CBSE Class 10 Mathematics Arithmetic Progressions MCQs Set 20

Answer: (a) 1

Explanation:
1. The gaps are equal: (4x − 1) − (x − 2) = (5x + 2) − (4x − 1).
2. 3x + 1 = x + 3, so x = 1.

Teacher's Note:
1. Equal gaps define an A.P.
2. Check: −1, 3, 7 has d = 4.

Answer: (d) −2

Explanation:
1. 2(a − 2) = a + 3a, so 2a − 4 = 4a.
2. −2a = 4, so a = −2.

Teacher's Note:
1. Use 2 × middle = first + third.
2. Check: −2, −4, −6 is an A.P.

Answer: (c) 9th

Explanation:
1. Here a = 72 and d = −9. 72 + (n − 1)(−9) = 0.
2. n − 1 = 8, so n = 9.

Teacher's Note:
1. Set the nth term equal to 0 and solve for n.
2. Check: 72 − 8 × 9 = 0.

Answer: (b) 21

Explanation:
1. −10 = 10 + (n − 1)(−1).
2. n − 1 = 20, so n = 21.

Teacher's Note:
1. The count includes both the first and the last term.
2. Check: 10 down to −10 is 21 numbers.

Answer: (a) 1

Explanation:
1. Put n = 1: a₁ = 3(1) − 2 = 1.

Teacher's Note:
1. Substitute n = 1 in the nth-term formula.
2. The next term is a₂ = 4, so d = 3.

Answer: (d) 695

Explanation:
1. a₁₀₀ = 7(100) − 5 = 700 − 5 = 695.

Teacher's Note:
1. Substitute n = 100 directly.
2. No need to find a and d first.

Answer: (c) 3

Explanation:
1. a₁ = 8 and a₂ = 11, so d = 3.
2. Or note that the coefficient of n is d.

Teacher's Note:
1. For aₙ = pn + q, the common difference is p.
2. The constant q is not the common difference.

Answer: (b) 5

Explanation:
1. a₁ = 8 and a₂ = 13, so d = 5.

Teacher's Note:
1. The coefficient of n is the common difference.
2. Order of terms in the formula does not matter.

Answer: (a) 5

Explanation:
1. 2(2x) = (x + 2) + (2x + 3).
2. 4x = 3x + 5, so x = 5.

Teacher's Note:
1. Use 2 × middle = sum of the outer two.
2. Check: 7, 10, 13 has d = 3.

Answer: (d) 3

Explanation:
1. 2(2x + k) = x + 3x + 6, so 4x + 2k = 4x + 6.
2. k = 3.

Teacher's Note:
1. The x terms cancel on both sides.
2. Check with x = 1: 1, 5, 9.

Answer: (c) \( \frac{7}{5} \)

Explanation:
1. 2a = \( \frac{4}{5} + 2 = \frac{14}{5} \).
2. a = \( \frac{7}{5} \).

Teacher's Note:
1. The middle term is the mean of the other two.
2. Check: gaps 7/5 − 4/5 = 3/5 and 2 − 7/5 = 3/5.

Answer: (b) 3

Explanation:
1. 2 × 7 = (2p − 1) + 3p, so 14 = 5p − 1.
2. p = 3.

Teacher's Note:
1. 2 × middle = sum of the outer terms.
2. Check: 5, 7, 9.

Answer: (a) 4

Explanation:
1. 26 = (2p + 1) + (5p − 3) = 7p − 2.
2. p = 4.

Teacher's Note:
1. Check by finding the three terms.
2. With p = 4: 9, 13, 17.

Answer: (d) p + 9q

Explanation:
1. a₁₀ = a + 9d = p + 9q.

Teacher's Note:
1. The nth term is a + (n − 1)d.
2. The 10th term has 9 gaps, not 10.

Answer: (c) 6th

Explanation:
1. d = −3. 14 + (n − 1)(−3) = −1.
2. n − 1 = 5, so n = 6.

Teacher's Note:
1. Subtract and divide: (14 − (−1)) ÷ 3 = 5 gaps.
2. Check: 14, 11, 8, 5, 2, −1.

Answer: (b) 17 and 21

Explanation:
1. d = 4, and 49 = −11 + 4(n − 1) gives n = 16 terms.
2. With 16 terms the middle terms are the 8th and 9th: a₈ = −11 + 28 = 17 and a₉ = −11 + 32 = 21.

Teacher's Note:
1. An even number of terms has two middle terms.
2. Check: the mean of 17 and 21 is 19, the mean of −11 and 49.