NCERT Based Practice Worksheet · AAI ATC Exam Prep
15
MCQs
10
One-Word
25
Total Qs
10+
Numericals
📝 Section A — Multiple Choice Questions
15 QsMCQ
1
The nth term of a G.P. with first term a and common ratio r is given by:
MCQ
(A) a + (n–1)r
(B) arn
(C) arn–1
(D) a·n·r
Answer: (C) arn–1
The general term of a G.P. is an = arn–1, where a is the first term and r is the common ratio. For example, for G.P. 5, 25, 125, …, a = 5, r = 5, so a10 = 5 × 59 = 510.
2
The sum of first n terms of a G.P. (r ≠ 1) is:
MCQ
(A) a(rn + 1)/(r + 1)
(B) a(rn – 1)/(r – 1)
(C) na
(D) a·rn/(r – 1)
Answer: (B) a(rn – 1)/(r – 1)
Sn = a(rn – 1)/(r – 1) when r ≠ 1. When r = 1, Sn = na. This is derived by multiplying the sum by r and subtracting.
3
In the G.P. 2, 8, 32, …, which term equals 131072?
MCQ
(A) 7th
(B) 8th
(C) 9th
(D) 10th
Answer: (C) 9th
Here a = 2, r = 4. So 131072 = 2 × 4n–1 → 65536 = 4n–1 → 48 = 4n–1 → n = 9.
4
If A.M. and G.M. of two positive numbers are 10 and 8 respectively, which of the following is the correct pair of numbers?
MCQ
(A) 6, 14
(B) 4, 16
(C) 2, 18
(D) 5, 15
Answer: (B) 4, 16
a + b = 20, ab = 64. Using (a–b)2 = (a+b)2 – 4ab = 400 – 256 = 144 → a–b = ±12. Solving: a = 16, b = 4 (or vice versa).
5
For two positive numbers A ≥ G, where A is the A.M. and G is the G.M., the expression A – G equals:
MCQ
(A) (√a – √b)2/2
(B) (√a + √b)2/2
(C) (a – b)2/4
(D) (a + b)/2 – ab
Answer: (A) (√a – √b)2/2
A – G = (a+b)/2 – √(ab) = (a + b – 2√(ab))/2 = (√a – √b)2/2 ≥ 0. This proves A ≥ G always.
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6
In a G.P., the 3rd term is 24 and the 6th term is 192. What is the 10th term?
MCQ · Numerical
(A) 1536
(B) 2048
(C) 3072
(D) 6144
Answer: (C) 3072
ar2 = 24 …(1), ar5 = 192 …(2). Dividing: r3 = 8 → r = 2. From (1): a = 6. So a10 = 6 × 29 = 6 × 512 = 3072.
7
How many terms of the G.P. 3, 3/2, 3/4, … are needed to give the sum 3069/512?
MCQ · Numerical
(A) 8
(B) 9
(C) 10
(D) 12
Answer: (C) 10
a = 3, r = 1/2. Sn = 3(1 – (1/2)n)/(1 – 1/2) = 6(1 – 1/2n). Setting equal to 3069/512: 1/2n = 1/1024 = 1/210 → n = 10.
8
A person has 2 parents, 4 grandparents, 8 great-grandparents, and so on. What is the total number of ancestors over 10 preceding generations?
MCQ · Numerical
(A) 1024
(B) 2046
(C) 2048
(D) 4096
Answer: (B) 2046
a = 2, r = 2, n = 10. S10 = 2(210 – 1)/(2 – 1) = 2(1024 – 1) = 2 × 1023 = 2046.
9
The sum of the first three terms of a G.P. is 13/12 and their product is –1. The common ratio is:
MCQ · Numerical
(A) –3/4 or –4/3
(B) 3/4 or 4/3
(C) –1/2 or –2
(D) 1/2 or 2
Answer: (A) –3/4 or –4/3
Let terms be a/r, a, ar. Product = a3 = –1 → a = –1. Substituting into the sum equation: –1/r – 1 – r = 13/12 → 12r2 + 25r + 12 = 0 → r = –3/4 or –4/3.
10
Three numbers between 1 and 256 are inserted so that the resulting sequence is a G.P. The common ratio is:
MCQ · Numerical
(A) ±2
(B) ±4
(C) ±8
(D) ±16
Answer: (B) ±4
1, G1, G2, G3, 256 is a G.P. with 5 terms. So 256 = 1 × r4 → r4 = 256 → r = ±4. For r = 4: G1 = 4, G2 = 16, G3 = 64.
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11
The 20th term of the sequence defined by an = (n–1)(2–n)(3+n) is: