CBSE 2025, Set 1 Solutions
CBSE Mathematics (2025)

Find: $\int \frac{x^2+1}{(x-1)^2(x+3)} dx$

If $\vec{a}$ and $\vec{b}$ are position vectors of point $A$ and point $B$ respectively, find the position vector of point $C$ on $BA$ produced such that $BC = 3BA$.

If $|\vec{a} + \vec{b}| = |\vec{a} - \vec{b}|$ for any two vectors $\vec{a}$ and $\vec{b}$, then vectors $\vec{a}$ and $\vec{b}$ are:

Case Study 1

Three friends A, B and C move out from the same location O at the same time in three different directions to reach their destinations. They move on straight paths and decide that A and B will meet C at his destination following straight paths from A to C and B to C such that $\overrightarrow{OA} = \vec{a}$, $\overrightarrow{OB} = \vec{b}$ and $\overrightarrow{OC} = 5\vec{a} - 2\vec{b}$.

Based upon the above information, answer the following questions:

(i) Complete the given figure to explain their entire movement plan along the respective vectors. [1 mark]

(ii) Find vectors $\overrightarrow{AC}$ and $\overrightarrow{BC}$. [1 mark]

(iii)(a) If $\vec{a} \cdot \vec{b} = 1$, distance OA = 1 km and distance OB = 2 km, then find the angle between $\overrightarrow{OA}$ and $\overrightarrow{OB}$. Also find $|\vec{a} \times \vec{b}|$. [2 marks]

OR

(iii)(b) If $\vec{a} = 2\hat{i} - \hat{j} + 4\hat{k}$ and $\vec{b} = \hat{j} - \hat{k}$, then find a unit vector perpendicular to both $(\vec{a} + \vec{b})$ and $(\vec{a} - \vec{b})$. [2 marks]

Let $|\vec{a}| = 5$ and $-2 \le \lambda \le 1$. Then, the range of $\lambda |\vec{a}|$ is:

Assertion (A): If $|\vec{a} \times \vec{b}|^2 + |\vec{a} \cdot \vec{b}|^2 = 256$ and $|\vec{b}| = 8$, then $|\vec{a}| = 2$. Reason (R): $\sin^2\theta + \cos^2\theta = 1$ and $|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin \theta$ and $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos \theta$.

Vector $\vec{r}$ is inclined at equal angles to the three axes $x, y$ and $z$. If magnitude of $\vec{r}$ is $5\sqrt{3}$ units, then find $\vec{r}$.

In a Linear Programming Problem, the objective function $Z = 5x + 4y$ needs to be maximised under constraints $3x + y \leq 6$, $x \leq 1$, $x, y \geq 0$. Express the LPP on the graph, shade the feasible region and mark the corner points.

In a Linear Programming Problem (LPP), the objective function $Z = 2x + 5y$ is to be maximised under the following constraints:

$x + y \le 4$, $3x + 3y \ge 18$, $x, y \ge 0$

Study the graph and select the correct option.

For a Linear Programming Problem (LPP), the given objective function is $Z = x + 2y$. The feasible region PQRS determined by the set of constraints is shown as a shaded region in the graph.

$P \equiv (\frac{3}{13}, \frac{24}{13})$, $Q \equiv (\frac{3}{2}, \frac{15}{4})$, $R \equiv (\frac{7}{2}, \frac{3}{4})$, $S \equiv (\frac{18}{7}, \frac{2}{7})$

Which of the following statements is correct?

Draw a rough sketch for the curve $y = 2 + |x + 1|$. Using integration, find the area of the region bounded by the curve $y = 2 + |x + 1|$, $x = -4$, $x = 3$ and $y = 0$.

A box has 4 green, 8 blue and 3 red pens. A student picks up a pen at random, checks its colour and replaces it in the box. He repeats this process 3 times. The probability that at least one pen picked was red is:

A person is Head of two independent selection committees I and II. If the probability of making a wrong selection in committee I is 0.03 and that in committee II is 0.01, then find the probability that the person makes the correct decision of selection: (i) in both committees (ii) in only one committee

Case Study - 3 A shop selling electronic items sells smartphones of only three reputed companies A, B and C because chances of their manufacturing a defective smartphone are only 5%, 4% and 2% respectively. In his inventory he has 25% smartphones from company A, 35% smartphones from company B and 40% smartphones from company C.

A person buys a smartphone from this shop. (i) Find the probability that it was defective.

A student wants to pair up natural numbers in such a way that they satisfy the equation $2x + y = 41$, $x, y \in N$. Find the domain and range of the relation. Check if the relation thus formed is reflexive, symmetric and transitive. Hence, state whether it is an equivalence relation or not.

