CBSE 2025, Set 3 Solutions
CBSE Mathematics (2025)

If the sides $AB$ and $AC$ of $\triangle ABC$ are represented by vectors $\hat{j}+\hat{k}$ and $3\hat{i}-\hat{j}+4\hat{k}$ respectively, then the length of the median through $A$ on $BC$ is :

Show that the area of a parallelogram whose diagonals are represented by $\vec{a}$ and $\vec{b}$ is given by $\frac{1}{2}|\vec{a}\times\vec{b}|$. Also find the area of a parallelogram whose diagonals are $2\hat{i}-\hat{j}+\hat{k}$ and $\hat{i}+3\hat{j}-\hat{k}$.

Assertion (A): In a Linear Programming Problem, if the feasible region is empty, then the Linear Programming Problem has no solution.\nReason (R): A feasible region is defined as the region that satisfies all the constraints.

If $\int e^{-3\log x}dx=f(x)+C,$ then $f(x)$ is :

Find: $\int \frac{\sqrt{x}}{1+\sqrt{x^{3}}} dx$

Find: $\int\frac{dx}{\sin x+\sin 2x}$

Case Study - 1 Some students are having a misconception while comparing decimals. For example, a student may mention that $78.56 > 78.9$ as $7856 > 789$. In order to assess this concept, a decimal comparison test was administered to the students of class VI through the following question: In the recently held Sports Day in the school, 5 students participated in a javelin throw competition. The distances to which they have thrown the javelin are shown below in the table:

The students were asked to identify who has thrown the javelin the farthest. Based on the test attempted by the students, the teacher concludes that $40\%$ of the students have the misconception in the concept of decimal comparison and the rest do not have the misconception. $80\%$ of the students having misconception answered Bijoy as the correct answer in the paper. $90\%$ of the students who are identified with not having misconception, did not answer Bijoy as their answer.

On the basis of the above information, answer the following question: (iii) (a) What is the probability that a student who answered as Bijoy is having misconception?

Case Study - 2 An engineer is designing a new metro rail network in a city.

Initially, two metro lines, Line A and Line B, each consisting of multiple stations are designed. The track for Line A is represented by $l_1: \frac{x-2}{3} = \frac{y+1}{-2} = \frac{z-3}{4}$ while the track for Line B is represented by $l_2: \frac{x-1}{2} = \frac{y-3}{1} = \frac{z+2}{-3}$.

Based on the above information, answer the following question: (iii) (a) To connect the stations, a pedestrian pathway perpendicular to the two metro lines is to be constructed which passes through point $(3, 2, 1)$. Determine the equation of the pedestrian walkway.

Case Study - 2 An engineer is designing a new metro rail network in a city.

Initially, two metro lines, Line A and Line B, each consisting of multiple stations are designed. The track for Line A is represented by $l_1: \frac{x-2}{3} = \frac{y+1}{-2} = \frac{z-3}{4}$ while the track for Line B is represented by $l_2: \frac{x-1}{2} = \frac{y-3}{1} = \frac{z+2}{-3}$.

Based on the above information, answer the following question: (iii) (b) Find the shortest distance between Line A and Line B.

Case Study - 1 Some students are having a misconception while comparing decimals. For example, a student may mention that $78.56 > 78.9$ as $7856 > 789$. In order to assess this concept, a decimal comparison test was administered to the students of class VI through the following question: In the recently held Sports Day in the school, 5 students participated in a javelin throw competition. The distances to which they have thrown the javelin are shown below in the table:

The students were asked to identify who has thrown the javelin the farthest. Based on the test attempted by the students, the teacher concludes that $40\%$ of the students have the misconception in the concept of decimal comparison and the rest do not have the misconception. $80\%$ of the students having misconception answered Bijoy as the correct answer in the paper. $90\%$ of the students who are identified with not having misconception, did not answer Bijoy as their answer.

On the basis of the above information, answer the following question: (iii) (b) What is the probability that a student who answered as Bijoy is amongst students who do not have the misconception?

Domain of $\sin^{-1}(2x^{2}-3)$ is :

The matrix $\begin{pmatrix}0&1&-2\\\\ -1&0&-7\\\\ 2&7&0\end{pmatrix}$ is a :

If $f(x)=\begin{cases}3x-2,&0<x\le1\\\\ 2x^{2}+ax,&1<x<2\end{cases}$ is continuous for $x\in(0,2)$, then $a$ is equal to :

If $y=\log_{2x}(\sqrt{2x})$, then $\frac{dy}{dx}$ is equal to :

If $f:\mathbb{N}\rightarrow \mathbb{W}$ is defined as\n$f(n) = \begin{cases}\frac{n}{2},&\text{if } n \text{ is even}\\\\ 0,&\text{if } n \text{ is odd}\end{cases}$\nthen $f$ is :

The coordinates of the foot of the perpendicular drawn from the point $A(-2,3,5)$ on the y-axis is :

If $A$ and $B$ are invertible matrices of order $3\times3$ such that $\det (A)=4$ and $\det [(AB)^{-1}]=\frac{1}{20}$ then $\det (B)$ is equal to :

For real $x$, let $f(x)=x^{3}+5x+1$. Then :

The values of $\lambda$ so that $f(x)=\sin x-\cos x-\lambda x+C$ decreases for all real values of $x$ are :

If $A$ and $B$ are square matrices of same order such that $AB=A$ and $BA=B$, then $A^{2}+B^{2}$ is equal to :

The area of the region enclosed by the curve $y=\sqrt{x}$ and the lines $x=0$ and $x=4$ and x-axis is :

The value of $\int_{0}^{1}\frac{dx}{e^{x}+e^{-x}}$ is :

