Mathematics · Class 12 Science

West Bengal Wbchse Class 12 Mathematics — Real Previous-Year Papers

with complete answers

Real previous-year board papers, year by year — the official exam pattern, the full question paper, and every question solved the concept-first way. Distinct from the chapter-wise textbook bank.

2019–2026
Years of papers: 7
Total Papers: 7
Real Board Papers: 0
Sample papers: 283
Real-paper Q & A: 0
Sample-paper Q & A

Real board-paper questions available, by year

53 Q20262 sets

44 Q2025complete

45 Q2024complete

45 Q2023complete

51 Q2022complete

—2021Not available

—2020Not available

45 Q2019complete

West Bengal HS (WBCHSE) Board 2026 · Set SEM3

Real board examination

Sets

This paper has 1 question on a topic removed in CBSE’s 2023-24 syllabus update (each marked Not in syllabus). It’s kept for historical accuracy — the exam really asked it that year — but isn’t in the current syllabus and doesn’t count toward a concept’s importance. Switch to to focus on what’s still examinable.

About this paper

The real Class-12 board examination held in 2026. Every question below is solved the concept-first way. Sample papers are labelled honestly — never shown as a past exam.

Total marks: 80
Questions: 44
Duration: 195 min
Sections: 14

The marks / questions / duration above are the official exam pattern. We currently have 40 of this paper’s questions (91% of the full paper), with 40 fully solved. Questions we couldn’t yet extract or verify are held — never shown as complete.

Sections & marks

Section	Type	Questions	Marks each	Total
ASection A	compulsory	10	1	10
BSection B	compulsory	2	2	4
BSection B	compulsory	2	2	4
BSection B	compulsory	6	2	12
BSection B	compulsory	2	2	4
BSection B	compulsory	2	2	4
CSection C	compulsory	2	4	8
CSection C	compulsory	2	4	8
CSection C	compulsory	4	4	16
CSection C	compulsory	2	4	8
CSection C	compulsory	2	4	8
DSection D	compulsory	4	5	20
DSection D	compulsory	2	5	10
DSection D	compulsory	2	5	10
Total		44		80

The question paper

The questions we hold for this paper, laid out by section. Solutions are on the Answers tab.

Board Examination

Mathematics

West Bengal HS (WBCHSE) Board 2026 · Set SEM3

Series/Set: SEM3Roll No. ________

Time Allowed: 3 hr 15 minMaximum Marks: 80

General Instructions

This question paper contains 44 questions divided into 14 sections — A, B, B, B, B, B, C, C, C, C, C, D, D, D.
Section A comprises 10 questions of 1 mark each (compulsory).
Section B comprises 2 questions of 2 marks each (compulsory).
Section B comprises 2 questions of 2 marks each (compulsory).
Section B comprises 6 questions of 2 marks each (compulsory).
Section B comprises 2 questions of 2 marks each (compulsory).
Section B comprises 2 questions of 2 marks each (compulsory).
Section C comprises 2 questions of 4 marks each (compulsory).
Section C comprises 2 questions of 4 marks each (compulsory).
Section C comprises 4 questions of 4 marks each (compulsory).
Section C comprises 2 questions of 4 marks each (compulsory).
Section C comprises 2 questions of 4 marks each (compulsory).
Section D comprises 4 questions of 5 marks each (compulsory).
Section D comprises 2 questions of 5 marks each (compulsory).
Section D comprises 2 questions of 5 marks each (compulsory).

Above is the official exam pattern. The questions printed below are those we currently hold for this paper.

Section A

compulsory · 1 mark each · 40 of 10 shown

Q1.

\sin^{-1} x + \sin^{-1} y = \dfrac{2\pi}{3}

, then the value of

\cos^{-1} x + \cos^{-1} y

(a) $\dfrac{\pi}{3}$
(b) $\dfrac{\pi}{6}$
(c) $\dfrac{2\pi}{3}$
(d) $\dfrac{5\pi}{3}$

[1]

Q2.

