TGT PGT Mathematics Objective Questions
Year: 2025
Score: 0/50
1. If A is a 3×3 matrix with det(A) = 5, what is det(2A)?
Correct Answer: b) 40
Explanation: For a 3×3 matrix, det(kA) = k³det(A). Here, k = 2, so det(2A) = 2³ * 5 = 8 * 5 = 40.
Explanation: For a 3×3 matrix, det(kA) = k³det(A). Here, k = 2, so det(2A) = 2³ * 5 = 8 * 5 = 40.
2. What is the limit of (sin x)/x as x approaches 0?
Correct Answer: b) 1
Explanation: This is a standard limit: lim(x→0) (sin x)/x = 1.
Explanation: This is a standard limit: lim(x→0) (sin x)/x = 1.
3. The number of ways to arrange 5 distinct books on a shelf is:
Correct Answer: a) 120
Explanation: Number of arrangements = 5! = 5 * 4 * 3 * 2 * 1 = 120.
Explanation: Number of arrangements = 5! = 5 * 4 * 3 * 2 * 1 = 120.
4. If f(x) = x² + 2x + 1, what is f'(2)?
Correct Answer: b) 6
Explanation: f(x) = x² + 2x + 1, f'(x) = 2x + 2. At x = 2, f'(2) = 2(2) + 2 = 6.
Explanation: f(x) = x² + 2x + 1, f'(x) = 2x + 2. At x = 2, f'(2) = 2(2) + 2 = 6.
5. The probability of getting exactly 2 heads in 3 coin tosses is:
Correct Answer: b) 3/8
Explanation: P(2 heads) = C(3,2) * (1/2)² * (1/2) = 3 * (1/4) * (1/2) = 3/8.
Explanation: P(2 heads) = C(3,2) * (1/2)² * (1/2) = 3 * (1/4) * (1/2) = 3/8.
6. The eigenvalues of a 2×2 identity matrix are:
Correct Answer: b) 1, 1
Explanation: For I₂, the characteristic equation is det(I – λI) = (1-λ)(1-λ) = 0, so λ = 1, 1.
Explanation: For I₂, the characteristic equation is det(I – λI) = (1-λ)(1-λ) = 0, so λ = 1, 1.
7. The sum of the series 1 + 1/2 + 1/4 + 1/8 + … is:
Correct Answer: b) 2
Explanation: Geometric series with a = 1, r = 1/2. Sum = a/(1-r) = 1/(1-1/2) = 2.
Explanation: Geometric series with a = 1, r = 1/2. Sum = a/(1-r) = 1/(1-1/2) = 2.
8. If z = 1 + i, what is |z|?
Correct Answer: b) √2
Explanation: |z| = √(1² + 1²) = √2.
Explanation: |z| = √(1² + 1²) = √2.
9. The equation of the tangent to y = x³ at x = 1 is:
Correct Answer: a) 3x – 2
Explanation: y’ = 3x², at x = 1, slope = 3. Point (1,1). Equation: y – 1 = 3(x – 1) → y = 3x – 2.
Explanation: y’ = 3x², at x = 1, slope = 3. Point (1,1). Equation: y – 1 = 3(x – 1) → y = 3x – 2.
10. The number of subsets of a set with 4 elements is:
Correct Answer: b) 16
Explanation: Number of subsets = 2⁴ = 16.
Explanation: Number of subsets = 2⁴ = 16.
11. If A = {1, 2, 3}, how many relations are possible on A?
Correct Answer: a) 2⁹
Explanation: Number of relations = 2^(n²), where n = 3. So, 2^(3²) = 2⁹.
Explanation: Number of relations = 2^(n²), where n = 3. So, 2^(3²) = 2⁹.
12. The value of ∫(0 to π) sin²x dx is:
Correct Answer: a) π/2
Explanation: Use identity sin²x = (1 – cos 2x)/2. Integral = π/2.
Explanation: Use identity sin²x = (1 – cos 2x)/2. Integral = π/2.
13. The rank of a 3×3 zero matrix is:
Correct Answer: a) 0
Explanation: A zero matrix has no non-zero rows, so rank = 0.
Explanation: A zero matrix has no non-zero rows, so rank = 0.
14. The solution to the differential equation dy/dx = y is:
Correct Answer: a) e^x
Explanation: dy/dx = y → y = Ce^x. General solution is y = e^x (with constant).
Explanation: dy/dx = y → y = Ce^x. General solution is y = e^x (with constant).
15. The number of onto functions from {1, 2, 3} to {1, 2} is:
Correct Answer: a) 6
Explanation: Number of onto functions = 2³ – 2 * 1³ = 8 – 2 = 6.
Explanation: Number of onto functions = 2³ – 2 * 1³ = 8 – 2 = 6.
16. If A and B are independent events with P(A) = 0.3, P(B) = 0.4, then P(A ∩ B) is:
Correct Answer: a) 0.12
Explanation: P(A ∩ B) = P(A) * P(B) = 0.3 * 0.4 = 0.12.
