CBSE Class 12 Mathematics Question Paper 2025

CBSE • Class 12 • 2025

Q.P. Code / कोड

65/1

SECTION -A (Multiple Choice Questions & Assertion-Reason - 1 Mark Each)

This section comprises of **18 multiple choice questions (MCQs)** (Q1 to Q18) and **2 Assertion-Reason based questions** (Q19, Q20) of 1 mark each.

A function \(f:R_{+}\rightarrow R\) (where \(R_{+}\) is the set of all non-negative real numbers) defined by \(f(x)=4x+3\) is:
1. one-one but not onto
2. onto but not one-one
3. both one-one and onto
4. neither one-one nor onto
If a matrix has 36 elements, the number of possible orders it can have, is:
1. 13
2. 3
3. 5
4. 9
Which of the following statements is true for the function \(f(x)=\begin{cases}x^{2}+3,&x\ne0\\ 1&,&x=0\end{cases}\) ?
1. \(f(x)\) is continuous and differentiable \(\forall x\in\mathbb{R}\)
2. \(f(x)\) is continuous \(\forall x\in\mathbb{R}\)
3. \(f(x)\) is continuous and differentiable \(\forall x\in\mathbb{R}-\{0\}\)
4. \(f(x)\) is discontinuous at infinitely many points
Let \(f(x)\) be a continuous function on [a, b] and differentiable on (a, b). Then, this function \(f(x)\) is strictly increasing in (a, b) if
1. \(f^{\prime}(x)<0\), \(\forall x\in(a,b)\)
2. \(f^{\prime}(x)>0\), \(\forall x\in(a,b)\)
3. \(f^{\prime}(x)=0\), \(\forall x\in(a,b)\)
4. \(f(x)>0\), \(\forall x\in(a,b)\)
If \(\begin{bmatrix}x+y&2\\ 5&xy\end{bmatrix}=\begin{bmatrix}6&2\\ 5&8\end{bmatrix},\) then the value of \((\frac{24}{x}+\frac{24}{y})\) is:
1. 7
2. 6
3. 8
4. 18
\(\int_{a}^{b}f(x)dx\) is equal to:
1. \(\int_{a}^{b}f(a-x)dx\)
2. \(\int_{a}^{b}f(a+b-x)dx\)
3. \(\int_{a}^{b}f(x-(a+b))dx\)
4. \(\int_{a}^{b}f((a-x)+(b-x))dx\)
Let \(\theta\) be the angle between two unit vectors \(\hat{a}\) and \(\hat{b}\) such that \(\sin\theta=\frac{3}{5}\). Then, \(\hat{a} \cdot \hat{b}\) is equal to:
1. \(\pm\frac{3}{5}\)
2. \(\pm\frac{3}{4}\)
3. \(\pm\frac{4}{5}\)
4. \(\pm\frac{4}{3}\)
The integrating factor of the differential equation \((1-x^{2})\frac{dy}{dx}+xy=ax,\) \(-1 \lt x\lt 1\) is:
1. \(\frac{1}{x^{2}-1}\)
2. \(\frac{1}{\sqrt{x^{2}-1}}\)
3. \(\frac{1}{1-x^{2}}\)
4. \(\frac{1}{\sqrt{1-x^{2}}}\)
If the direction cosines of a line are \(\sqrt{3}k, \sqrt{3}k\), \(\sqrt{3}k,\) then the value of k is:
1. \(\pm1\)
2. \(\pm\sqrt{3}\)
3. \(\pm3\)
4. \(\pm\frac{1}{3}\)
A linear programming problem deals with the optimization of a/an:
1. logarithmic function
2. linear function
3. quadratic function
4. exponential function
If \(P(A|B)=P(A^{\prime}|B)\), then which of the following statements is true?
1. \(P(A)=P(A^{\prime})\)
2. \(P(A)=2~P(B)\)
3. \(P(A\cap B)=\frac{1}{2}P(B)\)
4. \(P(A\cap B)=2~P(B)\)
\(\begin{vmatrix}x+1&x-1\\ x^{2}+x+1&x^{2}-x+1\end{vmatrix}\) is equal to:
1. \(2x^{3}\)
2. 2
3. 0
4. \(2x^{3}-2\)
The derivative of \(\sin(x^{2})\) w.r.t. x, at \(x=\sqrt{\pi}\) is :
1. 1
2. -1
3. \(-2\sqrt{\pi}\)
4. \(2\sqrt{\pi}\)
The order and degree of the differential equation \([1+(\frac{dy}{dx})^{2}]^{3}=\frac{d^{2}y}{dx^{2}}\) respectively are:
1. 1, 2
2. 2, 3
3. 2, 1
4. 2, 6
The vector with terminal point \(A(2,-3,5)\) and initial point \(B(3, 4, 7)\) is:
1. \(\hat{i}-\hat{j}+2\hat{k}\)
2. \(\hat{i}+\hat{j}+2\hat{k}\)
3. \(-\hat{i}-\hat{j}-2\hat{k}\)
4. \(-\hat{i}+\hat{j}-2\hat{k}\)
The distance of point \(P(a,b,c)\) from y-axis is :
1. b
2. \(b^{2}\)
3. \(\sqrt{a^{2}+c^{2}}\)
4. \(a^{2}+c^{2}\)
The number of corner points of the feasible region determined by constraints \(x\ge0, y\ge0, x+y\ge4\) is:
1. 0
2. 1
3. 2
4. 3
If A and B are two non-zero square matrices of same order such that \((A+B)^{2}=A^{2}+B^{2}\) then :
1. \(AB=O\)
2. \(AB=-BA\)
3. \(BA=O\)
4. \(AB=BA\)
Assertion (A): For matrix \(A=\begin{bmatrix}1&\cos\theta&1\\ -\cos\theta&1&\cos\theta\\ -1&-\cos\theta&1\end{bmatrix},\) where \(\theta\in[0,2\pi]\), \(|A|\in[2,4]\).

