CBSE Class 12 Mathematics Board Paper 4 — MCQs with Answers
Original paper · CBSE patternWritten by our subject team, not a reprint of an official sample paper.
- Questions
- 20
- Time
- 30 min
- Marking
- +1, no negative marking
- Mix
- 4 Easy · 8 Medium · 8 Hard
Board Paper 4 — CBSE Class 12 Mathematics is a full 20-question objective paper for Class 12 Mathematics, written in the CBSE board pattern and marked the way the board marks: +1 for a correct answer and 0 for a wrong one — no negative marking, exactly like the real board paper. It follows Section A of a real CBSE paper: Q1–Q16 are single-correct MCQs, Q17–Q18 are case-based questions built on a short source, and Q19–Q20 are assertion–reason items using the board's own four option strings.
The difficulty sits deliberately a notch above the real board — 4 Easy · 8 Medium · 8 Hard — because a paper you can clear comfortably tells you nothing on exam day. You get 30 minutes — about 90 seconds a question, the pace the real paper demands. 8 of the questions are solved in full below, with the correct option and a worked explanation for each; the remaining 12 are timed and scored when you sign in free.
Sit it under exam conditions rather than open-book. The score you get on a timed full-length paper is the only honest signal of whether your Mathematics revision is holding together across chapters — which is precisely what the board tests and what chapter-wise practice cannot tell you.
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.
- Q1Easy
The number of one-one functions from a set containing 3 elements to a set containing 4 elements is:
A.12B.64C.81D.24✓ CorrectAnswer: 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.
- 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✓ CorrectB.4 × 4C.3 × 4D.4 × 3Answer: 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.
- 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/cmB.12π cm²/cm✓ CorrectC.36π cm²/cmD.24π cm²/cmAnswer: 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.
- 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]]✓ CorrectD.[[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.
- Q5Medium
If A = [[1, 2], [4, 2]], then |2A⁻¹| equals:
A.−1/6B.−1/3C.2/3D.−2/3✓ CorrectAnswer: 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.
- Q6Medium
The function f(x) = x|x|, defined for all real x, is:
A.differentiable at x = 0 with f′(0) = 0✓ CorrectB.continuous but not differentiable at x = 0C.neither continuous nor differentiable at x = 0D.differentiable at x = 0 with f′(0) = 1Answer: 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.
- Q7Hard
The value of tan⁻¹(1/2) + tan⁻¹(1/5) + tan⁻¹(1/8) is:
A.π/2B.π/4✓ CorrectC.π/3D.3π/4Answer: 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.
- Q8Hard
If y = xˣ, then the value of d²y/dx² at x = 1 is:
A.0B.1C.2✓ CorrectD.3Answer: 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.
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Attempt this paper free →More Class 12 Mathematics board papers
- Board Paper 1 — 20 questions
- Board Paper 2 — 20 questions
- Board Paper 3 — 20 questions
- Board Paper 4 — you are here
- Board Paper 5 — 20 questions
Prefer chapter-by-chapter revision first? Class 12 Mathematics notes & chapter MCQs →
Questions students ask about this paper
Is this an official CBSE sample paper?
No — and that is deliberate. This is an original paper written in the CBSE board pattern by our subject team, not a copy of CBSE's official Sample Question Paper. It means you get fresh questions you have not already seen on a dozen other sites, with instant scoring and worked solutions rather than a PDF.
How is this Class 12 Mathematics paper marked?
+1 for a correct answer and 0 for a wrong one — no negative marking, exactly like the real board paper. So attempt every question — there is no penalty for a wrong answer on a board paper, and leaving a question blank can only cost you.
How long should this paper take?
30 minutes for 20 questions, roughly 90 seconds each. The timer runs whether or not you are watching it, which is the point: board marks are lost to pacing at least as often as to gaps in knowledge.
Do I need to pay to attempt it?
No. Sign in free with Google and the full paper opens with a live timer, automatic scoring and a worked solution for every question — including the ones not shown on this page.