26 free Mathematics MCQs written in the Class 12 Boards exam pattern, each with a step-by-step solution. Tap an option for instant feedback — no sign-up, no ads.
Revise with smart repetition →✍️ These are original practice questions written in the Class 12 Boards pattern for the Class 12 board exams (CBSE pattern) — not questions from any official past paper.
Q1. The derivative of sin x with respect to x is:
A. cos x — d/dx (sin x) = cos x.
Q2. The value of ∫ cos x dx is:
A. sin x + C — ∫ cos x dx = sin x + C, since the derivative of sin x is cos x.
Q3. The derivative of eˣ with respect to x is:
B. eˣ — The exponential function is its own derivative: d/dx (eˣ) = eˣ.
Q4. The order of a matrix with 2 rows and 3 columns is:
B. 2 × 3 — Order is written as (rows × columns), so a matrix with 2 rows and 3 columns is of order 2 × 3.
Q5. The value of ∫ (1/x) dx is:
B. ln|x| + C — ∫ (1/x) dx = ln|x| + C.
Q6. The derivative of any constant function is:
B. 0 — The derivative of a constant is 0, because a constant function does not change.
Q7. The value of lim (x→0) (sin x)/x is:
B. 1 — This is a standard limit: lim (x→0) (sin x)/x = 1.
Q8. The determinant of the 2×2 identity matrix is:
B. 1 — The identity matrix [[1,0],[0,1]] has determinant (1)(1) − (0)(0) = 1.
Q9. The derivative of x³ with respect to x is:
A. 3x² — By the power rule, d/dx (xⁿ) = n·xⁿ⁻¹, so d/dx (x³) = 3x².
Q10. If P(A) = 0.6, then the probability of 'not A' is:
B. 0.4 — P(not A) = 1 − P(A) = 1 − 0.6 = 0.4.
Q11. The function f: R → R defined by f(x) = x² is:
D. neither one-one nor onto — Since f(1) = f(-1) = 1, f is not one-one; and no negative real number is an output, so with codomain R it is not onto. Hence it is neither one-one nor onto.
Q12. The principal value of sin⁻¹(sin(2π/3)) is:
C. π/3 — Since 2π/3 lies outside [-π/2, π/2], use sin(2π/3) = sin(π - 2π/3) = sin(π/3) = √3/2. Thus sin⁻¹(√3/2) = π/3.
Q13. Every element on the main diagonal of a skew-symmetric matrix is equal to:
C. 0 — A skew-symmetric matrix satisfies aᵢⱼ = -aⱼᵢ. For diagonal entries aᵢᵢ = -aᵢᵢ, giving 2aᵢᵢ = 0, so every diagonal element is 0.
Q14. If A is a 3 × 3 matrix with |A| = 5, then the value of |2A| is:
D. 40 — For an n × n matrix, |kA| = kⁿ|A|. Here n = 3, so |2A| = 2³ × 5 = 8 × 5 = 40.
Q15. The function f(x) = |x| is:
A. continuous everywhere but not differentiable at x = 0 — |x| is continuous for all real x, but at x = 0 the left-hand derivative (-1) and right-hand derivative (+1) differ, so it is not differentiable there.
Q16. The function f(x) = x² - 4x + 3 has a local minimum at x =:
B. 2 — f'(x) = 2x - 4 = 0 gives x = 2, and f''(x) = 2 > 0, confirming a minimum at x = 2.
Q17. ∫ sec²x dx equals:
A. tan x + C — Since d/dx(tan x) = sec²x, the integral of sec²x is tan x + C.
Q18. The area of the region bounded by the line y = x, the x-axis, and the lines x = 0 and x = 2 is:
D. 2 — The area = ∫₀² x dx = [x²/2]₀² = 4/2 = 2 (equivalently, a right triangle of base 2 and height 2).
Q19. The order of the differential equation d²y/dx² + 3(dy/dx)³ + y = 0 is:
A. 2 — Order is the highest derivative present. Here the highest is d²y/dx², so the order is 2 (the cube affects degree, not order).
Q20. If a and b are two non-zero vectors such that a · b = 0, then a and b are:
B. perpendicular — a · b = |a||b|cosθ. Since a and b are non-zero, a · b = 0 forces cosθ = 0, i.e. θ = 90°, so the vectors are perpendicular.
Q21. The direction cosines of the y-axis are:
A. (0, 1, 0) — The y-axis makes 90°, 0°, 90° with the x, y, z axes respectively, giving direction cosines (cos90°, cos0°, cos90°) = (0, 1, 0).
Q22. In a linear programming problem, the feasible region is always:
C. a convex set — The feasible region is the intersection of half-planes defined by linear constraints, and such an intersection is always a convex set.
Q23. If A and B are independent events with P(A) = 0.5 and P(B) = 0.4, then P(A ∪ B) equals:
B. 0.7 — For independent events P(A ∩ B) = P(A)P(B) = 0.2, so P(A ∪ B) = P(A) + P(B) - P(A ∩ B) = 0.5 + 0.4 - 0.2 = 0.7.
Q24. If P(A ∩ B) = 0.2 and P(B) = 0.5, then the conditional probability P(A | B) is:
A. 0.4 — P(A | B) = P(A ∩ B) / P(B) = 0.2 / 0.5 = 0.4.
Q25. ∫ 1/(1 + x²) dx equals:
B. tan⁻¹x + C — The standard result is ∫ dx/(1 + x²) = tan⁻¹x + C, since d/dx(tan⁻¹x) = 1/(1 + x²).
Q26. The derivative of tan x with respect to x is:
B. sec²x — Writing tan x = sin x / cos x and applying the quotient rule gives d/dx(tan x) = sec²x.
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