Let the polished side of the mirror be along the line $\frac{x}{1} = \frac{1-y}{-2} = \frac{2z-4}{6}$. A point $P(1, 6, 3)$, some distance away from the mirror, has its image formed behind the mirror. Find the coordinates of the image point and the distance between the point $P$ and its image.

Find the distance of the point $P(2, 4, -1)$ from the line $\frac{x+5}{1} = \frac{y+3}{4} = \frac{z-6}{-9}$.

What is the total number of possible matrices of order $3 \times 3$ with each entry as $\sqrt{2}$ or $\sqrt{3}$?

Domain of $f(x) = \cos^{-1} x + \sin x$ is:

The matrix $A = \begin{bmatrix} \sqrt{3} & 0 & 0 \\ 0 & \sqrt{2} & 0 \\ 0 & 0 & \sqrt{5} \end{bmatrix}$ is a/an:

If A and B are two square matrices each of order 3 with $|A| = 3$ and $|B| = 5$, then $|2AB|$ is:

Let A be a square matrix of order 3. If $|A| = 5$, then $|\text{adj } A|$ is:

If $f(x) = \begin{cases} 1, & \text{if } x \leq 3 \\ ax + b, & \text{if } 3 < x < 5 \\ 7, & \text{if } 5 \leq x \end{cases}$ is continuous on $\mathbb{R}$, then the values of $a$ and $b$ are:

If $f(x) = -2x^8$, then the correct statement is:

A spherical ball has a variable diameter $\dfrac{5}{2}(3x + 1)$. The rate of change of its volume w.r.t. $x$, when $x = 1$, is:

If $f : \mathbb{R} \to \mathbb{R}$ is defined as $f(x) = 2x - \sin x$, then $f$ is:

$\displaystyle\int \dfrac{e^{9\log x} - e^{8\log x}}{e^{6\log x} - e^{5\log x}}\, dx$ is equal to:

For a function $f(x)$, which of the following holds true?

$\displaystyle\int \dfrac{e^x}{\sqrt{4 - e^{2x}}}\,dx$ is equal to:

If $P(A) = \dfrac{1}{7}$, $P(B) = \dfrac{5}{7}$ and $P(A \cap B) = \dfrac{4}{7}$, then $P(\bar{A} \mid B)$ is:

A coin is tossed and a card is selected at random from a well shuffled pack of 52 playing cards. The probability of getting head on the coin and a face card from the pack is:

Read the following statements: Assertion (A) and Reason (R). Select the correct answer.

Assertion (A): $f(x) = \begin{cases} x\sin\dfrac{1}{x}, & x \neq 0 \\ 0, & x = 0 \end{cases}$ is continuous at $x = 0$.

Reason (R): When $x \to 0$, $\sin\dfrac{1}{x}$ is a finite value between $-1$ and $1$.

A student tries to tie ropes, parallel to each other from one end of the wall to the other. If one rope is along the vector $3\hat{i} + 15\hat{j} + 6\hat{k}$ and the other is along the vector $2\hat{i} + 10\hat{j} + \lambda\hat{k}$, then the value of $\lambda$ is:

Let $f : A \to B$ be defined by $f(x) = \dfrac{x-2}{x-3}$, where $A = \mathbb{R} - \{3\}$ and $B = \mathbb{R} - \{1\}$. Discuss the bijectivity of the function.

If $A = \begin{bmatrix} 2 & 3 \\ -1 & 2 \end{bmatrix}$, then show that $A^2 - 4A + 7I = O$.

Differentiate $\left(\dfrac{5^x}{x^5}\right)$ with respect to $x$.

If $-2x^2 - 5xy + y^3 = 76$, then find $\dfrac{dy}{dx}$.

10 identical blocks are marked with '0' on two of them, '1' on three of them, '2' on four of them and '3' on one of them and put in a box. If $X$ denotes the number written on the block, then write the probability distribution of $X$ and calculate its mean.

In a village of 8000 people, 3000 go out of the village to work and 4000 are women. It is noted that 30% of women go out of the village to work. What is the probability that a randomly chosen individual is either a woman or a person working outside the village?

Show that the function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = 4x^3 - 5$, $\forall\, x \in \mathbb{R}$, is one-one and onto.

Let $R$ be a relation defined on the set $\mathbb{N}$ of natural numbers such that $R = \{(x, y) : xy \text{ is a perfect square of a natural number},\ x, y \in \mathbb{N}\}$. Determine if $R$ is an equivalence relation.

