Class 12 Mathematics — CBSE Board Sample Paper

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Class 12 Mathematics — CBSE Board Sample Paper — solved practice questions

8 Class 12 Board Sample Papers (CBSE) questions with step-by-step solutions. Attempt each, then reveal the answer.

1. Q1medium

   If for a square matrix A of order 3, A·(adj A) = [[2025, 0, 0], [0, 2025, 0], [0, 0, 2025]], then the value of |A| + |adj A| is equal to:

   • A1
   • B2025 + 1
   • C2025² + 45
   • D2025 + 2025²
   Show answer & solution

   Correct answer: (D) 2025 + 2025²

   Since A·(adj A) = |A|·I₃ = 2025 I₃, we get |A| = 2025. Also |adj A| = |A|^(3−1) = 2025². Hence |A| + |adj A| = 2025 + 2025².

2. Q2medium

   Assume X, Y, Z, W and P are matrices of order 2×n, 3×k, 2×p, n×3 and p×k, respectively. Then the restriction on n, k and p so that PY + WY will be defined are:

   • Ak = 3, p = n
   • Bk is arbitrary, p = 2
   • Cp is arbitrary, k = 3
   • Dk = 2, p = 3
   Show answer & solution

   Correct answer: (A) k = 3, p = n

   For PY (p×k times 3×k) to be defined, k = 3, giving order p×3; WY (n×3 times 3×k) has order n×k. For the sum PY + WY the orders must match, so p = n. Hence k = 3 and p = n.

3. Q3easy

   The interval in which the function f defined by f(x) = eˣ is strictly increasing, is

   • A[1, ∞)
   • B(−∞, 0)
   • C(−∞, ∞)
   • D(0, ∞)
   Show answer & solution

   Correct answer: (C) (−∞, ∞)

   f'(x) = eˣ > 0 for every real x, so f is strictly increasing on the whole real line (−∞, ∞).

4. Q4easy

   If A and B are non-singular matrices of the same order with |A| = 5, then the value of |B⁻¹AB|² is equal to

   • A5
   • B5²
   • C5⁴
   • D5⁵
   Show answer & solution

   Correct answer: (B) 5²

   |B⁻¹AB| = |B⁻¹||A||B| = |A| = 5, so |B⁻¹AB|² = 5².

5. Q5medium

   The value of 'n', such that the differential equation xⁿ (dy/dx) = y(logₑ y − logₑ x + 1); (x, y ∈ ℝ⁺) is homogeneous, is

   • A0
   • B1
   • C2
   • D3
   Show answer & solution

   Correct answer: (B) 1

   Writing dy/dx = (y/xⁿ)(logₑ(y/x) + 1), the right-hand side is a homogeneous function of degree 0 only when n = 1.

6. Q6medium

   If the points (x₁, y₁), (x₂, y₂) and (x₁ + x₂, y₁ + y₂) are collinear, then x₁y₂ is equal to

   • Ax₂y₁
   • Bx₁y₁
   • Cx₂y₂
   • Dx₁x₂
   Show answer & solution

   Correct answer: (A) x₂y₁

   The collinearity determinant condition reduces to x₁y₂ − x₂y₁ = 0, hence x₁y₂ = x₂y₁.

7. Q7medium

   If A = [[0, 1, c], [−1, a, b], [−2, 3, 0]] is a skew-symmetric matrix, then the value of a − b − c is

   • A1
   • B2
   • C3
   • D4
   Show answer & solution

   Correct answer: (A) 1

   In a skew-symmetric matrix the diagonal entries are zero (a = 0) and aᵢⱼ = −aⱼᵢ, giving c = 2 and b = −3. Hence a − b − c = 0 − (−3) − 2 = 1.

8. Q8medium

   For any two events A and B, if P(Ā) = 1/2, P(B̄) = 2/3 and P(A ∩ B) = 1/4, then P(Ā/B̄) equals:

   • A3/8
   • B8/9
   • C5/8
   • D1/4
   Show answer & solution

   Correct answer: (C) 5/8

   Here P(A) = 1/2 and P(B) = 1/3, so P(A∪B) = 1/2 + 1/3 − 1/4 = 7/12. Then P(Ā/B̄) = P((A∪B)')/P(B') = (1 − 7/12)/(2/3) = (5/12)/(2/3) = 5/8.

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