So, ∠CFE + 100° = 180°. Thus, ∠CFE = 80°.

Identify and verify theorems related to angles formed when two parallel lines are intersected by a transversal.

Q523: When two parallel lines are intersected by a transversal, what is the relationship between corresponding angles?

Corresponding angles are supplementary.

Corresponding angles are equal.

Corresponding angles sum to 90°.

Corresponding angles are vertically opposite.

Explanation: Corresponding angles are formed on the same side of the transversal and in corresponding positions (e.g., both above the parallel lines and to the left of the transversal). When lines are parallel, these angles are always equal.

Q524: What is the relationship between alternate interior angles when two parallel lines are intersected by a transversal?

Alternate interior angles are supplementary.

Alternate interior angles are equal.

Alternate interior angles sum to 180°.

Alternate interior angles are complementary.

Explanation: Alternate interior angles are formed on opposite sides of the transversal and between the two parallel lines. When lines are parallel, these angles are always equal.

Q525: When two parallel lines are intersected by a transversal, what is the relationship between consecutive interior (co-interior) angles?

Consecutive interior angles are equal.

Consecutive interior angles are supplementary (sum to 180°).

Consecutive interior angles sum to 90°.

Consecutive interior angles are vertically opposite.

Explanation: Consecutive interior angles are formed on the same side of the transversal and between the two parallel lines. When lines are parallel, these angles are always supplementary, meaning their sum is 180°.

Q526: In a diagram where two parallel lines are cut by a transversal, if angle A is 70°, and angle B is its corresponding angle, what is the measure of angle B?

110°

70°

90°

180°

Explanation: Angle A and angle B are corresponding angles. When two parallel lines are intersected by a transversal, corresponding angles are equal. Therefore, if angle A is 70°, angle B is also 70°.

Q527: If two parallel lines are intersected by a transversal and an alternate interior angle 'X' is 55°, what is the measure of its alternate interior angle 'Y'?

125°

55°

90°

180°

Explanation: Angle X and angle Y are alternate interior angles. When two parallel lines are intersected by a transversal, alternate interior angles are equal. Therefore, if angle X is 55°, angle Y is also 55°.

Q528: Consider two parallel lines cut by a transversal. If one consecutive interior angle 'M' is 100°, what is the measure of the other consecutive interior angle 'N'?

100°

80°

180°

90°

Explanation: Angle M and angle N are consecutive interior angles. When two parallel lines are intersected by a transversal, consecutive interior angles are supplementary, meaning their sum is 180°. So, if M is 100°, then N = 180° - 100° = 80°.

Q529: Lines AB and CD are parallel. A transversal line EF intersects AB at G and CD at H. If ∠EGB = 70°, what is the measure of ∠GHD?

70°

110°

90°

180°

Explanation: Angles EGB and GHD are corresponding angles. Since lines AB and CD are parallel, corresponding angles are equal. Therefore, ∠GHD = ∠EGB = 70°.

Q530: Lines PQ and RS are parallel. A transversal line TU intersects PQ at V and RS at W. If ∠QVW = 60°, what is the measure of ∠VWS?

60°

120°

90°

180°

Explanation: Angles QVW and VWS are alternate interior angles. Since lines PQ and RS are parallel, alternate interior angles are equal. Therefore, ∠VWS = ∠QVW = 60°.

Q531: Lines KL and MN are parallel. A transversal line OP intersects KL at Q and MN at R. If ∠LQR = 130°, what is the measure of ∠QRN?

50°

130°

90°

180°

Explanation: Angles LQR and QRN are consecutive interior (or allied) angles. Since lines KL and MN are parallel, the sum of consecutive interior angles is 180°. Therefore, ∠QRN = 180° - ∠LQR = 180° - 130° = 50°.

Q532: Lines DE and FG are parallel. A transversal line HI intersects DE at J and FG at K. If ∠DJH = 80°, what is the measure of ∠FKJ?

80°

100°

90°

180°

Explanation: ∠DJH and ∠EJI are vertically opposite angles, so ∠EJI = ∠DJH = 80°. Angles EJI and FKJ are corresponding angles. Since DE and FG are parallel, corresponding angles are equal. Therefore, ∠FKJ = ∠EJI = 80°.

Q533: Lines ST and UV are parallel. A transversal line WX intersects ST at Y and UV at Z. If ∠TYW = 125°, what is the measure of ∠UYZ?

55°

125°

90°

180°

Explanation: Angles TYW and SYW are angles on a straight line. So, ∠SYW = 180° - ∠TYW = 180° - 125° = 55°. Angles SYW and UYZ are alternate interior angles. Since ST and UV are parallel, alternate interior angles are equal. Therefore, ∠UYZ = ∠SYW = 55°.

Q534: Lines MN and OP are parallel. A transversal line QR intersects MN at S and OP at T. If ∠MSQ = 70°, what is the measure of ∠PTO?

110°

70°

90°

180°

Explanation: Angles MSQ and STO are corresponding angles. Since MN and OP are parallel, corresponding angles are equal. So, ∠STO = ∠MSQ = 70°. Angles STO and PTO are angles on a straight line. Therefore, ∠PTO = 180° - ∠STO = 180° - 70° = 110°.

Q535: If line AB is parallel to line CD (AB || CD), and EF is a transversal. If angle EGB = (2x + 10)° and angle CHG = (3x - 20)°, find the value of x.

190

Explanation: Angle EGB and angle AGH are vertically opposite angles, so ∠AGH = ∠EGB = (2x + 10)°. Angle AGH and angle CHG are co-interior angles, which are supplementary, so ∠AGH + ∠CHG = 180°. Therefore, (2x + 10) + (3x - 20) = 180. Simplifying gives 5x - 10 = 180, so 5x = 190. Thus, x = 38.

Q536: If line AB is parallel to line CD (AB || CD), and EF is a transversal. If angle AGF = (3x + 20)° and angle DHE = (x + 80)°, find the value of x.

Explanation: Angle AGF and angle EGB are angles on a straight line, so ∠EGB = 180° - ∠AGF = 180° - (3x + 20)°. Angle EGB and angle DHE are alternate exterior angles, so they are equal: ∠EGB = ∠DHE. Therefore, 180 - (3x + 20) = x + 80. Simplifying gives 160 - 3x = x + 80. Rearranging terms, 80 = 4x. Thus, x = 20.

Q537: If line AB is parallel to line CD (AB || CD), and EF is a transversal. If angle BGE = (5x - 30)° and angle DHE is vertically opposite to angle GHC, and angle GHC = (2x + 60)°, find the value of x.

Explanation: Angle DHE and angle GHC are vertically opposite angles, so ∠DHE = ∠GHC = (2x + 60)°. Angle BGE and angle DHE are corresponding angles, so they are equal: ∠BGE = ∠DHE. Therefore, 5x - 30 = 2x + 60. Simplifying gives 3x = 90. Thus, x = 30.

Q538: If line AB is parallel to line CD (AB || CD), and EF is a transversal. If angle CGE = (2x + 10)° and angle GHF = (4x - 50)°, find the value of x.

Explanation: Angle CGE and angle BGH are vertically opposite angles, so ∠BGH = ∠CGE = (2x + 10)°. Angle BGH and angle GHF are alternate interior angles, so they are equal: ∠BGH = ∠GHF. Therefore, 2x + 10 = 4x - 50. Simplifying gives 60 = 2x. Thus, x = 30.

Q539: If line AB is parallel to line CD (AB || CD), and EF is a transversal. If angle BGE = (3x + 20)° and angle DHE = (x + 80)°, find the measure of angle DHE.

30°

70°

100°

110°

Explanation: Angle BGE and angle DHE are alternate exterior angles, so they are equal: ∠BGE = ∠DHE. Therefore, 3x + 20 = x + 80. Simplifying gives 2x = 60, so x = 30. To find the measure of angle DHE, substitute x = 30 into the expression (x + 80): ∠DHE = 30 + 80 = 110°.

Q540: If line AB is parallel to line CD (AB || CD), and EF is a transversal. Angle AGE = (3x - 10)° and angle DHF = (x + 50)°. Find the measure of angle CGE.

35°

95°

85°

115°

Explanation: Angle DHF and angle CHG are vertically opposite angles, so ∠CHG = ∠DHF = (x + 50)°. Angle AGE and angle CHG are co-interior angles, which are supplementary, so ∠AGE + ∠CHG = 180°. Therefore, (3x - 10) + (x + 50) = 180. Simplifying gives 4x + 40 = 180, so 4x = 140. Thus, x = 35. Angle AGE and angle CGE are angles on a straight line, so ∠CGE = 180° - ∠AGE. First, find ∠AGE = 3x - 10 = 3(35) - 10 = 105 - 10 = 95°. Then, ∠CGE = 180° - 95° = 85°.

Find the relationship between milliliters (ml) and cubic centimeters (cm³).

Q541: What is the equivalent of 1 milliliter (ml) in cubic centimeters (cm³)?

10 cm³

1 cm³

100 cm³

1000 cm³

Explanation: The fundamental relationship between volume and capacity states that 1 milliliter (ml) is exactly equal to 1 cubic centimeter (cm³).

Q542: If you have 1 cubic centimeter (cm³) of liquid, how many milliliters (ml) is that?

0.1 ml

10 ml

1 ml

100 ml

Explanation: The relationship is reciprocal: 1 cubic centimeter (cm³) is equal to 1 milliliter (ml).

Q543: A small bottle contains 5 ml of medicine. How many cubic centimeters (cm³) of medicine is this?

0.5 cm³

50 cm³

5 cm³

500 cm³

Explanation: Since 1 ml = 1 cm³, then 5 ml will be equal to 5 cm³.

Q544: A cube with a volume of 10 cm³ is filled with water. How many milliliters (ml) of water does it contain?

1 ml

100 ml

10 ml

1000 ml

Explanation: The relationship 1 cm³ = 1 ml applies directly. Therefore, 10 cm³ is equal to 10 ml.

Q545: Which of the following statements correctly expresses the relationship between milliliters and cubic centimeters?

1 ml = 10 cm³

1 ml = 100 cm³

1 ml = 1 cm³

1 ml = 1000 cm³

Explanation: The internationally accepted standard relationship is that 1 milliliter is equivalent to 1 cubic centimeter.

Q546: Identify the incorrect statement among the following:

1 cm³ = 1 ml

2 ml = 2 cm³

1 ml = 100 cm³

15 cm³ = 15 ml

Explanation: The correct relationship is 1 ml = 1 cm³. Therefore, 1 ml = 100 cm³ is an incorrect statement.

Q547: A rectangular prism with internal dimensions 10 cm, 5 cm, and 2 cm is perfectly filled with 100 ml of liquid. What is the volume of this liquid in cubic centimeters?

10 cm³

100 cm³

1000 cm³

10000 cm³

Explanation: The relationship between milliliters (ml) and cubic centimeters (cm³) is 1 ml = 1 cm³. Therefore, 100 ml is equal to 100 cm³.

Q548: A container shaped like a cuboid has internal dimensions of 8 cm length, 5 cm width, and 4 cm height. If it is completely filled with 160 ml of water, what is the volume of the water in cubic centimeters?

16 cm³

160 cm³

1600 cm³

16000 cm³

Explanation: The relationship between milliliters (ml) and cubic centimeters (cm³) is 1 ml = 1 cm³. Therefore, 160 ml is equal to 160 cm³.

Q549: A rectangular tank, measuring 20 cm by 10 cm by 5 cm internally, holds 1000 ml of oil when full. Express the volume of the oil in cubic centimeters.

100 cm³

1000 cm³

10000 cm³

10 cm³

Explanation: The relationship between milliliters (ml) and cubic centimeters (cm³) is 1 ml = 1 cm³. Therefore, 1000 ml is equal to 1000 cm³.

Q550: A small cuboid-shaped bottle with internal dimensions 4 cm x 2.5 cm x 1 cm is filled with 10 ml of perfume. What is the volume of the perfume in cubic centimeters?

1 cm³

10 cm³

100 cm³

0.1 cm³

Explanation: The relationship between milliliters (ml) and cubic centimeters (cm³) is 1 ml = 1 cm³. Therefore, 10 ml is equal to 10 cm³.

Q551: A rectangular box has internal dimensions of 6 cm, 3 cm, and 2.5 cm. When filled to capacity, it contains 45 ml of a chemical solution. What is the volume of the solution in cubic centimeters?

4.5 cm³

45 cm³

450 cm³

0.45 cm³

Explanation: The relationship between milliliters (ml) and cubic centimeters (cm³) is 1 ml = 1 cm³. Therefore, 45 ml is equal to 45 cm³.

Q552: A rectangular container, with a base area of 50 cm² and a height of 4 cm, is filled with 200 ml of milk. What is the volume of the milk in cubic centimeters?

20 cm³

200 cm³

2000 cm³

2 cm³

Explanation: The relationship between milliliters (ml) and cubic centimeters (cm³) is 1 ml = 1 cm³. Therefore, 200 ml is equal to 200 cm³.

Q553: A cuboid-shaped container for medicinal syrup has internal dimensions of 5 cm, 4 cm, and 3.5 cm. When filled, it holds 70 ml of syrup. What is the volume of the syrup in cubic centimeters?

7 cm³

700 cm³

70 cm³

0.7 cm³

Explanation: The relationship between milliliters (ml) and cubic centimeters (cm³) is 1 ml = 1 cm³. Therefore, 70 ml is equal to 70 cm³.

Q554: A rectangular tank has a base area of 50 cm². If an additional 200 ml of liquid is poured into the tank, what is the increase in the water level in centimeters?

0.25 cm

4 cm

10000 cm

40 cm

Explanation: First, convert the volume from milliliters to cubic centimeters: 1 ml = 1 cm³. So, 200 ml = 200 cm³. The increase in height can be calculated using the formula: Height = Volume / Base Area. Height = 200 cm³ / 50 cm² = 4 cm.

Q555: A rectangular container has a base area of 120 cm². If 360 ml of water is added, by how many centimeters will the water level rise?

3 cm

0.33 cm

30 cm

43200 cm

Explanation: Convert the volume: 360 ml = 360 cm³. Using Height = Volume / Base Area, we get Height = 360 cm³ / 120 cm² = 3 cm.

Q556: A tank has a rectangular base with an area of 75 cm². If 150 ml of liquid is poured into it, what will be the increase in the liquid's height?

0.5 cm

2 cm

20 cm

11250 cm

Explanation: Convert volume: 150 ml = 150 cm³. Height = Volume / Base Area = 150 cm³ / 75 cm² = 2 cm.

Q557: A rectangular tank has a base area of 20 cm². If 100 ml of liquid is added to it, what will be the rise in the liquid level in centimeters?

0.2 cm

2000 cm

5 cm

50 cm

Explanation: The volume of liquid is 100 ml, which is equal to 100 cm³. The increase in height is Volume / Base Area = 100 cm³ / 20 cm² = 5 cm.

Q558: A rectangular fish tank has a base area of 40 cm². If 240 ml of water is poured into it, how much will the water level rise in centimeters?

0.167 cm

9600 cm

6 cm

60 cm

Explanation: Volume conversion: 240 ml = 240 cm³. Height = Volume / Base Area = 240 cm³ / 40 cm² = 6 cm.

Q559: A rectangular container has a base area of 60 cm². If 180 ml of liquid is poured into it, what is the increase in the liquid's height in centimeters?

0.33 cm

3 cm

10800 cm

30 cm

Explanation: The volume of liquid added is 180 ml, which is equivalent to 180 cm³. The increase in height is calculated by dividing the volume by the base area: Height = 180 cm³ / 60 cm² = 3 cm.

Find the relationship between liters (l) and cubic centimeters (cm³).

Q560: Convert 5 liters to cubic centimeters.

500 cm³

5000 cm³

50 cm³

50000 cm³

Explanation: Since 1 liter = 1000 cubic centimeters, to convert 5 liters to cubic centimeters, multiply 5 by 1000. So, 5 l = 5 × 1000 cm³ = 5000 cm³.

Q561: What is 3.5 liters in cubic centimeters?

350 cm³

3500 cm³

35 cm³

35000 cm³

Explanation: To convert liters to cubic centimeters, multiply by 1000. So, 3.5 l = 3.5 × 1000 cm³ = 3500 cm³.

Q562: Convert 7000 cm³ to liters.

70 l

0.7 l

7 l

700 l

Explanation: Since 1000 cubic centimeters = 1 liter, to convert 7000 cm³ to liters, divide 7000 by 1000. So, 7000 cm³ = 7000 ÷ 1000 l = 7 l.

Q563: What is 2500 cm³ in liters?

2.5 l

25 l

0.25 l

250 l

Explanation: To convert cubic centimeters to liters, divide by 1000. So, 2500 cm³ = 2500 ÷ 1000 l = 2.5 l.

Q564: How many cubic centimeters are there in 0.8 liters?

80 cm³

8 cm³

800 cm³

8000 cm³

Explanation: To convert liters to cubic centimeters, multiply by 1000. So, 0.8 l = 0.8 × 1000 cm³ = 800 cm³.

Q565: A container has a volume of 400 cm³. What is its volume in liters?

40 l

0.4 l

4 l

0.04 l

Explanation: To convert cubic centimeters to liters, divide by 1000. So, 400 cm³ = 400 ÷ 1000 l = 0.4 l.

Q566: A cubical container has a side length of 10 cm. What is its capacity in liters?

10 L

100 L

1 L

0.1 L

Explanation: First, calculate the volume of the cube: Volume = side × side × side = 10 cm × 10 cm × 10 cm = 1000 cm³. Then, convert cubic centimeters to liters. Since 1000 cm³ = 1 L, the capacity is 1000 cm³ / 1000 = 1 L.

Q567: A rectangular tank has dimensions of 20 cm, 15 cm, and 10 cm. What is its capacity in liters?

30 L

3 L

300 L

0.3 L

Explanation: First, calculate the volume of the rectangular tank: Volume = length × width × height = 20 cm × 15 cm × 10 cm = 3000 cm³. Then, convert cubic centimeters to liters. Since 1000 cm³ = 1 L, the capacity is 3000 cm³ / 1000 = 3 L.

Q568: A container has a capacity of 5 liters. What is its volume in cubic centimeters?

5 cm³

50 cm³

500 cm³

5000 cm³

Explanation: To convert liters to cubic centimeters, multiply by 1000. Since 1 L = 1000 cm³, the volume is 5 L × 1000 cm³/L = 5000 cm³.

Q569: A rectangular fish tank has internal dimensions of length 25 cm, width 20 cm, and height 15 cm. How many liters of water can it hold when full?

75 L

7.5 L

750 L

0.75 L

Explanation: First, calculate the volume of the fish tank: Volume = length × width × height = 25 cm × 20 cm × 15 cm = 7500 cm³. Then, convert cubic centimeters to liters. Since 1000 cm³ = 1 L, the capacity is 7500 cm³ / 1000 = 7.5 L.

Q570: A rectangular water tank has a base area of 1200 cm². If water is poured into it up to a height of 25 cm, what is the volume of water in liters?

300 L

3 L

30 L

3000 L

Explanation: First, calculate the volume of water: Volume = base area × height = 1200 cm² × 25 cm = 30000 cm³. Then, convert cubic centimeters to liters. Since 1000 cm³ = 1 L, the volume of water is 30000 cm³ / 1000 = 30 L.

Q571: A rectangular container with a base of 50 cm by 20 cm can hold 4 liters of liquid. What is the height of the container in centimeters?

8 cm

40 cm

0.4 cm

4 cm

Explanation: First, convert the capacity from liters to cubic centimeters: 4 L = 4 × 1000 cm³ = 4000 cm³. Then, calculate the base area of the container: Base Area = length × width = 50 cm × 20 cm = 1000 cm². Finally, find the height using the formula Volume = Base Area × Height: Height = Volume / Base Area = 4000 cm³ / 1000 cm² = 4 cm.

Q572: A rectangular water tank has a base area of 1200 cm². If it contains 36 liters of water, what is the height of the water level in the tank?

3 cm

30 cm

300 cm

360 cm

Explanation: 1 liter = 1000 cm³. So, 36 liters = 36 × 1000 cm³ = 36000 cm³. The volume of water is given by Base Area × Height. Therefore, Height = Volume / Base Area = 36000 cm³ / 1200 cm² = 30 cm.

Q573: A rectangular tank with dimensions 80 cm length, 50 cm width, and 60 cm height is half-filled with water. How much more water, in liters, is needed to fill the tank completely?

60 L

120 L

240 L

1200 L

Explanation: Total volume of the tank = Length × Width × Height = 80 cm × 50 cm × 60 cm = 240000 cm³. Convert total volume to liters: 240000 cm³ / 1000 cm³/L = 240 L. Since the tank is half-filled, the remaining volume to fill is half of the total volume = 240 L / 2 = 120 L.

Q574: Container A is a cuboid with length 20 cm, width 15 cm, and height 10 cm. Container B is a cylindrical tank with a base area of 300 cm² and a height of 12 cm. Which container has a larger capacity and by how many liters?

Container A, by 0.6 L

Container B, by 0.6 L

Container A, by 600 cm³

Container B, by 6 L

Explanation: Capacity of Container A = 20 cm × 15 cm × 10 cm = 3000 cm³. Convert to liters: 3000 cm³ / 1000 cm³/L = 3 L. Capacity of Container B = Base Area × Height = 300 cm² × 12 cm = 3600 cm³. Convert to liters: 3600 cm³ / 1000 cm³/L = 3.6 L. Container B has a larger capacity by 3.6 L - 3 L = 0.6 L.

Q575: A rectangular water tank has a base area of 1500 cm². If it is filled with 18 liters of water and this represents 3/5 of its total capacity, what is the total height of the tank?

12 cm

20 cm

30 cm

36 cm

Explanation: Volume of water filled = 18 L = 18 × 1000 cm³ = 18000 cm³. Height of the water level = Volume / Base Area = 18000 cm³ / 1500 cm² = 12 cm. This 12 cm represents 3/5 of the total height of the tank. Let the total height be H. Then (3/5) × H = 12 cm. So, H = 12 cm × (5/3) = 4 cm × 5 = 20 cm.

Q576: A container initially holds 4.5 liters of water. This water is poured into an empty rectangular tank with a base area of 300 cm². What will be the height of the water level in the rectangular tank?

1.5 cm

15 cm

150 cm

45 cm

Explanation: Volume of water = 4.5 L = 4.5 × 1000 cm³ = 4500 cm³. When this water is poured into the rectangular tank, the height of the water level will be = Volume / Base Area = 4500 cm³ / 300 cm² = 15 cm.

Q577: A rectangular tank has a base of 40 cm by 25 cm. It contains water up to a height of 10 cm. If an additional 5 liters of water are added to the tank, what will be the new height of the water level?

10 cm

15 cm

10.5 cm

20 cm

Explanation: Base area of the tank = 40 cm × 25 cm = 1000 cm². Initial volume of water = Base Area × Initial Height = 1000 cm² × 10 cm = 10000 cm³. Additional water added = 5 liters = 5 × 1000 cm³ = 5000 cm³. New total volume of water = 10000 cm³ + 5000 cm³ = 15000 cm³. New height of water level = New Total Volume / Base Area = 15000 cm³ / 1000 cm² = 15 cm.

Find the relationship between liters (l) and cubic meters (m³).

Q578: What is 2 m³ expressed in liters?

20 L

200 L

2000 L

0.002 L

Explanation: To convert cubic meters (m³) to liters (l), multiply the volume in cubic meters by 1000. So, 2 m³ × 1000 = 2000 liters.

Q579: Convert 0.5 m³ to liters.

5 L

50 L

500 L

0.0005 L

Explanation: To convert cubic meters (m³) to liters (l), multiply the volume in cubic meters by 1000. So, 0.5 m³ × 1000 = 500 liters.

Q580: How many liters are there in 4.5 m³?

45 L

450 L

4500 L

0.0045 L

Explanation: To convert cubic meters (m³) to liters (l), multiply the volume in cubic meters by 1000. So, 4.5 m³ × 1000 = 4500 liters.

Q581: A water tank has a volume of 1.2 m³. How many liters of water can it hold?

12 L

120 L

1200 L

0.0012 L

Explanation: To convert cubic meters (m³) to liters (l), multiply the volume in cubic meters by 1000. So, 1.2 m³ × 1000 = 1200 liters.

Q582: If a container has a volume of 0.003 m³, what is its capacity in liters?

0.3 L

3 L

30 L

0.000003 L

Explanation: To convert cubic meters (m³) to liters (l), multiply the volume in cubic meters by 1000. So, 0.003 m³ × 1000 = 3 liters.

Q583: A swimming pool has a volume of 25 m³. How many liters of water are needed to fill it completely?

250 L

2.5 L

25000 L

0.025 L

Explanation: To convert cubic meters (m³) to liters (l), multiply the volume in cubic meters by 1000. So, 25 m³ × 1000 = 25000 liters.

Q584: A rectangular tank has a length of 2 m, a width of 1 m, and a height of 0.5 m. What is its full capacity in liters?

100 liters

500 liters

1000 liters

2000 liters

Explanation: The volume of the tank is length × width × height = 2 m × 1 m × 0.5 m = 1 m³. Since 1 m³ = 1000 liters, the full capacity is 1 × 1000 = 1000 liters.

Q585: A cuboid container has dimensions 50 cm, 20 cm, and 100 cm. What is its capacity in liters?

10 liters

100 liters

1000 liters

10000 liters

Explanation: First, convert dimensions to meters: 50 cm = 0.5 m, 20 cm = 0.2 m, 100 cm = 1 m. Volume = length × width × height = 0.5 m × 0.2 m × 1 m = 0.1 m³. Since 1 m³ = 1000 liters, the capacity is 0.1 × 1000 = 100 liters.

Q586: A water tank has dimensions 1.5 m length, 0.8 m width, and 50 cm height. What is its capacity in liters?

60 liters

600 liters

6000 liters

60000 liters

Explanation: First, convert the height to meters: 50 cm = 0.5 m. Volume = length × width × height = 1.5 m × 0.8 m × 0.5 m = 0.6 m³. Since 1 m³ = 1000 liters, the capacity is 0.6 × 1000 = 600 liters.

Q587: A large industrial water tank has dimensions 3 m × 2.5 m × 2 m. What is its capacity in liters?

1500 liters

15000 liters

150000 liters

150 liters

Explanation: The volume of the tank is length × width × height = 3 m × 2.5 m × 2 m = 15 m³. Since 1 m³ = 1000 liters, the capacity is 15 × 1000 = 15000 liters.

Q588: A storage tank for fuel has dimensions 4 m length, 1.5 m width, and 1.2 m height. How many liters of fuel can it hold when full?

720 liters

7200 liters

72000 liters

7.2 liters

Explanation: The volume of the tank is length × width × height = 4 m × 1.5 m × 1.2 m = 7.2 m³. Since 1 m³ = 1000 liters, the capacity is 7.2 × 1000 = 7200 liters.

Q589: A rectangular swimming pool has a length of 10 m, a width of 5 m, and a uniform depth of 1.5 m. What is its capacity in liters?

7500 liters

75000 liters

750 liters

750000 liters

Explanation: The volume of the swimming pool is length × width × depth = 10 m × 5 m × 1.5 m = 75 m³. Since 1 m³ = 1000 liters, the capacity is 75 × 1000 = 75000 liters.

Q590: A rectangular tank has a base area of 4 m². If it contains 12000 liters of water, determine the height of the water level in the tank in centimeters (cm).

30 cm

300 cm

3 cm

3000 cm

Explanation: First, convert the volume from liters to cubic meters: 12000 liters = 12000 / 1000 m³ = 12 m³. Then, use the formula Volume = Base Area × Height to find the height: Height = Volume / Base Area = 12 m³ / 4 m² = 3 m. Finally, convert the height from meters to centimeters: 3 m × 100 cm/m = 300 cm.

Q591: A rectangular tank has a base area of 2.5 m². If it contains 5000 liters of water, determine the height of the water level in the tank in meters (m).

0.5 m

200 m

2 m

20 m

Explanation: First, convert the volume from liters to cubic meters: 5000 liters = 5000 / 1000 m³ = 5 m³. Then, use the formula Volume = Base Area × Height to find the height: Height = Volume / Base Area = 5 m³ / 2.5 m² = 2 m.

Q592: A rectangular tank contains 9000 liters of water. If the water level is 3 m high, what is the base area of the tank in m²?

27 m²

30 m²

3 m²

0.3 m²

Explanation: First, convert the volume from liters to cubic meters: 9000 liters = 9000 / 1000 m³ = 9 m³. Then, use the formula Base Area = Volume / Height to find the base area: Base Area = 9 m³ / 3 m = 3 m².

Q593: A rectangular tank has a base area of 5 m². If the water level is 1.5 m high, how many liters of water does it contain?

75 liters

750 liters

7500 liters

75000 liters

Explanation: First, calculate the volume in cubic meters: Volume = Base Area × Height = 5 m² × 1.5 m = 7.5 m³. Then, convert the volume from cubic meters to liters: 7.5 m³ × 1000 liters/m³ = 7500 liters.

Q594: A rectangular tank has a length of 2 m and a width of 1.5 m. If it contains 6000 liters of water, determine the height of the water level in the tank in centimeters (cm).

20 cm

2 cm

200 cm

2000 cm

Explanation: First, calculate the base area of the tank: Base Area = Length × Width = 2 m × 1.5 m = 3 m². Then, convert the volume from liters to cubic meters: 6000 liters = 6000 / 1000 m³ = 6 m³. Next, find the height: Height = Volume / Base Area = 6 m³ / 3 m² = 2 m. Finally, convert the height from meters to centimeters: 2 m × 100 cm/m = 200 cm.

Q595: A rectangular tank with a base area of 2 m² has water filled to a height of 1.25 m. How many liters of water does it contain?

25 liters

250 liters

2500 liters

25000 liters

Explanation: First, calculate the volume in cubic meters: Volume = Base Area × Height = 2 m² × 1.25 m = 2.5 m³. Then, convert the volume from cubic meters to liters: 2.5 m³ × 1000 liters/m³ = 2500 liters.

Q596: A cylindrical tank has a base area of 6 m². If it contains 15000 liters of water, what is the height of the water level in meters (m)?

0.25 m

25 m

2.5 m

250 m

Explanation: First, convert the volume from liters to cubic meters: 15000 liters = 15000 / 1000 m³ = 15 m³. Then, use the formula Volume = Base Area × Height to find the height: Height = Volume / Base Area = 15 m³ / 6 m² = 2.5 m.

Solve problems involving liquid volumes.

Q597: Convert 500 ml to litres.

5 L

0.5 L

0.05 L

50 L

Explanation: To convert millilitres to litres, divide the volume in millilitres by 1000. So, 500 ml ÷ 1000 = 0.5 L.

Q598: Convert 2.5 litres to millilitres.

250 ml

25 ml

2500 ml

0.0025 ml

Explanation: To convert litres to millilitres, multiply the volume in litres by 1000. So, 2.5 L × 1000 = 2500 ml.

Q599: A bottle contains 780 ml of water. What is this volume in litres?

7.8 L

78 L

0.078 L

0.78 L

Explanation: To convert millilitres to litres, divide the volume in millilitres by 1000. So, 780 ml ÷ 1000 = 0.78 L.

Q600: A jug holds 3 litres of juice. How many millilitres is this?

300 ml

30 ml

3000 ml

0.003 ml

Explanation: To convert litres to millilitres, multiply the volume in litres by 1000. So, 3 L × 1000 = 3000 ml.

Q601: A syringe holds 25 ml of medicine. How many litres is this?

0.25 L

2.5 L

0.0025 L

0.025 L

Explanation: To convert millilitres to litres, divide the volume in millilitres by 1000. So, 25 ml ÷ 1000 = 0.025 L.

Q602: A small tank contains 1.75 litres of fuel. What is the volume in millilitres?

175 ml

17500 ml

1.75 ml

1750 ml

Explanation: To convert litres to millilitres, multiply the volume in litres by 1000. So, 1.75 L × 1000 = 1750 ml.

Q603: A rectangular container has dimensions 20 cm x 10 cm x 5 cm. How many 50 ml bottles can be completely filled from it?

200

Explanation: First, calculate the volume of the rectangular container: Volume = length × width × height = 20 cm × 10 cm × 5 cm = 1000 cm³. Since 1000 cm³ = 1 litre, the capacity of the container is 1 litre. Convert the capacity to millilitres: 1 litre = 1000 ml. Now, divide the total capacity by the volume of one small bottle: Number of bottles = 1000 ml / 50 ml = 20.

Q604: A rectangular tank is 50 cm long, 30 cm wide, and 20 cm high. If it is full of water, how many 250 ml cups can be filled from it?

120

150

1200

Explanation: First, calculate the volume of the rectangular tank: Volume = 50 cm × 30 cm × 20 cm = 30000 cm³. Convert the capacity to litres: 30000 cm³ = 30 litres. Convert the capacity to millilitres: 30 litres = 30 × 1000 ml = 30000 ml. Now, divide the total capacity by the volume of one cup: Number of cups = 30000 ml / 250 ml = 120.

Q605: A rectangular container has a base area of 400 cm² and a height of 25 cm. How many 100 ml bottles can be filled from it?

100

1000

Explanation: First, calculate the volume of the rectangular container: Volume = base area × height = 400 cm² × 25 cm = 10000 cm³. Convert the capacity to litres: 10000 cm³ = 10 litres. Convert the capacity to millilitres: 10 litres = 10 × 1000 ml = 10000 ml. Now, divide the total capacity by the volume of one small bottle: Number of bottles = 10000 ml / 100 ml = 100.

Q606: A rectangular tank measures 30 cm by 20 cm by 15 cm. If it is full of liquid, how many 75 ml containers can be completely filled?

120

150

Explanation: First, calculate the volume of the rectangular tank: Volume = 30 cm × 20 cm × 15 cm = 9000 cm³. Convert the capacity to litres: 9000 cm³ = 9 litres. Convert the capacity to millilitres: 9 litres = 9 × 1000 ml = 9000 ml. Now, divide the total capacity by the volume of one container: Number of containers = 9000 ml / 75 ml = 120.

Q607: A rectangular water tank has dimensions 60 cm x 40 cm x 25 cm. If it is full, how many 200 ml packets can be filled from it?

250

300

600

Explanation: First, calculate the volume of the rectangular tank: Volume = 60 cm × 40 cm × 25 cm = 60000 cm³. Convert the capacity to litres: 60000 cm³ = 60 litres. Convert the capacity to millilitres: 60 litres = 60 × 1000 ml = 60000 ml. Now, divide the total capacity by the volume of one packet: Number of packets = 60000 ml / 200 ml = 300.

Q608: A rectangular container with dimensions 40 cm, 30 cm, and 20 cm is filled with oil. How many 150 ml bottles can be filled from this container?

120

240

160

Explanation: First, calculate the volume of the rectangular container: Volume = 40 cm × 30 cm × 20 cm = 24000 cm³. Convert the capacity to litres: 24000 cm³ = 24 litres. Convert the capacity to millilitres: 24 litres = 24 × 1000 ml = 24000 ml. Now, divide the total capacity by the volume of one bottle: Number of bottles = 24000 ml / 150 ml = 160.

Q609: A rectangular tank has a capacity of 100 litres. It is partially filled with water. When an additional 30 litres of water are poured into it, the tank overflows by 5 litres. What was the initial volume of water in the tank?

125 L

75 L

65 L

25 L

Explanation: The volume of water that actually filled the tank is 30 L - 5 L = 25 L. The initial volume of water in the tank was its total capacity minus the volume needed to fill it, which is 100 L - 25 L = 75 L.

Q610: A water tank has a maximum capacity of 50 litres. It contains some water. When 20 litres of water are poured into it, 2 litres of water overflow. How many litres of water were initially in the tank?

68 L

28 L

32 L

18 L

Explanation: The volume of water that actually went into the tank is 20 L - 2 L = 18 L. The initial volume was the total capacity minus the volume that was added to fill it, so 50 L - 18 L = 32 L.

Q611: A large rectangular container has a total capacity of 250 litres. It is partially filled with liquid. When 70 litres of liquid are added, 15 litres overflow. What was the original volume of liquid in the container?

305 L

165 L

195 L

55 L

Explanation: The amount of liquid that entered the container to fill it was 70 L - 15 L = 55 L. Therefore, the initial volume of liquid was 250 L - 55 L = 195 L.

Q612: An empty tank has a capacity of 80 litres. It is partially filled with water. When 40 litres of water are further added, 10 litres overflow. What was the initial volume of water in the tank?

110 L

30 L

40 L

50 L

Explanation: The volume of water that contributed to filling the tank (before overflow) is 40 L - 10 L = 30 L. The initial volume of water was the total capacity minus this filled amount, which is 80 L - 30 L = 50 L.

Q613: A tank has a capacity of 120 litres. It currently holds some water. When 50 litres of water are added, 10 litres overflow. What was the original amount of water in the tank?

160 L

60 L

80 L

40 L

Explanation: The net volume of water that entered the tank to fill it was 50 L - 10 L = 40 L. Therefore, the initial volume of water was 120 L - 40 L = 80 L.

Q614: A storage tank has a maximum capacity of 300 litres. It is partially filled. When an additional 100 litres of liquid are poured in, 25 litres overflow. Calculate the initial volume of liquid in the tank.

375 L

175 L

225 L

75 L

Explanation: The volume of liquid that actually went into the tank to fill it was 100 L - 25 L = 75 L. Therefore, the initial volume of liquid in the tank was 300 L - 75 L = 225 L.

Identifies writing a number as a product of a number between 1 (inclusive) and 10, and a power of ten as scientific notation.

Q2765: What is 78,000,000 expressed in scientific notation?

78 × 10⁶

7.8 × 10⁷

0.78 × 10⁸

7.8 × 10⁻⁷

Explanation: To express 78,000,000 in scientific notation, we move the decimal point from the end to between the 7 and the 8. This means moving it 7 places to the left. Therefore, the number becomes 7.8 and the power of 10 becomes 7. So, 78,000,000 = 7.8 × 10⁷.

Q2766: Which of the following statements correctly expresses 78,000,000 in scientific notation?

7.8 × 10⁶

78.0 × 10⁶

7.8 × 10⁷

780 × 10⁵

Explanation: Scientific notation requires the number part (coefficient) to be between 1 (inclusive) and 10 (exclusive). For 78,000,000, we place the decimal after the first digit (7), making it 7.8. We moved the decimal 7 places to the left, so the exponent of 10 is 7. Thus, 7.8 × 10⁷ is the correct scientific notation.

Q2767: 78,000,000 can be written as 7.8 × 10ⁿ. What is the value of n?

-7

Explanation: To convert 78,000,000 to 7.8, the decimal point needs to be moved 7 places to the left. When moving the decimal point to the left, the exponent of 10 is positive and equal to the number of places moved. Therefore, n = 7.

Q2768: Which of the following expressions represents 78,000,000 but is NOT written in correct scientific notation?

7.8 × 10⁷

78 × 10⁶

7.8 × 10⁸

0.78 × 10⁷

Explanation: Scientific notation requires the number part (coefficient) to be between 1 (inclusive) and 10 (exclusive). In the expression 78 × 10⁶, the number 78 is not between 1 and 10, even though the expression equals 78,000,000. Therefore, 78 × 10⁶ is not in correct scientific notation.

Q2769: If 78,000,000 is written in scientific notation as a × 10ⁿ, what is the value of 'a'?

7.8

0.78

Explanation: In scientific notation (a × 10ⁿ), the coefficient 'a' must satisfy 1 ≤ a < 10. For 78,000,000, we place the decimal after the first digit, making 'a' = 7.8.

Q2770: When 78,000,000 is expressed in the form a × 10ⁿ, what is the value of the exponent 'n'?

-7

Explanation: To get 7.8 from 78,000,000, the decimal point is moved 7 places to the left. When the decimal is moved to the left for a large number, the exponent 'n' is positive and equals the number of places moved. Therefore, n = 7.

Q2771: Express 0.00000045 in scientific notation.

4.5 x 10⁷

4.5 x 10⁻⁷

45 x 10⁻⁸

0.45 x 10⁻⁶

Explanation: To write 0.00000045 in scientific notation, move the decimal point 7 places to the right to get 4.5. Since the original number is less than 1, the exponent will be negative. So, it is 4.5 x 10⁻⁷.

Q2772: Write 72,000,000 in scientific notation.

7.2 x 10⁻⁷

7.2 x 10⁷

72 x 10⁶

0.72 x 10⁸

Explanation: To write 72,000,000 in scientific notation, move the decimal point 7 places to the left to get 7.2. Since the original number is greater than 10, the exponent will be positive. So, it is 7.2 x 10⁷.

Q2773: The scientific notation of 0.0000123 is:

1.23 x 10⁵

1.23 x 10⁻⁵

12.3 x 10⁻⁶

0.123 x 10⁻⁴

Explanation: To write 0.0000123 in scientific notation, move the decimal point 5 places to the right to get 1.23. Since the original number is less than 1, the exponent will be negative. So, it is 1.23 x 10⁻⁵.

Q2774: What is 345,600,000,000 written in scientific notation?

3.456 x 10⁻¹¹

3.456 x 10¹¹

34.56 x 10¹⁰

0.3456 x 10¹²

Explanation: To write 345,600,000,000 in scientific notation, move the decimal point 11 places to the left to get 3.456. Since the original number is greater than 10, the exponent will be positive. So, it is 3.456 x 10¹¹.

Q2775: Which of the following is the correct scientific notation for 0.000000009?

9 x 10⁹

9 x 10⁻⁹

0.9 x 10⁻⁸

90 x 10⁻¹⁰

Explanation: To write 0.000000009 in scientific notation, move the decimal point 9 places to the right to get 9. Since the original number is less than 1, the exponent will be negative. So, it is 9 x 10⁻⁹.

Q2776: Express 1,000,000,000 in scientific notation.

10 x 10⁸

1 x 10⁹

1 x 10⁻⁹

0.1 x 10¹⁰

Explanation: To write 1,000,000,000 in scientific notation, move the decimal point 9 places to the left to get 1. Since the original number is greater than 10, the exponent will be positive. So, it is 1 x 10⁹.

Q2777: A number is expressed as $67.5 \times 10^4$. Which of the following is the correct scientific notation for this number?

$6.75 \times 10^3$

$6.75 \times 10^5$

$0.675 \times 10^6$

$675 \times 10^3$

Explanation: To write $67.5 \times 10^4$ in scientific notation, the first part of the number (67.5) must be between 1 (inclusive) and 10 (exclusive). We move the decimal point one place to the left in 67.5 to get 6.75. Since we moved the decimal one place to the left, we must increase the exponent of 10 by 1. So, $4+1=5$. Thus, the correct scientific notation is $6.75 \times 10^5$.

Q2778: Rewrite $0.25 \times 10^3$ in correct scientific notation.

$2.5 \times 10^4$

$2.5 \times 10^2$

$25 \times 10^1$

$0.025 \times 10^5$

Explanation: To write $0.25 \times 10^3$ in scientific notation, the first part of the number (0.25) must be between 1 and 10. We move the decimal point one place to the right in 0.25 to get 2.5. Since we moved the decimal one place to the right, we must decrease the exponent of 10 by 1. So, $3-1=2$. Thus, the correct scientific notation is $2.5 \times 10^2$.

Q2779: Express $345 \times 10^{-2}$ in correct scientific notation.

$3.45 \times 10^{-4}$

$3.45 \times 10^{-2}$

$3.45 \times 10^0$

$0.345 \times 10^1$

Explanation: To write $345 \times 10^{-2}$ in scientific notation, the first part of the number (345) must be between 1 and 10. We move the decimal point two places to the left in 345 to get 3.45. Since we moved the decimal two places to the left, we must increase the exponent of 10 by 2. So, $-2+2=0$. Thus, the correct scientific notation is $3.45 \times 10^0$.

Q2780: What is $0.007 \times 10^{-5}$ written in correct scientific notation?

$7 \times 10^{-2}$

$7 \times 10^{-8}$

$70 \times 10^{-9}$

$0.7 \times 10^{-7}$

Explanation: To write $0.007 \times 10^{-5}$ in scientific notation, the first part of the number (0.007) must be between 1 and 10. We move the decimal point three places to the right in 0.007 to get 7. Since we moved the decimal three places to the right, we must decrease the exponent of 10 by 3. So, $-5-3=-8$. Thus, the correct scientific notation is $7 \times 10^{-8}$.

Q2781: Convert $5600 \times 10^2$ into correct scientific notation.

$5.6 \times 10^1$

$5.6 \times 10^5$

$56 \times 10^4$

$0.56 \times 10^6$

Explanation: To write $5600 \times 10^2$ in scientific notation, the first part of the number (5600) must be between 1 and 10. We move the decimal point three places to the left in 5600 to get 5.6. Since we moved the decimal three places to the left, we must increase the exponent of 10 by 3. So, $2+3=5$. Thus, the correct scientific notation is $5.6 \times 10^5$.

Q2782: Rewrite $123.45 \times 10^{-3}$ in correct scientific notation.

$1.2345 \times 10^{-5}$

$1.2345 \times 10^{-1}$

$12.345 \times 10^{-2}$

$0.12345 \times 10^0$

Explanation: To write $123.45 \times 10^{-3}$ in scientific notation, the first part of the number (123.45) must be between 1 and 10. We move the decimal point two places to the left in 123.45 to get 1.2345. Since we moved the decimal two places to the left, we must increase the exponent of 10 by 2. So, $-3+2=-1$. Thus, the correct scientific notation is $1.2345 \times 10^{-1}$.

Q2783: A number is given as $0.00045 \times 10^7$. Rewrite it in correct scientific notation.

$4.5 \times 10^{11}$

$4.5 \times 10^3$

$45 \times 10^2$

$0.45 \times 10^4$

Explanation: To write $0.00045 \times 10^7$ in scientific notation, the first part of the number (0.00045) must be between 1 and 10. We move the decimal point four places to the right in 0.00045 to get 4.5. Since we moved the decimal four places to the right, we must decrease the exponent of 10 by 4. So, $7-4=3$. Thus, the correct scientific notation is $4.5 \times 10^3$.

Writes numbers greater than one in scientific notation.

Q2784: Which of the following represents 87,000 in scientific notation?

87 x 10^3

8.7 x 10^4

0.87 x 10^5

8.7 x 10^-4

Explanation: To write 87,000 in scientific notation, the decimal point (initially after the last zero) is moved to the left until there is only one non-zero digit before it. In 87,000, the decimal point is moved 4 places to the left to get 8.7. Since the decimal point was moved 4 places to the left, the exponent of 10 is positive 4. Therefore, 87,000 in scientific notation is 8.7 x 10^4.

Q2785: What is 52,300 written in scientific notation?

52.3 x 10^3

5.23 x 10^4

5.23 x 10^-4

0.523 x 10^5

Explanation: To write 52,300 in scientific notation, the decimal point is moved to the left until there is only one non-zero digit before it. Moving the decimal point 4 places to the left yields 5.23. Since the decimal point was moved 4 places to the left, the exponent of 10 is positive 4. Therefore, 52,300 in scientific notation is 5.23 x 10^4.

Q2786: Express 450,000 in scientific notation.

45 x 10^4

4.5 x 10^5

4.5 x 10^-5

0.45 x 10^6

Explanation: To express 450,000 in scientific notation, the decimal point is moved to the left until there is only one non-zero digit before it. Moving the decimal point 5 places to the left yields 4.5. Since the decimal point was moved 5 places to the left, the exponent of 10 is positive 5. Thus, 450,000 in scientific notation is 4.5 x 10^5.

Q2787: How would you write 3,200 in scientific notation?

32 x 10^2

3.2 x 10^3

3.2 x 10^-3

0.32 x 10^4

Explanation: To write 3,200 in scientific notation, the decimal point is moved to the left until there is only one non-zero digit before it. Moving the decimal point 3 places to the left yields 3.2. Since the decimal point was moved 3 places to the left, the exponent of 10 is positive 3. Therefore, 3,200 in scientific notation is 3.2 x 10^3.

Q2788: Select the correct scientific notation for 123,400.

12.34 x 10^4

1.234 x 10^5

1.234 x 10^-5

123.4 x 10^3

Explanation: To write 123,400 in scientific notation, the decimal point is moved to the left until there is only one non-zero digit before it. Moving the decimal point 5 places to the left yields 1.234. Since the decimal point was moved 5 places to the left, the exponent of 10 is positive 5. Hence, 123,400 in scientific notation is 1.234 x 10^5.

Q2789: Which option correctly shows 900,000 in scientific notation?

90 x 10^4

9.0 x 10^5

9 x 10^-5

0.9 x 10^6

Explanation: To write 900,000 in scientific notation, the decimal point is moved to the left until there is only one non-zero digit before it. Moving the decimal point 5 places to the left yields 9.0 (or simply 9). Since the decimal point was moved 5 places to the left, the exponent of 10 is positive 5. Therefore, 900,000 in scientific notation is 9.0 x 10^5.

Q2790: The estimated land area of Sri Lanka is 65,610 square kilometers. Write this area in scientific notation.

65.61 x 10^3

6.561 x 10^5

6.561 x 10^4

6.561 x 10^-4

Explanation: To write 65,610 in scientific notation, we move the decimal point from its implied position after the last digit (65,610.) to a position where there is only one non-zero digit before the decimal point (6.5610). We moved the decimal point 4 places to the left. Therefore, the exponent of 10 is 4. So, 65,610 = 6.561 x 10^4.

Q2791: The approximate population of a certain country is 23,400,000. Express this number in scientific notation.

23.4 x 10^6

2.34 x 10^8

2.34 x 10^7

2.34 x 10^-7

Explanation: To write 23,400,000 in scientific notation, we move the decimal point 7 places to the left, from 23,400,000. to 2.3400000. This gives us 2.34. Since we moved the decimal point 7 places to the left, the exponent of 10 is 7. Thus, 23,400,000 = 2.34 x 10^7.

Q2792: The total output of a factory last year was 780,000 units. Write this number in scientific notation.

78 x 10^4

7.8 x 10^6

7.8 x 10^5

7.8 x 10^-5

Explanation: To write 780,000 in scientific notation, we move the decimal point from 780,000. to 7.8. This involves moving the decimal point 5 places to the left. Therefore, the exponent of 10 is 5. So, 780,000 = 7.8 x 10^5.

Q2793: A large company reported an annual revenue of 1,234,500 LKR. Express this amount in scientific notation.

12.345 x 10^5

1.2345 x 10^7

1.2345 x 10^6

1.2345 x 10^-6

Explanation: To write 1,234,500 in scientific notation, we move the decimal point 6 places to the left, from 1,234,500. to 1.234500. This gives us 1.2345. Since we moved the decimal point 6 places to the left, the exponent of 10 is 6. Thus, 1,234,500 = 1.2345 x 10^6.

Q2794: A small library has 950 books. Write the number of books in scientific notation.

95 x 10^1

9.5 x 10^3

9.5 x 10^2

0.95 x 10^3

Explanation: To write 950 in scientific notation, we move the decimal point from 950. to 9.5. This involves moving the decimal point 2 places to the left. Therefore, the exponent of 10 is 2. So, 950 = 9.5 x 10^2.

Q2795: The total value of goods exported by a country in a month was $4,005,000. Write this value in scientific notation.

40.05 x 10^5

4.005 x 10^7

4.005 x 10^6

4.005 x 10^-6

Explanation: To write 4,005,000 in scientific notation, we move the decimal point 6 places to the left, from 4,005,000. to 4.005000. This gives us 4.005. Since we moved the decimal point 6 places to the left, the exponent of 10 is 6. Thus, 4,005,000 = 4.005 x 10^6.

Q2796: A factory produces 1,500 units of a product every day. If the factory operates for 300 days in a year, what is the total number of units produced annually, expressed in scientific notation?

4.5 x 10^4

45 x 10^4

4.5 x 10^5

4.5 x 10^6

Explanation: To find the total number of units produced annually, multiply the daily production by the number of operating days: 1500 units/day * 300 days/year = 450,000 units. To express 450,000 in scientific notation, move the decimal point 5 places to the left to get 4.5. Since the decimal was moved 5 places to the left, the exponent of 10 is 5. So, 450,000 = 4.5 x 10^5.

Q2797: A company sells 2,500 tickets for a concert. If each ticket costs Rs. 500, what is the total revenue in scientific notation?

1.25 x 10^5

12.5 x 10^5

1.25 x 10^6

1.25 x 10^7

Explanation: To find the total revenue, multiply the number of tickets by the cost per ticket: 2500 tickets * Rs. 500/ticket = Rs. 1,250,000. To express 1,250,000 in scientific notation, move the decimal point 6 places to the left to get 1.25. Since the decimal was moved 6 places to the left, the exponent of 10 is 6. So, 1,250,000 = 1.25 x 10^6.

Q2798: A rectangular field is 800 meters long and 150 meters wide. What is its area in square meters, expressed in scientific notation?

1.2 x 10^4

12 x 10^4

1.2 x 10^5

1.2 x 10^6

Explanation: To find the area of a rectangular field, multiply its length by its width: 800 m * 150 m = 120,000 m^2. To express 120,000 in scientific notation, move the decimal point 5 places to the left to get 1.2. Since the decimal was moved 5 places to the left, the exponent of 10 is 5. So, 120,000 = 1.2 x 10^5.

Q2799: A car travels 300 km per day. If it travels for 120 days in a year, what is the total distance covered in scientific notation?

3.6 x 10^3

36 x 10^3

3.6 x 10^4

3.6 x 10^5

Explanation: To find the total distance covered, multiply the daily distance by the number of travel days: 300 km/day * 120 days/year = 36,000 km. To express 36,000 in scientific notation, move the decimal point 4 places to the left to get 3.6. Since the decimal was moved 4 places to the left, the exponent of 10 is 4. So, 36,000 = 3.6 x 10^4.

Q2800: A machine produces 750 items per hour. If it operates for 80 hours, what is the total number of items produced, in scientific notation?

6.0 x 10^3

60 x 10^3

6.0 x 10^4

6.0 x 10^5

Explanation: To find the total number of items produced, multiply the hourly production by the number of operating hours: 750 items/hour * 80 hours = 60,000 items. To express 60,000 in scientific notation, move the decimal point 4 places to the left to get 6.0. Since the decimal was moved 4 places to the left, the exponent of 10 is 4. So, 60,000 = 6.0 x 10^4.

Q2801: A city has 40,000 households. If each household uses an average of 250 units of electricity per month, what is the total electricity consumption per month for the city, in scientific notation?

1.0 x 10^6

10 x 10^6

1.0 x 10^7

1.0 x 10^8

Explanation: To find the total electricity consumption, multiply the number of households by the average consumption per household: 40,000 households * 250 units/household = 10,000,000 units. To express 10,000,000 in scientific notation, move the decimal point 7 places to the left to get 1.0. Since the decimal was moved 7 places to the left, the exponent of 10 is 7. So, 10,000,000 = 1.0 x 10^7.

