---

Mathematics Question Paper Class 12 September Exam 2025 

Maximum Marks: 80
Time Allowed: 3 Hrs

---

Section – A (Each Question carries 1 Mark)

Q1. MCQ (Choose the correct option):

(i) Let $f: \mathbb{N} \to \mathbb{N}$ be defined as $f(x) = x^2$, then
(a) $f$ is one-one only
(b) $f$ is onto only
(c) $f$ is both one-one and onto
(d) $f$ is neither one-one nor onto

(ii) If $L$ is a set of all straight lines in a plane and $R$ is a relation on $L$ defined as

$$R = \{ (L_1, L_2): L_1 \parallel L_2 \}$$

then relation $R$ is:
(a) Reflexive only
(b) Symmetric only
(c) Transitive only
(d) Equivalence relation

(iii) Principal value of $\sin^{-1}\left(-\tfrac{\sqrt{3}}{2}\right)$ is:
(a) $\tfrac{5\pi}{6}$
(b) $\tfrac{\pi}{6}$
(c) $-\tfrac{\pi}{6}$
(d) $-\tfrac{5\pi}{6}$

(iv) If $A$ is a matrix of order $2 \times 3$ and $B$ is a matrix of order $4 \times 2$, then order of $A^T B$ is:
(a) $3 \times 4$
(b) $4 \times 3$
(c) $2 \times 2$
(d) $2 \times 4$

(v) Let $A$ be a square matrix of order $3 \times 3$ and $|2A| = 264$. Then $|A|$ equals:
(a) 12
(b) 8
(c) 3
(d) 4

(vi) If

$$\begin{bmatrix} k & 4 \\ -4 & -k \end{bmatrix}$$

is a singular matrix, then value of $k$ is:
(a) 0
(b) $\pm 4$
(c) 4
(d) -4

(vii) If the function defined by

$$f(x) = \begin{cases} \dfrac{\sin 5x}{x}, & x \neq 0 \\ k+2, & x=0 \end{cases}$$

is continuous at $x=0$, then value of $k$ is:
(a) 0
(b) 2
(c) -2
(d) -1

(viii) If $x=2at, y=at^2$, then $\tfrac{dy}{dx}$ is equal to:
(a) $t$
(b) $2at$
(c) $\tfrac{1}{t}$
(d) $2a$

(ix) If $f(x) = x^2 + 3x + 4$, then $f'(0)$ is:
(a) 6
(b) 3
(c) 4
(d) 9

(x) The interval for which the function $f(x) = x^2 - 6x + 3$ is strictly increasing is:
(a) (-3, 3)
(b) (3, ∞)
(c) (-3, ∞)
(d) (-∞, 3)

(xi) $\int_{-3}^{3} (x^5 + \sin x) \, dx$ equals:
(a) $5^5 + \sin 3$
(b) 3
(c) 0
(d) 6

(xii) $\int \dfrac{(\log x)^2}{x} \, dx$ equals:
(a) $\dfrac{1}{x} + C$
(b) $\log x + C$
(c) $\dfrac{(\log x)^2}{2} + C$
(d) $\dfrac{(\log x)^3}{3} + C$

(xiii) The number of arbitrary constants in the particular solution of a differential equation of fourth order is:
(a) 0
(b) 4
(c) 3
(d) 2

(xiv) The inequality $|a+b| \leq |a| + |b|$ is called:
(a) Cauchy–Schwarz Inequality
(b) Rolle’s Theorem
(c) Triangle Inequality
(d) L.M.V. Theorem

(xv) If $\vec{a}$ is any vector, then $\vec{a} \times \vec{a}$ is:
(a) $\vec{0}$
(b) $a^2$
(c) $\vec{a}$
(d) $\hat{i}$

(xvi) If lines

$$\dfrac{x-1}{k} = \dfrac{y-3}{2} = \dfrac{z+6}{a}, \quad \dfrac{x-1}{1} = \dfrac{y-3}{2} = \dfrac{z+6}{2}$$

are perpendicular, then value of $k$ is:
(a) 0
(b) 2
(c) 1
(d) -2
(e) 3

(xvii) The vector form of line $\dfrac{x-5}{3} = \dfrac{y+4}{-1} = \dfrac{z+8}{6}$ is:
(a) $\vec{r} = (3\hat{i} + 7\hat{j} + 6\hat{k}) + \lambda (5\hat{i} + 4\hat{j} + 8\hat{k})$
(b) $\vec{r} = (5\hat{i} - 4\hat{j} - 8\hat{k}) + \lambda (3\hat{i} + 7\hat{j} + 6\hat{k})$
(c) $\vec{r} = (5\hat{i} + 4\hat{j} + 8\hat{k}) + \lambda (3\hat{i} + 7\hat{j} + 6\hat{k})$
(d) $\vec{r} = (3\hat{i} + 7\hat{j} + 6\hat{k}) + \lambda (5\hat{i} - 4\hat{j} - 8\hat{k})$

(xviii) The maximum value of $Z = 3x+5y$ subject to constraints $x+y \leq 4, x,y \geq 0$ is:
(a) 0
(b) 12
(c) 20
(d) 28

(xix) The probability of obtaining an even prime number on each die, when a pair of dice is rolled is:
(a) 0
(b) $\tfrac{1}{12}$
(c) $\tfrac{1}{6}$
(d) $\tfrac{1}{36}$

(xx) If $P(A) = \tfrac{7}{13}, P(B) = \tfrac{9}{13}, P(A \cap B) = \tfrac{4}{13}$, then $P(A \cup B)$ is:
(a) $\tfrac{7}{13}$
(b) $\tfrac{7}{9}$
(c) $\tfrac{4}{9}$
(d) $\tfrac{4}{7}$

---

Section – B (Each question carries 2 Marks)

Q2. If $A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}$, find $f(A)$ where $f(x) = x^2 - 5x + 7$.

