CBSE Class 12 Mathematics Board Paper 4 — MCQs with Answers

8 solved questions from this paper

Answer and worked explanation shown for each. The remaining 12 are timed and scored when you sign in free.

1. Q1Easy

   The number of one-one functions from a set containing 3 elements to a set containing 4 elements is:

   A.12

   B.64

   C.81

   D.24✓ Correct

   Answer: D. 24

   Explanation: A one-one function must send the three elements to distinct images, so there are 4 choices for the first, 3 for the second and 2 for the third, giving 4 × 3 × 2 = 24. The count 4³ = 64 counts all functions, one-one or not, because it allows images to repeat.

2. Q2Easy

   If A is a matrix of order 3 × 4 and B is a matrix of order 4 × 3, then the order of AB is:

   A.3 × 3✓ Correct

   B.4 × 4

   C.3 × 4

   D.4 × 3

   Answer: A. 3 × 3

   Explanation: The product of an m × n matrix with an n × p matrix has order m × p, the matching inner dimensions being consumed by the multiplication. Here 3 × 4 times 4 × 3 gives 3 × 3. The order 4 × 4 belongs to BA, which is a different product.

3. Q3Easy

   The rate of change of the area of a circle with respect to its radius, when the radius is 6 cm, is:

   A.6π cm²/cm

   B.12π cm²/cm✓ Correct

   C.36π cm²/cm

   D.24π cm²/cm

   Answer: B. 12π cm²/cm

   Explanation: With A = πr², the derivative is dA/dr = 2πr, which at r = 6 cm equals 12π cm² per cm. The value 36π is the area itself at that radius, not its rate of change with respect to the radius.

4. Q4Medium

   If A = [[2, −1], [3, 4]] is written as the sum of a symmetric and a skew-symmetric matrix, the skew-symmetric part is:

   A.[[0, 2], [−2, 0]]

   B.[[2, 1], [1, 4]]

   C.[[0, −2], [2, 0]]✓ Correct

   D.[[0, −4], [4, 0]]

   Answer: C. [[0, −2], [2, 0]]

   Explanation: Every square matrix splits as ½(A + A′) + ½(A − A′), the second summand being skew-symmetric. With A′ = [[2, 3], [−1, 4]], A − A′ = [[0, −4], [4, 0]], and halving gives [[0, −2], [2, 0]]. Skipping the factor ½ leaves [[0, −4], [4, 0]], which is twice the required matrix.

5. Q5Medium

   If A = [[1, 2], [4, 2]], then |2A⁻¹| equals:

   A.−1/6

   B.−1/3

   C.2/3

   D.−2/3✓ Correct

   Answer: D. −2/3

   Explanation: First |A| = (1)(2) − (2)(4) = −6, so |A⁻¹| = 1/|A| = −1/6. For a matrix of order 2, |kM| = k²|M|, hence |2A⁻¹| = 4 × (−1/6) = −2/3. Forgetting to square the scalar leaves −1/3.

6. Q6Medium

   The function f(x) = x|x|, defined for all real x, is:

   A.differentiable at x = 0 with f′(0) = 0✓ Correct

   B.continuous but not differentiable at x = 0

   C.neither continuous nor differentiable at x = 0

   D.differentiable at x = 0 with f′(0) = 1

   Answer: A. differentiable at x = 0 with f′(0) = 0

   Explanation: For x ≥ 0 the function is x² and for x < 0 it is −x², and both one-sided derivatives at the origin are 0, so f is differentiable there with f′(0) = 0; indeed f′(x) = 2|x| everywhere. The extra factor of x smooths out the corner that |x| alone would have. It is the second derivative that fails to exist at 0, not the first.

7. Q7Hard

   The value of tan⁻¹(1/2) + tan⁻¹(1/5) + tan⁻¹(1/8) is:

   A.π/2

   B.π/4✓ Correct

   C.π/3

   D.3π/4

   Answer: B. π/4

   Explanation: Combine the first two using tan⁻¹x + tan⁻¹y = tan⁻¹((x + y)/(1 − xy)), valid here since xy < 1: tan⁻¹(1/2) + tan⁻¹(1/5) = tan⁻¹((7/10)/(9/10)) = tan⁻¹(7/9). Adding tan⁻¹(1/8) gives tan⁻¹((65/72)/(65/72)) = tan⁻¹1 = π/4. Adding the three arguments directly, as if tan⁻¹ were linear, is what inflates the answer to a larger multiple of π.

8. Q8Hard

   If y = xˣ, then the value of d²y/dx² at x = 1 is:

   A.0

   B.1

   C.2✓ Correct

   D.3

   Answer: C. 2

   Explanation: Write y = e^(x ln x), so y′ = xˣ(1 + ln x). Differentiating again by the product rule gives y″ = xˣ(1 + ln x)² + xˣ·(1/x), which at x = 1 equals 1(1)² + 1 = 2. Differentiating xˣ as though it were a power function, giving x·x^(x−1), collapses the answer to 1.

More Class 12 Mathematics board papers

• Board Paper 1 — 20 questions
• Board Paper 2 — 20 questions
• Board Paper 3 — 20 questions
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• Board Paper 5 — 20 questions

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