The value of

2\tan^{-1}\sqrt{x} - \cos^{-1}\left(\dfrac{1-x}{1+x}\right)

(a) $0$
(b) $1$
(c) $\dfrac{1}{3}$
(d) $\dfrac{1}{2}$

[1]

Q3.

A = [a_{ij}]

is a

2 \times 2

matrix, where

a_{ij} = \dfrac{1}{2}(i + 2j)^2

, then

A

(a) $\begin{bmatrix} \dfrac{9}{2} & \dfrac{25}{2} \\ 8 & 18 \end{bmatrix}$
(b) $\begin{bmatrix} 9 & \dfrac{25}{2} \\ 8 & 18 \end{bmatrix}$
(c) $\begin{bmatrix} \dfrac{9}{2} & \dfrac{25}{2} \\ 8 & 9 \end{bmatrix}$
(d) $\begin{bmatrix} \dfrac{9}{2} & \dfrac{15}{2} \\ 4 & 18 \end{bmatrix}$

[1]

Q4.

A = \begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}

and

f(x) = x^2 - 2x - 5

, then

f(A)

is equal to

(a) $\begin{bmatrix} -2 & 0 \\ 0 & -2 \end{bmatrix}$
(b) $\begin{bmatrix} -3 & 0 \\ 0 & -3 \end{bmatrix}$
(c) $\begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix}$
(d) $\begin{bmatrix} -3 & 0 \\ 0 & -2 \end{bmatrix}$

[1]

Q5.

Let

A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}

and

B = \begin{bmatrix} -1 & 1 \\ -2 & 0 \end{bmatrix}

. Match the matrix on the left column with the matrix on the right column. Then choose the correct option. Left column:

(i) $A + A^T$ ;
(ii) $(A + B)^T$ ;
(iii) $(AB)^T$ ;
(iv) $B + B^T$ . Right column:
(a) $\begin{bmatrix} -2 & -1 \\ -1 & 0 \end{bmatrix}$ ;
(b) $\begin{bmatrix} 2 & 5 \\ 5 & 8 \end{bmatrix}$ ;
(c) $\begin{bmatrix} 0 & 1 \\ 3 & 4 \end{bmatrix}$ ;
(d) $\begin{bmatrix} -5 & -11 \\ 1 & 3 \end{bmatrix}$ .
(a) (i)-(a), (ii)-(c), (iii)-(d), (iv)-(b)
(b) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
(c) (i)-(b), (ii)-(c), (iii)-(d), (iv)-(a)
(d) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)

[1]

Q6.

y = \dfrac{\log_e x}{x}

then the maximum value of

y

(a) $e$
(b) $e^2$
(c) $\dfrac{1}{e}$
(d) $\dfrac{1}{e^2}$

[1]

Page 1 of 5

Q7.

Statement-I: f(x) = 3 + |x - 3| has a local minimum value 3. Statement-II: f(x) = sin x has infinite number of maximum and minimum values. Which of the following options is correct? (a) Statement-I is true, Statement-II is false (b) Statement-I is false, Statement-II is true (c) Statements-I and II both are true (d) Statements-I and II both are false

[1]

Q8.

If P(A) = (1)/(4), P(B) = (1)/(3) and P(A - B) = (1)/(6), then the events A and B are mutually (a) exclusive (b) independent (c) dependent (d) exhaustive

[1]

Q9.

A biased coin is tossed n times. The probability of getting a head is p(0 < p < 1), then the probability that the rth(r < n) head will appear in the nth tossing is (a) ⁿ⁻¹Cr-1 pr qn-r (b) ⁿCr pr qn-r (c) pr (d) pn-r

[1]

Q10.

If three events X, Y, Z are mutually exclusive and exhaustive where P(X) = (2)/(3) P(Y), P(Z) = (1)/(3) P(Y), then P(Z) is equal to (a) (1)/(6) (b) (1)/(3) (c) (5)/(6) (d) (2)/(3)

[1]

Q11.