Let $2x + 5y - 1 = 0$ and $3x + 2y - 7 = 0$ represent the equations of two lines on which ants are moving on the ground. Using matrix method, find a point common to the paths of the ants.

A shopkeeper sells 50 Chemistry, 60 Physics and 35 Maths books on Day I and sells 40 Chemistry, 45 Physics and 50 Maths books on Day II. If the selling price for each book is ₹150 (Chemistry), ₹175 (Physics) and ₹180 (Maths), then find his total sale in two days using matrix method. If the cost price of all the books together is ₹35,000, what profit did he earn after the sale of two days?

Differentiate $y = \sqrt{\log\left\{\sin\left(\dfrac{x^3}{3} - 1\right)\right\}}$ with respect to $x$.

Amongst all pairs of positive integers with product as 289, find which pair adds up to the least sum.

Find the shortest distance between the lines: $\vec{r} = (2\hat{i} - \hat{j} + 3\hat{k}) + \lambda(\hat{i} - 2\hat{j} + 3\hat{k})$ $\vec{r} = (\hat{i} + 4\hat{k}) + \mu(3\hat{i} - 6\hat{j} + 9\hat{k})$

Find: $\displaystyle\int \dfrac{x^2 + 1}{(x^2 + 2)(2x^2 + 1)}\,dx$

Evaluate: $\displaystyle\int_0^{\pi} \dfrac{x\tan x}{\sec x + \tan x}\,dx$

The scalar product of the vector $\vec{a} = \hat{i} - \hat{j} + 2\hat{k}$ with a unit vector along the sum of vectors $\vec{b} = 2\hat{i} - 4\hat{j} + 5\hat{k}$ and $\vec{c} = \lambda\hat{i} - 2\hat{j} - 3\hat{k}$ is equal to 1. Find the value of $\lambda$.

In a Linear Programming Problem for objective function $Z = 18x + 10y$ subject to constraints $4x + y \geq 20$, $2x + 3y \geq 30$, $x, y \geq 0$, find the minimum value of $Z$.

A woman discovered a scratch along a straight line on a circular table top of radius 8 cm. She divided the table top into 4 equal quadrants and discovered the scratch passing through the origin inclined at an angle $\dfrac{\pi}{4}$ anticlockwise along the positive direction of the x-axis. Find the area of the region enclosed by the x-axis, the scratch and the circular table top in the first quadrant, using integration.

Solve the differential equation $\dfrac{dy}{dx} = \cos x - 2y$.

Find the point Q on the line $\dfrac{2x+4}{6} = \dfrac{y+1}{2} = \dfrac{-2z+6}{-4}$ at a distance of $3\sqrt{2}$ from the point $P(1, 2, 3)$.

Find the image of the point $(-1, 5, 2)$ in the line $\dfrac{2x-4}{2} = \dfrac{y}{2} = \dfrac{2-z}{3}$. Also find the length of the line segment joining the given point and its image.

Case Study 2

Camphor is a waxy, colourless solid with a strong aroma that evaporates through sublimation when left in open air at room temperature.

A cylindrical camphor tablet whose height equals its radius ($r$) evaporates such that the rate of reduction of its volume is proportional to its total surface area: $\dfrac{dV}{dt} = kS$, where $V$ is volume, $S$ is surface area, and $t$ is time in hours.

Based upon the above information, answer the following questions:

(i) Write the order and degree of the given differential equation. [1 mark]

(ii) Substituting $V = \pi r^3$ and $S = 2\pi r^2$, the differential equation $\dfrac{dr}{dt} = \dfrac{2}{3}k$ is obtained. Solve it, given that $r(0) = 5$ mm. [1 mark]

(iii)(a) If $r = 3$ mm when $t = 1$ hour, find the value of $k$. Hence find $t$ when $r = 0$ mm. [2 marks]

OR

(iii)(b) If $r = 1$ mm when $t = 1$ hour, find the value of $k$. Hence find $t$ when $r = 0$ mm. [2 marks]

Case Study 3

Based on results of regular medical check-ups in a hospital, it was found that out of 1000 people, 700 were very healthy, 200 maintained average health and 100 had a poor health record.

Let $A_1$: people with good health, $A_2$: people with average health, $A_3$: people with poor health.

During a pandemic, the chances of people from categories $A_1$, $A_2$ and $A_3$ contracting the disease are 25%, 35% and 50% respectively.