Explanation: P(A ∩ B) = P(A) * P(B) = 0.3 * 0.4 = 0.12.
17. The area of the region bounded by y = x² and y = 4 is:
Correct Answer: a) 16/3
Explanation: Points of intersection at x = ±2. Area = ∫(-2 to 2) (4 – x²) dx = 16/3.
Explanation: Points of intersection at x = ±2. Area = ∫(-2 to 2) (4 – x²) dx = 16/3.
18. The number of solutions to x² – 5|x| + 6 = 0 is:
Correct Answer: d) 4
Explanation: Let y = |x|. Then y² – 5y + 6 = 0 → y = 2, 3. So, x = ±2, ±3.
Explanation: Let y = |x|. Then y² – 5y + 6 = 0 → y = 2, 3. So, x = ±2, ±3.
19. The value of cos(π/12) is:
Correct Answer: a) (√3 + 1)/(2√2)
Explanation: cos(π/12) = cos(15°) = (√3 + 1)/(2√2) using angle formulas.
Explanation: cos(π/12) = cos(15°) = (√3 + 1)/(2√2) using angle formulas.
20. The determinant of a skew-symmetric matrix of odd order is:
Correct Answer: c) Zero
Explanation: For a skew-symmetric matrix of odd order, det(A) = 0.
Explanation: For a skew-symmetric matrix of odd order, det(A) = 0.
21. The number of derangements of {1, 2, 3, 4} is:
Correct Answer: a) 9
Explanation: !4 = 4! * (1 – 1/1! + 1/2! – 1/3! + 1/4!) = 24 * (1 – 1 + 1/2 – 1/6 + 1/24) = 9.
Explanation: !4 = 4! * (1 – 1/1! + 1/2! – 1/3! + 1/4!) = 24 * (1 – 1 + 1/2 – 1/6 + 1/24) = 9.
22. If f(x) = e^x, then the nth derivative fⁿ(x) is:
Correct Answer: a) e^x
Explanation: The nth derivative of e^x is e^x.
Explanation: The nth derivative of e^x is e^x.
23. The variance of a binomial distribution with n = 10, p = 0.3 is:
Correct Answer: a) 2.1
Explanation: Variance = np(1-p) = 10 * 0.3 * 0.7 = 2.1.
Explanation: Variance = np(1-p) = 10 * 0.3 * 0.7 = 2.1.
24. The radius of convergence of the series Σ (x^n)/n! is:
Correct Answer: c) ∞
Explanation: Ratio test: lim |a_{n+1}/a_n| = lim (1/(n+1)) = 0. Radius = ∞.
Explanation: Ratio test: lim |a_{n+1}/a_n| = lim (1/(n+1)) = 0. Radius = ∞.
25. The number of edges in a complete graph K₅ is:
Correct Answer: a) 10
Explanation: Number of edges in K₅ = C(5,2) = 10.
Explanation: Number of edges in K₅ = C(5,2) = 10.
26. The value of lim(x→∞) (1 + 1/x)^x is:
Correct Answer: b) e
Explanation: lim(x→∞) (1 + 1/x)^x = e.
Explanation: lim(x→∞) (1 + 1/x)^x = e.
27. If A is an invertible matrix, then (A⁻¹)ᵀ is:
Correct Answer: a) (Aᵀ)⁻¹
Explanation: (A⁻¹)ᵀ = (Aᵀ)⁻¹.
Explanation: (A⁻¹)ᵀ = (Aᵀ)⁻¹.
28. The sum of the first n odd natural numbers is:
Correct Answer: b) n²
Explanation: Sum = 1 + 3 + … + (2n-1) = n².
Explanation: Sum = 1 + 3 + … + (2n-1) = n².
29. The equation of the normal to y = x² at (1,1) is:
Correct Answer: a) -x/2 + 3/2
Explanation: Slope of tangent = 2x = 2 at x = 1. Normal slope = -1/2. Equation: y – 1 = (-1/2)(x – 1) → y = -x/2 + 3/2.
Explanation: Slope of tangent = 2x = 2 at x = 1. Normal slope = -1/2. Equation: y – 1 = (-1/2)(x – 1) → y = -x/2 + 3/2.
30. The number of divisors of 360 is:
Correct Answer: b) 24
Explanation: Prime factorization: 360 = 2³ * 3² * 5. Number of divisors = (3+1)(2+1)(1+1) = 4*3*2 = 24.
Explanation: Prime factorization: 360 = 2³ * 3² * 5. Number of divisors = (3+1)(2+1)(1+1) = 4*3*2 = 24.
31. The value of ∫(0 to 1) x e^x dx is:
Correct Answer: a) e – 1
Explanation: Use integration by parts: ∫ x e^x dx = x e^x – ∫ e^x dx. Evaluate from 0 to 1: e – 1.
Explanation: Use integration by parts: ∫ x e^x dx = x e^x – ∫ e^x dx. Evaluate from 0 to 1: e – 1.