Writes numbers less than one in scientific notation.

Q2802: What is 0.05 expressed in scientific notation?

5 x 10²

5 x 10⁻²

0.5 x 10⁻¹

50 x 10⁻³

Explanation: To write 0.05 in scientific notation, move the decimal point two places to the right until it is after the first non-zero digit (5). This gives 5. Since the decimal point was moved two places to the right, the exponent of 10 will be -2. So, 0.05 = 5 x 10⁻².

Q2803: Convert 0.008 into scientific notation.

8 x 10³

0.8 x 10⁻²

8 x 10⁻³

80 x 10⁻⁴

Explanation: To express 0.008 in scientific notation, move the decimal point three places to the right to get 8. The number of places moved to the right determines the negative exponent of 10. Thus, 0.008 = 8 x 10⁻³.

Q2804: Which of the following is the correct scientific notation for 0.02?

2 x 10²

20 x 10⁻³

2 x 10⁻²

0.2 x 10⁻¹

Explanation: To write 0.02 in scientific notation, move the decimal point two places to the right to get 2. Since we moved the decimal point to the right, the exponent will be negative, and the number of places moved determines its magnitude. Thus, 0.02 = 2 x 10⁻².

Q2805: Express 0.004 in scientific notation.

4 x 10³

0.4 x 10⁻²

40 x 10⁻⁴

4 x 10⁻³

Explanation: To express 0.004 in scientific notation, move the decimal point three places to the right to place it after the first non-zero digit (4). This results in 4. Since the decimal point was moved three places to the right, the exponent of 10 is -3. So, 0.004 = 4 x 10⁻³.

Q2806: How would you write 0.007 in scientific notation?

7 x 10³

7 x 10⁻³

0.7 x 10⁻²

70 x 10⁻⁴

Explanation: To write 0.007 in scientific notation, move the decimal point three places to the right until it is after the first non-zero digit (7). This results in 7. Since the decimal point was moved three places to the right, the exponent of 10 will be -3. So, 0.007 = 7 x 10⁻³.

Q2807: What is 0.09 in scientific notation?

9 x 10²

0.9 x 10⁻¹

9 x 10⁻²

90 x 10⁻³

Explanation: To write 0.09 in scientific notation, move the decimal point two places to the right until it is after the first non-zero digit (9). This gives 9. Since the decimal point was moved two places to the right, the exponent of 10 will be -2. So, 0.09 = 9 x 10⁻².

Q2808: Convert 0.0000573 into scientific notation.

5.73 x 10⁵

0.573 x 10⁻⁴

5.73 x 10⁻⁵

57.3 x 10⁻⁶

Explanation: To convert 0.0000573 to scientific notation, move the decimal point to the right until there is a single non-zero digit before the decimal point. This means moving it 5 places to the right to get 5.73. Since the original number is less than 1 and we moved the decimal point 5 places to the right, the exponent of 10 will be -5. Thus, 0.0000573 = 5.73 x 10⁻⁵.

Q2809: Express 0.00308 in scientific notation.

3.08 x 10³

3.08 x 10⁻³

30.8 x 10⁻⁴

0.308 x 10⁻²

Explanation: To express 0.00308 in scientific notation, we move the decimal point to the right until it is after the first non-zero digit. We move it 3 places to the right, making the number 3.08. Since the original number is less than 1 and we moved the decimal point 3 places to the right, the exponent of 10 is -3. So, 0.00308 = 3.08 x 10⁻³.

Q2810: What is 0.000000125 in scientific notation?

1.25 x 10⁷

1.25 x 10⁻⁷

12.5 x 10⁻⁸

0.125 x 10⁻⁶

Explanation: To write 0.000000125 in scientific notation, we shift the decimal point to the right until it is after the first non-zero digit (1). This requires moving the decimal point 7 places to the right, resulting in 1.25. Since the original number is less than 1, the exponent for 10 is negative, corresponding to the number of shifts. So, 0.000000125 = 1.25 x 10⁻⁷.

Q2811: Convert 0.000901 to scientific notation.

9.01 x 10⁴

90.1 x 10⁻⁵

9.01 x 10⁻⁴

0.901 x 10⁻³

Explanation: To convert 0.000901 to scientific notation, move the decimal point to the right until it is after the first non-zero digit (9). This requires moving it 4 places to the right, giving 9.01. Since the original number is less than 1 and we moved the decimal point 4 places to the right, the exponent of 10 is -4. So, 0.000901 = 9.01 x 10⁻⁴.

Q2812: Write 0.0000045 in scientific notation.

4.5 x 10⁶

45 x 10⁻⁷

4.5 x 10⁻⁶

0.45 x 10⁻⁵

Explanation: To write 0.0000045 in scientific notation, we need to move the decimal point to the right until it is after the first non-zero digit, which is 4. This requires moving the decimal point 6 places to the right, resulting in 4.5. Since the original number is less than 1, the exponent of 10 will be -6. Therefore, 0.0000045 = 4.5 x 10⁻⁶.

Q2813: Which of the following is the correct scientific notation for 0.0002078?

2.078 x 10⁴

20.78 x 10⁻⁵

2.078 x 10⁻⁴

0.2078 x 10⁻³

Explanation: To convert 0.0002078 to scientific notation, we must move the decimal point to the right until it is positioned after the first non-zero digit (2). This means moving it 4 places to the right, resulting in 2.078. Since the original number is less than 1, the exponent of 10 will be negative, corresponding to the 4 places moved. Thus, 0.0002078 = 2.078 x 10⁻⁴.

Q2814: Calculate 0.04 / 20 and express the answer in scientific notation.

2 x 10^3

2 x 10^-3

0.2 x 10^-2

20 x 10^-4

Explanation: First, perform the division: 0.04 / 20 = 0.002. To express 0.002 in scientific notation, move the decimal point 3 places to the right to get 2.0. Since the decimal point was moved to the right for a number less than one, the exponent of 10 is negative. Therefore, 0.002 = 2 x 10^-3.

Q2815: Calculate 0.005 x 0.02 and express the answer in scientific notation.

1 x 10^4

1 x 10^-4

0.1 x 10^-3

10 x 10^-5

Explanation: First, perform the multiplication: 0.005 x 0.02 = 0.0001. To express 0.0001 in scientific notation, move the decimal point 4 places to the right to get 1.0. Since the decimal point was moved to the right, the exponent of 10 is negative. Therefore, 0.0001 = 1 x 10^-4.

Q2816: Calculate 0.003 - 0.0025 and express the answer in scientific notation.

5 x 10^4

0.5 x 10^-3

5 x 10^-4

2.5 x 10^-4

Explanation: First, perform the subtraction: 0.003 - 0.0025 = 0.0005. To express 0.0005 in scientific notation, move the decimal point 4 places to the right to get 5.0. Since the decimal point was moved to the right, the exponent of 10 is negative. Therefore, 0.0005 = 5 x 10^-4.

Q2817: Simplify (3 x 10^-2) / (6 x 10^2) and express the answer in scientific notation.

5 x 10^5

0.5 x 10^-4

5 x 10^-4

5 x 10^-5

Explanation: First, simplify the expression: (3 x 10^-2) / (6 x 10^2) = (3/6) x (10^-2 / 10^2) = 0.5 x 10^(-2-2) = 0.5 x 10^-4. To express 0.5 x 10^-4 in proper scientific notation, move the decimal point 1 place to the right in 0.5 to get 5.0, and decrease the exponent by 1. Therefore, 0.5 x 10^-4 = 5 x 10^-5.

Q2818: Calculate (0.002)^2 and express the answer in scientific notation.

4 x 10^6

4 x 10^-4

0.4 x 10^-5

4 x 10^-6

Explanation: First, perform the calculation: (0.002)^2 = 0.002 x 0.002 = 0.000004. To express 0.000004 in scientific notation, move the decimal point 6 places to the right to get 4.0. Since the decimal point was moved to the right, the exponent of 10 is negative. Therefore, 0.000004 = 4 x 10^-6.

Q2819: Calculate 0.00015 + 0.00002 and express the answer in scientific notation.

1.7 x 10^4

0.17 x 10^-3

1.7 x 10^-4

17 x 10^-5

Explanation: First, perform the addition: 0.00015 + 0.00002 = 0.00017. To express 0.00017 in scientific notation, move the decimal point 4 places to the right to get 1.7. Since the decimal point was moved to the right, the exponent of 10 is negative. Therefore, 0.00017 = 1.7 x 10^-4.

Q2820: A machine produces a metal wire of 0.000008 meters length per second. Express the length produced in 0.5 seconds in scientific notation.

4 x 10^6 m

0.4 x 10^-5 m

4 x 10^-7 m

4 x 10^-6 m

Explanation: First, calculate the total length produced: 0.000008 meters/second * 0.5 seconds = 0.000004 meters. To express 0.000004 in scientific notation, move the decimal point 6 places to the right to get 4.0. Since the decimal point was moved to the right, the exponent of 10 is negative. Therefore, the length is 4 x 10^-6 meters.

Rounds off whole numbers to the nearest ten, hundred, and thousand.

Q2821: Round 628 to the nearest ten.

620

630

600

629

Explanation: To round 628 to the nearest ten, look at the units digit. Since the units digit (8) is 5 or greater, round up the tens digit. The 2 in the tens place becomes 3, and the units digit becomes 0. So, 628 rounded to the nearest ten is 630.

Q2822: Round 3,451 to the nearest hundred.

3,400

3,500

3,000

3,450

Explanation: To round 3,451 to the nearest hundred, look at the tens digit. Since the tens digit (5) is 5 or greater, round up the hundreds digit. The 4 in the hundreds place becomes 5, and the tens and units digits become 0. So, 3,451 rounded to the nearest hundred is 3,500.

Q2823: Round 8,172 to the nearest thousand.

9,000

8,100

8,000

8,200

Explanation: To round 8,172 to the nearest thousand, look at the hundreds digit. Since the hundreds digit (1) is less than 5, round down the thousands digit. The 8 in the thousands place remains 8, and all digits to its right become 0. So, 8,172 rounded to the nearest thousand is 8,000.

Q2824: Round 795 to the nearest ten.

790

800

700

799

Explanation: To round 795 to the nearest ten, look at the units digit. Since the units digit (5) is 5 or greater, round up the tens digit. The 9 in the tens place becomes 10, which means the 7 in the hundreds place also increases by 1. So, 79 becomes 80, and the units digit becomes 0. Therefore, 795 rounded to the nearest ten is 800.

Q2825: Round 1,963 to the nearest hundred.

1,900

2,000

1,000

1,970

Explanation: To round 1,963 to the nearest hundred, look at the tens digit. Since the tens digit (6) is 5 or greater, round up the hundreds digit. The 9 in the hundreds place becomes 10, which means the 1 in the thousands place also increases by 1. So, 19 becomes 20, and the tens and units digits become 0. Therefore, 1,963 rounded to the nearest hundred is 2,000.

Q2826: Round 5,500 to the nearest thousand.

5,000

6,000

5,500

5,600

Explanation: To round 5,500 to the nearest thousand, look at the hundreds digit. Since the hundreds digit (5) is 5 or greater, round up the thousands digit. The 5 in the thousands place becomes 6, and all digits to its right become 0. So, 5,500 rounded to the nearest thousand is 6,000.

Q2827: A farmer counted 3,248 mangoes from his orchard. Round this number to the nearest hundred and then to the nearest thousand. State both rounded values.

3,250 and 3,000

3,200 and 4,000

3,200 and 3,000

3,240 and 3,000

Explanation: To round 3,248 to the nearest hundred, look at the tens digit (4). Since 4 is less than 5, keep the hundreds digit as 2 and replace the tens and units digits with zeros, giving 3,200. To round 3,248 to the nearest thousand, look at the hundreds digit (2). Since 2 is less than 5, keep the thousands digit as 3 and replace the hundreds, tens, and units digits with zeros, giving 3,000. So the correct answer is 3,200 and 3,000.

Q2828: A factory produced 6,783 toys last month. Round this number to the nearest ten and then to the nearest hundred. State both rounded values.

6,780 and 6,700

6,790 and 6,800

6,780 and 6,800

6,780 and 6,790

Explanation: To round 6,783 to the nearest ten, look at the units digit (3). Since 3 is less than 5, keep the tens digit as 8 and replace the units digit with zero, giving 6,780. To round 6,783 to the nearest hundred, look at the tens digit (8). Since 8 is 5 or greater, round up the hundreds digit (7) to 8 and replace the tens and units digits with zeros, giving 6,800. So the correct answer is 6,780 and 6,800.

Q2829: The population of a small town is 5,819. Round this number to the nearest hundred and then to the nearest thousand. State both rounded values.

5,800 and 5,000

5,900 and 6,000

5,800 and 6,000

5,820 and 6,000

Explanation: To round 5,819 to the nearest hundred, look at the tens digit (1). Since 1 is less than 5, keep the hundreds digit as 8 and replace the tens and units digits with zeros, giving 5,800. To round 5,819 to the nearest thousand, look at the hundreds digit (8). Since 8 is 5 or greater, round up the thousands digit (5) to 6 and replace the hundreds, tens, and units digits with zeros, giving 6,000. So the correct answer is 5,800 and 6,000.

Q2830: A school library has 1,452 books. Round this number to the nearest ten and then to the nearest thousand. State both rounded values.

1,450 and 2,000

1,460 and 1,000

1,450 and 1,000

1,400 and 1,000

Explanation: To round 1,452 to the nearest ten, look at the units digit (2). Since 2 is less than 5, keep the tens digit as 5 and replace the units digit with zero, giving 1,450. To round 1,452 to the nearest thousand, look at the hundreds digit (4). Since 4 is less than 5, keep the thousands digit as 1 and replace the hundreds, tens, and units digits with zeros, giving 1,000. So the correct answer is 1,450 and 1,000.

Q2831: A stadium can hold 7,550 spectators. Round this number to the nearest hundred and then to the nearest thousand. State both rounded values.

7,500 and 7,000

7,500 and 8,000

7,600 and 8,000

7,600 and 7,000

Explanation: To round 7,550 to the nearest hundred, look at the tens digit (5). Since 5 is 5 or greater, round up the hundreds digit (5) to 6 and replace the tens and units digits with zeros, giving 7,600. To round 7,550 to the nearest thousand, look at the hundreds digit (5). Since 5 is 5 or greater, round up the thousands digit (7) to 8 and replace the hundreds, tens, and units digits with zeros, giving 8,000. So the correct answer is 7,600 and 8,000.

Q2832: A collection of stamps contains 10,296 items. Round this number to the nearest ten and then to the nearest hundred. State both rounded values.

10,290 and 10,200

10,300 and 10,200

10,300 and 10,300

10,290 and 10,300

Explanation: To round 10,296 to the nearest ten, look at the units digit (6). Since 6 is 5 or greater, round up the tens digit (9). Rounding 9 up results in 10, so the 9 becomes 0 and 1 is carried over to the hundreds digit (2), making it 3. The units digit becomes 0. This gives 10,300. To round 10,296 to the nearest hundred, look at the tens digit (9). Since 9 is 5 or greater, round up the hundreds digit (2) to 3 and replace the tens and units digits with zeros, giving 10,300. So the correct answer is 10,300 and 10,300.

Q2833: The total number of visitors to a national park last year was 2,875. Round this number to the nearest ten and then to the nearest hundred. State both rounded values.

2,870 and 2,800

2,880 and 2,900

2,880 and 2,800

2,870 and 2,900

Explanation: To round 2,875 to the nearest ten, look at the units digit (5). Since 5 is 5 or greater, round up the tens digit (7) to 8 and replace the units digit with zero, giving 2,880. To round 2,875 to the nearest hundred, look at the tens digit (7). Since 7 is 5 or greater, round up the hundreds digit (8) to 9 and replace the tens and units digits with zeros, giving 2,900. So the correct answer is 2,880 and 2,900.

Q2834: A charity collected Rs. 9,049 in donations. Round this amount to the nearest hundred and then to the nearest thousand. State both rounded values.

9,050 and 9,000

9,100 and 10,000

9,000 and 10,000

9,000 and 9,000

Explanation: To round 9,049 to the nearest hundred, look at the tens digit (4). Since 4 is less than 5, keep the hundreds digit as 0 and replace the tens and units digits with zeros, giving 9,000. To round 9,049 to the nearest thousand, look at the hundreds digit (0). Since 0 is less than 5, keep the thousands digit as 9 and replace the hundreds, tens, and units digits with zeros, giving 9,000. So the correct answer is 9,000 and 9,000.

Q2835: A secret whole number, when rounded to the nearest hundred, becomes 5,700. When the same secret whole number is rounded to the nearest thousand, it becomes 6,000. Which of the following ranges represents all possible whole numbers that fit both rounding conditions?

5,600 to 5,700

5,650 to 5,749

5,500 to 6,499

5,650 to 5,750

Explanation: A number rounded to the nearest hundred to 5,700 must be in the range [5650, 5749]. A number rounded to the nearest thousand to 6,000 must be in the range [5500, 6499]. The intersection of these two ranges is [5650, 5749].

Q2836: A secret whole number, when rounded to the nearest hundred, becomes 5,700. When the same secret whole number is rounded to the nearest thousand, it becomes 6,000. Which of the following numbers could be the secret whole number?

5,649

5,750

5,675

5,549

Q2837: A secret whole number, when rounded to the nearest hundred, becomes 5,700. When the same secret whole number is rounded to the nearest thousand, it becomes 6,000. What is the smallest possible whole number that fits both rounding conditions?

5,600

5,500

5,650

5,700

Explanation: For a number to round to 5,700 to the nearest hundred, the smallest possible whole number is 5,650. For a number to round to 6,000 to the nearest thousand, the smallest possible whole number is 5,500. To satisfy both, the smallest number must be the greater of these two lower bounds, which is 5,650.

Q2838: A secret whole number, when rounded to the nearest hundred, becomes 5,700. When the same secret whole number is rounded to the nearest thousand, it becomes 6,000. What is the largest possible whole number that fits both rounding conditions?

5,700

5,749

5,750

6,499

Explanation: For a number to round to 5,700 to the nearest hundred, the largest possible whole number is 5,749. For a number to round to 6,000 to the nearest thousand, the largest possible whole number is 6,499. To satisfy both, the largest number must be the smaller of these two upper bounds, which is 5,749.

Q2839: A secret whole number, when rounded to the nearest hundred, becomes 5,700. When the same secret whole number is rounded to the nearest thousand, it becomes 6,000. How many different whole numbers could be the secret number?

100

1,000

Explanation: The range of numbers that satisfy both conditions is from 5,650 to 5,749, inclusive. To find the count of whole numbers in this range, we calculate (Largest number - Smallest number) + 1, which is (5749 - 5650) + 1 = 99 + 1 = 100.

Q2840: A secret whole number, when rounded to the nearest hundred, becomes 4,700. When the same secret whole number is rounded to the nearest thousand, it becomes 5,000. Which of the following numbers could be the secret whole number?

4,649

4,750

4,720

4,599

Explanation: A number rounded to the nearest hundred to 4,700 must be in the range [4650, 4749]. A number rounded to the nearest thousand to 5,000 must be in the range [4500, 5499]. The intersection of these two ranges is [4650, 4749]. Only 4,720 falls within this range.

Rounds off a decimal number to a given decimal place.

Q2841: Round off 7.6 to the nearest whole number.

7.0

Explanation: To round 7.6 to the nearest whole number, look at the first decimal place. Since it is 6 (which is 5 or greater), round up the whole number. So, 7 becomes 8.

Q2842: Round off 12.3 to the nearest whole number.

12.0

123

Explanation: To round 12.3 to the nearest whole number, look at the first decimal place. Since it is 3 (which is less than 5), keep the whole number as it is. So, 12.3 becomes 12.

Q2843: Round off 5.48 to the nearest whole number.

5.5

548

Explanation: To round 5.48 to the nearest whole number, look at the first decimal place. It is 4 (which is less than 5). Therefore, the whole number remains 5.

Q2844: Round off 9.72 to the nearest whole number.

9.7

972

Explanation: To round 9.72 to the nearest whole number, look at the first decimal place. It is 7 (which is 5 or greater). So, round up the whole number from 9 to 10.

Q2845: Round off 3.45 to one decimal place.

3.4

3.5

3.40

Explanation: To round 3.45 to one decimal place, look at the second decimal place. It is 5 (which is 5 or greater). So, round up the first decimal place. 4 becomes 5. The number becomes 3.5.

Q2846: Round off 15.81 to one decimal place.

15.9

15.8

15.80

Explanation: To round 15.81 to one decimal place, look at the second decimal place. It is 1 (which is less than 5). So, keep the first decimal place as it is. The number becomes 15.8.

Q2847: What is the value of 3.456 + 2.17 when rounded to two decimal places?

5.62

5.63

5.626

5.630

Explanation: First, perform the addition: 3.456 + 2.17 = 5.626. To round 5.626 to two decimal places, we look at the third decimal digit, which is 6. Since 6 is 5 or greater, we round up the second decimal digit (2) by 1. So, 5.626 rounded to two decimal places is 5.63.

Q2848: Subtract 3.123 from 7.89 and round the result to two decimal places.

4.76

4.767

4.77

4.770

Explanation: First, perform the subtraction: 7.89 - 3.123 = 4.767. To round 4.767 to two decimal places, we look at the third decimal digit, which is 7. Since 7 is 5 or greater, we round up the second decimal digit (6) by 1. So, 4.767 rounded to two decimal places is 4.77.

Q2849: Calculate 1.234 + 5.678 and round the answer to three decimal places.

6.91

6.92

6.912

6.913

Explanation: First, perform the addition: 1.234 + 5.678 = 6.912. To round 6.912 to three decimal places, we need to consider the fourth decimal digit. Since there is no fourth digit, or it can be considered zero, the number is already expressed to three decimal places and does not require further rounding up or down. So, 6.912 rounded to three decimal places is 6.912.

Q2850: What is the result of 10.5 - 4.789, rounded to three decimal places?

5.71

5.711

5.712

5.710

Explanation: First, perform the subtraction: 10.5 - 4.789 = 5.711. To round 5.711 to three decimal places, we need to consider the fourth decimal digit. Since there is no fourth digit, or it can be considered zero, the number is already expressed to three decimal places and does not require further rounding up or down. So, 5.711 rounded to three decimal places is 5.711.

Q2851: Add 0.997 and 0.054, then round the sum to two decimal places.

1.04

1.051

1.05

1.06

Explanation: First, perform the addition: 0.997 + 0.054 = 1.051. To round 1.051 to two decimal places, we look at the third decimal digit, which is 1. Since 1 is less than 5, we keep the second decimal digit (5) as it is. So, 1.051 rounded to two decimal places is 1.05.

Q2852: Find the difference between 12.345 and 6.78, and round the answer to two decimal places.

5.56

5.565

5.57

5.570

Explanation: First, perform the subtraction: 12.345 - 6.78 = 5.565. To round 5.565 to two decimal places, we look at the third decimal digit, which is 5. Since 5 is 5 or greater, we round up the second decimal digit (6) by 1. So, 5.565 rounded to two decimal places is 5.57.

Q2853: A 4.5-meter long fabric costs Rs. 875.25. What is the cost of one meter of fabric, rounded to two decimal places?

Rs. 194.5

Rs. 194.49

Rs. 194.50

Rs. 194.51

Explanation: To find the cost of one meter, divide the total cost by the total length: 875.25 ÷ 4.5 = 194.5. When rounded to two decimal places, this becomes 194.50.

Q2854: A gardener uses fertilizer at a rate of 0.28 kg per square meter. If the garden has an area of 35.75 square meters, what is the total amount of fertilizer needed, rounded to one decimal place?

10.01 kg

10.0 kg

10.1 kg

9.9 kg

Explanation: To find the total amount of fertilizer, multiply the rate by the area: 0.28 kg/m² × 35.75 m² = 10.01 kg. When rounded to one decimal place, this becomes 10.0 kg.

Q2855: A car travels a distance of 185.3 km in 2.75 hours. What is its average speed in km/h, rounded to one decimal place?

67.3 km/h

67.38 km/h

67.4 km/h

67.5 km/h

Explanation: To find the average speed, divide the total distance by the total time: 185.3 km ÷ 2.75 hours = 67.3818... km/h. When rounded to one decimal place, this becomes 67.4 km/h.

Q2856: A tailor uses 3.15 meters of fabric for one shirt. If he makes 12 shirts, what is the total length of fabric used, rounded to the nearest meter?

37 meters

37.8 meters

38 meters

39 meters

Explanation: To find the total length of fabric, multiply the fabric per shirt by the number of shirts: 3.15 m/shirt × 12 shirts = 37.8 m. When rounded to the nearest meter, this becomes 38 m.

Q2857: A total of 5.8 liters of juice is to be equally distributed among 8 glasses. How much juice will each glass contain, rounded to two decimal places?

0.72 L

0.725 L

0.73 L

0.7 L

Explanation: To find the amount of juice per glass, divide the total juice by the number of glasses: 5.8 L ÷ 8 = 0.725 L. When rounded to two decimal places, this becomes 0.73 L.

Q2858: A rectangular plot of land is 15.6 meters long and 8.2 meters wide. If the cost of fencing is Rs. 125.50 per meter, what is the total cost to fence the entire plot, rounded to the nearest rupee?

Rs. 5979

Rs. 5979.80

Rs. 5980

Rs. 5981

Explanation: First, calculate the perimeter of the plot: 2 × (15.6 m + 8.2 m) = 2 × 23.8 m = 47.6 m. Then, calculate the total cost: 47.6 m × Rs. 125.50/m = Rs. 5979.80. When rounded to the nearest rupee, this becomes Rs. 5980.

Solves problems related to rounding off.

Q2859: Round 3.14159 to two decimal places.

3.1

3.15

3.14

3.141

Explanation: To round 3.14159 to two decimal places, we look at the third decimal place. The third digit is 1. Since 1 is less than 5, we keep the second decimal digit as it is. So, 3.14159 rounded to two decimal places is 3.14.

Q2860: Round 0.876 to one decimal place.

0.8

0.87

0.9

0.7

Explanation: To round 0.876 to one decimal place, we look at the second decimal place. The second digit is 7. Since 7 is 5 or greater, we round up the first decimal digit. So, 0.876 rounded to one decimal place is 0.9.

Q2861: Round 12.055 to two decimal places.

12.05

12.06

12.10

12.00

Explanation: To round 12.055 to two decimal places, we look at the third decimal place. The third digit is 5. Since 5 is 5 or greater, we round up the second decimal digit. So, 12.055 rounded to two decimal places is 12.06.

Q2862: Round 5.996 to two decimal places.

5.99

6.0

6.00

5.90

Explanation: To round 5.996 to two decimal places, we look at the third decimal place. The third digit is 6. Since 6 is 5 or greater, we round up the second decimal digit (9). Rounding up 9 results in 10, so we carry over 1 to the first decimal digit, which becomes 0 (from 9+1=10, 0 remains, 1 carries over). Then 5.9 becomes 6.0. Therefore, 5.996 rounded to two decimal places is 6.00.

Q2863: Round 7.2348 to three decimal places.

7.234

7.240

7.235

7.23

Explanation: To round 7.2348 to three decimal places, we look at the fourth decimal place. The fourth digit is 8. Since 8 is 5 or greater, we round up the third decimal digit. So, 7.2348 rounded to three decimal places is 7.235.

Q2864: Round 0.0051 to two decimal places.

0.00

0.005

0.010

0.01

Explanation: To round 0.0051 to two decimal places, we look at the third decimal place. The third digit is 5. Since 5 is 5 or greater, we round up the second decimal digit. The second decimal digit is 0, so rounding it up makes it 1. Thus, 0.0051 rounded to two decimal places is 0.01.