If -x² xy xz xy -y² yz xz yz -z² = λ x² y² z², then the value of λ is equal to (a) 1 (b) 2 (c) 3 (d) 4

[1]

Q12.

The system of equations kx + y + z = 1, x + ky + z = k and x + y + kz = k² will have unique solution when (a) k ≠ 1 (b) k ≠ 2 (c) k ≠ 1, k ≠ -2 (d) k ≠ 0

[1]

Q13.

Let f(x) = (x² - 1)/(x³ - 1), when x ≠ 1, is continuous at x = 1. Then the value of f(1) is (a) 1 (b) (1)/(3) (c) (2)/(3) (d) 2

[1]

Q14.

If xm yⁿ = (x + y)m+n, then the value of (dy)/(dx) is (a) 0 (b) (y)/(x) (c) (x + y)/(xy) (d) xy

[1]

Q15.

If f(2) = 4, f'(2) = 4, then the value of limx → 2 (x f(2) - 2 f(x))/(x - 2) is (a) -2 (b) 2 (c) 3 (d) -4

[1]

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Q16.

If X = 3 0 0 \0 3 0 \0 0 3 , then X⁵ will be (a) 36X (b) 50X (c) 90X (d) 81X

[1]

Q17.

Let S be a 2 × m ordered and T be a 3 × n ordered matrix, and conformable for product TS matrix of order p × 4. Then the values of m, n and p are (a) m = 3, n = 2, p = 4 (b) m = 4, n = 2, p = 3 (c) m = 3, n = 4, p = 2 (d) m = 4, n = 3, p = 2

[1]

Q18.

The values of a, b and c for which the matrix 1 a+b-c a+b+c \1 2 a-b+c \9 5 3 will be symmetric are (a) a = 3, b = 2, c = 4 (b) a = 2, b = 3, c = 1 (c) a = 1, b = 2, c = 3 (d) a = 0, b = 1, c = 3

[1]

Q19.

If A is a square matrix of order 3 and |A| = 7, then the value of |2AT| is (a) 32 (b) 28 (c) 16 (d) 56

[1]

Q20.

If the inverse of a matrix A of order 3 × 3 exists and |A| = 5, then the value of |adj A| is (a) 20 (b) 15 (c) 5 (d) 25

[1]

Q21.

The volume of a spherical balloon increases at the rate of 10cm³/sec. The rate of change of its surface area when its radius is 16 cm, is (a) 1·5cm²/sec (b) 1·8cm²/sec (c) 2cm²/sec (d) 1·25cm²/sec

[1]

Q22.

Which one of the following is correct for all values of x if x ∈ (0, 1)? (a) ex < 1 + x (b) loge (1 + x) < x (c) sin x > x (d) loge x > x

[1]

Q23.

Let f(x) = (x)/(1 + |x|). Then f(x) is monotonically increasing in the interval (where mathbbR is the set of all real numbers). (a) mathbbR (b) mathbbR - \-1\ (c) (-1, 1) (d) (-∞, 0)

[1]

Q24.

If the straight line y = x and the curve xy = k² cut at a right angle, then (k is a real constant) (a) k = 0 (b) k = ± 1 (c) -∞ < k < ∞ (d) 0 ≤ k < ∞

[1]

Q25.

If the straight line lx - my + n = 0 touches the parabola y² = 4ax then (a) am² = nl (b) an² = ml (c) al² = mn (d) mn = al

⚠ This question is not in the current syllabus — This question tests the tangency condition for a straight line and the parabola $y^2=4ax$ (conic sections / coordinate g

[1]

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Q26.

If ρ be a relation on the set of all integers mathbbZ and ρ = \(x, y) : |x - y| ≤ 5, x, y ∈ mathbbZ\ then the relation ρ is (a) Reflexive and symmetric (b) Reflexive and Transitive (c) Transitive and symmetric (d) Equivalence

[1]

Q27.