Q3. Differentiate $\cos^{-1}\left(\dfrac{1-x^2}{1+x^2}\right)$ w.r.t $x$.

Q4. Evaluate: $\int e^x [\cot x + \log(\sin x)] dx$.

Q5. Find the area of the region bounded by ellipse $\dfrac{x^2}{4} + \dfrac{y^2}{9} = 1$.

Q6. A man 2 m high walks at uniform speed 5 m/s away from a lamp post 6 m high. Find the rate at which length of shadow increases.

Q7. A particle moves along curve $6y = x^3 + 2$. Find points where $y$-coordinate is changing 8 times as fast as $x$-coordinate.

Q8. Find the projection of $(\hat{i} + 3\hat{j} + \hat{k})$ on $(7\hat{i} - \hat{j} + 8\hat{k})$.

---

Section – C (Each question carries 4 Marks)

Q9. Show that the function $f: \mathbb{R} \to \mathbb{R}, f(x) = \tfrac{3x-6}{7}$ is one-one and onto.

Q10. (a) If $A = \begin{bmatrix} 4 \\ 3 \end{bmatrix}, B = \begin{bmatrix} -1 & 2 \end{bmatrix}$, verify that $(AB)^T = B^T A^T$.
(b) Using determinants, find equation of line joining (1,2) and (3,6).

Q11. Find $\tfrac{dy}{dx}$ when $x^y + y^x = 2$.

Q12. If $x = \tan\left(\tfrac{y}{a}\log y\right)$, show that

$$(1 + x^2)\frac{dy}{dx} + (2x-a)\frac{dy}{dx} = 0$$

Q13. (a) Evaluate: $\int_{2}^{5} |x-3| dx$
(b) Evaluate: $\int_{0}^{\pi/2} [2\log\cos x - \log \sin 2x] dx$
(c) Evaluate: $\int (\tan x + \sqrt{\cot x}) dx$

---

Section – D (Each question carries 6 Marks)

Q14. Solve LPP graphically: Max/Min $Z = 8x + 5y - 2$ subject to constraints:

$$x+y \leq 10, \; x+2y \geq 6, \; 3x-y \leq 9, \; y \leq 9, \; x,y \geq 0$$

Q15. A problem is given to 3 students whose chances of solving it are $\tfrac{1}{3}, \tfrac{1}{5}, \tfrac{1}{7}$. Find probability that:
(i) Problem is solved
(ii) Exactly one solves

OR

A factory has two machines A and B. Machine A produces 60% of items, B produces 40%. 2% of A’s items and 1% of B’s items are defective. If one item is chosen at random and found defective, find probability it was produced by B.

Q16. Solve system by Matrix Method:

$$5x + 2y + z = 3, \quad 2x + 3z = 1, \quad 3x - 2y + 4z = -1$$

OR

(a) Express $\begin{bmatrix} 3 & 5 \\ 4 & 2 \end{bmatrix}$ as sum of symmetric and skew-symmetric matrices.
(b) If $A = \begin{bmatrix} 3 & -1 \\ -1 & 2 \end{bmatrix}$, show $A^2 - 5A + 7I = 0$. Hence find $A^{-1}$.

Q17. (a) Find general solution of DE:

$$\frac{dy}{dx} = (4+y^2)(1+3x^2)$$

(b) Solve DE: $\tfrac{dy}{dx} + 3y = \cos 2x$

(c) Show that height of cylinder of maximum volume inscribed in sphere radius $R$ is $\tfrac{2R}{\sqrt{3}}$. Also find max volume.

Q18. (a) Express $\vec{a} = 5\hat{i} - 2\hat{j} + 5\hat{k}$ as sum of two vectors, one parallel and one perpendicular to $\vec{b} = 3\hat{i} + \hat{k}$.

(b) Find area of parallelogram with diagonals $\vec{d}_1 = \hat{i} - 2\hat{j} + 3\hat{k}, \vec{d}_2 = 3\hat{i} + 2\hat{j} + \hat{k}$.

Q19. Find equation of line through (1,2,3) and parallel to line

$$\dfrac{x-2}{1} = \dfrac{y+3}{7} = \dfrac{z-6}{3}$$

Q20. (b) Find the shortest distance between lines:

$$\vec{r}_1 = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda (\hat{i} - 3\hat{j} + 2\hat{k})$$

and

$$\vec{r}_2 = (4\hat{i} + 5\hat{j} + 6\hat{k}) + \mu (2\hat{i} + 3\hat{j} + \hat{k})$$

---