CBSE Class 12 Mathematics Board Paper 2 — 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 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.

2. 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.

3. 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.

4. 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.

5. 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.

6. 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.

7. 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.

8. 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.

More Class 12 Mathematics board papers

• Board Paper 1 — 20 questions
• Board Paper 2 — you are here
• Board Paper 3 — 20 questions
• Board Paper 4 — 20 questions
• Board Paper 5 — 20 questions

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