Let mathbbR be the set of real numbers and f : mathbbR → mathbbR, g : mathbbR → mathbbR are two mappings such that f(x) = |x| - x², g(x) = 2x + 3; ∀ x ∈ mathbbR, then the value of (g ° f)(-3) is (a) 9 (b) -9 (c) 6 (d) -6

[1]

Q28.

Statement (Q): f : mathbbR → mathbbR is a function defined as f(x) = [x], greatest integer function, f(x) is not onto function. Reason (R): A function F : X → Y is one-one if F(a) = F(b) ⇒ a = b. Alternatives: (a) (Q) and (R) both are true, and (R) is a correct explanation of (Q) (b) (Q) and (R) both are true, but (R) is not a correct explanation of (Q) (c) (Q) is true but (R) is false (d) (Q) is false but (R) is true

[1]

Q29.

Let mathbbR be the set of real numbers and f : mathbbR → mathbbR be given by f(x) = 2x - 3. Then the value of f⁻¹(0) is (a) -3 (b) (3)/(2) (c) 3 (d) ± 3

[1]

Q30.

The principal value of cot⁻¹(-dfrac1√(3)) is (a) (2π)/(3) (b) -(π)/(3) (c) (π)/(3) (d) (π)/(6)

[1]

Q31.

A and B are two independent events. The probability of occurrence of exactly one of the two events is (a) P(A / B) (b) P(B / A) (c) P(A) + P(B) - P(AB) (d) P(A) + P(B) - 2P(AB)

[1]

Q32.

A person regularly watches on TV either the Discovery channel or a Sports channel at night. The probability of watching Sports channel is (4)/(5). The probability of him falling asleep while watching the Discovery channel is (3)/(4) and in case of Sports channel this probability is (1)/(4). Then the probability of watching the Discovery channel if some day the person falls asleep is (a) (3)/(5) (b) (4)/(5) (c) (4)/(7) (d) (3)/(7)

[1]

Q33.

In a probability distribution, the mean of the random variable X is (6)/(5) and mean of X² is 2, then the standard deviation of X is (a) dfrac√(7)5 (b) dfrac√(14)5 (c) dfrac7√(5) (d) √((6)/(5))

[1]

Q34.

If the probability that exactly one of the two events A and B occurs is x and the probability that both A and B occur is y then P(A) + P(B) is equal to (a) x + y (b) x + 2y (c) 2x + 2y (d) 2x + y

[1]

Page 4 of 5

Q35.

The following represents a probability distribution of a random variable X: | X = xᵢ | 0 | 1 | 2 | 3 | 4 | |---|---|---|---|---|---| | P(X = xᵢ) | k | 2k | 3k | 4k | 5k | Then P(X ≥ 2) is (a) (1)/(5) (b) (2)/(5) (c) (3)/(5) (d) (4)/(5)

[1]

Q36.

If f(x) = logₓ (loge x), then the value of f'(e) is (a) e (b) (2)/(e) (c) (1)/(e) (d) 0

[1]

Q37.

If x = sin⁻¹ t, y = √(1 - t²), then the value of (d² y)/(dx²) at t = 1 is (a) 1 (b) 0 (c) (1)/(2) (d) -1

[1]

Q38.

If (dx)/(dy) = l and (d² x)/(dy²) = m, then the value of (d² y)/(dx²) is (a) -(m)/(l³) (b) (m)/(l³) (c) (1)/(m) (d) 0

[1]

Q39.

Let f(x) = x² + ax + b, x < 1 x, x ≥ 1 . If f(x) is differentiable at x = 1, then (a - b) is equal to (a) 0 (b) -2 (c) -6 (d) -3

[1]

Q40.

The points of discontinuity of the function f(x) = (x² + 4x + 3)/(x³ + 3x² - x - 3) are (a) x = 1, -1, -3 (b) x = -1, -3 (c) x = 1, -3 (d) x = 1, -1

[1]

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