Reason (R): \(\cos\theta\in[-1,1]\) \(\forall\theta\in[0,2\pi].\)

(Choose the correct option from the standard codes A, B, C, D)
Assertion (A): A line in space cannot be drawn perpendicular to x, y and z axes simultaneously.

Reason (R): For any line making angles, \(\alpha, \beta, \gamma\) with the positive directions of X, y and Z axes respectively, \(\cos^{2}\alpha+\cos^{2}\beta+\cos^{2}\gamma=1\).

(Choose the correct option from the standard codes A, B, C, D)

SECTION -B (Very Short Answer Type Questions - 2 Marks Each)

This section comprises of **5 Very Short Answer (VSA)** type questions of 2 marks each. (Internal choice has been provided in 2 questions)

(a) Check whether the function \(f(x)=x^{2}|x|\) is differentiable at \(x=0\) or not.
OR
(b) If \(y=\sqrt{\tan\sqrt{x}}\), prove that \(\sqrt{x}\frac{dy}{dx}=\frac{1+y^{4}}{4y}\). (2)
Show that the function \(f(x)=4x^{3}-18x^{2}+27x-7\) has neither maxima nor minima. (2)
(a) Find: \(\int x\sqrt{1+2x}dx\).
OR
(b) Evaluate: \(\int_{0}^{\frac{\pi^{2}}{4}}\frac{\sin\sqrt{x}}{\sqrt{x}}dx\). (2)
If \(\vec{a}\) and \(\vec{b}\) are two non-zero vectors such that \((\vec{a}+\vec{b})\perp\vec{a}\) and \((2\vec{a}+\vec{b})\perp\vec{b}\), then prove that \(|\vec{b}|=\sqrt{2}|\vec{a}|\). (2)
In the given figure, ABCD is a parallelogram. If \(\vec{AB}=2\hat{i}-4\hat{j}+5\hat{k}\) and \(\vec{DB}=3\hat{i}-6\hat{j}+2\hat{k}\), then find \(\vec{AD}\) and hence find the area of parallelogram ABCD. (2)

SECTION -C (Short Answer Type Questions - 3 Marks Each)

This section comprises of **6 Short Answer (SA)** type questions of 3 marks each. (Internal choice has been provided in 3 questions)

(a) A relation R on set \(A=\{1,2,3,4,5\}\) is defined as \(R=\{(x,y):|x^{2}-y^{2}|<8\}\). Check whether the relation R is reflexive, symmetric and transitive.
OR
(b) A function \(f\) is defined from \(R\rightarrow R\) as \(f(x)=ax+b\), such that \(f(1)=1\) and \(f(2)=3\). Find function \(f(x)\). Hence, check whether function \(f(x)\) is one-one and onto or not. (3)
(a) If \(\sqrt{1-x^{2}}+\sqrt{1-y^{2}}=a(x-y),\) prove that \(\frac{dy}{dx}=\sqrt{\frac{1-y^{2}}{1-x^{2}}}.\)
OR
(b) If \(y=(\tan x)^{x},\) then find \(\frac{dy}{dx}\). (3)
(a) Find: \(\int\frac{x^{2}}{(x^{2}+4)(x^{2}+9)}dx\).
OR
(b) Evaluate: \(\int_{1}^{3}(|x-1|+|x-2|+|x-3|)dx\). (3)
Find the particular solution of the differential equation given by \(x^{2}\frac{dy}{dx}-xy=x^{2}\cos^{2}(\frac{y}{2x})\) given that when \(x=1\), \(y=\frac{\pi}{2}\). (3)
Solve the following linear programming problem graphically: Maximise \(Z=500x+300y\), subject to constraints \(x+2y\le12\), \(2x+y\le12\), \(4x+5y\ge20\), \(x\ge0\), \(y\ge0\). (3)
E and F are two independent events such that \(P(\overline{E})=0\cdot6\) and \(P(E\cup F)=0\cdot6\). Find \(P(F)\) and \(P(\overline{E}\cup\overline{F})\). (3)

