Q1. Let A be a 3x3 non-singular matrix such that A * adj(A) = adj(A) * A. If |adj(A)| = 25, then |5A⁻¹| is equal to:
Correct Answer: Option A (= adj(A) * A. If |adj(A)| = 25, then |5A⁻¹| is equal to:)
Explanation: Detailed explanation will be updated shortly.
Q2. The area of the region bounded by the curve y = |x² - 9| and the line y = 7 is given by:
Correct Answer: Option A (∫[-4, 4] (16 - x²) dx)
Explanation: Detailed explanation will be updated shortly.
Q3. A function f: R → R is defined as f(x) = (x² - ax + 1) / (x² + ax + 1). If the function is surjective (onto), then the range of 'a' is:
Correct Answer: Option A (a ∈ R)
Explanation: Detailed explanation will be updated shortly.
Q4. The value of the integral ∫[0, π/2] log( (4 + 3sin(x)) / (4 + 3cos(x)) ) dx is:
Correct Answer: Option A (0)
Explanation: Detailed explanation will be updated shortly.
Q5. Two lines L₁: (x-1)/2 = (y-2)/3 = (z-3)/k and L₂: (x-1)/k = (y-2)/2 = (z-3)/3 are coplanar. Which of the following is a possible value for k?
Correct Answer: Option A (2 only)
Explanation: Detailed explanation will be updated shortly.
Q6. If the variance of a random variable X, which follows a binomial distribution B(n, p), is 4 and the probability of 2 successes is 5 times the probability of 3 successes, then the value of 'n' is:
Correct Answer: Option A (24)
Explanation: Detailed explanation will be updated shortly.
Q7. The derivative of f(x) = tan⁻¹( (√(1+x²) - 1) / x ) with respect to g(x) = sin⁻¹( 2x / (1+x²) ) at x = 1/√3 is:
Correct Answer: Option A (1/8)
Explanation: Detailed explanation will be updated shortly.
Q8. Consider the differential equation (x² + y²) dy = xy dx. The general solution is given by:
Correct Answer: Option A (y = C * e^(x²/2y²))
Explanation: Detailed explanation will be updated shortly.
Q9. Let vectors a = i + j - k, b = i - j + k. A vector c is such that a x c = b and a . c = 3. Then |c|² is equal to:
Correct Answer: Option A (9/2)
Explanation: Detailed explanation will be updated shortly.
Q10. For a Linear Programming Problem, the objective function is Z = 5x + 3y. The vertices of the feasible region are (0,10), (5,5), (15,15), and (0,20). Which of the following statements is true?
Correct Answer: Option A (Maximum value of Z occurs only at (15,15).)
Explanation: Detailed explanation will be updated shortly.
Q11. The value of sin⁻¹(12/13) + cos⁻¹(4/5) + tan⁻¹(63/16) is:
Correct Answer: Option A (π/2)
Explanation: Detailed explanation will be updated shortly.
Q12. If A = [[cosθ, -sinθ], [sinθ, cosθ]], then the matrix A⁻⁵⁰ when θ = π/12 is:
Correct Answer: Option A ([[√3/2, 1/2], [-1/2, √3/2]])
Explanation: Detailed explanation will be updated shortly.
Q13. The equation of the plane containing the line (x-1)/2 = (y+1)/-1 = z/3 and perpendicular to the plane x + y + z = 5 is:
Correct Answer: Option A (4x - y - 3z - 5 = 0)
Explanation: Detailed explanation will be updated shortly.
Q14. A function f(x) is defined as f(x) = [x²] - [x]², for x ≥ 0, where [.] denotes the greatest integer function. The points of discontinuity for f(x) in the interval [0, 2) are:
Correct Answer: Option A (All integers)
Explanation: Detailed explanation will be updated shortly.
Q15. The value of the integral ∫ e^(tan⁻¹x) * ( (1 + x + x²) / (1 + x²) ) dx is:
Correct Answer: Option A (x * e^(tan⁻¹x) + C)
Explanation: Detailed explanation will be updated shortly.
Q16. If ω is a complex cube root of unity and x, y, z are non-zero numbers such that x+y+z = a, x+yω+zω² = b, and x+yω²+zω = c, then the value of |a|²+|b|²+|c|² is:
Correct Answer: Option A (3(|x|²+|y|²+|z|²))
Explanation: Detailed explanation will be updated shortly.
Q17. The rate of change of the volume of a sphere is constant. If the radius is initially 3 units and after 3 seconds it is 6 units, the radius 'r' at time 't' is given by:
Correct Answer: Option A (r = (63t + 27)^(1/3))
Explanation: Detailed explanation will be updated shortly.
Q18. A man is known to speak the truth 4 out of 5 times. He throws a die and reports that it is a 'six'. The probability that it is actually a six is:
Correct Answer: Option A (4/9)
Explanation: Detailed explanation will be updated shortly.
Q19. The order and degree (if defined) of the differential equation y = x(dy/dx) + √(a²(dy/dx)² + b²) are respectively:
Correct Answer: Option A (1, 2)
Explanation: Detailed explanation will be updated shortly.
Q20. If a, b, c are the p-th, q-th, and r-th terms of a geometric progression (G.P.), then the value of the determinant |[log a, p, 1], [log b, q, 1], [log c, r, 1]| is:
Correct Answer: Option A (1)
Explanation: Detailed explanation will be updated shortly.