32. The number of ways to choose 3 items from 5 is:
Correct Answer: a) 10
Explanation: C(5,3) = 5!/(3!2!) = 10.
Explanation: C(5,3) = 5!/(3!2!) = 10.
33. The roots of the equation x² – 4x + 4 = 0 are:
Correct Answer: a) 2, 2
Explanation: x² – 4x + 4 = (x – 2)² = 0 → x = 2 (repeated root).
Explanation: x² – 4x + 4 = (x – 2)² = 0 → x = 2 (repeated root).
34. The value of tan⁻¹(1) is:
Correct Answer: b) π/4
Explanation: tan(π/4) = 1, so tan⁻¹(1) = π/4.
Explanation: tan(π/4) = 1, so tan⁻¹(1) = π/4.
35. The trace of a 3×3 matrix with eigenvalues 1, 2, 3 is:
Correct Answer: a) 6
Explanation: Trace = sum of eigenvalues = 1 + 2 + 3 = 6.
Explanation: Trace = sum of eigenvalues = 1 + 2 + 3 = 6.
36. The probability that a leap year has 53 Sundays is:
Correct Answer: b) 2/7
Explanation: Leap year has 366 days = 52 weeks + 2 days. Probability of Sunday = 2/7.
Explanation: Leap year has 366 days = 52 weeks + 2 days. Probability of Sunday = 2/7.
37. The value of lim(x→0) (1 – cos x)/x² is:
Correct Answer: a) 1/2
Explanation: Use identity: (1 – cos x)/x² = (2 sin²(x/2))/x² = 1/2.
Explanation: Use identity: (1 – cos x)/x² = (2 sin²(x/2))/x² = 1/2.
38. The number of solutions to sin x = x/2 in [0, 2π] is:
Correct Answer: c) 3
Explanation: Analyze f(x) = sin x – x/2. It has three roots in [0, 2π].
Explanation: Analyze f(x) = sin x – x/2. It has three roots in [0, 2π].
39. The sum of the series Σ (1/n²) from n=1 to ∞ is:
Correct Answer: a) π²/6
Explanation: Basel problem: Σ (1/n²) = π²/6.
Explanation: Basel problem: Σ (1/n²) = π²/6.
40. The equation of the circle with center (2, 3) and radius 4 is:
Correct Answer: a) (x-2)² + (y-3)² = 16
Explanation: (x-h)² + (y-k)² = r² → (x-2)² + (y-3)² = 4² = 16.
Explanation: (x-h)² + (y-k)² = r² → (x-2)² + (y-3)² = 4² = 16.
41. The value of log₂(8) is:
Correct Answer: b) 3
Explanation: log₂(8) = log₂(2³) = 3.
Explanation: log₂(8) = log₂(2³) = 3.
42. The number of ways to form a committee of 3 from 5 people is:
Correct Answer: a) 10
Explanation: C(5,3) = 10.
Explanation: C(5,3) = 10.
43. The value of ∫(0 to π/2) cos x dx is:
Correct Answer: b) 1
Explanation: ∫ cos x dx = sin x. From 0 to π/2: sin(π/2) – sin(0) = 1.
Explanation: ∫ cos x dx = sin x. From 0 to π/2: sin(π/2) – sin(0) = 1.
44. The number of permutations of the word “MATH” is:
Correct Answer: a) 24
Explanation: 4 distinct letters → 4! = 24.
Explanation: 4 distinct letters → 4! = 24.
45. The value of det([1 2; 3 4]) is:
Correct Answer: a) -2
Explanation: det = 1*4 – 2*3 = 4 – 6 = -2.
Explanation: det = 1*4 – 2*3 = 4 – 6 = -2.
46. The sum of the angles in a triangle is:
Correct Answer: a) 180°
Explanation: Sum of angles in a triangle = 180°.
Explanation: Sum of angles in a triangle = 180°.
47. The value of e⁰ is:
Correct Answer: b) 1
Explanation: e⁰ = 1.
Explanation: e⁰ = 1.
48. The number of zeros at the end of 100! is:
Correct Answer: a) 24
Explanation: Number of zeros = ⌊100/5⌋ + ⌊100/25⌋ = 20 + 4 = 24.
Explanation: Number of zeros = ⌊100/5⌋ + ⌊100/25⌋ = 20 + 4 = 24.
49. The value of sin²θ + cos²θ is:
Correct Answer: b) 1
Explanation: Trigonometric identity: sin²θ + cos²θ = 1.
Explanation: Trigonometric identity: sin²θ + cos²θ = 1.
50. The number of ways to distribute 3 identical balls into 4 distinct boxes is:
Correct Answer: a) 10
Explanation: Stars and bars: C(3+4-1, 4-1) = C(6,3) = 20. Adjust for identical balls: C(3+4-1,3) = 10.
Explanation: Stars and bars: C(3+4-1, 4-1) = C(6,3) = 20. Adjust for identical balls: C(3+4-1,3) = 10.