SECTION-D (Long Answer Type Questions - 5 Marks Each)

This section comprises of **4 Long Answer (LA)** type questions of 5 marks each. (Internal choice has been provided in 2 questions)

(a) If \(A=\begin{bmatrix}1&-2&0\\ 2&-1&-1\\ 0&-2&1\end{bmatrix},\) find \(A^{-1}\) and use it to solve the following system of equations: \(x-2y=10\), \(2x-y-z=8\), \(-2y+z=7\).
OR
(b) If \(A=\begin{bmatrix}-1&a&2\\ 1&2&x\\ 3&1&1\end{bmatrix}\) and \(A^{-1}=\begin{bmatrix}1&-1&1\\ -8&7&-5\\ b&y&3\end{bmatrix},\) find the value of \((a+x)-(b+y)\). (5)
(a) Evaluate: \(\int_{0}^{\frac{\pi}{4}}\frac{\sin x+\cos x}{9+16\sin 2x}dx\).
OR
(b) Evaluate: \(\int_{0}^{\frac{\pi}{2}}\sin 2x\tan^{-1}(\sin x)dx\). (5)
Using integration, find the area of the ellipse \(\frac{x^{2}}{16}+\frac{y^{2}}{4}=1,\) included between the lines \(x=-2\) and \(x=2\). (5)
The image of point \(P(x,y,z)\) with respect to line \(\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}\) is \(P^{\prime}(1,0,7)\). Find the coordinates of point P. (5)

SECTION-E (Case Study Based Questions - 4 Marks Each)

This section comprises of **3 case study based questions** of 4 marks each. (Internal choice has been provided in 2 questions)

Case Study 1: Over Speed Violation Detection (OSVD)

The traffic police has installed Over Speed Violation Detection (OSVD) system at various locations in a city. These cameras can capture a speeding vehicle from a distance of 300 m and even function in the dark. A camera is installed on a pole at the height of 5 m. It detects a car travelling away from the pole at the speed of \(20\text{ m/s}\). At any point, \(x\text{ m}\) away from the base of the pole, the angle of elevation of the speed camera from the car C is \(\theta\). [Image illustrating Average Speed Detection and RADAR Speed Detection]

On the basis of the above information, answer the following questions:
1. Express \(\tan \theta\) (based on context) in terms of height of the camera installed on the pole and \(x\). (1 mark)
2. Find \(\frac{d\theta}{dx}\). (1 mark)
3. (a) Find the rate of change of angle of elevation with respect to time at an instant when the car is 50 m away from the pole. (2 marks)
  OR
  (b) If the rate of change of angle of elevation with respect to time of another car at a distance of 50 m from the base of the pole is \(\frac{3}{101}\text{ rad/s}\), then find the speed of the car. (2 marks)
(4)
Case Study 2: Air Turbulence (Probability)

According to recent research, air turbulence has increased in various regions around the world due to climate change. Turbulence makes flights bumpy and often delays the flights. Assume that, an airplane observes severe turbulence, moderate turbulence or light turbulence with equal probabilities. Further, the chance of an airplane reaching late to the destination are 55%, 37% and 17% due to severe, moderate and light turbulence respectively. [Image illustrating Turbulence intensity]

On the basis of the above information, answer the following questions:
1. Find the probability that an airplane reached its destination late. (2 marks)
2. If the airplane reached its destination late, find the probability that it was due to moderate turbulence. (2 marks)
(4)
Case Study 3: [Title is missing in source document]

The preamble starts: A doctor has been appointed in a village... (Preamble is incomplete in the source document.)

Based upon the above information, answer the following questions:
1. Find the probability that the doctor visits: (Question is incomplete in source document) (1 mark)
2. [Part 2 of Q38, 1 mark - Missing content]
3. (a) [Part 3(a) of Q38, 2 marks - Missing content]
  OR
  (b) [Part 3(b) of Q38, 2 marks - Missing content]
(4)