Q2865: Calculate 7.25 × 3.8 and round the answer to 3 significant figures.

27.5

27.6

28.0

27.55

Explanation: First, calculate the product: 7.25 × 3.8 = 27.55. To round 27.55 to 3 significant figures, we look at the fourth digit (which is 5). Since it is 5 or greater, we round up the third digit. So, 27.55 rounded to 3 significant figures is 27.6.

Q2866: Calculate 58.7 ÷ 4.5 and round the answer to 2 significant figures.

13.0

13.04

Explanation: First, calculate the quotient: 58.7 ÷ 4.5 = 13.044... To round 13.044... to 2 significant figures, we look at the third digit (which is 0). Since it is less than 5, we keep the second digit as it is. So, 13.044... rounded to 2 significant figures is 13.

Q2867: Calculate 0.45 × 12.3 and round the answer to 1 significant figure.

5.5

0.5

Explanation: First, calculate the product: 0.45 × 12.3 = 5.535. To round 5.535 to 1 significant figure, we look at the second digit (which is 5). Since it is 5 or greater, we round up the first digit. So, 5.535 rounded to 1 significant figure is 6.

Q2868: Calculate 125.8 ÷ 0.65 and round the answer to 3 significant figures.

193

193.5

194

190

Explanation: First, calculate the quotient: 125.8 ÷ 0.65 = 193.538... To round 193.538... to 3 significant figures, we look at the fourth digit (which is 5). Since it is 5 or greater, we round up the third digit. So, 193.538... rounded to 3 significant figures is 194.

Q2869: A rectangle has a length of 15.6 cm and a width of 8.2 cm. Calculate its area and round the answer to 4 significant figures.

127.9

128.0

127.92

128

Explanation: First, calculate the area: Area = length × width = 15.6 cm × 8.2 cm = 127.92 cm². To round 127.92 to 4 significant figures, we look at the fifth digit (which is 2). Since it is less than 5, we keep the fourth digit as it is. So, 127.92 rounded to 4 significant figures is 127.9.

Q2870: A car travels 345 km in 3.8 hours. Calculate its average speed in km/h and round the answer to 2 significant figures.

90.8

90.79

Explanation: First, calculate the average speed: Speed = Distance / Time = 345 km / 3.8 hours = 90.789... km/h. To round 90.789... to 2 significant figures, we look at the third digit (which is 7). Since it is 5 or greater, we round up the second digit. So, 90.789... rounded to 2 significant figures is 91.

Q2871: A number, when rounded to the nearest whole number, is 15. What is the range of possible values for the original number?

14 ≤ x < 15

14.5 ≤ x < 15.5

15 ≤ x < 16

14.5 ≤ x ≤ 15.5

Explanation: When a number is rounded to the nearest whole number, the actual value lies halfway between the rounded number and the next/previous whole number. For 15, the lower bound is 15 - 0.5 = 14.5, and the upper bound is 15 + 0.5 = 15.5. The lower bound is inclusive, and the upper bound is exclusive.

Q2872: A number is 3.8 when rounded to one decimal place. What is the range of possible original values?

3.7 ≤ x < 3.8

3.75 ≤ x < 3.85

3.8 ≤ x < 3.9

3.75 ≤ x ≤ 3.85

Explanation: When a number is rounded to one decimal place, the actual value lies halfway between the rounded number and the next/previous value at the second decimal place. For 3.8, the lower bound is 3.8 - 0.05 = 3.75, and the upper bound is 3.8 + 0.05 = 3.85. The lower bound is inclusive, and the upper bound is exclusive.

Q2873: A measurement is given as 12.47 cm, rounded to two decimal places. What is the range of its actual length?

12.46 ≤ x < 12.47

12.465 ≤ x < 12.475

12.47 ≤ x < 12.48

12.465 ≤ x ≤ 12.475

Explanation: When a number is rounded to two decimal places, the actual value lies halfway between the rounded number and the next/previous value at the third decimal place. For 12.47, the lower bound is 12.47 - 0.005 = 12.465, and the upper bound is 12.47 + 0.005 = 12.475. The lower bound is inclusive, and the upper bound is exclusive.

Q2874: The population of a town is 8000 when rounded to one significant figure. What is the range of possible actual populations?

7000 ≤ x < 9000

7500 ≤ x < 8500

7999 ≤ x < 8000

7500 ≤ x ≤ 8500

Explanation: When a number is rounded to one significant figure, the actual value lies halfway between the rounded number and the next/previous number at that magnitude. For 8000 (1 s.f.), the rounding unit is 1000. So, the lower bound is 8000 - 500 = 7500, and the upper bound is 8000 + 500 = 8500. The lower bound is inclusive, and the upper bound is exclusive.

Q2875: A number is 0.045 when rounded to two significant figures. What is the range of possible original values?

0.040 ≤ x < 0.050

0.0445 ≤ x < 0.0455

0.044 ≤ x < 0.045

0.0445 ≤ x ≤ 0.0455

Explanation: When a number is rounded to two significant figures (0.045), the last significant digit is 5, which is in the thousandths place (0.001). The rounding unit is therefore 0.001. Half of this is 0.0005. So, the lower bound is 0.045 - 0.0005 = 0.0445, and the upper bound is 0.045 + 0.0005 = 0.0455. The lower bound is inclusive, and the upper bound is exclusive.

Q2876: The number of students in a school is 670 when rounded to the nearest ten. What is the range of possible actual student counts?

660 ≤ x < 670

665 ≤ x < 675

670 ≤ x < 680

665 ≤ x ≤ 675

Explanation: When a number is rounded to the nearest ten, the actual value lies halfway between the rounded number and the next/previous multiple of ten. For 670, the lower bound is 670 - 5 = 665, and the upper bound is 670 + 5 = 675. The lower bound is inclusive, and the upper bound is exclusive.

Identifies and explains direct proportions with examples.

Q2877: Which of the following statements represents a direct proportional relationship?

The time taken to travel a fixed distance and the speed of travel.

The total cost of apples and the number of apples bought.

The area of a square and its side length.

The number of workers and the time taken to complete a fixed amount of work.

Explanation: In a direct proportional relationship, as one quantity increases, the other quantity increases by a constant factor. The total cost of apples and the number of apples bought directly relates because the cost per apple is constant. (Cost = k × Number of apples).

Q2878: If the total rainfall (R) in a certain area is directly proportional to the number of rainy days (D), and 75 mm of rain falls in 5 rainy days, what is the constant of proportionality (k)?

75 mm

5 mm

15 mm/day

375 mm

Explanation: For a direct proportion, R = kD, which means k = R/D. Given R = 75 mm and D = 5 days, k = 75/5 = 15 mm/day.

Q2879: Which of the following tables shows a direct proportional relationship between x and y?

x | 1 | 2 | 3 y | 3 | 4 | 5

x | 1 | 2 | 3 y | 6 | 3 | 2

x | 2 | 3 | 4 y | 6 | 9 | 12

x | 2 | 4 | 6 y | 4 | 6 | 8

Explanation: In a direct proportional relationship, the ratio y/x (or x/y) must be constant. Only in option C, y/x = 6/2 = 3, 9/3 = 3, 12/4 = 3. Thus, k=3.

Q2880: The table below shows a direct proportional relationship between the number of hours worked (H) and the amount earned (E). What is the constant of proportionality (k)? Hours (H) | 2 | 4 | 6 Earned (E) | 1000 | 2000 | 3000

1000

500

2000

Explanation: For a direct proportion, E = kH, so k = E/H. Using any pair from the table: k = 1000/2 = 500. This means Rs. 500 is earned per hour.

Q2881: The amount of fuel consumed by a car (F) is directly proportional to the distance traveled (D). If a car consumes 10 liters of fuel to travel 100 km, what is the constant of proportionality (k)?

10 liters

100 km

0.1 liters/km

1000 liters.km

Explanation: For a direct proportion, F = kD, which means k = F/D. Given F = 10 liters and D = 100 km, k = 10/100 = 0.1 liters/km.

Q2882: A factory produces 'P' items in 'T' hours. The table shows the production data. What is the constant of proportionality (k) and the relationship between P and T? Time (T hours) | 2 | 4 | 6 Items (P) | 50 | 100 | 150

k = 50, P = 50T

k = 25, P = 25T

k = 2, P = 2T

k = 1/25, P = T/25

Explanation: For a direct proportion, P = kT, so k = P/T. Using any pair: k = 50/2 = 25. Therefore, the relationship is P = 25T.

Q2883: If 5 pens cost Rs. 100, what is the cost of 8 such pens?

Rs. 140

Rs. 160

Rs. 200

Rs. 62.50

Explanation: The cost of pens is directly proportional to the number of pens. First, find the cost of one pen: Rs. 100 / 5 pens = Rs. 20 per pen. Then, multiply the cost per pen by the desired number of pens: 8 pens * Rs. 20/pen = Rs. 160.

Q2884: A car travels 150 km in 3 hours. How far will it travel in 5 hours at the same constant speed?

200 km

225 km

250 km

90 km

Explanation: Distance is directly proportional to time when speed is constant. First, find the speed of the car: 150 km / 3 hours = 50 km/h. Then, multiply the speed by the new time: 50 km/h * 5 hours = 250 km.

Q2885: A machine produces 200 items in 4 hours. How many items will it produce in 7 hours, working at the same rate?

280 items

350 items

400 items

114 items

Explanation: The number of items produced is directly proportional to the time worked. First, find the production rate: 200 items / 4 hours = 50 items/hour. Then, multiply the rate by the new time: 50 items/hour * 7 hours = 350 items.

Q2886: 3 kg of sugar costs Rs. 450. What is the cost of 7 kg of sugar?

Rs. 750

Rs. 1050

Rs. 1350

Rs. 193

Explanation: The cost of sugar is directly proportional to its weight. First, find the cost of one kg of sugar: Rs. 450 / 3 kg = Rs. 150 per kg. Then, multiply the cost per kg by the desired weight: 7 kg * Rs. 150/kg = Rs. 1050.

Q2887: A recipe for 6 servings requires 300g of flour. How much flour is needed for 10 servings?

400g

500g

600g

180g

Explanation: The amount of flour is directly proportional to the number of servings. First, find the amount of flour per serving: 300g / 6 servings = 50g per serving. Then, multiply the flour per serving by the new number of servings: 10 servings * 50g/serving = 500g.

Q2888: A worker earns Rs. 1200 for 4 hours of work. How much will they earn for 6 hours of work at the same rate?

Rs. 1600

Rs. 1800

Rs. 2400

Rs. 800

Explanation: Earnings are directly proportional to the hours worked. First, find the hourly rate: Rs. 1200 / 4 hours = Rs. 300 per hour. Then, multiply the hourly rate by the new hours worked: 6 hours * Rs. 300/hour = Rs. 1800.

Q2889: A fabric shop sells a certain type of fabric where 2.5 meters cost Rs. 800. If a customer wants to buy 400 cm of this fabric, what would be the total cost? The shop offers a 10% discount on purchases over Rs. 1500.

Rs. 768

Rs. 1380

Rs. 1280

Rs. 1600

Explanation: First, convert 2.5 meters to centimeters: 2.5 m = 250 cm. The cost per centimeter is Rs. 800 / 250 cm = Rs. 3.20 per cm. For 400 cm, the cost is 400 cm * Rs. 3.20/cm = Rs. 1280. The 10% discount is extraneous information because the total cost (Rs. 1280) is not over Rs. 1500, so the discount does not apply. The cost of fabric is directly proportional to its length.

Q2890: One can of paint covers an area of 15 square meters. If a wall is 4 meters high and 7.5 meters long, how many cans of paint are needed to paint this wall? The paint is available in 1-liter cans, and each can costs Rs. 950.

1 can

2 cans

3 cans

4 cans

Explanation: First, calculate the area of the wall: Area = length × height = 7.5 m × 4 m = 30 square meters. Since one can covers 15 square meters, the number of cans needed is directly proportional to the area. So, 30 square meters / 15 square meters/can = 2 cans. The information about the can size and cost is extraneous for determining the number of cans.

Q2891: A water pump fills 300 liters of water in 5 minutes. How long will it take to fill a tank with a capacity of 1.2 m³? (Assume 1 m³ = 1000 liters).

15 minutes

20 minutes

25 minutes

30 minutes

Explanation: First, convert the tank capacity from cubic meters to liters: 1.2 m³ * 1000 liters/m³ = 1200 liters. The pump fills at a rate of 300 liters in 5 minutes, which is 60 liters per minute (300/5). The time taken is directly proportional to the volume of water. Therefore, to fill 1200 liters, it will take 1200 liters / 60 liters/minute = 20 minutes.

Q2892: A recipe for a cake serving 8 people requires 360g of flour and 180ml of milk. If you want to make a cake for 12 people, how much flour and milk do you need? The recipe also requires 3 eggs and 2 teaspoons of baking powder.

480g flour, 240ml milk

540g flour, 270ml milk

720g flour, 360ml milk

600g flour, 300ml milk

Explanation: The quantities of ingredients are directly proportional to the number of servings. The scaling factor is 12 people / 8 people = 1.5. So, for flour: 360g * 1.5 = 540g. For milk: 180ml * 1.5 = 270ml. The information about eggs and baking powder is extraneous to the question asked.

Q2893: On a map, a distance of 4 cm represents an actual distance of 20 km. If two cities are 12 cm apart on the map, what is the actual distance between them in meters?

60 km

6000 meters

60,000 meters

120,000 meters

Explanation: The map distance is directly proportional to the actual distance. The scale is 4 cm : 20 km. This means 1 cm represents 20 km / 4 = 5 km. For two cities 12 cm apart on the map, the actual distance is 12 cm * 5 km/cm = 60 km. To convert kilometers to meters, multiply by 1000: 60 km * 1000 m/km = 60,000 meters.

Q2894: A worker earns Rs. 550 for 4 hours of work. If they work 7 hours a day, 5 days a week, how much will they earn in a week? The company also provides a transport allowance of Rs. 1500 per week.

Rs. 6312.50

Rs. 4812.50

Rs. 3850

Rs. 5350

Explanation: First, calculate the hourly wage: Rs. 550 / 4 hours = Rs. 137.50 per hour. Next, calculate the total hours worked per week: 7 hours/day * 5 days/week = 35 hours per week. Weekly earnings are directly proportional to the total hours worked. So, 35 hours * Rs. 137.50/hour = Rs. 4812.50. The transport allowance is extraneous information as the question asks for earnings from work, not total income including allowances.

Solves problems related to direct proportions by applying the unitary method.

Q2895: If 5 identical pens cost Rs. 100, what is the cost of 7 such pens?

Rs. 20

Rs. 140

Rs. 50

Rs. 700

Explanation: First, find the cost of one pen. Cost of 1 pen = Rs. 100 / 5 = Rs. 20. Then, multiply the cost of one pen by the required number of pens: Cost of 7 pens = Rs. 20 × 7 = Rs. 140.

Q2896: If 8 exercise books cost Rs. 480, how much would 11 exercise books cost?

Rs. 60

Rs. 491

Rs. 660

Rs. 5280

Explanation: First, find the cost of one exercise book. Cost of 1 book = Rs. 480 / 8 = Rs. 60. Then, multiply the cost of one book by the required number of books: Cost of 11 books = Rs. 60 × 11 = Rs. 660.

Q2897: 12 apples cost Rs. 360. What is the cost of 9 apples?

Rs. 30

Rs. 40

Rs. 270

Rs. 3240

Explanation: First, find the cost of one apple. Cost of 1 apple = Rs. 360 / 12 = Rs. 30. Then, multiply the cost of one apple by the required number of apples: Cost of 9 apples = Rs. 30 × 9 = Rs. 270.

Q2898: 10 kg of rice costs Rs. 1200. What is the cost of 15 kg of rice?

Rs. 120

Rs. 80

Rs. 1800

Rs. 18000

Explanation: First, find the cost of one kilogram of rice. Cost of 1 kg of rice = Rs. 1200 / 10 = Rs. 120. Then, multiply the cost of one kg by the required number of kilograms: Cost of 15 kg of rice = Rs. 120 × 15 = Rs. 1800.

Q2899: A dozen (12) eggs cost Rs. 420. What is the cost of 8 eggs?

Rs. 35

Rs. 105

Rs. 280

Rs. 3360

Explanation: First, find the cost of one egg. Cost of 1 egg = Rs. 420 / 12 = Rs. 35. Then, multiply the cost of one egg by the required number of eggs: Cost of 8 eggs = Rs. 35 × 8 = Rs. 280.

Q2900: 25 meters of cloth cost Rs. 2500. What would be the cost of 18 meters of the same cloth?

Rs. 100

Rs. 2518

Rs. 1800

Rs. 45000

Explanation: First, find the cost of one meter of cloth. Cost of 1 meter = Rs. 2500 / 25 = Rs. 100. Then, multiply the cost of one meter by the required number of meters: Cost of 18 meters = Rs. 100 × 18 = Rs. 1800.

Q2901: A car travels 120 km using 8 liters of petrol. How many liters of petrol are needed to travel 210 km?

12 liters

14 liters

16 liters

18 liters

Explanation: First, find the petrol consumption per km. 8 liters / 120 km = 1/15 liters per km. Then, for 210 km, the petrol needed is (1/15) * 210 = 14 liters.

Q2902: A recipe for 6 servings requires 240g of flour. How much flour is needed for 15 servings?

480g

249g

600g

96g

Explanation: First, find the amount of flour needed per serving. 240g / 6 servings = 40g per serving. Then, for 15 servings, the flour needed is 40g * 15 = 600g.

Q2903: 5 identical books cost Rs. 800. How much will 12 such books cost?

Rs. 1600

Rs. 807

Rs. 1920

Rs. 333

Explanation: First, find the cost of one book. Rs. 800 / 5 books = Rs. 160 per book. Then, for 12 books, the cost is Rs. 160 * 12 = Rs. 1920.

Q2904: A factory produces 450 toys in 3 hours. How many toys can it produce in 7 hours, assuming a constant rate?

900 toys

454 toys

1350 toys

1050 toys

Explanation: First, find the production rate per hour. 450 toys / 3 hours = 150 toys per hour. Then, for 7 hours, the number of toys produced is 150 * 7 = 1050 toys.

Q2905: 5 kg of sugar costs Rs. 1250. How much will 3 kg of sugar cost?

Rs. 1000

Rs. 1248

Rs. 750

Rs. 2083

Explanation: First, find the cost of 1 kg of sugar. Rs. 1250 / 5 kg = Rs. 250 per kg. Then, for 3 kg of sugar, the cost is Rs. 250 * 3 = Rs. 750.

Q2906: A painter earns Rs. 3600 for working 8 hours. How much will he earn if he works 13 hours?

Rs. 3605

Rs. 5400

Rs. 2215

Rs. 5850

Explanation: First, find the painter's hourly earning rate. Rs. 3600 / 8 hours = Rs. 450 per hour. Then, for 13 hours, the earning is Rs. 450 * 13 = Rs. 5850.

Q2907: A 3.5 meter length of cloth costs Rs. 875. If a customer wants to buy cloth worth Rs. 1250, what length of cloth can they buy?

4.5 meters

5 meters

5.5 meters

6 meters

Explanation: First, find the cost of 1 meter of cloth: Rs. 875 / 3.5 m = Rs. 250 per meter. Then, find the length of cloth that can be bought for Rs. 1250: Rs. 1250 / Rs. 250 per meter = 5 meters.

Q2908: A factory machine produces 480 toys in 6 hours. If the machine operates for 8.5 hours, how many toys can it produce?

640 toys

680 toys

720 toys

760 toys

Explanation: First, find the number of toys produced in one hour: 480 toys / 6 hours = 80 toys per hour. Then, multiply the hourly rate by the new time: 80 toys/hour * 8.5 hours = 680 toys.

Q2909: A car consumes 12 liters of petrol to travel 180 km. If petrol costs Rs. 350 per liter, what is the total cost of petrol needed to travel 450 km?

Rs. 9750

Rs. 10500

Rs. 10250

Rs. 11250

Explanation: First, find the distance covered per liter of petrol: 180 km / 12 liters = 15 km/liter. Next, calculate the total liters needed for 450 km: 450 km / 15 km/liter = 30 liters. Finally, calculate the total cost: 30 liters * Rs. 350/liter = Rs. 10500.

Q2910: A baker uses 2.5 kg of flour to make 20 small cakes. If the baker has 8 kg of flour, how many small cakes can they make?

60 cakes

64 cakes

68 cakes

72 cakes

Explanation: First, find how many cakes can be made with 1 kg of flour: 20 cakes / 2.5 kg = 8 cakes per kg. Then, multiply this rate by the total flour available: 8 cakes/kg * 8 kg = 64 cakes.

Q2911: A worker earns Rs. 3600 for working 15 hours. If the worker wants to earn Rs. 6000, how many hours must they work?

20 hours

22.5 hours

25 hours

27.5 hours

Explanation: First, find the hourly earning rate: Rs. 3600 / 15 hours = Rs. 240 per hour. Then, divide the target earnings by the hourly rate to find the required hours: Rs. 6000 / Rs. 240 per hour = 25 hours.

Q2912: A 5-liter can of paint covers an area of 40 square meters. If the cost of one 5-liter can of paint is Rs. 2800, what would be the total cost to paint a wall with an area of 120 square meters?

Rs. 7800

Rs. 8400

Rs. 9200

Rs. 10000

Explanation: First, determine how many 5-liter cans of paint are needed: 120 sq m / 40 sq m per can = 3 cans. Then, calculate the total cost by multiplying the number of cans by the cost per can: 3 cans * Rs. 2800/can = Rs. 8400.

Solves problems related to direct proportions by using the definition of proportion.

Q2913: If x and y are directly proportional, and x = 4 when y = 12, what is the value of y when x = 7?

Explanation: Since x and y are directly proportional, the ratio y/x is constant. So, y₁/x₁ = y₂/x₂. Given x₁=4, y₁=12, and x₂=7. Substituting these values: 12/4 = y₂/7. This simplifies to 3 = y₂/7. Multiplying both sides by 7 gives y₂ = 3 * 7 = 21.

Q2914: The cost of 5 notebooks is Rs. 150. What is the cost of 8 such notebooks?

180

210

240

270

Explanation: Let C be the cost and N be the number of notebooks. Since cost is directly proportional to the number of notebooks, C/N is constant. So, C₁/N₁ = C₂/N₂. Given C₁=150, N₁=5, and N₂=8. Substituting these values: 150/5 = C₂/8. This simplifies to 30 = C₂/8. Multiplying both sides by 8 gives C₂ = 30 * 8 = Rs. 240.

Q2915: A car travels 180 km in 3 hours. How far will it travel in 5 hours at the same speed?

240 km

270 km

300 km

360 km

Explanation: Distance (D) is directly proportional to time (T) when speed is constant. So, D/T is constant. D₁/T₁ = D₂/T₂. Given D₁=180, T₁=3, and T₂=5. Substituting these values: 180/3 = D₂/5. This simplifies to 60 = D₂/5. Multiplying both sides by 5 gives D₂ = 60 * 5 = 300 km.

Q2916: 6 identical machines can produce 480 toys in an hour. How many toys can 9 such machines produce in an hour?

320 toys

600 toys

720 toys

840 toys

Explanation: The number of toys (T) produced is directly proportional to the number of machines (M) when time is constant. So, T/M is constant. T₁/M₁ = T₂/M₂. Given T₁=480, M₁=6, and M₂=9. Substituting these values: 480/6 = T₂/9. This simplifies to 80 = T₂/9. Multiplying both sides by 9 gives T₂ = 80 * 9 = 720 toys.

Q2917: If p is directly proportional to q, and p = 15 when q = 5. What is the value of p when q = 8?

Explanation: Since p is directly proportional to q, the ratio p/q is constant. So, p₁/q₁ = p₂/q₂. Given p₁=15, q₁=5, and q₂=8. Substituting these values: 15/5 = p₂/8. This simplifies to 3 = p₂/8. Multiplying both sides by 8 gives p₂ = 3 * 8 = 24.

Q2918: The amount of flour needed for 12 cakes is 300g. How much flour is needed for 20 cakes?

400g

450g

500g

600g

Explanation: The amount of flour (F) is directly proportional to the number of cakes (C). So, F/C is constant. F₁/C₁ = F₂/C₂. Given F₁=300, C₁=12, and C₂=20. Substituting these values: 300/12 = F₂/20. This simplifies to 25 = F₂/20. Multiplying both sides by 20 gives F₂ = 25 * 20 = 500g.

Q2919: If 5 pens cost Rs. 100, how much would 12 pens cost?

Rs. 200

Rs. 240

Rs. 180

Rs. 120

Q2920: A car travels 150 km in 3 hours at a constant speed. How far will it travel in 5 hours?

180 km

200 km

250 km

300 km

Explanation: Distance traveled is directly proportional to the time taken when speed is constant. First, find the speed: 150 km / 3 hours = 50 km/h. Then, calculate the distance for 5 hours: 5 hours * 50 km/h = 250 km.

Q2921: A machine produces 200 items in 4 hours. How many items will it produce in 7 hours, assuming a constant production rate?

280 items

300 items

350 items

400 items

Explanation: The number of items produced is directly proportional to the time taken. First, find the production rate: 200 items / 4 hours = 50 items/hour. Then, calculate the items for 7 hours: 7 hours * 50 items/hour = 350 items.

Q2922: To bake 6 cakes, 3 cups of flour are needed. How many cups of flour are needed to bake 10 cakes?

4 cups

5 cups

6 cups

7 cups

Explanation: The amount of flour is directly proportional to the number of cakes. First, find the flour needed per cake: 3 cups / 6 cakes = 0.5 cups per cake. Then, calculate the flour for 10 cakes: 10 cakes * 0.5 cups/cake = 5 cups.

Q2923: If 2 liters of a certain liquid weigh 4 kg, what is the weight of 5 liters of the same liquid?

8 kg

9 kg

10 kg

12 kg

Explanation: The weight of the liquid is directly proportional to its volume. First, find the weight per liter: 4 kg / 2 liters = 2 kg per liter. Then, calculate the weight for 5 liters: 5 liters * 2 kg/liter = 10 kg.

Q2924: If 10 US dollars (USD) is equivalent to 3000 Sri Lankan Rupees (LKR), how many LKR would you get for 15 USD?

4000 LKR

4200 LKR

4500 LKR

5000 LKR

Explanation: The amount in LKR is directly proportional to the amount in USD. First, find the exchange rate for 1 USD: 3000 LKR / 10 USD = 300 LKR per USD. Then, calculate the LKR for 15 USD: 15 USD * 300 LKR/USD = 4500 LKR.

Q2925: 10 pens cost Rs. 300. If the number of pens bought increases by 20%, what is the new total cost?

Rs. 320

Rs. 360

Rs. 280

Rs. 300

Explanation: Since the cost is directly proportional to the number of pens, if the number of pens increases by 20%, the total cost will also increase by 20%. Original cost = Rs. 300. Increase in cost = 20% of Rs. 300 = (20/100) * 300 = Rs. 60. New total cost = Rs. 300 + Rs. 60 = Rs. 360.

Q2926: A car travels 240 km in 4 hours at a constant speed. If the time taken for a journey is reduced by 25%, how much distance can it cover in the new time?

210 km

180 km

160 km

240 km

Explanation: Since distance is directly proportional to time (at constant speed), if the time is reduced by 25%, the distance covered will also be reduced by 25%. Original distance = 240 km. Reduction in distance = 25% of 240 km = (25/100) * 240 = 60 km. New distance = 240 km - 60 km = 180 km.

