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CBSE Class 12 Mathematics Board Paper 2 — 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 2 — 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 function f : R → R defined by f(x) = 3 − 4x is:
A.one-one but not onto
B.onto but not one-one
C.both one-one and onto✓ Correct
D.neither one-one nor onto
Answer: C. both one-one and onto
Explanation: If 3 − 4x₁ = 3 − 4x₂ then x₁ = x₂, so f is one-one, and every real y is the image of x = (3 − y)/4, so f is onto. A line of non-zero slope always covers all of R. The 'not onto' options describe functions such as x², whose graph misses part of the codomain.
Q2Easy
If A is a square matrix of order 3 with |A| = 3, then |2A| equals:
A.6
B.24✓ Correct
C.12
D.9
Answer: B. 24
Explanation: For an n × n matrix, |kA| = kⁿ|A|, since the scalar comes out once from each row. With n = 3, |2A| = 2³ × 3 = 24. Taking the factor out only once, as |2A| = 2|A| = 6, is the common slip.
Q3Easy
For x ∈ (−1, 1), d/dx (sin⁻¹x + cos⁻¹x) equals:
A.1/√(1 − x²)
B.0✓ Correct
C.2/√(1 − x²)
D.1
Answer: B. 0
Explanation: For every x in (−1, 1), sin⁻¹x + cos⁻¹x = π/2, a constant, and the derivative of a constant is 0. Differentiating term by term confirms it: 1/√(1 − x²) − 1/√(1 − x²) = 0. Adding the two derivatives instead of subtracting produces the doubled expression.
Q4Medium
The principal value of cos⁻¹(cos 7π/6) is:
A.5π/6✓ Correct
B.7π/6
C.π/6
D.−5π/6
Answer: A. 5π/6
Explanation: cos⁻¹ returns a value in [0, π], and 7π/6 lies outside that range. Since cos(7π/6) = −√3/2 and cos(5π/6) = −√3/2 with 5π/6 inside the range, the principal value is 5π/6. Echoing the input 7π/6 ignores the range restriction that defines a principal value.
Q5Medium
If A and B are symmetric matrices of the same order, then AB − BA is always:
A.a symmetric matrix
B.a zero matrix
C.an identity matrix
D.a skew-symmetric matrix✓ Correct
Answer: D. a skew-symmetric matrix
Explanation: Transposing, (AB − BA)′ = B′A′ − A′B′ = BA − AB = −(AB − BA), which is precisely the definition of a skew-symmetric matrix. It need not be zero, because symmetric matrices generally do not commute. Assuming the product of two symmetric matrices is symmetric is the usual error — AB is symmetric only in the special case AB = BA.
Q6Medium
The number of points in the open interval (0, 6) at which f(x) = |x − 3| + |x − 5| fails to be differentiable is:
A.2✓ Correct
B.1
C.0
D.3
Answer: A. 2
Explanation: A sum of modulus functions has corners exactly where the arguments vanish, here at x = 3 and x = 5, both of which lie inside (0, 6). Between and outside those points f reduces to a straight line and is perfectly smooth, so exactly two points fail. Assuming the two corners somehow cancel gives the answer 0.
Q7Hard
The system x + y + z = 6, x + 2y + 3z = 14, x + 2y + λz = μ has infinitely many solutions when:
A.λ = 3 and μ ≠ 14
B.λ ≠ 3 and μ = 14
C.λ = 3 and μ = 14✓ Correct
D.λ ≠ 3 and μ ≠ 14
Answer: C. λ = 3 and μ = 14
Explanation: Subtracting the second equation from the third eliminates x and y and leaves (λ − 3)z = μ − 14. Infinitely many solutions require that to reduce to the identity 0 = 0, so λ = 3 and μ = 14. With λ = 3 but μ ≠ 14 the same step gives 0 equal to a non-zero number, which means no solution at all.
Q8Hard
The tangent to the curve y = x³ − 6x² + 9x + 4 is parallel to the x-axis at certain points. The sum of the y-coordinates of all such points is:
A.8
B.4
C.16
D.12✓ Correct
Answer: D. 12
Explanation: A tangent parallel to the x-axis needs dy/dx = 0, and 3x² − 12x + 9 = 3(x − 1)(x − 3) vanishes at x = 1 and x = 3. The corresponding y-values are 8 and 4, whose sum is 12. Reporting only the larger value 8 answers for one point instead of both.

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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.

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