Maths MCQ practice

 

This is a comprehensive mock test based on the CBSE Class 12 Mathematics syllabus and the structure of the papers you provided. Since you are teaching grades 10-12 and preparing for KVS/NVS exams, these are designed to match that "Board Level" difficulty. CBSE Class 12 Math: Mock Section A Time: 60 Minutes | Marks: 30 Multiple Choice Questions (1 Mark Each) 1. Let R be a relation on the set L of all lines in a plane defined by R = \{(L_1, L_2) : L_1 \perp L_2\}. Then R is: (A) Reflexive (B) Symmetric (C) Transitive (D) Equivalence 2. The principal value of \cos^{-1}\left(\cos \frac{7\pi}{6}\right) is: (A) \frac{7\pi}{6} (B) \frac{5\pi}{6} (C) \frac{\pi}{6} (D) \frac{-\pi}{6} 3. If A is a square matrix such that A^2 = A, then (I + A)^3 - 7A is equal to: (A) A (B) I - A (C) I (D) 3A 4. If A = \begin{bmatrix} \alpha & 0 \\ 1 & 1 \end{bmatrix} and B = \begin{bmatrix} 1 & 0 \\ 5 & 1 \end{bmatrix}, then value of \alpha for which A^2 = B is: (A) 1 (B) -1 (C) 4 (D) No real values 5. The value of \begin{vmatrix} 0 & a-b & a-c \\ b-a & 0 & b-c \\ c-a & c-b & 0 \end{vmatrix} is: (A) a+b+c (B) abc (C) 0 (D) 1 6. If f(x) = \begin{cases} \frac{1-\cos 4x}{8x^2}, & x \neq 0 \\ k, & x = 0 \end{cases} is continuous at x = 0, then k is: (A) 1 (B) 2 (C) 1/2 (D) 4 7. The derivative of \log_{10} x with respect to x is: (A) \frac{1}{x} (B) \frac{1}{x \log_e 10} (C) \frac{\log_{10} e}{x} (D) Both (B) and (C) 8. The rate of change of the area of a circle with respect to its radius r at r = 6 cm is: (A) 10\pi (B) 12\pi (C) 8\pi (D) 11\pi 9. The interval in which y = x^2 e^{-x} is increasing is: (A) (-\infty, \infty) (B) (-2, 0) (C) (2, \infty) (D) (0, 2) 10. \int \frac{dx}{\sin^2 x \cos^2 x} is equal to: (A) \tan x + \cot x + C (B) \tan x - \cot x + C (C) \tan x \cot x + C (D) \tan x - \cot 2x + C 11. The value of \int_{-\pi/2}^{\pi/2} \sin^7 x \, dx is: (A) 0 (B) 1 (C) 2 (D) \pi 12. The order and degree of the differential equation \left(\frac{d^2y}{dx^2}\right)^3 + \left(\frac{dy}{dx}\right)^2 + \sin\left(\frac{dy}{dx}\right) + 1 = 0 are: (A) Order 2, Degree 3 (B) Order 2, Degree 2 (C) Order 2, Degree not defined (D) Order 3, Degree 2 13. The integrating factor of the differential equation x \frac{dy}{dx} - y = 2x^2 is: (A) e^{-x} (B) e^{-y} (C) 1/x (D) x 14. If |\vec{a}| = 10, |\vec{b}| = 2 and \vec{a} \cdot \vec{b} = 12, then |\vec{a} \times \vec{b}| is: (A) 5 (B) 10 (C) 14 (D) 16 15. The vector projection of \vec{a} = 2\hat{i} + 3\hat{j} + 2\hat{k} on \vec{b} = \hat{i} + 2\hat{j} + \hat{k} is: (A) \frac{10}{\sqrt{6}} (B) \frac{10}{6}(\hat{i} + 2\hat{j} + \hat{k}) (C) \frac{5}{3}(\hat{i} + 2\hat{j} + \hat{k}) (D) Both (B) and (C) 16. The distance of the plane 2x - 3y + 4z - 6 = 0 from the origin is: (A) \frac{6}{\sqrt{29}} (B) \frac{-6}{\sqrt{29}} (C) \frac{6}{29} (D) \sqrt{29} 17. If P(A) = 0.4, P(B) = 0.8 and P(B|A) = 0.6, then P(A \cup B) is: (A) 0.24 (B) 0.96 (C) 0.48 (D) 0.16 18. The corner points of the feasible region for an LPP are (0,0), (5,0), (3,4) and (0,5). The maximum value of Z = 4x + 3y is: (A) 20 (B) 24 (C) 15 (D) 18 19. A bag contains 3 red and 5 black balls. If two balls are drawn at random without replacement, the probability that both are red is: (A) 3/28 (B) 9/64 (C) 3/56 (D) 3/8 20. If A is a 3 \times 3 matrix and |A| = 5, then |adj A| is: (A) 5 (B) 25 (C) 125 (D) 15 21. The direction cosines of the y-axis are: (A) (1, 0, 0) (B) (0, 1, 0) (C) (0, 0, 1) (D) (1, 1, 1) 22. If \vec{a} and \vec{b} are unit vectors and \theta is the angle between them, then \vec{a} + \vec{b} is a unit vector if \theta is: (A) \pi/4 (B) \pi/3 (C) \pi/2 (D) 2\pi/3 23. The function f(x) = |x| at x = 0 is: (A) Continuous and differentiable (B) Neither continuous nor differentiable (C) Continuous but not differentiable (D) None of these 24. The area bounded by the curve y = \cos x between x=0 and x=2\pi is: (A) 0 units (B) 2 units (C) 4 units (D) 1 unit 25. The general solution of \frac{dy}{dx} = e^{x+y} is: (A) e^x + e^{-y} = C (B) e^x - e^{-y} = C (C) e^{-x} + e^y = C (D) e^x + e^y = C Assertion-Reason Questions (1 Mark Each) Directions: (A) Both A and R are true and R is the correct explanation of A. (B) Both A and R are true but R is NOT the correct explanation. (C) A is true, R is false. (D) A is false, R is true. 26. Assertion (A): The function f(x) = x^3 - 3x^2 + 3x - 100 is increasing on \mathbb{R}. Reason (R): f'(x) \geq 0 for all x \in \mathbb{R}. 27. Assertion (A): The angle between the vectors \hat{i} and \hat{j} is 90^\circ. Reason (R): The dot product of two perpendicular vectors is zero. 28. Assertion (A): The value of \int_0^2 |x-1| \, dx is 1. Reason (R): \int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx. 29. Assertion (A): If P(A) = 0.3 and P(B) = 0.4, then P(A \cap B) = 0.12. Reason (R): For any two events A and B, P(A \cap B) = P(A) \cdot P(B). 30. Assertion (A): A square matrix A is invertible if and only if A is non-singular. Reason (R): |A| = 0 for a singular matrix. Answer Key for your reference: 1.B, 2.B, 3.C, 4.D, 5.C, 6.A, 7.D, 8.B, 9.D, 10.B, 11.A, 12.C, 13.C, 14.D, 15.D, 16.A, 17.B, 18.B, 19.A, 20.B, 21.B, 22.D, 23.C, 24.C, 25.A, 26.A, 27.A, 28.A, 29.D (R is only true for independent events), 30.B. Would you like me to provide the step-by-step solutions for any of these specific questions?