Based upon the above information, answer the following questions:

(i) A person was tested randomly. What is the probability that he/she has contracted the disease? [2 marks]

(ii) Given that the person has not contracted the disease, what is the probability that the person is from category $A_2$? [2 marks]

Evaluate: $\int \frac{x+5}{(x+6)^2} e^x dx$

The order and degree of the following differential equation are, respectively: $-\frac{d^4y}{dx^4} + 2e^{dy/dx} + y^2 = 0$

The solution for the differential equation $\log(\frac{dy}{dx}) = 3x + 4y$ is:

$\int \frac{x+5}{(x+6)^2} e^x dx$ is equal to:

Assertion (A): Let $f(x) = e^x$ and $g(x) = \log x$. Then $(f+g)(x) = e^x + \log x$, where domain of $(f+g)$ is $R$. Reason (R): $Dom(f+g) = Dom(f) \cap Dom(g)$.

If $(x)^y = (y)^x$, then find $\frac{dy}{dx}$.

Determine the values of $x$ for which $f(x) = \frac{x-4}{x+1}$, $x \ne -1$ is an increasing or a decreasing function.

Determine if the lines $\vec{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda(3\hat{i} - \hat{j})$ and $\vec{r} = (4\hat{i} - \hat{k}) + \mu(2\hat{i} + 3\hat{k})$ intersect with each other.

Differentiate $y = \sin^{-1}(3x - 4x^3)$ w.r.t. x, if $x \in [-\frac{1}{2}, \frac{1}{2}]$.

Differentiate $y = \cos^{-1}(\frac{1 - x^2}{1 + x^2})$ with respect to x, when $x \in (0, 1)$.

Show that the function $f: N \rightarrow N$, where $N$ is a set of natural numbers, given by $f(n) = \begin{cases} n - 1, & \text{if } n \text{ is even} \\ n + 1, & \text{if } n \text{ is odd} \end{cases}$ is a bijection.

Let the position vectors of the points A, B and C be $3\hat{i} - \hat{j} - 2\hat{k}$, $\hat{i} + 2\hat{j} - \hat{k}$ and $\hat{i} + 5\hat{j} + 3\hat{k}$ respectively. Find the vector and cartesian equations of the line passing through A and parallel to line BC.

Case Study - 1 Three students, Neha, Rani and Sam go to a market to purchase stationery items. Neha buys 4 pens, 3 notepads and 2 erasers and pays ₹60. Rani buys 2 pens, 4 notepads and 6 erasers for ₹90. Sam pays ₹70 for 6 pens, 2 notepads and 3 erasers.

Based upon the above information, answer the following question: (i) Form the equations required to solve the problem of finding the price of each item, and express it in the matrix form $AX = B$.

Case Study - 1 Three students, Neha, Rani and Sam go to a market to purchase stationery items. Neha buys 4 pens, 3 notepads and 2 erasers and pays ₹60. Rani buys 2 pens, 4 notepads and 6 erasers for ₹90. Sam pays ₹70 for 6 pens, 2 notepads and 3 erasers.

Based upon the above information, answer the following question: (ii) Find $|A|$ and confirm if it is possible to find $A^{-1}$.

Case Study - 1 Three students, Neha, Rani and Sam go to a market to purchase stationery items. Neha buys 4 pens, 3 notepads and 2 erasers and pays ₹60. Rani buys 2 pens, 4 notepads and 6 erasers for ₹90. Sam pays ₹70 for 6 pens, 2 notepads and 3 erasers.

Based upon the above information, answer the following question: (iii) (a) Find $A^{-1}$, if possible, and write the formula to find $X$.

Case Study - 1 Three students, Neha, Rani and Sam go to a market to purchase stationery items. Neha buys 4 pens, 3 notepads and 2 erasers and pays ₹60. Rani buys 2 pens, 4 notepads and 6 erasers for ₹90. Sam pays ₹70 for 6 pens, 2 notepads and 3 erasers.

Based upon the above information, answer the following question: (iii) (b) Find $A^2 - 8I$ where $I$ is an identity matrix.

Case Study - 2 A ladder of fixed length 'h' is to be placed along the wall such that it is free to move along the height of the wall.

Based upon the above information, answer the following question: (ii) Find the derivative of the area ($A$) with respect to the height on the wall ($x$), and find its critical point.

Case Study - 2 A ladder of fixed length 'h' is to be placed along the wall such that it is free to move along the height of the wall.

Based upon the above information, answer the following question: (iii) (b) If the foot of the ladder whose length is $5 \text{ m}$, is being pulled towards the wall such that the rate of decrease of distance ($y$) is $2 \text{ m/s}$. then at what rate is the height on the wall ($x$) increasing, when the foot of the ladder is $3 \text{ m}$ away from the wall?

Solve the differential equation $(1+x^2)\frac{dy}{dx} + 2xy - 4x^2 = 0$ subject to initial condition $y(0) = 0$.

