Q1. Let R be a relation on the set L of all lines in a plane, defined by R = {(l₁, l₂) : l₁ is perpendicular to l₂}. Then the relation R is:
Correct Answer: Option A (Reflexive and Symmetric)
Explanation: Detailed explanation will be updated shortly.
Q2. The domain of the function f(x) = sin⁻¹(x² - 6x + 10) is:
Correct Answer: Option A ([2, 4])
Explanation: Detailed explanation will be updated shortly.
Q3. If A is a square matrix of order 3 such that A³ = O (null matrix), then the inverse of (I - A) is:
Correct Answer: Option A (is:)
Explanation: Detailed explanation will be updated shortly.
Q4. Let A be a 3x3 matrix with det(A) = 4. The value of det(adj(2A)) is:
Correct Answer: Option A (= 4. The value of det(adj(2A)) is:)
Explanation: Detailed explanation will be updated shortly.
Q5. The integral ∫ eˣ [ (x² + 3x + 3) / (x+2)² ] dx is equal to:
Correct Answer: Option A (eˣ (x / (x+2)) + C)
Explanation: Detailed explanation will be updated shortly.
Q6. The number of points in the interval (0, 2π) where the function f(x) = |sin(x) - √3 cos(x)| is not differentiable is:
Correct Answer: Option A (1)
Explanation: Detailed explanation will be updated shortly.
Q7. Water is leaking from a conical funnel at a rate of 5 cm³/s. If the radius of the base of the funnel is 10 cm and the height is 20 cm, find the rate at which the slant height of the water is decreasing when the water is 8 cm from the top.
Correct Answer: Option A (5 / (18√5 π) cm/s)
Explanation: Detailed explanation will be updated shortly.
Q8. The system of linear equations x + y + z = 2, 2x + y - z = 3, and 3x + 2y + kz = 4 has a unique solution if:
Correct Answer: Option A (k ≠ 0)
Explanation: Detailed explanation will be updated shortly.
Q9. The value of the definite integral ∫[π/8, 3π/8] (sin²x) / (sin²x + cos²(π/2 - x)) dx is:
Correct Answer: Option A (π/8)
Explanation: Detailed explanation will be updated shortly.
Q10. The area (in sq. units) of the region bounded by the curves y = x|x| and the line y = 4 is:
Correct Answer: Option A (8)
Explanation: Detailed explanation will be updated shortly.
Q11. The solution of the differential equation (x² + 1) dy/dx + 2xy = 1/(x² + 1) with y(0) = 0 is:
Correct Answer: Option A (y(x² + 1) = tan⁻¹(x))
Explanation: Detailed explanation will be updated shortly.
Q12. If the vectors a = î - ĵ + k̂, b = î + 2ĵ - k̂, and c = λî + ĵ + λk̂ are coplanar, then the sum of all possible values of λ is:
Correct Answer: Option A (2)
Explanation: Detailed explanation will be updated shortly.
Q13. The length of the perpendicular from the point P(1, 2, 3) to the line (x-6)/3 = (y-7)/2 = (z-7)/-2 is:
Correct Answer: Option A (7)
Explanation: Detailed explanation will be updated shortly.
Q14. The image of the point (1, 3, 4) in the plane 2x - y + z + 3 = 0 is:
Correct Answer: Option A ((-3, 5, 2))
Explanation: Detailed explanation will be updated shortly.
Q15. The feasible region for an LPP is shown in the figure. Let Z = 3x - 4y be the objective function. The minimum value of Z occurs at:
Correct Answer: Option A ((0, 0))
Explanation: Detailed explanation will be updated shortly.
Q16. A factory has two machines, A and B. Past records show that machine A produces 60% of the items of output and machine B produces 40%. Further, 2% of the items produced by machine A and 1% of items produced by machine B were defective. If an item is drawn at random and is found to be defective, what is the probability that it was produced by machine B?
Correct Answer: Option A (1/3)
Explanation: Detailed explanation will be updated shortly.
Q17. The shortest distance between the lines r = (î + 2ĵ + k̂) + λ(î - ĵ + k̂) and r = (2î - ĵ - k̂) + μ(2î + ĵ + 2k̂) is:
Correct Answer: Option A (3/√2)
Explanation: Detailed explanation will be updated shortly.
Q18. For a random variable X, the probability distribution is given as P(X=k) = C(2/3)ᵏ for k = 0, 1, 2, 3... Then the value of C is:
Correct Answer: Option A (1/3)
Explanation: Detailed explanation will be updated shortly.
Q19. If f: R → R is defined by f(x) = x / (x² + 1), then the value of f(f(2)) is:
Correct Answer: Option A (2/5)
Explanation: Detailed explanation will be updated shortly.
Q20. The maximum area (in sq. units) of a rectangle that can be inscribed in a semi-circle of radius 'r' with one of its sides on the diameter is:
Correct Answer: Option A (r²)
Explanation: Detailed explanation will be updated shortly.