Q2927: 15 workers can paint 75 chairs in a day. If the number of workers decreases by 20%, how many chairs can be painted in a day by the remaining workers?

70 chairs

60 chairs

55 chairs

75 chairs

Explanation: The number of chairs painted is directly proportional to the number of workers. Original number of workers = 15. Decrease in workers = 20% of 15 = (20/100) * 15 = 3 workers. New number of workers = 15 - 3 = 12 workers. Since 15 workers paint 75 chairs, 1 worker paints 75/15 = 5 chairs. So, 12 workers will paint 12 * 5 = 60 chairs.

Q2928: A machine produces 200 units in 5 hours. If the production time is increased by 30%, what is the percentage increase in the number of units produced?

20%

30%

50%

10%

Explanation: Since the number of units produced is directly proportional to the production time, if the production time increases by 30%, the number of units produced will also increase by the same percentage, i.e., 30%.

Q2929: A car travels 120 km using 8 litres of petrol. If the price of petrol increases by 25%, and a person wants to spend the same amount of money on petrol, what is the percentage decrease in the distance they can travel?

25%

20%

15%

10%

Explanation: Let the original price per litre be P. Original cost for 8 litres = 8P. New price per litre = P * (1 + 0.25) = 1.25P. To spend 8P, the new quantity of petrol that can be bought = 8P / 1.25P = 6.4 litres. Distance per litre = 120 km / 8 litres = 15 km/litre. New distance = 6.4 litres * 15 km/litre = 96 km. Decrease in distance = 120 km - 96 km = 24 km. Percentage decrease = (24 / 120) * 100% = 20%.

Q2930: 4 kg of sugar costs Rs. 800. If the price per kg of sugar decreases by 20%, how much more sugar (in kg) can be bought for Rs. 800?

0.8 kg

1 kg

1.2 kg

0.5 kg

Explanation: Original price per kg = Rs. 800 / 4 kg = Rs. 200/kg. Decrease in price per kg = 20% of Rs. 200 = (20/100) * 200 = Rs. 40. New price per kg = Rs. 200 - Rs. 40 = Rs. 160/kg. New quantity of sugar for Rs. 800 = Rs. 800 / Rs. 160/kg = 5 kg. More sugar that can be bought = 5 kg - 4 kg = 1 kg.

Writes the relationship between two directly proportional quantities in the form y = kx.

Q2931: Quantity A is directly proportional to quantity B. When A = 12, B = 3. What is the relationship between A and B?

B = 4A

A = 4B

A = 12B

A = 3B

Explanation: If A is directly proportional to B, then A = kB, where k is the constant of proportionality. Given A = 12 and B = 3, we substitute these values into the equation: 12 = k * 3. Dividing both sides by 3 gives k = 4. Therefore, the relationship is A = 4B.

Q2932: X is directly proportional to Y. If X = 45 when Y = 9, what is the relationship between X and Y?

Y = 5X

X = 9Y

X = 5Y

X = Y/5

Explanation: Since X is directly proportional to Y, we can write X = kY. Using the given values X = 45 and Y = 9, we find k: 45 = k * 9. Dividing by 9 gives k = 5. So, the relationship is X = 5Y.

Q2933: P is directly proportional to Q. When P = 20, Q = 4. Determine the constant of proportionality (k) and write the relationship.

k = 4, P = 4Q

k = 5, P = 5Q

k = 1/5, P = Q/5

k = 20, P = 20Q

Explanation: Since P is directly proportional to Q, P = kQ. Given P = 20 and Q = 4, substitute these into the equation: 20 = k * 4. Dividing by 4 gives k = 5. So, the relationship is P = 5Q.

Q2934: The cost (C) of apples is directly proportional to their weight (W). If 5 kg of apples cost Rs. 250, write the relationship between C and W.

W = 50C

C = W + 245

C = 50W

C = 5W

Explanation: Since C is directly proportional to W, we can write C = kW. Given C = 250 when W = 5, we substitute these values: 250 = k * 5. Dividing by 5 gives k = 50. Therefore, the relationship is C = 50W.

Q2935: Let M be directly proportional to N. When M = 18, N = 6. What is the constant of proportionality (k) and the relationship between M and N?

k = 1/3, M = N/3

k = 3, M = 3N

k = 6, M = 6N

k = 12, M = N + 12

Explanation: If M is directly proportional to N, then M = kN. Given M = 18 and N = 6, substitute these values: 18 = k * 6. Dividing by 6 gives k = 3. So, the constant of proportionality is 3, and the relationship is M = 3N.

Q2936: The distance (D) travelled is directly proportional to the time (T) taken. When T = 4 hours, D = 240 km. What is the relationship between D and T?

T = 60D

D = T + 236

D = 240T

D = 60T

Explanation: If D is directly proportional to T, then D = kT. Given D = 240 km and T = 4 hours, substitute these values: 240 = k * 4. Dividing by 4 gives k = 60. Therefore, the relationship is D = 60T.

Q2937: The cost of a certain type of fabric is directly proportional to its length. If 3 meters of fabric cost Rs. 450, what is the cost of 7 meters of the same fabric?

Rs. 900

Rs. 1050

Rs. 1350

Rs. 150

Explanation: Let C be the cost and L be the length. Since C is directly proportional to L, C = kL. Given C = 450 when L = 3, we find k = 450/3 = 150. So the relationship is C = 150L. For L = 7 meters, C = 150 * 7 = Rs. 1050.

Q2938: The distance a car travels is directly proportional to the time taken, assuming constant speed. If a car travels 180 km in 3 hours, how far can it travel in 5 hours?

270 km

300 km

108 km

360 km

Explanation: Let D be the distance and T be the time. Since D is directly proportional to T, D = kT. Given D = 180 when T = 3, we find k = 180/3 = 60. So the relationship is D = 60T. For T = 5 hours, D = 60 * 5 = 300 km.

Q2939: The amount of sugar (in kg) that can be bought is directly proportional to the amount of money spent (in Rs.). If 2 kg of sugar costs Rs. 260, how many kilograms of sugar can be bought for Rs. 650?

4 kg

5 kg

3 kg

6 kg

Explanation: Let S be the amount of sugar and M be the money spent. Since S is directly proportional to M, S = kM. Given S = 2 when M = 260, we find k = 2/260 = 1/130. So the relationship is S = (1/130)M. For M = 650, S = (1/130) * 650 = 5 kg.

Q2940: A worker's daily wage is directly proportional to the number of hours worked. If a worker earns Rs. 1200 for working 6 hours, how much will they earn for working 9 hours?

Rs. 1500

Rs. 1600

Rs. 1800

Rs. 2000

Explanation: Let W be the wage and H be the hours worked. Since W is directly proportional to H, W = kH. Given W = 1200 when H = 6, we find k = 1200/6 = 200. So the relationship is W = 200H. For H = 9 hours, W = 200 * 9 = Rs. 1800.

Q2941: The volume of water flowing from a tap is directly proportional to the time the tap is open. If 40 liters of water flows in 5 minutes, how many liters will flow in 8 minutes?

50 liters

60 liters

64 liters

72 liters

Explanation: Let V be the volume of water and T be the time. Since V is directly proportional to T, V = kT. Given V = 40 when T = 5, we find k = 40/5 = 8. So the relationship is V = 8T. For T = 8 minutes, V = 8 * 8 = 64 liters.

Q2942: The pressure exerted by a fluid at a certain depth is directly proportional to the depth. If the pressure at a depth of 5 meters is 50 kPa, what is the pressure at a depth of 9 meters?

80 kPa

90 kPa

100 kPa

45 kPa

Explanation: Let P be the pressure and D be the depth. Since P is directly proportional to D, P = kD. Given P = 50 when D = 5, we find k = 50/5 = 10. So the relationship is P = 10D. For D = 9 meters, P = 10 * 9 = 90 kPa.

Q2943: The table below shows corresponding values for two quantities, x and y. | x | y | |---|---| | 2 | 8 | | 5 | 20 | | 7 | 28 | Which of the following equations correctly represents the relationship between x and y?

y = x + 6

y = 2x

y = 4x

y = x/4

Explanation: For direct proportionality, the ratio y/x must be constant. For (2, 8), y/x = 8/2 = 4. For (5, 20), y/x = 20/5 = 4. For (7, 28), y/x = 28/7 = 4. Since the ratio y/x is constant (k = 4), x and y are directly proportional, and the relationship is y = 4x.

Q2944: The table below shows corresponding values for two quantities, x and y. | x | y | |---|---| | 3 | 1 | | 6 | 2 | | 9 | 3 | Which of the following equations correctly represents the relationship between x and y?

y = 3x

y = x - 2

y = (1/3)x

y = 3/x

Explanation: For direct proportionality, the ratio y/x must be constant. For (3, 1), y/x = 1/3. For (6, 2), y/x = 2/6 = 1/3. For (9, 3), y/x = 3/9 = 1/3. Since the ratio y/x is constant (k = 1/3), x and y are directly proportional, and the relationship is y = (1/3)x.

Q2945: The table below shows corresponding values for two quantities, x and y. | x | y | |---|---| | 4 | 10 | | 8 | 20 | | 12 | 30 | Which of the following equations correctly represents the relationship between x and y?

y = x + 6

y = 0.4x

y = 2.5x

y = 2x

Explanation: For direct proportionality, the ratio y/x must be constant. For (4, 10), y/x = 10/4 = 2.5. For (8, 20), y/x = 20/8 = 2.5. For (12, 30), y/x = 30/12 = 2.5. Since the ratio y/x is constant (k = 2.5), x and y are directly proportional, and the relationship is y = 2.5x.

Q2946: The table below shows corresponding values for two quantities, x and y. | x | y | |---|---| | 5 | 15 | | 8 | 24 | | 10 | 30 | Which of the following equations correctly represents the relationship between x and y?

y = x + 10

y = x/3

y = 2x

y = 3x

Explanation: For direct proportionality, the ratio y/x must be constant. For (5, 15), y/x = 15/5 = 3. For (8, 24), y/x = 24/8 = 3. For (10, 30), y/x = 30/10 = 3. Since the ratio y/x is constant (k = 3), x and y are directly proportional, and the relationship is y = 3x.

Q2947: The table below shows corresponding values for two quantities, x and y. | x | y | |---|---| | 10 | 4 | | 25 | 10 | | 40 | 16 | Which of the following equations correctly represents the relationship between x and y?

y = x - 6

y = 2.5x

y = 0.4x

y = x/4

Explanation: For direct proportionality, the ratio y/x must be constant. For (10, 4), y/x = 4/10 = 0.4. For (25, 10), y/x = 10/25 = 0.4. For (40, 16), y/x = 16/40 = 0.4. Since the ratio y/x is constant (k = 0.4), x and y are directly proportional, and the relationship is y = 0.4x.

Q2948: The table below shows corresponding values for two quantities, x and y. | x | y | |-----|----| | 1.5 | 6 | | 2.5 | 10 | | 4 | 16 | Which of the following equations correctly represents the relationship between x and y?

y = x + 4.5

y = 0.25x

y = 2x

y = 4x

Explanation: For direct proportionality, the ratio y/x must be constant. For (1.5, 6), y/x = 6/1.5 = 4. For (2.5, 10), y/x = 10/2.5 = 4. For (4, 16), y/x = 16/4 = 4. Since the ratio y/x is constant (k = 4), x and y are directly proportional, and the relationship is y = 4x.

Solves problems involving foreign currency conversion using direct proportions.

Q2949: If the exchange rate is 1 USD = LKR 300, how much is $50 in Sri Lankan Rupees?

LKR 150

LKR 15000

LKR 350

LKR 1500

Explanation: To convert USD to LKR, multiply the USD amount by the exchange rate. 50 USD * 300 LKR/USD = 15000 LKR.

Q2950: If 1 USD = LKR 300, how many US Dollars can you get for LKR 60000?

$18000000

$200

$20

$60300

Explanation: To convert LKR to USD, divide the LKR amount by the exchange rate. 60000 LKR / 300 LKR/USD = 200 USD.

Q2951: An exchange rate is 1 EUR = LKR 330. How many Sri Lankan Rupees are 75 Euros worth?

LKR 247500

LKR 24750

LKR 225

LKR 405

Explanation: To convert EUR to LKR, multiply the EUR amount by the exchange rate. 75 EUR * 330 LKR/EUR = 24750 LKR.

Q2952: If 1 EUR = LKR 330, how many Euros can you get for LKR 16500?

55 EUR

50 EUR

5445000 EUR

16170 EUR

Explanation: To convert LKR to EUR, divide the LKR amount by the exchange rate. 16500 LKR / 330 LKR/EUR = 50 EUR.

Q2953: The exchange rate between Australian Dollars (AUD) and Sri Lankan Rupees (LKR) is 1 AUD = LKR 200. How much is 120 AUD in LKR?

LKR 2400

LKR 24000

LKR 320

LKR 0.6

Explanation: To convert AUD to LKR, multiply the AUD amount by the exchange rate. 120 AUD * 200 LKR/AUD = 24000 LKR.

Q2954: Using the exchange rate 1 AUD = LKR 200, convert LKR 40000 into Australian Dollars.

8000000 AUD

200 AUD

20 AUD

40200 AUD

Explanation: To convert LKR to AUD, divide the LKR amount by the exchange rate. 40000 LKR / 200 LKR/AUD = 200 AUD.

Q2955: A person has LKR 15,000. The exchange rate is 1 USD = LKR 300. If they buy a souvenir for USD 40, how much US Dollars will they have left?

USD 50

USD 10

LKR 14,960

USD 40

Explanation: First, convert the initial LKR amount to USD: 15,000 LKR / 300 LKR/USD = 50 USD. Then, subtract the cost of the souvenir: 50 USD - 40 USD = 10 USD. So, 10 US Dollars will be left.

Q2956: A tourist has USD 60. The exchange rate is 1 USD = LKR 300. He wants to buy a local craft item priced at LKR 18,500. Can he afford it, and if not, how much more LKR does he need?

Yes, he has LKR 500 left.

No, he needs LKR 500 more.

No, he needs LKR 18,000 more.

Yes, he has exactly enough.

Explanation: First, convert the tourist's USD to LKR: 60 USD * 300 LKR/USD = 18,000 LKR. The item costs LKR 18,500. Since 18,000 LKR < 18,500 LKR, he cannot afford it. He needs 18,500 LKR - 18,000 LKR = 500 LKR more.

Q2957: A student has LKR 19,800. The exchange rate is 1 EUR = LKR 330. She wants to buy two books, one for EUR 35 and another for EUR 20. How much money (in EUR) will she have left after buying both books?

EUR 60

EUR 5

EUR 55

LKR 1,650

Explanation: First, convert the student's LKR to EUR: 19,800 LKR / 330 LKR/EUR = 60 EUR. Then, calculate the total cost of the books: 35 EUR + 20 EUR = 55 EUR. Finally, find the remaining amount: 60 EUR - 55 EUR = 5 EUR.

Q2958: A person has LKR 10,000. The exchange rate is 1 AUD = LKR 200. He wants to buy a hat for AUD 30 and a scarf for AUD 25. How much more LKR does he need to buy both items?

LKR 11,000

LKR 1,000

AUD 55

LKR 5,000

Explanation: First, calculate the total cost of both items in AUD: 30 AUD + 25 AUD = 55 AUD. Then, convert this total cost to LKR: 55 AUD * 200 LKR/AUD = 11,000 LKR. Since he only has LKR 10,000, he needs 11,000 LKR - 10,000 LKR = 1,000 LKR more.

Q2959: A person has LKR 22,800. The exchange rate is 1 GBP = LKR 380. How many items, each costing GBP 12, can they buy?

60 items

5 items

4 items

380 items

Explanation: First, convert the LKR amount to GBP: 22,800 LKR / 380 LKR/GBP = 60 GBP. Then, divide the total GBP by the cost per item to find how many items can be bought: 60 GBP / 12 GBP/item = 5 items.

Q2960: A traveler buys a gift for USD 25 and another souvenir for LKR 5,000. If the exchange rate is 1 USD = LKR 300, what is the total cost in LKR?

LKR 7,500

LKR 12,500

LKR 5,025

USD 41.67

Explanation: First, convert the cost of the gift from USD to LKR: 25 USD * 300 LKR/USD = 7,500 LKR. Then, add this to the cost of the souvenir in LKR to find the total cost: 7,500 LKR + 5,000 LKR = 12,500 LKR.

Q2961: Kamala wants to exchange LKR 50,000 for US Dollars (USD). Bank A offers an exchange rate of 1 USD = LKR 305.00. Bank B offers an exchange rate of 1 USD = LKR 302.50. Which bank offers a more advantageous rate for Kamala, and how much more USD would she receive from the better option compared to the other?

Bank A; USD 1.33

Bank B; USD 1.33

Bank A; USD 2.50

Bank B; USD 2.50

Explanation: To find the more advantageous bank, we calculate how much USD Kamala would receive from each bank for LKR 50,000. For Bank A: USD = LKR 50,000 / 305.00 = 163.93 USD (approx.) For Bank B: USD = LKR 50,000 / 302.50 = 165.26 USD (approx.) Bank B is more advantageous because Kamala receives more USD (165.26 USD > 163.93 USD). The difference is 165.26 - 163.93 = 1.33 USD.

Q2962: Nimal has 200 Euros (EUR) and wants to convert them to Sri Lankan Rupees (LKR). Exchange Center X buys EUR at LKR 345.00 per EUR. Exchange Center Y buys EUR at LKR 348.50 per EUR. Which exchange center offers a more advantageous rate for Nimal, and how much more LKR would he receive from the better option compared to the other?

Center X; LKR 700

Center Y; LKR 700

Center X; LKR 3.50

Center Y; LKR 3.50

Explanation: To find the more advantageous exchange center, we calculate how much LKR Nimal would receive from each center for 200 EUR. For Center X: LKR = 200 EUR * 345.00 LKR/EUR = 69,000 LKR For Center Y: LKR = 200 EUR * 348.50 LKR/EUR = 69,700 LKR Center Y is more advantageous because Nimal receives more LKR (69,700 LKR > 69,000 LKR). The difference is 69,700 - 69,000 = 700 LKR.

Q2963: A university student needs to pay a registration fee of USD 150. They have two options: 1. Convert LKR directly to USD at a rate of 1 USD = LKR 308.00. 2. Convert LKR to EUR first at 1 EUR = LKR 340.00, then convert EUR to USD at 1 EUR = 1.10 USD. Which option requires less Sri Lankan Rupees, and what is the minimum LKR amount needed for the fee?

Option 1; LKR 46,200.00

Option 2; LKR 46,363.64

Option 1; LKR 45,000.00

Option 2; LKR 46,750.00

Explanation: We need to calculate the LKR cost for each option to get USD 150. Option 1: Direct conversion LKR = 150 USD * 308.00 LKR/USD = 46,200 LKR Option 2: Two-step conversion First, find EUR needed for USD 150: EUR = 150 USD / 1.10 USD/EUR = 136.3636... EUR Then, find LKR for this EUR amount: LKR = 136.3636... EUR * 340.00 LKR/EUR = 46,363.64 LKR (approx.) Comparing the costs, Option 1 (LKR 46,200) is cheaper than Option 2 (LKR 46,363.64). So, the minimum LKR amount needed is LKR 46,200.

Q2964: A tourist wants to convert LKR 100,000 to Great British Pounds (GBP). Scenario A: Direct conversion at a bank, where 1 GBP = LKR 390.00. Scenario B: Convert LKR to USD first at 1 USD = LKR 300.00, then convert USD to GBP at 1 GBP = 1.25 USD. However, the second conversion (USD to GBP) incurs a 2% commission on the USD amount converted. How much GBP would the tourist receive in each scenario, and which scenario yields more GBP?

Scenario A: 256.41 GBP; Scenario B: 266.67 GBP; Scenario B is better

Scenario A: 256.41 GBP; Scenario B: 261.33 GBP; Scenario B is better

Scenario A: 261.33 GBP; Scenario B: 256.41 GBP; Scenario A is better

Scenario A: 256.41 GBP; Scenario B: 259.00 GBP; Scenario B is better

Explanation: Scenario A: Direct conversion GBP = LKR 100,000 / 390.00 LKR/GBP = 256.41 GBP (approx.) Scenario B: Two-step conversion with commission 1. LKR to USD: USD = LKR 100,000 / 300.00 LKR/USD = 333.33 USD (approx.) 2. Apply 2% commission on USD amount: Commission = 0.02 * 333.33 = 6.67 USD (approx.) USD remaining for conversion = 333.33 - 6.67 = 326.66 USD (approx.) 3. USD to GBP: GBP = 326.66 USD / 1.25 USD/GBP = 261.33 GBP (approx.) Comparing the two, Scenario B (261.33 GBP) yields more GBP than Scenario A (256.41 GBP).

Q2965: A father wants to send USD 500 to his son studying abroad. He has two money transfer services available: Service P: Charges LKR 307.50 for 1 USD. Service Q: Charges LKR 306.00 for 1 USD, plus a fixed service fee of LKR 500. Which service would be more economical (require less LKR) for the father to send USD 500, and what is the total LKR cost for that service?

Service P; LKR 153,750.00

Service Q; LKR 153,000.00

Service Q; LKR 153,500.00

Service P; LKR 154,250.00

Explanation: We calculate the total LKR cost for sending USD 500 through each service. Service P: Total LKR = 500 USD * 307.50 LKR/USD = 153,750 LKR Service Q: Total LKR = (500 USD * 306.00 LKR/USD) + 500 LKR (fixed fee) = 153,000 LKR + 500 LKR = 153,500 LKR Comparing the costs, Service Q (LKR 153,500) is more economical than Service P (LKR 153,750). The total LKR cost for Service Q is LKR 153,500.

Q2966: A popular smartphone model costs USD 800 in Country A and EUR 750 in Country B. The current exchange rates are 1 USD = LKR 305.00 and 1 EUR = LKR 340.00. In which country is the smartphone cheaper for a buyer from Sri Lanka, and what is the price difference in LKR?

Country A; LKR 11,000.00

Country B; LKR 11,000.00

Country A; LKR 1,000.00

Country B; LKR 1,000.00

Explanation: To compare the prices, we convert both prices to LKR. Cost in Country A: LKR = 800 USD * 305.00 LKR/USD = 244,000 LKR Cost in Country B: LKR = 750 EUR * 340.00 LKR/EUR = 255,000 LKR The smartphone is cheaper in Country A (LKR 244,000) compared to Country B (LKR 255,000). The price difference is 255,000 LKR - 244,000 LKR = 11,000 LKR.

Identifies and uses the keys of a scientific calculator, including basic operations, percentage, square, and square root.

Q2967: What is the value of 13² when calculated using a scientific calculator?

169

163

Explanation: To find the square of 13, press '1', '3', then the 'x²' key, and finally '='. The result is 169.

Q2968: Using a scientific calculator, find the value of √144.

14.4

144

Explanation: To find the square root of 144, press the '√' key, then '1', '4', '4', and finally '='. The result is 12.

Q2969: Calculate 25% of 300 using a scientific calculator.

7500

Explanation: To find 25% of 300, enter '3', '0', '0', then press '×', then '2', '5', and finally the '%' key, then '='. The result is 75. (Alternatively, 300 * 25 / 100).

Q2970: What is the result of calculating 18² using a scientific calculator?

324

182

Explanation: To find the square of 18, press '1', '8', then the 'x²' key, and finally '='. The result is 324.

Q2971: Find the value of √400 using a scientific calculator.

200

400

Explanation: To find the square root of 400, press the '√' key, then '4', '0', '0', and finally '='. The result is 20.

Q2972: What is 15% of 600 when calculated using a scientific calculator?

9000

Explanation: To find 15% of 600, enter '6', '0', '0', then press '×', then '1', '5', and finally the '%' key, then '='. The result is 90. (Alternatively, 600 * 15 / 100).

Q2973: Evaluate 15² - √(225) + 8 using a scientific calculator.

218

202

248

Explanation: First, calculate the square: 15² = 225. Next, calculate the square root: √(225) = 15. Then perform the subtraction and addition: 225 - 15 + 8 = 210 + 8 = 218.

Q2974: Evaluate 200 + 15% of 200 - √100 using a scientific calculator.

205

220

240

130

Explanation: First, calculate the percentage: 15% of 200 = 0.15 × 200 = 30. Next, calculate the square root: √100 = 10. Then perform the addition and subtraction: 200 + 30 - 10 = 230 - 10 = 220.

Q2975: Evaluate (3.5)² + 49 / 7 using a scientific calculator.

19.25

7.5

18.25

Explanation: First, calculate the square: (3.5)² = 12.25. Next, perform the division: 49 / 7 = 7. Finally, add the results: 12.25 + 7 = 19.25.

Q2976: Evaluate √(64 + 36) × 5 using a scientific calculator.

220

188

Explanation: First, perform the addition inside the square root: 64 + 36 = 100. Next, calculate the square root: √100 = 10. Finally, perform the multiplication: 10 × 5 = 50.

Q2977: Evaluate 12² - (√81 × 3) using a scientific calculator.

-99

117

132

405

Explanation: First, calculate the square: 12² = 144. Next, calculate the square root: √81 = 9. Then, perform the multiplication inside the parentheses: 9 × 3 = 27. Finally, perform the subtraction: 144 - 27 = 117.

Q2978: Evaluate 25% of 300 + 10² / 5 using a scientific calculator.

1445

Explanation: First, calculate the percentage: 25% of 300 = 0.25 × 300 = 75. Next, calculate the square: 10² = 100. Then, perform the division: 100 / 5 = 20. Finally, add the results: 75 + 20 = 95.

Q2979: Evaluate the expression `c = √(a² + b²) ` when `a = 6.5` and `b = 8.2` using a scientific calculator. Round your answer to two decimal places.

14.7

10.46

109.49

7.27

Explanation: Substitute the given values into the formula: `c = √(6.5² + 8.2²) `. First, calculate the squares: `6.5² = 42.25` and `8.2² = 67.24`. Then add them: `42.25 + 67.24 = 109.49`. Finally, find the square root: `√109.49 ≈ 10.4637`. Rounded to two decimal places, the answer is `10.46`.

Q2980: Using a scientific calculator, evaluate `A = πr² + √(r² + h²) ` when `r = 4.0` and `h = 3.0`. Use the π key on your calculator and round your answer to two decimal places.

50.27

55.27

21.00

20.27

Explanation: Substitute the values: `A = π(4.0)² + √(4.0² + 3.0²) `. Calculate the terms: `4.0² = 16`, `3.0² = 9`. So, `A = π(16) + √(16 + 9) = 16π + √25 = 16π + 5`. Using the π key, `16π ≈ 50.26548`. Adding 5, `A ≈ 50.26548 + 5 = 55.26548`. Rounded to two decimal places, the answer is `55.27`.

Q2981: Calculate the value of `X = 15.6² - √(72.25)` using a scientific calculator. Round your answer to two decimal places.

251.86

234.86

234.36

25.04

Explanation: First, calculate `15.6² = 243.36`. Next, find the square root of `72.25`, which is `√72.25 = 8.5`. Finally, subtract the second result from the first: `243.36 - 8.5 = 234.86`. This value is already exact to two decimal places.

Q2982: Evaluate the expression `Y = (P² + Q²) / Q` when `P = 5.3` and `Q = 2.1` using a scientific calculator. Round your answer to two decimal places.