Case Study - 3 A shop selling electronic items sells smartphones of only three reputed companies A, B and C because chances of their manufacturing a defective smartphone are only 5%, 4% and 2% respectively. In his inventory he has 25% smartphones from company A, 35% smartphones from company B and 40% smartphones from company C.

A person buys a smartphone from this shop. (ii) What is the probability that this defective smartphone was manufactured by company B?

Evaluate: $\int_0^{\pi/2} \frac{x}{\sin x + \cos x} dx$

Solve the differential equation: $x^2y dx - (x^3 + y^3)dy = 0$.

If $A = \begin{bmatrix} -1 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \\ \end{bmatrix}$, then $A$ is a/an:

The distance of the point $(1, 2, 3)$ from the plane $2x - 2y + z = 5$ is:

If the direction ratios of a line are proportional to $(2, -3, 6)$, then the direction cosines are:

If $P(A) = \frac{1}{3}$, $P(B) = \frac{1}{4}$ and $A$ and $B$ are independent events, then $P(A ∩ B)$ is:

If $\int_0^a f(x) dx = 5$ and $\int_0^a g(x) dx = 3$, then $\int_0^a [2f(x) - 3g(x)] dx$ is:

If $\tan^{-1}(x^2 - y^2) = a$, where 'a' is a constant, then $\frac{dy}{dx}$ is:

If $y = a \cos(\log x) + b \sin(\log x)$, then $x^2 \frac{d^2y}{dx^2} + x \frac{dy}{dx}$ is:

Let both $AB'$ and $B'A$ be defined for matrices A and B. If order of A is $n × m$, then the order of B is:

If $f(x) = \begin{cases} \frac{\log(1 + ax) + \log(1 - bx)}{x}, & x \ne 0 \\ k, & x = 0 \end{cases}$ is continuous at $x = 0$, then the value of $k$ is:

Sum of two skew-symmetric matrices of same order is always a/an:

Let $f(x) = |x|, x \in R$. Then, which of the following statements is incorrect?

Let $f'(x) = 3(x^2 + 2x) - \frac{4}{x^3} + 5$, $f(1)=0$. Then $f(x)$ is:

The area of the region bounded by the curve $y^2 = x$ between $x = 0$ and $x = 1$ is:

Differentiate $\sqrt{e^{\sqrt{2x}}}$ with respect to $e^{\sqrt{2x}}$ for $x > 0$.

Let $A = \begin{bmatrix} 1 \\ 4 \\ -2 \end{bmatrix}$ and $C = \begin{bmatrix} 3 & 4 & 2 \\ 12 & 16 & 8 \\ -6 & -8 & -4 \end{bmatrix}$ be two matrices. Then, find the matrix B if $AB = C$.

Case Study - 2 A ladder of fixed length 'h' is to be placed along the wall such that it is free to move along the height of the wall.

Based upon the above information, answer the following question: (i) Express the distance ($y$) between the wall and foot of the ladder in terms of '$h$' and height ($x$) on the wall at a certain instant. Also, write an expression in terms of $h$ and $x$ for the area ($A$) of the right triangle, as seen from the side by an observer.

Consider the Linear Programming Problem, where the objective function $Z = x + 4y$ needs to be minimized subject to constraints $2x + y \ge 1000$ $x + 2y \ge 800$ $x, y \ge 0$ Draw a neat graph of the feasible region and find the minimum value of $Z$.

Case Study - 2 A ladder of fixed length 'h' is to be placed along the wall such that it is free to move along the height of the wall.

Based upon the above information, answer the following question: (iii) (a) Show that the area ($A$) of the right triangle is maximum at the critical point.

Find the domain of $f(x) = sin^{-1}(-x^2)$.

The given graph illustrates:

(The graph passes through the origin, has horizontal asymptotes at $y = \pi/2$ and $y = -\pi/2$, and is defined for all real x)

Read the following statements: Assertion (A) and Reason (R). Select the correct answer.

Assertion (A): Set of values of $\sec^{-1}\!\left(\dfrac{\sqrt{3}}{2}\right)$ is a null set.

Reason (R): $\sec^{-1} x$ is defined for $x \in \mathbb{R} - (-1, 1)$.

Evaluate: $\left[ sec^{-1}(-\sqrt{2}) - tan^{-1}\left(\frac{1}{\sqrt{3}}\right) \right]$

The following graph is a combination of:\n\n

If $\begin{bmatrix} 2x-1 & 3x \\ 0 & y^2-1 \end{bmatrix} = \begin{bmatrix} x+3 & 12 \\ 0 & 35 \end{bmatrix}$, then the value of $(x - y)$ is: