---

PSEB CLASS – XII (2025–26) MATHEMATICS SAMPLE PAPER SET 2

Maximum Marks: 80
Time Allowed: 3 Hours

General Instructions:

1. All questions are compulsory.

2. The question paper consists of 18 questions divided into 4 sections A, B, C, and D.

3. Section A comprises 1 question of 20 multiple-choice type questions carrying 1 mark each.

4. Section B comprises 7 questions of 2 marks each.

5. Section C comprises 7 questions of 4 marks each.

6. Section D comprises 3 questions of 6 marks each.

7. Internal choice has been provided in some questions. Attempt only one of the choices.

8. Use of calculators is not permitted.

---

Section – A (Each Question 1 Mark, Total 20 Marks)

Q1. MCQ (Choose the correct option):

(i) If $f(x) = \dfrac{1}{x+1}$, then domain of $f$ is:
(a) $\mathbb{R}$
(b) $\mathbb{R} - \{-1\}$
(c) $\mathbb{N}$
(d) None of these

(ii) Principal value of $\cos^{-1}\left(-\tfrac{1}{2}\right)$ is:
(a) $\dfrac{2\pi}{3}$
(b) $\dfrac{\pi}{3}$
(c) $-\dfrac{\pi}{3}$
(d) $-\dfrac{2\pi}{3}$

(iii) If $A$ is a $2\times2$ matrix with $|A| = 5$, then $|3A|$ is:
(a) 15
(b) 30
(c) 45
(d) 75

(iv) If adjoint of $A = \begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix}$, then trace of adj$A$ is:
(a) 6
(b) 5
(c) 8
(d) 4

(v) If $f(x) = x^3$, then $f'(2)$ is:
(a) 4
(b) 6
(c) 12
(d) 3

(vi) The point at which the function $f(x) = |x-2|$ is not differentiable is:
(a) 0
(b) 1
(c) 2
(d) 3

(vii) $\lim_{x\to 0} \dfrac{\sin 7x}{x}$ equals:
(a) 0
(b) 1
(c) 7
(d) ∞

(viii) If $y = \sin^{-1} x$, then $\dfrac{dy}{dx}$ is:
(a) $\dfrac{1}{\sqrt{1-x^2}}$
(b) $\sqrt{1-x^2}$
(c) $\dfrac{1}{1+x^2}$
(d) None

(ix) The interval of monotonic increase of $f(x) = x^3 - 9x$ is:
(a) (-∞, -√3) ∪ (√3, ∞)
(b) (-√3, √3)
(c) (-∞, ∞)
(d) None

(x) $\int_0^\pi \sin^2 x \, dx$ equals:
(a) $\pi/2$
(b) $\pi$
(c) 0
(d) 1

(xi) $\int \dfrac{1}{x^2+1} dx$ is:
(a) $\tan^{-1}x + C$
(b) $\sin^{-1}x + C$
(c) $\cos^{-1}x + C$
(d) None

(xii) Area under the curve $y = x^2$ between $x=0$ and $x=2$ is:
(a) 4
(b) 8/3
(c) 16/3
(d) 2

(xiii) If $x = a\cos t, y = b\sin t$, then $\dfrac{dy}{dx}$ is:
(a) $\dfrac{a}{b}\cot t$
(b) $\dfrac{b}{a}\cot t$
(c) $-\dfrac{a}{b}\tan t$
(d) $-\dfrac{b}{a}\tan t$

(xiv) If mean of binomial distribution $B(n, p)$ is 12 and variance is 3, then $n$ is:
(a) 12
(b) 15
(c) 20
(d) 25

(xv) If events A and B are independent, with $P(A) = 0.5, P(B) = 0.3$, then $P(A\cap B)$ is:
(a) 0.15
(b) 0.20
(c) 0.80
(d) 0.10

(xvi) The vector $\vec{a} = 2\hat{i} - \hat{j} + 2\hat{k}$ has magnitude:
(a) 2
(b) 3
(c) √9
(d) 3

(xvii) Equation of line through (1,2,3) parallel to vector $2\hat{i} - \hat{j} + \hat{k}$ is:
(a) $\dfrac{x-1}{2} = \dfrac{y-2}{-1} = \dfrac{z-3}{1}$
(b) $\dfrac{x+1}{2} = \dfrac{y+2}{-1} = \dfrac{z+3}{1}$
(c) $\dfrac{x-1}{-2} = \dfrac{y-2}{1} = \dfrac{z-3}{-1}$
(d) None

(xviii) Distance between parallel planes $2x+3y+6z=4$ and $2x+3y+6z=7$ is:
(a) 3/7
(b) 7/3
(c) 1/√49
(d) 3/7

(xix) If $A$ and $B$ are two events such that $P(A) = 0.4, P(B)=0.5, P(A\cup B)=0.7$, then $P(A\cap B)$ is:
(a) 0.2
(b) 0.1
(c) 0.3
(d) 0.9

(xx) The feasible region of an LPP is always:
(a) Convex
(b) Concave
(c) Circular
(d) None

---

Section – B (Each Question 2 Marks, Total 14 Marks)

Q2. If $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, find $A^{-1}$.

Q3. Find value of $k$ if matrix $\begin{bmatrix} k & 2 \\ 6 & 8 \end{bmatrix}$ is singular.

Q4. If $y = \sin^{-1}\sqrt{x}$, find $\dfrac{dy}{dx}$.

Q5. If $f(x) = x^3 - 3x + 1$, find intervals of monotonicity.

Q6. Evaluate $\int (e^x + x^2) dx$.

Q7. Find mean of distribution:

| x | 0 | 1 | 2 | 3 | 4 |
|-----|---|---|---|---|
| f | 5 |10 |10 | 5 | 0 |

Q8. Find projection of vector $\vec{a} = 2\hat{i} + \hat{j} + 2\hat{k}$ on $\vec{b} = \hat{i} + 2\hat{j} + 2\hat{k}$.

---

Section – C (Each Question 4 Marks, Total 28 Marks)

Q9. Show that function $f(x) = \dfrac{x-1}{x+1}, x \neq -1$ is one-one and onto.

Q10. If $A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}, B = \begin{bmatrix} 1 & 2 \\ 3 & 0 \end{bmatrix}$, verify $(AB)^T = B^T A^T$.

Q11. Find derivative of $y = x^x$.

Q12. Evaluate $\int_0^1 \dfrac{x^2}{1+x^2} dx$.

Q13. Find area under parabola $y^2 = 4x$ between $x=0$ and $x=4$.

Q14. Solve differential equation $\dfrac{dy}{dx} + y = e^x$.

Q15. A card is drawn at random from a well-shuffled pack of 52 cards. Find probability that it is (i) a red card, (ii) a king or queen, (iii) not a face card.

---

Section – D (Each Question 6 Marks, Total 18 Marks)

Q16. Solve system of equations by matrix method:

$$x+y+z=6, \quad 2x-y+3z=14, \quad x+2y+z=10$$

Q17. Find the equation of the plane passing through points (1,0,0), (0,2,0), (0,0,3).

OR

Find shortest distance between lines:

$$\vec{r} = (2\hat{i}+3\hat{j}+4\hat{k}) + \lambda(\hat{i} - \hat{j} + 2\hat{k}), \quad \vec{r} = (5\hat{i}+2\hat{j}+\hat{k}) + \mu(2\hat{i}+3\hat{j}+\hat{k})$$

Q18. Solve the following Linear Programming Problem graphically:
Maximize $Z = 3x+2y$ subject to constraints:

$$x+y \leq 6, \quad x \geq 0, \quad y \geq 0$$

---