17.48

15.48

13.48

19.67

Explanation: Substitute the values: `Y = (5.3² + 2.1²) / 2.1`. Calculate the squares: `5.3² = 28.09` and `2.1² = 4.41`. Add them: `28.09 + 4.41 = 32.5`. Finally, divide by `Q`: `32.5 / 2.1 ≈ 15.47619`. Rounded to two decimal places, the answer is `15.48`.

Q2983: The height `h` of a cone can be found using the formula `h = √(l² - r²) `. If `l = 10` and `r = 6`, calculate `h` and then find the volume `V` of the cone using `V = (1/3)πr²h`. Use the π key and round your final answer to two decimal places.

226.19

376.99

301.59

100.53

Explanation: First, calculate `h`: `h = √(10² - 6²) = √(100 - 36) = √64 = 8`. Now, substitute `h = 8` and `r = 6` into the volume formula: `V = (1/3)π(6²)(8) = (1/3)π(36)(8)`. Simplify: `V = (1/3) * 36 * 8 * π = 12 * 8 * π = 96π`. Using the π key, `96π ≈ 301.59289`. Rounded to two decimal places, the answer is `301.59`.

Q2984: Using a scientific calculator, evaluate `Z = (X² + Y) / √(X + Y)` when `X = 3.2` and `Y = 6.8`. Round your answer to three decimal places.

1.704

5.388

5.325

3.162

Explanation: First, calculate the numerator: `X² + Y = 3.2² + 6.8 = 10.24 + 6.8 = 17.04`. Next, calculate the denominator: `√(X + Y) = √(3.2 + 6.8) = √10`. Finally, divide the numerator by the denominator: `17.04 / √10 ≈ 17.04 / 3.162277 ≈ 5.38817`. Rounded to three decimal places, the answer is `5.388`.

Q2985: Find the value of `M = √(P² + Q² + R²) ` when `P = 2.5`, `Q = 3.1`, and `R = 4.0` using a scientific calculator. Round your answer to two decimal places.

31.86

5.64

9.60

5.02

Explanation: Substitute the values into the formula: `M = √(2.5² + 3.1² + 4.0²) `. Calculate the squares: `2.5² = 6.25`, `3.1² = 9.61`, and `4.0² = 16`. Add them: `6.25 + 9.61 + 16 = 31.86`. Finally, find the square root: `√31.86 ≈ 5.64446`. Rounded to two decimal places, the answer is `5.64`.

Checks the accuracy of answers using the scientific calculator.

Q2986: A student calculated 35 x 18 manually and got 630. Which of the following statements correctly verifies this answer using a scientific calculator?

Incorrect, the calculator shows 620.

Correct, the calculator shows 630.

Incorrect, the calculator shows 630.

Correct, the calculator shows 63.

Explanation: The manual calculation 35 x 18 = 630 is correct. When this operation is entered into a scientific calculator, it also displays 630. Therefore, the statement 'Correct, the calculator shows 630' is the accurate verification.

Q2987: A student calculated 252 / 7 manually and got 34. Which of the following statements correctly verifies this answer using a scientific calculator?

Correct, the calculator shows 34.

Incorrect, the calculator shows 36.

Correct, the calculator shows 36.

Incorrect, the calculator shows 35.

Explanation: The actual calculation 252 / 7 = 36. The student's manual answer of 34 is incorrect. When this operation is entered into a scientific calculator, it displays 36. Therefore, the statement 'Incorrect, the calculator shows 36' is the accurate verification.

Q2988: A student calculated 145 + 287 manually and got 432. Which of the following statements correctly verifies this answer using a scientific calculator?

Incorrect, the calculator shows 432.

Correct, the calculator shows 432.

Correct, the calculator shows 422.

Incorrect, the calculator shows 422.

Explanation: The manual calculation 145 + 287 = 432 is correct. When this operation is entered into a scientific calculator, it also displays 432. Therefore, the statement 'Correct, the calculator shows 432' is the accurate verification.

Q2989: A student calculated 503 - 178 manually and got 335. Which of the following statements correctly verifies this answer using a scientific calculator?

Correct, the calculator shows 335.

Incorrect, the calculator shows 325.

Incorrect, the calculator shows 335.

Correct, the calculator shows 325.

Explanation: The actual calculation 503 - 178 = 325. The student's manual answer of 335 is incorrect. When this operation is entered into a scientific calculator, it displays 325. Therefore, the statement 'Incorrect, the calculator shows 325' is the accurate verification.

Q2990: A student calculated 64 x 23 manually and got 1452. Which of the following statements correctly verifies this answer using a scientific calculator?

Incorrect, the calculator shows 1472.

Correct, the calculator shows 1452.

Incorrect, the calculator shows 1452.

Correct, the calculator shows 1462.

Explanation: The actual calculation 64 x 23 = 1472. The student's manual answer of 1452 is incorrect. When this operation is entered into a scientific calculator, it displays 1472. Therefore, the statement 'Incorrect, the calculator shows 1472' is the accurate verification.

Q2991: A student calculated 468 / 9 manually and got 52. Which of the following statements correctly verifies this answer using a scientific calculator?

Correct, the calculator shows 52.

Incorrect, the calculator shows 51.

Correct, the calculator shows 51.

Incorrect, the calculator shows 52.

Explanation: The manual calculation 468 / 9 = 52 is correct. When this operation is entered into a scientific calculator, it also displays 52. Therefore, the statement 'Correct, the calculator shows 52' is the accurate verification.

Q2992: You are asked to calculate the value of 10 + 5 * 2. You perform a manual calculation and arrive at 30. When you verify your answer using a scientific calculator, you get 20. What is the correct answer, and what is the most likely common error that led to your manual answer?

30; You correctly applied the order of operations.

20; You forgot to perform multiplication before addition.

15; You added 10 and 5, then multiplied by 2 incorrectly.

25; You added all numbers without considering multiplication.

Explanation: According to the order of operations (BODMAS/PEMDAS), multiplication must be performed before addition. So, 5 * 2 = 10, then 10 + 10 = 20. A common error is performing operations from left to right without respecting the order of operations, leading to (10 + 5) * 2 = 30.

Q2993: You are asked to calculate the value of 24 - 12 / 3. You perform a manual calculation and arrive at 4. When you verify your answer using a scientific calculator, you get 20. What is the correct answer, and what is the most likely common error that led to your manual answer?

4; You correctly applied the order of operations.

20; You forgot to perform division before subtraction.

16; You subtracted 24 from 12 and then divided.

36; You added 24 and 12, then divided by 3.

Explanation: According to the order of operations (BODMAS/PEMDAS), division must be performed before subtraction. So, 12 / 3 = 4, then 24 - 4 = 20. A common error is performing operations from left to right without respecting the order of operations, leading to (24 - 12) / 3 = 12 / 3 = 4.

Q2994: You are asked to calculate the value of (7 + 3) * 4. You perform a manual calculation and arrive at 19. When you verify your answer using a scientific calculator, you get 40. What is the correct answer, and what is the most likely common error that led to your manual answer?

19; You correctly applied the order of operations.

40; You ignored the parentheses and performed multiplication first.

28; You multiplied 7 by 4 and then added 3.

14; You added 7 and 3, then divided by 4.

Explanation: According to the order of operations (BODMAS/PEMDAS), operations inside parentheses must be performed first. So, (7 + 3) = 10, then 10 * 4 = 40. A common error is ignoring the parentheses and applying multiplication before addition, leading to 7 + (3 * 4) = 7 + 12 = 19.

Q2995: You are asked to calculate the value of (30 - 6) / 3. You perform a manual calculation and arrive at 28. When you verify your answer using a scientific calculator, you get 8. What is the correct answer, and what is the most likely common error that led to your manual answer?

28; You correctly applied the order of operations.

8; You ignored the parentheses and performed division first.

18; You subtracted 30 from 6, then divided by 3.

10; You added 30 and 6, then divided by 3.

Explanation: According to the order of operations (BODMAS/PEMDAS), operations inside parentheses must be performed first. So, (30 - 6) = 24, then 24 / 3 = 8. A common error is ignoring the parentheses and applying division before subtraction, leading to 30 - (6 / 3) = 30 - 2 = 28.

Q2996: You are asked to calculate the value of 15 - 3 * 2 + 7. You perform a manual calculation and arrive at 31. When you verify your answer using a scientific calculator, you get 16. What is the correct answer, and what is the most likely common error that led to your manual answer?

31; You correctly applied the order of operations.

16; You performed subtraction before multiplication.

10; You added 2 and 7 first, then multiplied by 3, and subtracted from 15.

6; You only performed the multiplication and ignored other operations.

Explanation: According to the order of operations (BODMAS/PEMDAS), multiplication must be performed first. So, 3 * 2 = 6. Then, perform subtraction and addition from left to right: 15 - 6 = 9, and 9 + 7 = 16. A common error is performing operations strictly from left to right without prioritizing multiplication, leading to (15 - 3) * 2 + 7 = 12 * 2 + 7 = 24 + 7 = 31.

Q2997: You are asked to calculate the value of 30 / 5 + 2 * 3. You perform a manual calculation and arrive at 24. When you verify your answer using a scientific calculator, you get 12. What is the correct answer, and what is the most likely common error that led to your manual answer?

24; You correctly applied the order of operations.

12; You performed addition before multiplication or division.

3; You performed division and multiplication, then subtracted the results.

18; You added 5 and 2 first, then performed other operations.

Explanation: According to the order of operations (BODMAS/PEMDAS), division and multiplication must be performed before addition. So, 30 / 5 = 6 and 2 * 3 = 6. Then, 6 + 6 = 12. A common error is performing division and addition from left to right, then multiplying the result, leading to (30 / 5 + 2) * 3 = (6 + 2) * 3 = 8 * 3 = 24.

Q2998: You are asked to calculate the value of 8 + 6 / 2 - 1. You perform a manual calculation and arrive at 6. When you verify your answer using a scientific calculator, you get 10. What is the correct answer, and what is the most likely common error that led to your manual answer?

6; You correctly applied the order of operations.

10; You performed addition before division.

7; You performed division, then added 8 and subtracted 1.

13; You added 8 and 6, then added 2 and 1.

Explanation: According to the order of operations (BODMAS/PEMDAS), division must be performed first. So, 6 / 2 = 3. Then, perform addition and subtraction from left to right: 8 + 3 = 11, and 11 - 1 = 10. A common error is performing operations strictly from left to right, leading to (8 + 6) / 2 - 1 = 14 / 2 - 1 = 7 - 1 = 6.

Q2999: A student calculates the area of a circle with a radius of 7 cm using π ≈ 22/7 and obtains 154 cm². When the same calculation is performed using a scientific calculator, the result is approximately 153.938 cm² (to 3 decimal places). What is the most accurate explanation for the difference between the two answers?

The student made a calculation error in the manual method.

The scientific calculator is faulty and provides an incorrect answer.

The difference is due to the approximation of π (22/7) used in the manual calculation, which is less precise than the calculator's internal value of π.

The student used the wrong formula for the area of a circle.

Explanation: The manual calculation used an approximation of π (22/7, which is 3.142857...), while the calculator uses a more precise value of π (approximately 3.14159265...). This slight difference in the value of π leads to the observed difference in the final area.

Q3000: A student calculates the volume of a cuboid with dimensions 4.5 cm, 6 cm, and 8.2 cm manually and gets 221.4 cm³. Using a scientific calculator, the volume is also found to be exactly 221.4 cm³. What can be concluded from this comparison?

The student made a minor rounding error in the manual calculation.

The student's manual calculation was perfectly accurate.

The calculator must have made a mistake in its computation.

The student used an incorrect formula for the volume of a cuboid.

Explanation: Since both the manual calculation and the calculator's calculation yielded the exact same result (221.4 cm³), it confirms that the student's manual calculation was accurate, with no errors or approximations leading to a difference. (4.5 * 6 * 8.2 = 221.4).

Q3001: A student calculates the simple interest on Rs. 5000 at an annual rate of 8% for 3 years manually and gets Rs. 1200. Using a scientific calculator, the simple interest is also found to be Rs. 1200. What does this comparison indicate?

The student made a conceptual error in understanding simple interest.

The calculator has a bug and shows the same incorrect answer.

The student's manual calculation of simple interest is correct.

The manual calculation involved an approximation that luckily matched.

Explanation: Simple interest (I) is calculated using the formula I = (P * R * T) / 100. For P=5000, R=8, T=3, I = (5000 * 8 * 3) / 100 = 120000 / 100 = Rs. 1200. Since both manual and calculator results match, the manual calculation is correct.

Q3002: A student calculates (2.5)³ manually and obtains 15.625. Using a scientific calculator, the value of (2.5)³ is also found to be exactly 15.625. What conclusion can be drawn regarding the student's manual calculation?

The student made a significant error, but the calculator coincidentally produced the same value.

The student's manual calculation of the power was precise and correct.

The calculator rounded the answer to match the student's incorrect manual calculation.

The problem involved irrational numbers, causing an inherent difference.

Explanation: (2.5)³ = 2.5 * 2.5 * 2.5. This calculation results in exactly 15.625. Since the manual result matches the calculator's precise result, the manual calculation was perfectly correct and precise.

Q3003: A student manually calculates √7 and rounds the answer to two decimal places, getting 2.64. When checking with a scientific calculator, the value of √7 is approximately 2.64575... What is the primary reason for the difference between the student's answer and the calculator's answer?

The student made a fundamental calculation error in determining the square root.

The calculator is set to a different base system, causing the discrepancy.

The student's answer is a rounded approximation of the irrational number √7, while the calculator provides a more precise value.

The scientific calculator itself performs an incorrect square root operation.

Explanation: √7 is an irrational number, meaning its decimal representation is non-terminating and non-repeating. The student's manual calculation provided an approximation (2.64) by rounding or truncating. The scientific calculator, however, displays a more precise value with many more decimal places, leading to the observed difference.

Q3004: A student manually calculates the sum of fractions (3/4) + (1/5) and obtains 0.95. Using a scientific calculator, the sum is also found to be exactly 0.95. What does this comparison reveal about the student's manual calculation?

The student made a rounding error that coincidentally resulted in the correct final answer.

The student had a conceptual misunderstanding of fraction addition.

The student's manual calculation for adding the fractions was entirely correct.

The calculator automatically corrected a mistake made by the student.

Explanation: To add fractions, a common denominator is found: 3/4 = 15/20 and 1/5 = 4/20. So, (15/20) + (4/20) = 19/20. Converting to decimal, 19/20 = 0.95. Since the manual calculation matches the calculator's exact result, the student's calculation was correct.

Q3005: A student calculates the volume of a sphere with a radius of 3 cm. They use the formula V = (4/3)πr³ and approximate π as 3.14. Their manual answer is 113.04 cm³. When they use a scientific calculator with its internal π value, the result is approximately 113.097 cm³ (to 3 decimal places). What is the main reason for the slight difference?

The student applied the wrong formula for the volume of a sphere.

The scientific calculator itself made a computational error.

The difference is due to the approximation of π (3.14) in the manual calculation compared to the calculator's more precise value.

The student made a significant multiplication error in their manual calculation.

Explanation: The difference arises because the student used an approximated value of π (3.14) in the manual calculation, which is less precise than the more accurate internal value of π used by the scientific calculator. This approximation leads to a slight deviation in the final volume.

Identifies the laws of indices for multiplying and dividing powers.

Q3006: Simplify $3^4 \times 3^2$.

$3^8$

$3^6$

$9^6$

$3^2$

Explanation: When multiplying powers with the same base, we add the exponents. So, $3^4 \times 3^2 = 3^{(4+2)} = 3^6$. This is based on the law $a^m \times a^n = a^{m+n}$.

Q3007: Simplify $5^7 \div 5^3$.

$5^{10}$

$5^4$

$1^4$

$5^{21}$

Explanation: When dividing powers with the same base, we subtract the exponents. So, $5^7 \div 5^3 = 5^{(7-3)} = 5^4$. This is based on the law $a^m \div a^n = a^{m-n}$.

Q3008: Simplify $x^5 \times x^3$.

$x^{15}$

$x^8$

$2x^8$

$x^2$

Explanation: When multiplying powers with the same base, we add the exponents. So, $x^5 \times x^3 = x^{(5+3)} = x^8$. This is based on the law $a^m \times a^n = a^{m+n}$.

Q3009: Simplify $y^9 \div y^2$.

$y^{11}$

$y^7$

$y^{18}$

$1^7$

Explanation: When dividing powers with the same base, we subtract the exponents. So, $y^9 \div y^2 = y^{(9-2)} = y^7$. This is based on the law $a^m \div a^n = a^{m-n}$.

Q3010: What is the simplified form of $2^6 \times 2^3$?

$2^{18}$

$2^9$

$4^9$

$2^3$

Explanation: According to the multiplication law of indices, $a^m \times a^n = a^{m+n}$. Therefore, $2^6 \times 2^3 = 2^{(6+3)} = 2^9$.

Q3011: Simplify the expression $7^{10} \div 7^4$.

$7^{14}$

$7^6$

$7^{40}$

$1^6$

Explanation: According to the division law of indices, $a^m \div a^n = a^{m-n}$. Therefore, $7^{10} \div 7^4 = 7^{(10-4)} = 7^6$.

Q3012: Simplify: (x^5 * x^3) / x^2

x^8

x^6

x^4

x^7

Explanation: According to the laws of indices, when multiplying powers with the same base, we add the exponents (x^m * x^n = x^(m+n)). When dividing powers with the same base, we subtract the exponents (x^m / x^n = x^(m-n)). So, (x^5 * x^3) / x^2 = x^(5+3) / x^2 = x^8 / x^2 = x^(8-2) = x^6.

Q3013: Simplify: (2a^4 * 3b^2 * a^3) / (6b * a^5)

a^2b^2

a^2b

a^7b

1/b

Explanation: First, multiply the coefficients: 2 * 3 = 6. Combine like bases in the numerator: a^4 * a^3 = a^(4+3) = a^7. So the numerator is 6a^7b^2. The expression becomes (6a^7b^2) / (6a^5b). Now, divide the coefficients: 6 / 6 = 1. For base a: a^7 / a^5 = a^(7-5) = a^2. For base b: b^2 / b^1 = b^(2-1) = b^1 = b. Thus, the simplified expression is 1 * a^2 * b = a^2b.

Q3014: Simplify: (4p^2q^3 * 2p^4) / (8pq^2)

p^5q^5

p^5q

p^6q

p^3q

Explanation: First, simplify the numerator: Multiply coefficients: 4 * 2 = 8. Combine p terms: p^2 * p^4 = p^(2+4) = p^6. The numerator becomes 8p^6q^3. The expression is now (8p^6q^3) / (8pq^2). Now, divide the coefficients: 8 / 8 = 1. For base p: p^6 / p^1 = p^(6-1) = p^5. For base q: q^3 / q^2 = q^(3-2) = q^1 = q. Thus, the simplified expression is 1 * p^5 * q = p^5q.

Q3015: Simplify: (10m^6n^3) / (2m^2n) * (m^3n^2)

5m^7n^5

5m^7n^4

5m^11n^6

5m^6n^4

Explanation: This expression involves division followed by multiplication. First, perform the division: (10m^6n^3) / (2m^2n). Divide coefficients: 10 / 2 = 5. For base m: m^6 / m^2 = m^(6-2) = m^4. For base n: n^3 / n^1 = n^(3-1) = n^2. So, the result of the division is 5m^4n^2. Now, multiply this result by m^3n^2: (5m^4n^2) * (m^3n^2). Multiply coefficients: 5 * 1 = 5. For base m: m^4 * m^3 = m^(4+3) = m^7. For base n: n^2 * n^2 = n^(2+2) = n^4. Thus, the simplified expression is 5m^7n^4.

Q3016: Simplify: (15y^7z^4) / (3y^3z^2 * 5yz)

y^4z^2

y^3z

y^3z^2

Explanation: First, simplify the denominator: 3y^3z^2 * 5yz. Multiply coefficients: 3 * 5 = 15. Combine y terms: y^3 * y^1 = y^(3+1) = y^4. Combine z terms: z^2 * z^1 = z^(2+1) = z^3. So the denominator becomes 15y^4z^3. The expression is now (15y^7z^4) / (15y^4z^3). Now, perform the division. Divide coefficients: 15 / 15 = 1. For base y: y^7 / y^4 = y^(7-4) = y^3. For base z: z^4 / z^3 = z^(4-3) = z^1 = z. Thus, the simplified expression is 1 * y^3 * z = y^3z.

Q3017: Simplify: (6x^4y^2 * 3x^2y^5) / (9x^3y^4)

2x^3y^7

2x^3y^3

x^3y^3

2x^6y^7

Explanation: First, simplify the numerator: 6x^4y^2 * 3x^2y^5. Multiply coefficients: 6 * 3 = 18. Combine x terms: x^4 * x^2 = x^(4+2) = x^6. Combine y terms: y^2 * y^5 = y^(2+5) = y^7. So the numerator becomes 18x^6y^7. The expression is now (18x^6y^7) / (9x^3y^4). Now, perform the division. Divide coefficients: 18 / 9 = 2. For base x: x^6 / x^3 = x^(6-3) = x^3. For base y: y^7 / y^4 = y^(7-4) = y^3. Thus, the simplified expression is 2x^3y^3.

Q3018: If 3^x * 3^4 = 3^9, what is the value of x?

2.25

Explanation: According to the law of indices for multiplication, when bases are the same, exponents are added: a^m * a^n = a^(m+n). So, 3^(x+4) = 3^9. Equating the exponents, x+4 = 9, which means x = 9 - 4 = 5.

Q3019: If 5^7 / 5^x = 5^3, what is the value of x?

-4

2.33

Explanation: According to the law of indices for division, when bases are the same, exponents are subtracted: a^m / a^n = a^(m-n). So, 5^(7-x) = 5^3. Equating the exponents, 7-x = 3, which means x = 7 - 3 = 4.

Q3020: If 2^5 * 2^x / 2^3 = 2^10, what is the value of x?

Explanation: Apply the laws of indices: for multiplication, add exponents (5+x); for division, subtract exponents (-3). So, 2^(5+x-3) = 2^10. This simplifies to 2^(2+x) = 2^10. Equating the exponents, 2+x = 10, which gives x = 10 - 2 = 8.

Q3021: If 4^x * 2^3 = 2^11, what is the value of x?

Explanation: First, express 4 with base 2: 4 = 2^2. So, 4^x becomes (2^2)^x = 2^(2x). The equation becomes 2^(2x) * 2^3 = 2^11. Applying the multiplication law, 2^(2x+3) = 2^11. Equating the exponents, 2x+3 = 11. Subtracting 3 from both sides gives 2x = 8. Dividing by 2, x = 4.

Q3022: If 9^5 / 3^x = 3^6, what is the value of x?

-4

Explanation: First, express 9 with base 3: 9 = 3^2. So, 9^5 becomes (3^2)^5 = 3^(2*5) = 3^10. The equation becomes 3^10 / 3^x = 3^6. Applying the division law, 3^(10-x) = 3^6. Equating the exponents, 10-x = 6. Subtracting 10 from both sides gives -x = 6 - 10, so -x = -4. Therefore, x = 4.

Q3023: If (27^2 * 3^y) / 9^3 = 3^8, what is the value of y?

-4

Explanation: Convert all bases to 3: 27 = 3^3, so 27^2 = (3^3)^2 = 3^6. And 9 = 3^2, so 9^3 = (3^2)^3 = 3^6. Substitute these into the equation: (3^6 * 3^y) / 3^6 = 3^8. Apply the multiplication and division laws of indices: 3^(6+y-6) = 3^8. This simplifies to 3^y = 3^8. Equating the exponents, y = 8.

Identifies the law of indices for finding the power of a power.

Q3024: Simplify $(2^3)^4$.

$2^7$

$2^{12}$

$2^{34}$

$2^{64}$

Explanation: According to the law of indices for the power of a power, $(a^m)^n = a^{m \times n}$. Therefore, $(2^3)^4 = 2^{3 \times 4} = 2^{12}$.

Q3025: What is the simplified form of $(5^2)^3$?

$5^5$

$5^6$

$5^{23}$

$5^8$

Explanation: Applying the law of indices for the power of a power, $(a^m)^n = a^{m \times n}$. So, $(5^2)^3 = 5^{2 \times 3} = 5^6$.

Q3026: Simplify $(10^5)^2$.

$10^7$

$10^{10}$

$10^{25}$

$10^{52}$

Explanation: Using the power of a power rule, $(a^m)^n = a^{m \times n}$. Thus, $(10^5)^2 = 10^{5 \times 2} = 10^{10}$.

Q3027: Which of the following is the simplified form of $(3^6)^2$?

$3^8$

$3^{12}$

$3^{36}$

$3^{62}$

Explanation: The law of indices for power of a power states $(a^m)^n = a^{m \times n}$. So, $(3^6)^2 = 3^{6 \times 2} = 3^{12}$.

Q3028: Which of the following is equal to $(7^4)^3$?

$7^7$

$7^{43}$

$7^{12}$

$7^{64}$

Explanation: To simplify a power of a power, we multiply the exponents. So, $(7^4)^3 = 7^{4 \times 3} = 7^{12}$.

Q3029: Simplify $(4^2)^5$.

$4^7$

$4^{25}$

$4^{10}$

$4^{32}$

Explanation: The law for the power of a power is $(a^m)^n = a^{m \times n}$. Applying this, $(4^2)^5 = 4^{2 \times 5} = 4^{10}$.

Q3030: Simplify $(x^3)^4 \times x^2$.

$x^9$

$x^{14}$

$x^{24}$

$x^{11}$

Explanation: To simplify $(x^3)^4 \times x^2$, first apply the power of a power rule: $(x^3)^4 = x^{3 \times 4} = x^{12}$. Then, apply the multiplication rule for indices with the same base: $x^{12} \times x^2 = x^{12+2} = x^{14}$.

Q3031: Simplify $(y^5)^2 \div y^3$.

$y^4$

$y^{13}$

$y^7$

$y^{10/3}$

Explanation: To simplify $(y^5)^2 \div y^3$, first apply the power of a power rule: $(y^5)^2 = y^{5 \times 2} = y^{10}$. Then, apply the division rule for indices with the same base: $y^{10} \div y^3 = y^{10-3} = y^7$.

Q3032: Simplify $(2a^3)^2 \times a^4$.

$2a^9$

$4a^{10}$

$4a^{24}$

$2a^{10}$

Explanation: To simplify $(2a^3)^2 \times a^4$, first apply the power of a power rule to both the coefficient and the variable: $(2a^3)^2 = 2^2 \times (a^3)^2 = 4a^{3 \times 2} = 4a^6$. Then, apply the multiplication rule for indices with the same base: $4a^6 \times a^4 = 4a^{6+4} = 4a^{10}$.

Q3033: Simplify $(m^6)^3 / m^{10}$.

$m^{-1}$

$m^{28}$

$m^8$

$m^{18}$

Explanation: To simplify $(m^6)^3 / m^{10}$, first apply the power of a power rule: $(m^6)^3 = m^{6 \times 3} = m^{18}$. Then, apply the division rule for indices with the same base: $m^{18} / m^{10} = m^{18-10} = m^8$.

Q3034: Simplify $(p^2)^3 \times (p^4)^2$.

$p^{11}$

$p^{48}$

$p^{14}$

$p^{24}$

Explanation: To simplify $(p^2)^3 \times (p^4)^2$, first apply the power of a power rule to each term: $(p^2)^3 = p^{2 \times 3} = p^6$ and $(p^4)^2 = p^{4 \times 2} = p^8$. Then, apply the multiplication rule for indices with the same base: $p^6 \times p^8 = p^{6+8} = p^{14}$.

Q3035: Simplify $\frac{(n^4)^3 \times n^5}{n^2}$.

$n^{10}$

$n^{58}$

$n^{19}$

$n^{15}$

Explanation: To simplify $\frac{(n^4)^3 \times n^5}{n^2}$, first apply the power of a power rule: $(n^4)^3 = n^{4 \times 3} = n^{12}$. Next, simplify the numerator using the multiplication rule: $n^{12} \times n^5 = n^{12+5} = n^{17}$. Finally, apply the division rule: $\frac{n^{17}}{n^2} = n^{17-2} = n^{15}$.

Q3036: If $(2^{3x})^4 = 2^{36}$, find the value of $x$.

32/3

Explanation: According to the law of indices $(a^m)^n = a^{mn}$, we can write $(2^{3x})^4$ as $2^{3x \times 4} = 2^{12x}$. So, $2^{12x} = 2^{36}$. Since the bases are equal, the exponents must be equal. $12x = 36$ $x = \frac{36}{12}$ $x = 3$.

Q3037: If $(5^{m-1})^3 = 5^{15}$, find the value of $m$.

Explanation: Using the law of indices $(a^m)^n = a^{mn}$, we have: $5^{(m-1) \times 3} = 5^{15}$ $5^{3m-3} = 5^{15}$ Equating the exponents: $3m - 3 = 15$ $3m = 15 + 3$ $3m = 18$ $m = \frac{18}{3}$ $m = 6$.

Q3038: If $(y^2)^k = y^{14}$, find the value of $k$.

Explanation: Applying the law of indices $(a^m)^n = a^{mn}$: $y^{2 \times k} = y^{14}$ $y^{2k} = y^{14}$ Since the bases are the same, equate the exponents: $2k = 14$ $k = \frac{14}{2}$ $k = 7$.

Q3039: If $(3^{2p+1})^2 = 3^{10}$, find the value of $p$.

3.5

Explanation: Using the law of indices $(a^m)^n = a^{mn}$: $3^{(2p+1) \times 2} = 3^{10}$ $3^{4p+2} = 3^{10}$ Equating the exponents: $4p + 2 = 10$ $4p = 10 - 2$ $4p = 8$ $p = \frac{8}{4}$ $p = 2$.

Q3040: If $(10^{4a})^3 = 10^{60}$, find the value of $a$.

57/4

Explanation: According to the law of indices $(a^m)^n = a^{mn}$: $10^{4a \times 3} = 10^{60}$ $10^{12a} = 10^{60}$ Since the bases are equal, the exponents must be equal: $12a = 60$ $a = \frac{60}{12}$ $a = 5$.

Q3041: If $(7^x)^5 = 7^{30}$, find the value of $x$.

150

Explanation: Using the law of indices $(a^m)^n = a^{mn}$: $7^{x \times 5} = 7^{30}$ $7^{5x} = 7^{30}$ Equating the exponents: $5x = 30$ $x = \frac{30}{5}$ $x = 6$.

Recognizes and uses the zero index (a⁰ = 1) and negative indices (a⁻ⁿ = 1/aⁿ).

Q3042: Evaluate 7⁰.

-7

Explanation: According to the rule a⁰ = 1, any non-zero number raised to the power of zero is 1. Therefore, 7⁰ = 1.

Q3043: Evaluate 4⁻¹.

-4

1/4

1/16

Explanation: According to the rule a⁻ⁿ = 1/aⁿ, a number raised to a negative power is the reciprocal of the number raised to the positive power. Therefore, 4⁻¹ = 1/4¹ = 1/4.

Q3044: Evaluate (1/3)⁻².

-9

1/9

Explanation: According to the rule (a/b)⁻ⁿ = (b/a)ⁿ, a fraction raised to a negative power is the reciprocal of the fraction raised to the positive power. Therefore, (1/3)⁻² = (3/1)² = 3² = 9.

Q3045: Evaluate 5⁰ + 2⁻¹.

1/2

3/2

Explanation: First, evaluate each term: 5⁰ = 1 (any non-zero number to the power of zero is 1) and 2⁻¹ = 1/2 (a⁻ⁿ = 1/aⁿ). Then, add the results: 1 + 1/2 = 3/2.

Q3046: Evaluate (-5)⁰.

-1

-5

Explanation: According to the rule a⁰ = 1, any non-zero number, whether positive or negative, raised to the power of zero is 1. Therefore, (-5)⁰ = 1.

Q3047: Evaluate (2/5)⁻².

-25/4

4/25

25/4

10/4

Explanation: According to the rule (a/b)⁻ⁿ = (b/a)ⁿ, a fraction raised to a negative power is the reciprocal of the fraction raised to the positive power. Therefore, (2/5)⁻² = (5/2)² = 5²/2² = 25/4.

Q3048: Simplify x³ * x⁻² * y⁰

x⁵

1/x

x¹y⁰

Explanation: To simplify x³ * x⁻² * y⁰, we apply the index law a^m * a^n = a^(m+n) for the terms with base x: x^(3 + (-2)) = x¹. For y⁰, we use the zero index law a⁰ = 1. So, x¹ * 1 = x.

Q3049: Simplify (a⁵b⁻²) / (a³b⁻¹)

a²b

a²/b

a⁸b⁻³

a²b⁻³

Explanation: To simplify (a⁵b⁻²) / (a³b⁻¹), we apply the index law a^m / a^n = a^(m-n) for each base. For 'a': a^(5-3) = a². For 'b': b^(-2 - (-1)) = b^(-2+1) = b⁻¹. Then, using a⁻ⁿ = 1/aⁿ, b⁻¹ becomes 1/b. So, a² * (1/b) = a²/b.

Q3050: Simplify (2x⁻³y⁰)²

4x⁶

4/x⁶

4x⁻⁶y

2x⁶

Explanation: To simplify (2x⁻³y⁰)², we apply the index law (abc)^n = a^n b^n c^n. This gives 2² * (x⁻³)² * (y⁰)². First, 2² = 4. For x, apply (a^m)^n = a^(mn): (x⁻³)² = x^(-3*2) = x⁻⁶. For y, apply a⁰ = 1, so y⁰ = 1, and (y⁰)² = 1² = 1. Combining these, we get 4 * x⁻⁶ * 1 = 4x⁻⁶. Finally, using a⁻ⁿ = 1/aⁿ, we get 4/x⁶.

Q3051: Simplify (m⁴n⁰) / (m⁻²n⁻³)

m²n⁻³

m⁶n⁻³

m⁶n³

m²n³

Explanation: To simplify (m⁴n⁰) / (m⁻²n⁻³), we apply the index law a^m / a^n = a^(m-n) for each base. For 'm': m^(4 - (-2)) = m^(4+2) = m⁶. For 'n': n^(0 - (-3)) = n^(0+3) = n³. So, the simplified expression is m⁶n³. Note that n⁰ = 1 is implicitly handled by n^(0-(-3)).

Q3052: Simplify 3⁻¹ * (x⁻²y)⁻²

3x⁴y⁻²

1/(3x⁴y²)

x⁴ / (3y²)

x⁴y² / 3

Explanation: First, simplify 3⁻¹ to 1/3 using a⁻ⁿ = 1/aⁿ. Next, simplify (x⁻²y)⁻² using (ab)^n = a^n b^n and (a^m)^n = a^(mn). This gives (x⁻²)⁻² * y⁻² = x⁴ * y⁻². Now multiply the simplified parts: (1/3) * x⁴ * y⁻² = (1/3) * x⁴ * (1/y²) = x⁴ / (3y²).

Q3053: Simplify (p⁻²q³) * (p⁰q⁻¹) / (pq⁻²)

p⁻¹q²

q⁴/p³

p⁻³q⁶

p⁻³q⁰

Explanation: First, simplify the numerator: (p⁻²q³) * (p⁰q⁻¹). Using a⁰=1, p⁰=1. So, p⁻²q³ * 1q⁻¹ = p⁻²q^(3-1) = p⁻²q². Now, divide this by the denominator (pq⁻²): (p⁻²q²) / (p¹q⁻²). Using a^m/a^n = a^(m-n), for 'p': p^(-2-1) = p⁻³. For 'q': q^(2 - (-2)) = q^(2+2) = q⁴. Combining these, we get p⁻³q⁴. Finally, using a⁻ⁿ = 1/aⁿ, p⁻³q⁴ = q⁴/p³.

Q3054: If 2ⁿ = 1/32, what is the value of n?

-5

1/5

-1/5

Explanation: To solve 2ⁿ = 1/32, first express 1/32 as a power of 2. We know that 32 = 2⁵. Therefore, 1/32 can be written as 1/2⁵. Using the rule of negative indices, 1/aⁿ = a⁻ⁿ, we can write 1/2⁵ as 2⁻⁵. So the equation becomes 2ⁿ = 2⁻⁵. Since the bases are equal, the exponents must be equal. Thus, n = -5.

Q3055: If p⁻² = 16, what are the possible values of p?

±4

1/4

±1/4

-4

Explanation: Given p⁻² = 16. Using the negative index rule, a⁻ⁿ = 1/aⁿ, we can rewrite p⁻² as 1/p². So, 1/p² = 16. To find p², invert both sides: p² = 1/16. To find p, take the square root of both sides. Remember that the square root of a positive number has both a positive and a negative solution. So, p = ±√(1/16) = ±1/4.

Q3056: Evaluate (3⁻¹ + 2⁻¹)⁰.

5/6

1/6

Explanation: According to the zero index rule, any non-zero number raised to the power of 0 is equal to 1. In this expression, (3⁻¹ + 2⁻¹) is (1/3 + 1/2) which is 5/6. Since 5/6 is a non-zero number, (5/6)⁰ = 1. Therefore, (3⁻¹ + 2⁻¹)⁰ = 1.

Q3057: Simplify (x⁻³y²) / (x²y⁻¹).

x⁻¹y

y³/x⁵

x⁵y⁻³

x⁻⁵y

Explanation: When dividing powers with the same base, subtract the exponents. For x: x⁻³ / x² = x⁽⁻³⁻²⁾ = x⁻⁵. For y: y² / y⁻¹ = y⁽²⁻⁽⁻¹⁾⁾ = y⁽²⁺¹⁾ = y³. So the simplified expression is x⁻⁵y³. Using the negative index rule (a⁻ⁿ = 1/aⁿ), x⁻⁵ can be written as 1/x⁵. Therefore, x⁻⁵y³ = y³/x⁵.

Q3058: If 3ˣ = 1/81, find the value of x.

-4

1/4

-1/4

Explanation: To solve 3ˣ = 1/81, first express 1/81 as a power of 3. We know that 81 = 3⁴. Therefore, 1/81 can be written as 1/3⁴. Using the rule of negative indices, 1/aⁿ = a⁻ⁿ, we can write 1/3⁴ as 3⁻⁴. So the equation becomes 3ˣ = 3⁻⁴. Since the bases are equal, the exponents must be equal. Thus, x = -4.

Q3059: If (1/5)⁻³ = 5ⁿ, what is the value of n?

-3

1/3

-1/3

Explanation: Given the equation (1/5)⁻³ = 5ⁿ. Using the negative index rule for fractions, (1/a)⁻ⁿ = aⁿ, we can rewrite (1/5)⁻³ as 5³. So the equation becomes 5³ = 5ⁿ. Since the bases are equal, the exponents must be equal. Therefore, n = 3.

Applies the laws of indices to simplify expressions.

Q3060: Simplify: x^3 * x^5

x^15

x^8

x^2

2x^8

Explanation: When multiplying powers with the same base, you add the exponents. So, x^(3+5) = x^8.

Q3061: Simplify: a^2 * a^7

a^14

a^9

a^5

2a^9

Explanation: According to the law of indices for multiplication, when bases are the same, exponents are added. So, a^(2+7) = a^9.

Q3062: Simplify: y^7 / y^3

y^10

y^4

y^21

y^(7/3)

Explanation: When dividing powers with the same base, you subtract the exponents. So, y^(7-3) = y^4.

Q3063: Simplify: b^9 / b^2

b^11

b^7

b^18

b^(9/2)

Explanation: According to the law of indices for division, when bases are the same, exponents are subtracted. So, b^(9-2) = b^7.

Q3064: Simplify: (m^4)^2

m^6

m^8

m^16

2m^4

Explanation: When raising a power to another power, you multiply the exponents. So, m^(4*2) = m^8.

Q3065: Simplify: (p^3)^5

p^8

p^15

p^243

5p^3

Explanation: According to the law of indices for power of a power, you multiply the exponents. So, p^(3*5) = p^15.

Q3066: Simplify (x^3)^2 * x^4.

x^9

x^10

x^24

x^12

Explanation: First, apply the power of a power law: (x^3)^2 = x^(3*2) = x^6. Then, apply the product law: x^6 * x^4 = x^(6+4) = x^10.

Q3067: Simplify (2a^2)^3 * a^5.

2a^11

8a^11

6a^11

8a^30

Explanation: First, apply the power of a product and power of a power laws: (2a^2)^3 = 2^3 * (a^2)^3 = 8 * a^(2*3) = 8a^6. Then, apply the product law: 8a^6 * a^5 = 8 * a^(6+5) = 8a^11.

Q3068: Simplify (y^8 / y^2)^3.

y^6

y^18

y^30

y^48

Explanation: First, apply the quotient law inside the parenthesis: y^8 / y^2 = y^(8-2) = y^6. Then, apply the power of a power law: (y^6)^3 = y^(6*3) = y^18.

Q3069: Simplify (2x^3)^2 * (x^5 / x^2).

2x^9

4x^9

4x^11

4x^18

Explanation: First, simplify (2x^3)^2 using power of a product and power of a power laws: 2^2 * (x^3)^2 = 4x^6. Next, simplify (x^5 / x^2) using the quotient law: x^(5-2) = x^3. Finally, multiply the simplified terms using the product law: 4x^6 * x^3 = 4x^(6+3) = 4x^9.

Q3070: Simplify (4a^2b)^2 / (2a^3b).

2ab

8ab

8a^7b^3

8a^12b

Explanation: First, expand the numerator (4a^2b)^2 using the power of a product and power of a power laws: 4^2 * (a^2)^2 * b^2 = 16a^4b^2. Then, divide by the denominator 2a^3b. Apply the quotient law for coefficients and each variable: (16/2) * (a^4/a^3) * (b^2/b^1) = 8 * a^(4-3) * b^(2-1) = 8a^1b^1 = 8ab.

Q3071: Simplify (p^5q^2)^3 / (p^6q^4).

p^9q^2

p^21q^10

p^2q

p^9q^6

Explanation: First, expand the numerator (p^5q^2)^3 using the power of a product and power of a power laws: (p^5)^3 * (q^2)^3 = p^(5*3) * q^(2*3) = p^15q^6. Then, divide by the denominator p^6q^4. Apply the quotient law for each variable: (p^15/p^6) * (q^6/q^4) = p^(15-6) * q^(6-4) = p^9q^2.

Q3072: Simplify (x^4y^3)^2 * (x^2y) / x^5.

x^5y^7

x^15y^7

x^8y^6

x^10y^7

Explanation: First, expand (x^4y^3)^2 using the power of a product and power of a power laws: (x^4)^2 * (y^3)^2 = x^8y^6. Then, multiply by x^2y: x^8y^6 * x^2y = x^(8+2)y^(6+1) = x^10y^7. Finally, divide by x^5 using the quotient law: x^10y^7 / x^5 = x^(10-5)y^7 = x^5y^7.

Q3073: Simplify: (x^2 y^-1)^3 * (x^-1 y^2)^2

x^8 y^1

x^4 y

x^4 y^-1

x^4 y^7

Explanation: First, apply the 'power of a power' law ((a^m)^n = a^(mn)) to each term: (x^2 y^-1)^3 = x^(2*3) y^(-1*3) = x^6 y^-3 (x^-1 y^2)^2 = x^(-1*2) y^(2*2) = x^-2 y^4 Now, multiply these simplified terms using the 'product' law (a^m * a^n = a^(m+n)): x^6 y^-3 * x^-2 y^4 = x^(6 + (-2)) y^(-3 + 4) = x^4 y^1 = x^4 y

Q3074: Simplify: (12a^5 b^0) / (3a^-2 b^3)

4a^3 b^3

4a^7 / b^3

4a^7 b^3

(4a^3) / b^3

Explanation: First, evaluate the term with a zero exponent: b^0 = 1. The expression becomes: (12a^5 * 1) / (3a^-2 b^3) = (12a^5) / (3a^-2 b^3). Now, simplify the numerical coefficients: 12 / 3 = 4. Apply the 'quotient' law (a^m / a^n = a^(m-n)) for 'a' terms: a^5 / a^-2 = a^(5 - (-2)) = a^(5 + 2) = a^7. For 'b' terms, b^3 in the denominator is equivalent to b^-3 in the numerator: So, the simplified expression is 4a^7b^-3, which can also be written as (4a^7) / b^3.

Q3075: Simplify: ( (2x^3)^-2 * y^4 ) / ( 4x^-1 y^2 )

y^2 / (16x^5)

y^6 / (16x^7)

x^5 y^2 / 16

1 / (16x^5 y^2)

Explanation: First, simplify the numerator: (2x^3)^-2 * y^4 = 2^-2 * (x^3)^-2 * y^4 = (1/4) * x^-6 * y^4 Now, the expression is: ( (1/4) * x^-6 * y^4 ) / ( 4x^-1 y^2 ) Simplify numerical coefficients: (1/4) / 4 = 1/16. Apply the 'quotient' law for 'x' terms: x^-6 / x^-1 = x^(-6 - (-1)) = x^(-6 + 1) = x^-5. Apply the 'quotient' law for 'y' terms: y^4 / y^2 = y^(4-2) = y^2. Combine all terms: (1/16) * x^-5 * y^2 = y^2 / (16x^5).

Q3076: Simplify: ( (3p^2 q^-1)^-2 ) / ( (p^-1 q^3)^2 )

p^4 q^8 / 9

1 / (9p^2 q^4)

p^2 q^4 / 9

q^4 / (9p^2)

Explanation: First, simplify the numerator using the 'power of a product' and 'power of a power' laws: (3p^2 q^-1)^-2 = 3^-2 * (p^2)^-2 * (q^-1)^-2 = (1/9) * p^-4 * q^2 Next, simplify the denominator: (p^-1 q^3)^2 = (p^-1)^2 * (q^3)^2 = p^-2 * q^6 Now, divide the simplified numerator by the simplified denominator: ( (1/9) * p^-4 * q^2 ) / ( p^-2 * q^6 ) Apply the 'quotient' law for 'p' terms: p^-4 / p^-2 = p^(-4 - (-2)) = p^(-4 + 2) = p^-2. Apply the 'quotient' law for 'q' terms: q^2 / q^6 = q^(2 - 6) = q^-4. Combine all terms: (1/9) * p^-2 * q^-4 = 1 / (9p^2 q^4).

Q3077: Simplify: ( (5x^0 y^-3)^2 ) / ( 25x^2 y^-1 )

1 / (x^2 y^5)

x^2 y^5

y^5 / x^2

1 / (y^5)

Explanation: First, simplify the numerator: Recall that x^0 = 1. So, (5x^0 y^-3)^2 = (5 * 1 * y^-3)^2 = (5y^-3)^2. Apply the 'power of a product' law: (5y^-3)^2 = 5^2 * (y^-3)^2 = 25 * y^(-3*2) = 25y^-6. Now, the expression is: (25y^-6) / (25x^2 y^-1). Simplify numerical coefficients: 25 / 25 = 1. Apply the 'quotient' law for 'x' terms: 1 / x^2 = x^-2. Apply the 'quotient' law for 'y' terms: y^-6 / y^-1 = y^(-6 - (-1)) = y^(-6 + 1) = y^-5. Combine all terms: 1 * x^-2 * y^-5 = 1 / (x^2 y^5).

Q3078: Simplify: ( (a^3 b^-2) / (a^-1 b^3) )^-2

a^8 / b^10

b^10 / a^8

a^2 b^-5

1 / (a^8 b^10)

Explanation: First, simplify the expression inside the parenthesis using the 'quotient' law (a^m / a^n = a^(m-n)): For 'a' terms: a^3 / a^-1 = a^(3 - (-1)) = a^(3 + 1) = a^4. For 'b' terms: b^-2 / b^3 = b^(-2 - 3) = b^-5. So, the expression inside the parenthesis simplifies to a^4 b^-5. Now, apply the outer exponent of -2 using the 'power of a product' and 'power of a power' laws: (a^4 b^-5)^-2 = (a^4)^-2 * (b^-5)^-2 = a^(4 * -2) * b^(-5 * -2) = a^-8 * b^10. Finally, rewrite with positive exponents: b^10 / a^8.

Q3079: Simplify: ( (2xy)^-2 * (x^3 y)^2 ) / ( 8x^-3 y^0 )

x^7 / 32

x / 32

32x^7

x^7 / 2

Explanation: First, simplify the numerator: (2xy)^-2 = 2^-2 x^-2 y^-2 = (1/4)x^-2 y^-2 (x^3 y)^2 = (x^3)^2 y^2 = x^6 y^2 Multiplying these: (1/4)x^-2 y^-2 * x^6 y^2 = (1/4)x^(-2+6) y^(-2+2) = (1/4)x^4 y^0 = (1/4)x^4 * 1 = (1/4)x^4. Next, simplify the denominator: 8x^-3 y^0 = 8x^-3 * 1 = 8x^-3. Now, divide the simplified numerator by the simplified denominator: ( (1/4)x^4 ) / ( 8x^-3 ) Simplify numerical coefficients: (1/4) / 8 = 1/32. Apply the 'quotient' law for 'x' terms: x^4 / x^-3 = x^(4 - (-3)) = x^(4 + 3) = x^7. Combine all terms: (1/32)x^7 = x^7 / 32.

Q3080: Simplify: ( (m^-3 n^2)^-1 * (m^2 n^-1)^3 ) / ( m^-5 n^4 )

m^4 / n^10

m^14 / n^9

m^14 n^-1

n^9 / m^14

Explanation: First, simplify the terms in the numerator: (m^-3 n^2)^-1 = m^(-3 * -1) n^(2 * -1) = m^3 n^-2 (m^2 n^-1)^3 = m^(2 * 3) n^(-1 * 3) = m^6 n^-3 Multiply these simplified terms in the numerator: m^3 n^-2 * m^6 n^-3 = m^(3+6) n^(-2-3) = m^9 n^-5 Now, divide the simplified numerator by the denominator: (m^9 n^-5) / (m^-5 n^4) Apply the 'quotient' law for 'm' terms: m^9 / m^-5 = m^(9 - (-5)) = m^(9 + 5) = m^14. Apply the 'quotient' law for 'n' terms: n^-5 / n^4 = n^(-5 - 4) = n^-9. Combine all terms: m^14 n^-9 = m^14 / n^9.

Identifies and constructs the four basic loci.

Q3081: What construction represents the locus of points equidistant from two given points A and B?

A circle with center A.

The perpendicular bisector of the line segment AB.

The angle bisector of the angle formed by AB and any other line.

A line parallel to the line segment AB.

Explanation: The locus of points equidistant from two given points is the perpendicular bisector of the line segment joining those two points.

Q3082: What is the locus of a point that moves such that its distance from a fixed point O is always constant, say 'r'?

A straight line passing through O.

A circle with center O and radius r.

The perpendicular bisector of any line segment through O.

An angle bisector.

Explanation: The locus of points at a constant distance from a fixed point is a circle with the fixed point as its center and the constant distance as its radius.

Q3083: To find the locus of points equidistant from two intersecting lines P and Q, which construction method would you use using only a ruler and compass?

Drawing a perpendicular from the intersection point to one of the lines.

Constructing the angle bisector of the angle formed by the two lines.

Drawing a line parallel to one of the lines.

Constructing a circle centered at the intersection point.

Explanation: The locus of points equidistant from two intersecting lines is the angle bisector(s) of the angles formed by the lines.

Q3084: What is the locus of points that are at a constant distance 'd' from a given straight line L?

A single line parallel to L.

A circle with any point on L as its center.

A pair of lines parallel to L, one on each side, at distance 'd'.

The perpendicular bisector of a segment on L.

Explanation: The locus of points at a constant distance 'd' from a given straight line L consists of two lines parallel to L, one on each side, at that distance 'd'.

Q3085: A point P moves such that it is always 5 cm away from a fixed point O. What type of locus is this?

A straight line.

A circle.

A perpendicular bisector.

An angle bisector.

Explanation: A point moving at a constant distance from a fixed point traces a circle. In this case, the fixed point O is the center and 5 cm is the radius.

Q3086: A point P moves such that it is always equidistant from two parallel lines L1 and L2. What is the locus of P?

A circle between L1 and L2.

A line perpendicular to L1 and L2.

A line parallel to L1 and L2, exactly midway between them.

Two lines, one parallel to L1 and one parallel to L2.

Explanation: The locus of points equidistant from two parallel lines is a third parallel line that lies exactly midway between them.

Q3087: What is the locus of points equidistant from two sides of a triangle, for example, sides AB and AC of triangle ABC?

The perpendicular bisector of side BC.

The angle bisector of angle BAC.

The median from vertex A to side BC.

The altitude from vertex A to side BC.

Explanation: The locus of points equidistant from two intersecting lines (which are the sides of the triangle in this case) is the angle bisector of the angle formed by those two lines. Therefore, for sides AB and AC, it is the angle bisector of angle BAC.

Q3088: To construct the locus of points that are equidistant from sides PQ and PR of triangle PQR, which of the following constructions must be performed?

Bisect side QR.

Draw a perpendicular from P to QR.

Bisect angle QPR.

Bisect angle PQR.

Explanation: The locus of points equidistant from two sides of a triangle is the angle bisector of the angle formed by those two sides. Therefore, for sides PQ and PR, we need to bisect angle QPR.

Q3089: If a point X lies on the locus of points equidistant from sides DE and DF of triangle DEF, what must be true about point X?

X is equidistant from vertices D and E.

X lies on the angle bisector of angle EDF.

X is equidistant from vertices E and F.

X lies on the perpendicular bisector of side EF.

Explanation: The locus of points equidistant from two sides of a triangle is the angle bisector of the angle formed by those two sides. Thus, if X is on this locus, it must be on the angle bisector of angle EDF.

Q3090: In triangle XYZ, if you want to find a point that is exactly the same perpendicular distance from side XY as it is from side XZ, what construction would you perform?

Construct the perpendicular bisector of YZ.

Construct the angle bisector of angle YXZ.

Construct the median from X to YZ.

Construct the altitude from X to YZ.

Explanation: The set of all points equidistant from two intersecting lines (sides XY and XZ) forms the angle bisector of the angle between them. Therefore, to find such a point, you would construct the angle bisector of angle YXZ.

Q3091: Which of the following statements about the locus of points equidistant from two sides, say AB and AC, of a triangle ABC is INCORRECT?

It is the angle bisector of angle BAC.

Every point on this locus has an equal perpendicular distance to AB and AC.

It is the perpendicular bisector of side BC.

It passes through vertex A.

Explanation: The locus of points equidistant from two sides of a triangle is indeed the angle bisector of the angle between them, and it passes through the common vertex. The statement that it is the perpendicular bisector of BC is incorrect; that describes the locus of points equidistant from vertices B and C.

Q3092: In triangle ABC, if the locus of points equidistant from sides AB and AC is constructed, and also the locus of points equidistant from sides BC and BA is constructed, what is the name of the point where these two loci intersect?

The circumcenter

The centroid

The incenter

The orthocenter

Explanation: The locus of points equidistant from two sides of a triangle is an angle bisector. When two angle bisectors of a triangle intersect, their intersection point is the incenter, which is equidistant from all three sides of the triangle.

