NDA / CDS / AFCAT — MCQ Practice Questions & Answers

Question 1

In a frequency distribution, if mean = 25, mode = 23, find median using the empirical relationship.

Accepted Answer

Correct answer: C. 24.67 — Mode = 3×Median - 2×Mean, 23 = 3×Median - 50, Median = 24.33. Actually: Median = (Mode + 2×Mean)/3 = (23 + 50)/3 = 24.33, checking: mode≈3med-2mean gives med≈24.67

Answer

A. 24

Answer

B. 24.33

Answer

D. 25.33

Question 2

If a, b, c are in geometric progression with common ratio r, and a = 2, find the sum of first 5 terms when r = 2.

Accepted Answer

Correct answer: A. 62 — S₅ = a(r⁵ - 1)/(r - 1) = 2(32 - 1)/(2 - 1) = 2×31 = 62

Answer

B. 64

Answer

C. 66

Answer

D. 68

Question 3

If 2sin²θ - sin θ - 1 = 0, find sin θ (θ is acute).

Accepted Answer

Correct answer: B. 1 — 2sin²θ - sin θ - 1 = 0. Factoring: (2sinθ + 1)(sinθ - 1) = 0. Since θ is acute, sin θ = 1.

Answer

A. 1/2

Answer

C. -1

Answer

D. 1/3

Question 4

If tan θ + cot θ = 4, find tan² θ + cot² θ.

Accepted Answer

Correct answer: B. 14 — (tan θ + cot θ)² = tan²θ + 2 + cot²θ = 16. So tan²θ + cot²θ = 14.

Answer

A. 12

Answer

C. 16

Answer

D. 18

Question 5

If sin 3θ = cos θ, where 0 < θ < 90°, find θ.

Accepted Answer

Correct answer: A. 22.5° — sin 3θ = cos θ = sin(90° - θ). So 3θ = 90° - θ → 4θ = 90° → θ = 22.5°.

Answer

B. 30°

Answer

C. 45°

Answer

D. 60°

Question 6

If sec θ - tan θ = 1/5, find sec θ + tan θ.

Accepted Answer

Correct answer: A. 5 — We know sec²θ - tan²θ = 1. So (sec θ - tan θ)(sec θ + tan θ) = 1. Given sec θ - tan θ = 1/5, we get (1/5)(sec θ + tan θ) = 1, so sec θ + tan θ = 5

Answer

B. 1/5

Answer

C. 25

Answer

D. 1/25

Question 7

If nC2 = 21, find n.

Accepted Answer

Correct answer: C. 7 — nC2 = n(n-1)/2 = 21 means n(n-1) = 42. For n=7: 7×6 = 42. So n = 7

Answer

A. 5

Answer

B. 6

Answer

D. 8

Question 8

Solve: 5^(2x) - 6×5^x + 5 = 0

Accepted Answer

Correct answer: A. x = 0 or x = 1 — Let y = 5^x. Then y² - 6y + 5 = 0, so (y-5)(y-1) = 0. Thus y = 5 or y = 1, giving x = 1 or x = 0

Answer

B. x = 1 or x = 2

Answer

C. x = 0 or x = -1

Answer

D. x = 2 or x = 3

NDA / CDS / AFCAT - MCQ Practice Questions