These essential MCQ questions are selected from the most important topics in CBSE Class 12 Mathematics 2026 as per the latest NCERT syllabus. Sections include Relations & Functions, Matrices & Determinants, Differential Calculus, Integral Calculus, Vectors & 3D Geometry, and Probability. For truly unlimited daily MCQ practice, visit Vooo AI Education.
📊 CBSE Class 12 Mathematics
1A function f: A → B is said to be bijective if it is:
Answer: C — Both one-one and onto
A bijective function is both injective (one-one: each element in domain maps to a unique element in codomain) and surjective (onto: every element in codomain has at least one pre-image). A bijective function has a perfect inverse function. Only bijective functions can be inverted. One-one alone is injective; onto alone is surjective. A bijection establishes a one-to-one correspondence between two sets.
A bijective function is both injective (one-one: each element in domain maps to a unique element in codomain) and surjective (onto: every element in codomain has at least one pre-image). A bijective function has a perfect inverse function. Only bijective functions can be inverted. One-one alone is injective; onto alone is surjective. A bijection establishes a one-to-one correspondence between two sets.
2If A is a 3×3 matrix and |A| = 5, then |2A| equals:
Answer: C — 40
For an n×n matrix A, |kA| = kⁿ |A|. Here A is 3×3 (n=3) and k=2: |2A| = 2³ × |A| = 8 × 5 = 40. Each row of A gets multiplied by k when we compute kA, so the determinant (being multilinear) gets multiplied by k once per row, giving kⁿ total. This is a key property: for a 2×2 matrix, |2A| = 4|A|; for 3×3, |2A| = 8|A|.
For an n×n matrix A, |kA| = kⁿ |A|. Here A is 3×3 (n=3) and k=2: |2A| = 2³ × |A| = 8 × 5 = 40. Each row of A gets multiplied by k when we compute kA, so the determinant (being multilinear) gets multiplied by k once per row, giving kⁿ total. This is a key property: for a 2×2 matrix, |2A| = 4|A|; for 3×3, |2A| = 8|A|.
3The derivative of sin⁻¹(x) with respect to x is:
Answer: B — 1/√(1-x²)
d/dx[sin⁻¹(x)] = 1/√(1-x²), for |x| < 1. Similarly: d/dx[cos⁻¹(x)] = -1/√(1-x²); d/dx[tan⁻¹(x)] = 1/(1+x²); d/dx[cot⁻¹(x)] = -1/(1+x²). These are standard inverse trigonometric derivatives that must be memorised for Class 12. They are derived using implicit differentiation of the corresponding trigonometric function.
d/dx[sin⁻¹(x)] = 1/√(1-x²), for |x| < 1. Similarly: d/dx[cos⁻¹(x)] = -1/√(1-x²); d/dx[tan⁻¹(x)] = 1/(1+x²); d/dx[cot⁻¹(x)] = -1/(1+x²). These are standard inverse trigonometric derivatives that must be memorised for Class 12. They are derived using implicit differentiation of the corresponding trigonometric function.
4∫eˣ(f(x) + f'(x))dx equals:
Answer: A — eˣ f(x) + C
This is a standard integration formula: ∫eˣ[f(x) + f'(x)]dx = eˣf(x) + C. Proof by differentiation: d/dx[eˣf(x)] = eˣf(x) + eˣf'(x) = eˣ[f(x) + f'(x)]. This formula is widely used in CBSE Class 12. Example: ∫eˣ(sin x + cos x)dx = eˣ sin x + C, where f(x) = sin x and f'(x) = cos x.
This is a standard integration formula: ∫eˣ[f(x) + f'(x)]dx = eˣf(x) + C. Proof by differentiation: d/dx[eˣf(x)] = eˣf(x) + eˣf'(x) = eˣ[f(x) + f'(x)]. This formula is widely used in CBSE Class 12. Example: ∫eˣ(sin x + cos x)dx = eˣ sin x + C, where f(x) = sin x and f'(x) = cos x.
5The angle between two vectors a⃗ and b⃗ is 60°, |a⃗| = 2, |b⃗| = 3. Then a⃗·b⃗ equals:
Answer: B — 3
a⃗·b⃗ = |a⃗||b⃗|cosθ = 2 × 3 × cos60° = 6 × (1/2) = 3. The dot product formula connects magnitude and angle between vectors. If a⃗·b⃗ = 0, vectors are perpendicular. If θ = 0°, dot product = |a⃗||b⃗| (maximum). The dot product is a scalar. It is used to find the angle between vectors, work done (W = F⃗·d⃗), and projections of one vector on another.
a⃗·b⃗ = |a⃗||b⃗|cosθ = 2 × 3 × cos60° = 6 × (1/2) = 3. The dot product formula connects magnitude and angle between vectors. If a⃗·b⃗ = 0, vectors are perpendicular. If θ = 0°, dot product = |a⃗||b⃗| (maximum). The dot product is a scalar. It is used to find the angle between vectors, work done (W = F⃗·d⃗), and projections of one vector on another.