Q3093: Which of the following constructions must be done to find a point P that is equidistant from two given points A and B, and also equidistant from two intersecting lines L1 and L2?

Only the perpendicular bisector of AB.

Only the angle bisector of the angle between L1 and L2.

The perpendicular bisector of AB AND the angle bisector of the angle between L1 and L2.

A line parallel to AB AND a line perpendicular to L1.

Explanation: To find a point equidistant from two points A and B, we construct the perpendicular bisector of the line segment AB. To find a point equidistant from two intersecting lines L1 and L2, we construct the angle bisector of the angle formed by L1 and L2. For a point to satisfy both conditions, it must lie on the intersection of these two loci.

Q3094: What is the correct sequence of steps to construct a point P that is equidistant from points A and B, and also equidistant from intersecting lines L1 and L2?

Construct the perpendicular bisector of AB, then construct a line parallel to L1, and their intersection is P.

Construct the angle bisector of the angle between L1 and L2, then construct a circle centered at A passing through B, and their intersection is P.

Construct the perpendicular bisector of AB. Then, construct the angle bisector of the angle formed by L1 and L2. The point P is the intersection of these two loci.

Draw a line segment from A to L1, and another from B to L2, and their intersection is P.

Explanation: The locus of points equidistant from two points A and B is their perpendicular bisector. The locus of points equidistant from two intersecting lines L1 and L2 is the angle bisector of the angle formed by them. To satisfy both conditions, point P must be at the intersection of these two specific loci.

Q3095: If point P is constructed such that it is equidistant from two points X and Y, and also equidistant from two intersecting lines M and N, which of the following statements is true about P?

P is equidistant from X and M.

P is equidistant from X, Y, M, and N.

P is equidistant from X and Y, and parallel to M.

P is equidistant from M and N only.

Explanation: A point equidistant from two points X and Y lies on the perpendicular bisector of XY. A point equidistant from two intersecting lines M and N lies on the angle bisector of the angle formed by M and N. Therefore, a point P satisfying both conditions must be equidistant from X and Y, AND equidistant from M and N. This means it is equidistant from all four entities: X, Y, M, and N.

Q3096: When constructing a point P that is equidistant from points A and B, which of the following is the *first* essential step?

Construct the angle bisector of any given angle.

Draw a line segment connecting A and B.

Construct the perpendicular bisector of the line segment AB.

Construct a circle with A as the center.

Explanation: To find a point equidistant from two points A and B, the essential first step is to construct the perpendicular bisector of the line segment AB. This line represents all points equidistant from A and B. While drawing AB is a preparatory step, the actual construction for the locus starts with the perpendicular bisector.

Q3097: Which geometric construction represents the locus of points equidistant from two intersecting lines P and Q?

The perpendicular bisector of the line segment PQ.

A line parallel to P.

The angle bisector of the angle formed by P and Q.

A circle passing through the intersection point of P and Q.

Explanation: The locus of points equidistant from two intersecting lines is the angle bisector of the angle formed by those lines. This line ensures that any point on it has equal perpendicular distance to both intersecting lines.

Q3098: What is the locus of points equidistant from two fixed points M and N?

The angle bisector of any angle formed by M and N.

A circle with M as the center passing through N.

A line segment connecting M and N.

The perpendicular bisector of the line segment MN.

Explanation: The locus of points equidistant from two fixed points M and N is defined as the perpendicular bisector of the line segment connecting M and N. Any point on this line is at the same distance from M and N.

Constructs a perpendicular to a straight line from a point on the line and from an external point.

Q3099: When constructing a perpendicular to a straight line at a point P on the line, what is the *first* step after placing the compass needle at P?

Draw an arc above the line.

Draw arcs on both sides of P, intersecting the line.

Open the compass to a large radius.

Measure the length of the line.

Explanation: The first step is to place the compass needle at P and draw arcs on both sides of P to intersect the line, creating two points equidistant from P.

Q3100: In constructing a perpendicular to a line at a point P on it, what type of radius should be used when drawing the initial arcs from P to intersect the line on both sides?

A radius equal to the length of the line.

Any convenient radius.

A radius greater than the length of the line.

A radius exactly half the length of the line.

Explanation: For the initial arcs from point P, any convenient radius can be used as long as it intersects the line on both sides of P.

Q3101: After marking two points A and B equidistant from P on the line, what is the requirement for the radius when drawing arcs from A and B to intersect above/below the line?

It must be equal to the radius used for the first arcs.

It must be less than the distance AP.

It must be greater than the distance AP (or BP).

It must be exactly double the distance AB.

Explanation: The radius used for drawing arcs from A and B must be greater than the distance AP (or BP) to ensure that the arcs intersect away from the line.

Q3102: When constructing a perpendicular to a line at point P, after drawing arcs from P to intersect the line at A and B, what do points A and B represent?

The endpoints of the given line.

Two points on the line equidistant from P.

Two points that define the length of the perpendicular.

Points for drawing parallel lines.

Explanation: Points A and B are created by drawing arcs from P with the same radius, meaning they are equidistant from P on the line.

Q3103: In the construction of a perpendicular to a line at point P, what is the significance of the point where the two arcs drawn from A and B (equidistant from P) intersect?

It is the midpoint of the line segment AB.

It is a point equidistant from A and B.

It is the center of the line.

It defines the angle of 45 degrees.

Explanation: The intersection point of the arcs drawn from A and B is equidistant from both A and B. This property is crucial for forming the perpendicular bisector of segment AB, which passes through P and is perpendicular to the line.

Q3104: After drawing the two intersecting arcs from points A and B (equidistant from P), what is the final step to complete the construction of the perpendicular at P?

Draw a line connecting A and B.

Draw a line from A to the intersection point of the arcs.

Draw a line from P through the intersection point of the arcs.

Measure the angle formed by the arcs.

Explanation: The final step is to use a straightedge to draw a straight line from point P through the intersection point of the two arcs, which forms the perpendicular.

Q3105: What is the first step to construct a perpendicular to a line segment PQ at a point A on PQ?

Draw an arc from A intersecting PQ at one point.

Draw an arc from A intersecting PQ at two distinct points.

Draw an arc from an external point B.

Draw a line through A not intersecting PQ.

Explanation: The first step to construct a perpendicular at a point on a line is to place the compass point at the given point (A) and draw an arc that intersects the line segment (PQ) at two distinct points. This creates two equidistant points from A on the line, which are essential for the subsequent steps of the construction.

Q3106: To construct a perpendicular from an external point B to a line segment PQ, what is the initial action with the compass?

Place the compass point at A (on PQ) and draw an arc.

Place the compass point at B and draw an arc intersecting PQ at two distinct points.

Draw a line from B to PQ.

Measure the distance from B to PQ.

Explanation: To construct a perpendicular from an external point (B) to a line segment (PQ), the initial action is to place the compass point at the external point (B) and open the compass to a radius large enough to draw an arc that intersects the line segment (PQ) at two distinct points. These two intersection points will serve as the bases for drawing further arcs to find the perpendicular.

Q3107: If you are asked to construct a perpendicular to a line segment PQ that passes through a point A *on* PQ, which construction method would you use?

Perpendicular from an external point.

Perpendicular at a point on the line.

Angle bisector.

Line segment bisector.

Explanation: The question specifically states that the perpendicular must pass through a point A *on* the line segment PQ. This describes the construction of a perpendicular at a specific point on a given line, which is one of the two key constructions for perpendiculars.

Q3108: A student correctly constructs a perpendicular to a line segment PQ at point A on PQ. What is the angle formed between the constructed perpendicular and the line segment PQ?

45 degrees.

90 degrees.

60 degrees.

180 degrees.

Explanation: By definition, a perpendicular line or segment intersects another line or segment at a 90-degree angle. Therefore, when a perpendicular is correctly constructed, the angle formed at the point of intersection is always 90 degrees.

Q3109: When constructing a perpendicular to PQ at point A on PQ, a student places the compass point at A and draws an arc that intersects PQ at only *one* point. What is the most likely error?

The compass point was not at A.

The radius of the arc was too small.

The line segment PQ was not long enough.

The student used a ruler instead of a compass.

Explanation: When constructing a perpendicular at a point on a line, the first step requires drawing an arc from the point (A) that intersects the line (PQ) at *two* distinct points. If the arc only intersects at one point, it means the radius used was too small, preventing it from extending far enough to cross the line on both sides of A.

Q3110: To construct a perpendicular from an external point B to line segment PQ, a student places the compass point at B and draws an arc, but it does not intersect PQ. What should the student do?

Decrease the radius of the compass.

Increase the radius of the compass to intersect PQ at two points.

Move point B closer to PQ.

Draw a parallel line to PQ through B.

Explanation: For a perpendicular construction from an external point (B) to a line (PQ), the initial arc drawn from B must intersect the line PQ at two distinct points. If it doesn't intersect, it means the radius of the compass is too small. To correct this, the student needs to increase the compass radius until the arc successfully crosses the line twice.

Q3111: What geometric construction is achieved when a perpendicular line segment is drawn from vertex A to the side BC of a triangle ABC?

Median

Altitude

Angle bisector

Perpendicular bisector

Explanation: An altitude of a triangle is a line segment from a vertex to the opposite side, forming a right angle with that side.

Q3112: When constructing the altitude from vertex A to side BC of triangle ABC, what is the initial step using a compass?

Place the compass needle at B and draw an arc.

Place the compass needle at the midpoint of BC.

Place the compass needle at A and draw an arc intersecting side BC at two points.

Draw arcs from vertices B and C to intersect each other.

Explanation: The first step to construct a perpendicular from an external point (A) to a line (BC) is to place the compass at the external point and draw an arc that intersects the line at two distinct points.

Q3113: After placing the compass needle at vertex A and drawing an arc that intersects side BC at two points, P and Q, what is the next step to construct the altitude?

Draw a line segment from A to the midpoint of PQ.

With the compass needle at P, draw an arc, and with the compass needle at Q, draw another arc to intersect the first arc.

Measure the length of PQ and draw a line segment from A to the midpoint of BC.

Draw a line segment from A to point P.

Explanation: To find the perpendicular, after establishing two points on the line segment, arcs are drawn from these two points to intersect, creating a point that, when connected to the original vertex, forms the perpendicular.

Q3114: The constructed altitude from vertex A to side BC is always ____ to side BC.

Parallel

Bisecting

Perpendicular

Equal in length

Explanation: By definition and construction method, an altitude is a line segment from a vertex perpendicular to the opposite side (or its extension).

Q3115: If triangle ABC is an obtuse-angled triangle with the obtuse angle at B, where would the foot of the altitude from vertex A to side BC lie?

Exactly at point B.

On the segment BC.

Outside the segment BC, on the extension of BC.

It cannot be constructed in an obtuse triangle.

Explanation: In an obtuse-angled triangle, the altitude from a vertex of an acute angle to the opposite side (if that side is adjacent to the obtuse angle) falls outside the triangle, requiring the extension of the side.

Q3116: Which pair of tools is essential for accurately constructing an altitude in a triangle?

Protractor and ruler

Set square and ruler

Compass and straightedge

Divider and ruler

Explanation: Geometric constructions, including altitudes, are fundamentally performed using only a compass for drawing arcs and a straightedge for drawing straight lines.

Constructs the perpendicular bisector of a straight line segment.

Q3117: When constructing the perpendicular bisector of a straight line segment PQ using a compass and straightedge, what is the crucial initial step after drawing the segment?

Measure the length of PQ and mark its midpoint using a ruler.

Place the compass point on P, open it to a radius greater than half the length of PQ, and draw arcs above and below the line. Repeat this from Q with the same radius.

Draw a line perpendicular to PQ passing through point P.

Using a protractor, draw a 90-degree angle at the midpoint of PQ.

Explanation: The standard construction method for a perpendicular bisector involves drawing intersecting arcs from both ends of the line segment with a radius greater than half the segment's length. This ensures the arcs intersect at two distinct points, which when connected by a straightedge, form the perpendicular bisector. Measuring with a ruler or using a protractor are not valid construction methods using only a compass and straightedge.

Q3118: When constructing the perpendicular bisector of a line segment PQ, what is the crucial condition for the radius of the arcs drawn from P and Q?

Exactly half the length of PQ.

Greater than half the length of PQ.

Equal to the full length of PQ.

Any arbitrary radius will work as long as it's the same for both arcs.

Explanation: To ensure the arcs drawn from P and Q intersect at two distinct points, the radius chosen must be greater than half the length of the line segment PQ. If the radius is exactly half, the arcs will only touch at the midpoint, making it difficult to find two distinct intersection points. If it's less than half, the arcs won't intersect at all.

Q3119: After drawing the intersecting arcs from P and Q, and then connecting their intersection points with a straightedge, what is the property of the line segment formed?

It passes through the midpoint of PQ but is not necessarily perpendicular.

It is perpendicular to PQ but does not necessarily bisect it.

It is perpendicular to PQ and bisects PQ at its midpoint.

It bisects the angle formed at point P.

Explanation: The line segment constructed by joining the intersection points of the arcs created with the same radius from the endpoints of PQ is by definition the perpendicular bisector. This means it is both perpendicular to PQ and passes through its exact midpoint, effectively bisecting it.

Q3120: What would be the outcome if you use different compass radii when drawing arcs from point P and point Q to construct the perpendicular bisector?

The construction will still result in the correct perpendicular bisector.

It will result in a line that is perpendicular to PQ but does not bisect it.

It will result in a line that bisects PQ but is not perpendicular to it.

The arcs may not intersect at all, or if they do, the line connecting the intersection points will not be the perpendicular bisector.

Explanation: For a correct perpendicular bisector construction, the radii of the arcs drawn from P and Q must be identical. This ensures that the intersection points are equidistant from P and Q, which is a fundamental property of the perpendicular bisector. Using different radii would lead to incorrect intersection points or no intersection at all, thus failing to construct the perpendicular bisector.

Q3121: After drawing two pairs of intersecting arcs (one above and one below PQ) from points P and Q with the same radius, what is the final step to complete the construction of the perpendicular bisector?

Measure the segment PQ and mark its exact midpoint.

Extend the arcs until they intersect the line segment PQ.

Use a straightedge to draw a straight line connecting the two intersection points of the arcs.

Draw a curved line connecting the two intersection points.

Explanation: The two intersection points created by the arcs are equidistant from both P and Q. By connecting these two points with a straightedge, we construct the locus of points equidistant from P and Q, which is precisely the perpendicular bisector of the segment PQ.

Q3122: The benchmark specifies using 'only a compass and a straightedge' for constructing the perpendicular bisector of a line segment. Which of the following sets of tools adheres to this instruction?

Ruler and protractor.

Compass and straightedge.

Compass, ruler, and protractor.

Set square and ruler.

Explanation: Geometric constructions, by definition, rely solely on a compass (for drawing circles/arcs) and a straightedge (for drawing straight lines). Tools like rulers (for measuring lengths), protractors (for measuring angles), or set squares (for drawing specific angles without construction) are not permitted in pure geometric constructions unless explicitly allowed. The benchmark specifically limits tools to 'only a compass and a straightedge.'

Q3123: Points A(1, 2) and B(7, 2) are plotted on a Cartesian plane. A line segment AB is drawn. What is the equation of the perpendicular bisector of the line segment AB?

y = 2

x = 4

x = 3

y = 4

Explanation: The line segment AB connects A(1, 2) and B(7, 2). Since the y-coordinates are the same, it is a horizontal line. The midpoint of AB is ((1+7)/2, (2+2)/2) = (4, 2). The perpendicular bisector of a horizontal line is a vertical line passing through its midpoint. Therefore, the equation of the perpendicular bisector is x = 4.

Q3124: On a Cartesian plane, points A(1, 2) and B(7, 2) are plotted, and the line segment AB is drawn. What are the coordinates of the point where the perpendicular bisector of AB intersects the line segment AB?

(1, 2)

(7, 2)

(4, 2)

(4, 4)

Explanation: The perpendicular bisector intersects the line segment AB at its midpoint. The coordinates of the midpoint of A(1, 2) and B(7, 2) are calculated as ((1+7)/2, (2+2)/2) = (8/2, 4/2) = (4, 2).

Q3125: Points A(1, 2) and B(7, 2) are given. What is the gradient of the perpendicular bisector of the line segment AB?

Undefined

-1

Explanation: The line segment AB connects A(1, 2) and B(7, 2). Since the y-coordinates are the same, AB is a horizontal line. The gradient of a horizontal line is 0. A line perpendicular to a horizontal line is a vertical line. The gradient of a vertical line is undefined.

Q3126: If P is any point on the perpendicular bisector of the line segment joining A(1, 2) and B(7, 2), which of the following statements is always true?

P is the midpoint of AB

PA > PB

PA = PB

PA < PB

Explanation: By definition, the perpendicular bisector is the locus of all points equidistant from the two endpoints of the line segment. Therefore, for any point P on the perpendicular bisector, the distance from P to A is equal to the distance from P to B (PA = PB).

Q3127: The perpendicular bisector of the line segment AB, where A(1, 2) and B(7, 2), passes through the point (k, 10). What is the value of k?

Explanation: As determined in previous questions, the equation of the perpendicular bisector of AB is x = 4. If this line passes through the point (k, 10), then the x-coordinate of this point must satisfy the equation. Therefore, k = 4.

Q3128: Points A(1, 2) and B(7, 2) are plotted on a Cartesian plane. The line segment AB is drawn. Which of the following is NOT a step in constructing the perpendicular bisector of AB using only a compass and a straightedge?

With A as center, draw an arc with a radius greater than half of AB.

With B as center, using the same radius, draw another arc to intersect the first arc.

Measure the length of AB and divide it by two to find the midpoint.

Draw a straight line connecting the two intersection points of the arcs.

Explanation: Construction of a perpendicular bisector involves using a compass to draw arcs from each endpoint with a radius greater than half the line segment's length, and then drawing a line through the intersection points of these arcs. Measuring the length of AB and dividing it by two is a calculation, not a geometric construction step using only a compass and straightedge.

Q3129: To construct a point equidistant from the three vertices of a triangle ABC, which lines should be constructed?

Angle bisectors of the angles

Perpendicular bisectors of the sides

Medians from the vertices

Altitudes from the vertices

Explanation: The locus of points equidistant from two fixed points is the perpendicular bisector of the line segment joining them. For a point to be equidistant from all three vertices of a triangle, it must lie on the perpendicular bisector of all three sides. The intersection of these perpendicular bisectors is the circumcenter, which is equidistant from the vertices.

Q3130: The point constructed to be equidistant from the three vertices of a triangle is known as the:

Incenter

Circumcenter

Centroid

Orthocenter

Explanation: The circumcenter is the point where the perpendicular bisectors of the sides of a triangle intersect. By definition, this point is equidistant from all three vertices of the triangle, and it is the center of the circumcircle that passes through all three vertices.

Q3131: When constructing the point equidistant from the vertices A, B, and C of a triangle using a compass and straightedge, which of the following is a necessary first step?

Construct the angle bisector of angle A

Construct the perpendicular bisector of side AB

Draw a median from vertex A to the midpoint of BC

Draw an altitude from vertex A to side BC

Explanation: To find a point equidistant from A, B, and C, we need to find a point that is equidistant from A and B, AND equidistant from B and C. The locus of points equidistant from two points (e.g., A and B) is the perpendicular bisector of the line segment joining them. Therefore, constructing the perpendicular bisector of any side (like AB) is a crucial first step.

Q3132: A key property used in finding the point equidistant from the three vertices of a triangle is that any point on the perpendicular bisector of a line segment is:

Equidistant from the two endpoints of the segment

Equidistant from the sides of the triangle

Always inside the triangle

Forms a 90-degree angle with one of the vertices

Explanation: The definition of a perpendicular bisector is the locus of all points equidistant from the two endpoints of the line segment it bisects. Therefore, by constructing the perpendicular bisectors of the sides of a triangle, we find a point that is simultaneously equidistant from A and B, from B and C, and from C and A, thus making it equidistant from all three vertices.

Q3133: To find the point equidistant from the three vertices of a triangle, you must construct the intersection of:

At least two angle bisectors

At least two perpendicular bisectors of the sides

All three medians

All three altitudes

Explanation: The point equidistant from the three vertices of a triangle is the circumcenter. The circumcenter is the intersection point of the perpendicular bisectors of the triangle's sides. While all three perpendicular bisectors are concurrent at this point, only two are theoretically sufficient to locate it, as the third must also pass through the same point.

Q3134: After constructing the point equidistant from the vertices A, B, and C of a triangle, if you draw a circle with this point as the center and the distance to any vertex as the radius, what will be observed?

The circle will touch all three sides of the triangle

The circle will pass through all three vertices A, B, and C

The circle will pass through the midpoints of the sides

The circle will be completely outside the triangle

Explanation: The point equidistant from the three vertices is the circumcenter. By definition, the circumcenter is the center of the circumcircle, which is a circle that passes through all three vertices of the triangle. Therefore, if a circle is drawn with the circumcenter as its center and the distance to any vertex as its radius, it will necessarily pass through all three vertices.

Constructs angles of 60°, 120°, 90°, 45°, 30° and copies a given angle.

Q3135: When constructing a 60° angle at point A on a line segment, after drawing an arc centered at A that intersects the line at point P, what is the correct next step?

Change the compass radius and draw another arc from A.

Place the compass point at P and, using the same radius, draw an arc to intersect the first arc.

Use a protractor to mark 60° from point A.

Draw a line segment from P to A.

Explanation: To construct a 60° angle, after drawing the initial arc from the vertex, the crucial next step is to place the compass point on the intersection of the arc and the line segment (P) and, *without changing the radius*, draw a second arc to intersect the first one. This creates an equilateral triangle, ensuring the angle is 60°.

Q3136: Using only a compass and straightedge, if you draw an arc from the vertex, then without changing the radius, draw a second arc from where the first arc intersects the base line, what angle will be formed by joining the vertex to the intersection of the two arcs?

90°

45°

60°

120°

Explanation: This procedure describes the standard construction for a 60° angle. By keeping the radius the same for both arcs, an equilateral triangle is formed, and all angles in an equilateral triangle are 60°.

Q3137: A student constructs an initial arc for a 60° angle. However, they accidentally change the compass radius before drawing the second arc from the point where the first arc intersects the base line. What will be the consequence?

The angle formed will still be exactly 60° due to the initial arc.

The angle formed will be either 30° or 90°.

The construction will not result in a 60° angle.

The angle will be 180° because the radius was changed.

Explanation: For a 60° angle construction, it is crucial that the compass radius remains constant for both the initial arc from the vertex and the subsequent arc from the intersection point on the base line. Changing the radius will alter the geometry and prevent the formation of an equilateral triangle, thus not resulting in a 60° angle.

Q3138: When copying a given angle (e.g., a 70° angle) using only a compass and straightedge, what is the *first* required action on the new sheet where the angle is to be copied?

Draw an arc on the original angle.

Use a protractor to measure the original angle.

Draw a new base line and mark a point on it to be the new vertex.

Draw a circle with a large radius on the new sheet.

Explanation: Before any arcs or measurements can be transferred, you must establish a base line and a new vertex on the sheet where the copied angle will reside. This provides the starting point for the new angle.

Q3139: After drawing an arc from the vertex of the original angle (intersecting both arms) and a corresponding arc of the *same radius* from the new vertex on the new base line, what is the next critical measurement to transfer to complete the angle copying?

The length of the initial arc.

The distance between the two points where the first arc intersects the arms of the original angle.

The perpendicular distance from the vertex to the arc.

The length of one of the arms of the original angle.

Explanation: To accurately copy an angle, after drawing identical arcs, you need to measure the 'chord length' of the original angle – that is, the distance between the two points where the arc intersects the arms of the original angle. This distance is then transferred to the new arc, which defines the second arm of the copied angle.

Q3140: A student is asked to copy an angle using only a compass and straightedge. After drawing the initial arc on the new base line, they use a ruler to measure the distance between the intersection points on the original angle's arms, then try to mark that length on the new arc. Why is this incorrect?

A ruler cannot accurately measure distances on an arc.

The task specifically requires using a compass for all measurements, not a ruler.

The distance between intersection points is irrelevant for copying an angle.

A protractor should have been used instead of a ruler.

Explanation: The instruction 'using only a compass and straightedge' implies that all measurements and transfers must be done with these two tools. While measuring the chord length is the correct step, using a ruler violates the specified construction method. A compass should be used to 'measure' (by setting its width) and transfer this chord length.

Q3141: What is the first step to construct a 90° angle at a point P on a line XY using only a compass and straightedge?

Draw a straight line through P perpendicular to XY.

Place the compass point at P and draw arcs intersecting XY on both sides of P.

Draw an arc from P intersecting XY at one point.

Measure 90° with a protractor at P.

Explanation: The first step in constructing a 90° angle at a point on a line using a compass is to establish two equidistant points on the line from the given point, which is done by drawing arcs on both sides of the point P intersecting the line XY.

Q3142: After constructing a 90° angle with vertex A (formed by rays AB and AC), what is the next compass and straightedge step to construct a 45° angle from vertex A?

Extend one of the rays (AB or AC) further.

Place the compass point on the arcs that define the 90° angle on AB and AC (or the points where an arc from A intersects AB and AC), and draw intersecting arcs in the interior of the angle.

Measure 45° with a protractor from A.

Draw an arc from A to bisect the line segment AB.

Explanation: To bisect an angle, one must place the compass point on the points where an arc from the vertex intersects the arms of the angle, and then draw intersecting arcs within the angle's interior. This creates the bisector for the 45° angle.

Q3143: A student successfully constructed a 90° angle at a point on a line and then accurately bisected it. What is the measure of the angle formed by the bisector and one of the original sides of the 90° angle?

30°

60°

45°

22.5°

Explanation: Bisecting an angle means dividing it into two equal halves. Therefore, bisecting a 90° angle results in two 45° angles.

Q3144: To construct a 90° angle at point P on line XY, after drawing an arc centered at P that intersects XY at two points (say A and B), what is the next correct step using the compass?

Draw a line from P to A.

Adjust the compass to 45° and draw an arc.

With the compass point at A and then at B, draw two intersecting arcs of the same radius (larger than AP) above point P.

Draw a full circle centered at P.

Explanation: To construct a perpendicular line (and thus a 90° angle), after establishing two equidistant points on the line, arcs of a larger radius are drawn from these points to intersect, defining the direction of the perpendicular.

Q3145: If you have successfully constructed a 90° angle at a point on a line, what geometric operation would you perform from the vertex of the 90° angle to accurately obtain a 45° angle?

Bisect one of the arms of the 90° angle.

Construct another 90° angle adjacent to the first one.

Bisect the 90° angle.

Draw a line segment of half the length of one arm.

Explanation: To obtain exactly half of an angle using only a compass and straightedge, the angle must be bisected. Bisecting a 90° angle yields a 45° angle.

Q3146: A student is constructing a 45° angle by first creating a 90° angle at point P on line L, and then bisecting it. Which of the following describes a correct action for bisecting the 90° angle?

Draw an arc from P with an arbitrary radius and connect it to a point on line L.

Measure half of the 90° angle with a protractor.

With the compass point at the two points where an arc from P intersects the arms of the 90° angle, draw intersecting arcs inside the angle.

Draw a line from P to the midpoint of the line segment that forms the base of the 90° angle.

Explanation: The standard method for bisecting an angle involves drawing an arc from the vertex that intersects both arms of the angle, then using those intersection points as centers to draw two more intersecting arcs within the angle. Connecting the vertex to this new intersection point creates the angle bisector.