Q1. Let S be the set of all non-singular 2x2 matrices. A relation R is defined on S as A R B if and only if det(A⁻¹B) = 1. This relation R is:
Correct Answer: Option A (Reflexive and Symmetric but not Transitive)
Explanation: Detailed explanation will be updated shortly.
Q2. If f(x) = sin⁻¹(sin x) and g(x) = cos⁻¹(cos x), then the value of f(10) + g(10) is:
Correct Answer: Option A (20)
Explanation: Detailed explanation will be updated shortly.
Q3. Let A be a 3x3 matrix with det(A) = 4. If B = adj(2A), then the value of det(B) is:
Correct Answer: Option A (= 4. If B = adj(2A), then the value of det(B) is:)
Explanation: Detailed explanation will be updated shortly.
Q4. The system of linear equations: x + ky + 3z = 0, 3x + ky - 2z = 0, 2x + 4y - 3z = 0 has a non-trivial solution. For this value of k, what is the value of x/y?
Correct Answer: Option A (5)
Explanation: Detailed explanation will be updated shortly.
Q5. The function g(x) = (x² - 1)|x² - 3x + 2| + cos(|x|) is non-differentiable at:
Correct Answer: Option A (x = 1 and x = 2)
Explanation: Detailed explanation will be updated shortly.
Q6. The coordinates of a moving particle are given by x = a(cos t + t sin t) and y = a(sin t - t cos t), where 'a' is a constant. The rate of change of the distance of the particle from the origin with respect to time 't' is:
Correct Answer: Option A (a)
Explanation: Detailed explanation will be updated shortly.
Q7. The integral ∫(x² - 1) / (x³√(2x⁴ - 2x² + 1)) dx is equal to:
Correct Answer: Option A (√(2x⁴ - 2x² + 1) / x² + C)
Explanation: Detailed explanation will be updated shortly.
Q8. Let f(x) be a continuous function on [0, 2] such that f(x) + f(2-x) = 2x² - 4x + 4. The value of the integral I = ∫₀² f(x) dx is:
Correct Answer: Option A (4/3)
Explanation: Detailed explanation will be updated shortly.
Q9. The area of the region bounded by the curve y = |logₑ(x)|, the x-axis, and the lines x = 1/e and x = e is:
Correct Answer: Option A (2)
Explanation: Detailed explanation will be updated shortly.
Q10. The equation of a curve passing through the point (1, π/4) is given by the differential equation (x cos(y/x) + y sin(y/x))y dx = (y sin(y/x) - x cos(y/x))x dy. The equation of the curve is:
Correct Answer: Option A (yx sec(y/x) = π/4)
Explanation: Detailed explanation will be updated shortly.
Q11. If ā and b̄ are two non-collinear unit vectors and |ā + b̄| = √3, then the value of the dot product (2ā - 5b̄) ⋅ (3ā + b̄) is:
Correct Answer: Option A (-11/2)
Explanation: Detailed explanation will be updated shortly.
Q12. A variable plane is at a constant distance 'p' from the origin and cuts the coordinate axes at points A, B, and C. The locus of the centroid of the tetrahedron OABC is:
Correct Answer: Option A (x⁻² + y⁻² + z⁻² = p⁻²)
Explanation: Detailed explanation will be updated shortly.
Q13. A bag contains 5 red and 3 black balls. A second bag contains 2 red and 6 black balls. One bag is chosen at random and a ball is drawn from it. The ball is found to be red and is not replaced. The probability that the next ball drawn from the same bag is also red is:
Correct Answer: Option A (5/7)
Explanation: Detailed explanation will be updated shortly.
Q14. The objective function is Z = Px + Qy. In an LPP, the maximum value of Z occurs at two adjacent corner points of the feasible region, (4, 3) and (5, 0). Which of the following is true?
Correct Answer: Option A (P = Q)
Explanation: Detailed explanation will be updated shortly.
Q15. If y = x^(x^x), then the value of dy/dx at x = 1 is:
Correct Answer: Option A (0)
Explanation: Detailed explanation will be updated shortly.
Q16. If A is an invertible matrix of order 3 and det(A) = 2, then det(A⁻¹ ⋅ adj(A)) is equal to:
Correct Answer: Option A (= 2, then det(A⁻¹ ⋅ adj(A)) is equal to:)
Explanation: Detailed explanation will be updated shortly.
Q17. The reflection of the point P(1, 2, 3) in the plane x - y + 2z = 9 is:
Correct Answer: Option A ((5/3, 4/3, 13/3))
Explanation: Detailed explanation will be updated shortly.
Q18. In a hurdle race, a runner must cross 8 hurdles. The probability of clearing each hurdle is 5/6. The probability that he will knock down fewer than 2 hurdles is:
Correct Answer: Option A ((5/6)⁸)
Explanation: Detailed explanation will be updated shortly.
Q19. Let the function f: R → R be defined by f(x) = x / (1 + |x|). Then f is:
Correct Answer: Option A (One-one and onto)
Explanation: Detailed explanation will be updated shortly.
Q20. The number of points in the interval (0, π) where the function f(x) = max{tan x, cot x} is non-differentiable is:
Correct Answer: Option A (1)
Explanation: Detailed explanation will be updated shortly.