6The direction cosines of a line making equal angles with all three coordinate axes are:
Answer: B — (1/√3, 1/√3, 1/√3)
If a line makes equal angles α with all three axes, then l = m = n = cosα. Direction cosines satisfy l² + m² + n² = 1. So 3cos²α = 1 → cosα = 1/√3. Direction cosines = (1/√3, 1/√3, 1/√3). Direction cosines (l, m, n) are the cosines of angles a line makes with the x, y, and z axes respectively. Note: (1,1,1) are direction ratios, not cosines, as they don't satisfy l²+m²+n²=1.
If a line makes equal angles α with all three axes, then l = m = n = cosα. Direction cosines satisfy l² + m² + n² = 1. So 3cos²α = 1 → cosα = 1/√3. Direction cosines = (1/√3, 1/√3, 1/√3). Direction cosines (l, m, n) are the cosines of angles a line makes with the x, y, and z axes respectively. Note: (1,1,1) are direction ratios, not cosines, as they don't satisfy l²+m²+n²=1.
7If P(A) = 0.3 and P(B|A) = 0.5, then P(A∩B) equals:
Answer: C — 0.15
P(B|A) = P(A∩B)/P(A) → P(A∩B) = P(B|A) × P(A) = 0.5 × 0.3 = 0.15. This is the multiplication theorem of probability. P(B|A) is the conditional probability of B given A has already occurred. Bayes' theorem extends this: P(A|B) = P(B|A)·P(A) / P(B). Conditional probability is fundamental in topics like Bayes' theorem and independent events [where P(A∩B) = P(A)·P(B)].
P(B|A) = P(A∩B)/P(A) → P(A∩B) = P(B|A) × P(A) = 0.5 × 0.3 = 0.15. This is the multiplication theorem of probability. P(B|A) is the conditional probability of B given A has already occurred. Bayes' theorem extends this: P(A|B) = P(B|A)·P(A) / P(B). Conditional probability is fundamental in topics like Bayes' theorem and independent events [where P(A∩B) = P(A)·P(B)].
8The maximum value of f(x) = sin x + cos x is:
Answer: C — √2
f(x) = sin x + cos x = √2 sin(x + π/4). Maximum value = √2 (when sin(x+π/4) = 1, i.e., x = π/4). Alternatively: f'(x) = cos x - sin x = 0 → tan x = 1 → x = π/4. f(π/4) = sin(π/4) + cos(π/4) = 1/√2 + 1/√2 = √2. f''(π/4) = -sin(π/4) - cos(π/4) = -√2 < 0, confirming maximum. Minimum value = -√2.
f(x) = sin x + cos x = √2 sin(x + π/4). Maximum value = √2 (when sin(x+π/4) = 1, i.e., x = π/4). Alternatively: f'(x) = cos x - sin x = 0 → tan x = 1 → x = π/4. f(π/4) = sin(π/4) + cos(π/4) = 1/√2 + 1/√2 = √2. f''(π/4) = -sin(π/4) - cos(π/4) = -√2 < 0, confirming maximum. Minimum value = -√2.
9The order of the differential equation d²y/dx² + (dy/dx)³ + y = 0 is:
Answer: B — 2
The order of a differential equation is the highest order derivative present. Here, d²y/dx² is the highest derivative, so the order is 2. The degree is the power of the highest order derivative (after clearing fractions/radicals) — here degree = 1 (d²y/dx² has power 1). Note: the (dy/dx)³ term has order 1 (not the highest). Always identify the highest derivative first to determine order.
The order of a differential equation is the highest order derivative present. Here, d²y/dx² is the highest derivative, so the order is 2. The degree is the power of the highest order derivative (after clearing fractions/radicals) — here degree = 1 (d²y/dx² has power 1). Note: the (dy/dx)³ term has order 1 (not the highest). Always identify the highest derivative first to determine order.
10The area of a parallelogram with sides a⃗ and b⃗ is given by:
Answer: B — |a⃗ × b⃗|
Area of parallelogram = |a⃗ × b⃗| = |a⃗||b⃗|sinθ, where θ is the angle between the vectors. The cross product gives a vector perpendicular to both a⃗ and b⃗, whose magnitude equals the area of the parallelogram. Area of triangle with sides a⃗ and b⃗ = (1/2)|a⃗ × b⃗|. If a⃗ × b⃗ = 0, vectors are parallel. The cross product is used in torque (τ = r × F) and angular momentum calculations.
Area of parallelogram = |a⃗ × b⃗| = |a⃗||b⃗|sinθ, where θ is the angle between the vectors. The cross product gives a vector perpendicular to both a⃗ and b⃗, whose magnitude equals the area of the parallelogram. Area of triangle with sides a⃗ and b⃗ = (1/2)|a⃗ × b⃗|. If a⃗ × b⃗ = 0, vectors are parallel. The cross product is used in torque (τ = r × F) and angular momentum calculations.
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