Q1. A function f: S → R is defined by f(A) = trace(A), where S is the set of all 2x2 real matrices such that A² = A. The function f is:
Correct Answer: Option A (= trace(A), where S is the set of all 2x2 real matrices such that A² = A. The function f is:)
Explanation: Detailed explanation will be updated shortly.
Q2. If A is a square matrix of order 3 with |A| = 4, what is the value of |adj(adj(2A))|?
Correct Answer: Option A ()|?)
Explanation: Detailed explanation will be updated shortly.
Q3. The number of real solutions to the equation `tan⁻¹(√(x(x+1))) + sin⁻¹(√(x²+x+1)) = π/2` is:
Correct Answer: Option A (0)
Explanation: Detailed explanation will be updated shortly.
Q4. Consider the following statements regarding the function f(x) = |x-1| + [x] for x ∈ (0, 2), where [.] denotes the greatest integer function.
Statement I: f(x) is discontinuous at x = 1.
Statement II: f(x) is not differentiable at x = 1.
Correct Answer: Option A (Both Statement I and Statement II are true.)
Explanation: Detailed explanation will be updated shortly.
Q5. The tangent to the curve y = ∫₀ˣ (t² - 3t + 2) dt at x = 3 makes an angle θ with the positive x-axis. The value of tan(θ) is:
Correct Answer: Option A (1)
Explanation: Detailed explanation will be updated shortly.
Q6. The integral I = ∫ e^(2x) * ( (2sin(x)cos(x) - 1) / sin²(x) ) dx is equal to:
Correct Answer: Option A (e^(2x) cot(x) + C)
Explanation: Detailed explanation will be updated shortly.
Q7. The area of the region bounded by the curves y = |x² - 4| and the line y = 5 is:
Correct Answer: Option A (36 sq. units)
Explanation: Detailed explanation will be updated shortly.
Q8. The solution of the differential equation (x² + y²)dy = xy dx can be represented by a family of:
Correct Answer: Option A (Circles centered at the origin)
Explanation: Detailed explanation will be updated shortly.
Q9. Let a, b, and c be three non-coplanar vectors. If r is a vector such that r ⋅ a = r ⋅ b = r ⋅ c = 0, then the vector r must be:
Correct Answer: Option A (A unit vector)
Explanation: Detailed explanation will be updated shortly.
Q10. A plane π passes through the point (1, 1, 1) and is parallel to the vectors b = î - ĵ + k̂ and c = î + ĵ - k̂. The perpendicular distance of the plane from the origin is:
Correct Answer: Option A (1/√2)
Explanation: Detailed explanation will be updated shortly.
Q11. In a Linear Programming Problem, the feasible region is a convex polygon bounded by the points (0, 10), (5, 5), (15, 15), and (0, 20). If the objective function is Z = px + 2y, where p > 0, and the maximum value of Z occurs at both (15, 15) and (0, 20), then the value of p is:
Correct Answer: Option A (2/3)
Explanation: Detailed explanation will be updated shortly.
Q12. A committee of 5 is to be formed from 6 men and 4 women. The probability that the committee has at least 3 men is:
Correct Answer: Option A (23/42)
Explanation: Detailed explanation will be updated shortly.
Q13. If f(x) = log( (1+x) / (1-x) ), then f( 2x / (1+x²) ) is equal to:
Correct Answer: Option A ([f(x)]²)
Explanation: Detailed explanation will be updated shortly.
Q14. Let matrix A = [[cos(θ), -sin(θ)], [sin(θ), cos(θ)]]. 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.
Q15. The value of the definite integral ∫₁ᵉ (log x)³ / x dx is:
Correct Answer: Option A (1/4)
Explanation: Detailed explanation will be updated shortly.
Q16. The order and degree of the differential equation representing the family of curves y² = 2c(x + √c), where c is a positive parameter, are respectively:
Correct Answer: Option A (1, 3)
Explanation: Detailed explanation will be updated shortly.
Q17. The shortest distance between the lines x/2 = y/2 = z/1 and (x+2)/-1 = (y-4)/8 = (z-5)/4 is:
Correct Answer: Option A (3√2)
Explanation: Detailed explanation will be updated shortly.
Q18. A random variable X has the following probability distribution:
| X | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| P(X) | k | 2k | 2k | k |
The variance of X is:
Correct Answer: Option A (1)
Explanation: Detailed explanation will be updated shortly.
Q19. The rate of change of the volume of a sphere is equal to the rate of change of its surface area. The radius of the sphere is:
Correct Answer: Option A (1 unit)
Explanation: Detailed explanation will be updated shortly.
Q20. If A and B are two events such that P(A) > 0 and P(B) ≠ 1, then P(A' | B') is equal to:
Correct Answer: Option A (> 0 and P(B) ≠ 1, then P(A' | B') is equal to:)
Explanation: Detailed explanation will be updated shortly.