The function $f$ defined by\n$f(x)=\begin{cases}x,&\text{if } x\le1\\\\ 5,&\text{if } x>1\end{cases}$\nis not continuous at :

The solution of the differential equation $\frac{dy}{dx}=\frac{-x}{y}$ represents family of :

If $f(x)=2x+\cos x$, then $f(x)$ :

The corner points of the feasible region of a Linear Programming Problem are $(0, 2)$, $(3, 0)$, $(6, 0)$, $(6, 8)$ and $(0, 5)$. If $Z=ax+by;$ ($a, b>0$) be the objective function, and maximum value of $Z$ is obtained at $(0, 2)$ and $(3, 0)$, then the relation between $a$ and $b$ is :

Assertion (A): If $A$ and $B$ are two events such that $P(A\cap B)=0$, then $A$ and $B$ are independent events.\nReason (R): Two events are independent if the occurrence of one does not effect the occurrence of the other.

If $A=\begin{pmatrix}2&-2\\\\ -2&2\end{pmatrix}$ and $A^{2}=kA$ then find the value of $k$.

Calculate the area of the region bounded by the curve $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$ and the x-axis using integration.

Find the least value of '$a$' so that $f(x)=2x^{2}-ax+3$ is an increasing function on $[2, 4]$.

If $f(x)=x+\frac{1}{x}$, $x\ge1$, show that $f$ is an increasing function.

A cylindrical water container has developed a leak at the bottom. The water is leaking at the rate of $5 \text{ cm}^{3}\text{/s}$ from the leak. If the radius of the container is $15 \text{ cm}$, find the rate at which the height of water is decreasing inside the container, when the height of water is $2 \text{ metres}$.

Find the distance of the point $(-1,-5,-10)$ from the point of intersection of the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and $\frac{x-4}{5}=\frac{y-1}{2}=z$.

If $f:\mathbb{R}^{+}\rightarrow \mathbb{R}$ is defined as $f(x)=\log_{a}x$ ($a>0$ and $a\ne1$), prove that $f$ is a bijection.\n($\mathbb{R}^{+}$ is a set of all positive real numbers.)

Let $A=\{1,2,3\}$ and $B=\{4,5,6\}$. A relation $R$ from $A$ to $B$ is defined as $R=\{(x,y):x+y=6, x\in A, y\in B\}$.\n(i) Write all elements of $R$.\n(ii) Is $R$ a function? Justify.\n(iii) Determine domain and range of $R$.

The probability distribution of a random variable $X$ is given by :\n\n| $X$ | $0$ | $1$ | $2$ | $3$ |\n|---|---|---|---|---|\n| $P(X)$ | $p$ | $\frac{p}{3}$ | $\frac{p}{6}$ | $\frac{p}{12}$ |\n\n(i) Determine the value of $p$.\n(ii) Calculate $P(X\ge1)$.\n(iii) Calculate expectation of $X$, i.e. $E(X)$.

In a city, a survey was conducted among residents about their preferred mode of commuting. It was found that $50\%$ people preferred using public transport, $35\%$ preferred using a bicycle and $20\%$ use both public transport and a bicycle. If a person is selected at random, find the probability that :\n(i) The person uses only public transport.\n(ii) The person uses a bicycle, given that they also use the public transport.\n(iii) The person uses neither public transport nor a bicycle.

Find $k$ so that \n$f(x)=\begin{cases}\frac{x^{2}-2x-3}{x+1},&x\ne-1\\\\ k,&x=-1\end{cases}$ \nis continuous at $x=-1$.

Check the differentiability of function $f(x)=x|x|$ at $x=0$.

The relation between the height of the plant ($y\text{ cm}$) with respect to exposure to sunlight is governed by the equation $y=4x-\frac{1}{2}x^{2}$, where $x$ is the number of days exposed to sunlight. (i) Find the rate of growth of the plant with respect to sunlight. (ii) In how many days will the plant attain its maximum height ? What is the maximum height ?

Find the equation of a line in vector and cartesian form which passes through the point $(1, 2, -4)$ and is perpendicular to the lines $\frac{x-8}{3}=\frac{y+19}{-16}=\frac{z-10}{7}$ and $\vec{r}=15\hat{i}+29\hat{j}+5\hat{k}+\mu(3\hat{i}+8\hat{j}-5\hat{k})$.

For the given graph of a Linear Programming Problem, write all the constraints satisfying the given feasible region.\n\n

Evaluate: $\int_{0}^{3/2}|x \cos \pi x| dx$

If $A$ is a $3\times3$ invertible matrix, show that for any scalar $k\ne0$, $(kA)^{-1}=\frac{1}{k}A^{-1}$. Hence calculate $(3A)^{-1}$, where $A=\begin{pmatrix}2&-1&1\\\\ -1&2&-1\\\\ 1&-1&2\end{pmatrix}$.

Case Study - 3 During a heavy gaming session, the temperature of a student's laptop processor increases significantly. After the session, the processor begins to cool down, and the rate of cooling is proportional to the difference between the processor's temperature and the room temperature ($25^\circ\text{C}$). Initially the processor's temperature is $85^\circ\text{C}$. The rate of cooling is defined by the equation $\frac{d}{dt}(T(t))=-k(T(t)-25)$, where $T(t)$ represents the temperature of the processor at time $t$ (in minutes) and $k$ is a constant.

Based on the above information, answer the following question: (i) Find the expression for the temperature of the processor $T(t)$ given that $T(0)=85^\circ\text{C}$.

Case Study - 3 During a heavy gaming session, the temperature of a student's laptop processor increases significantly. After the session, the processor begins to cool down, and the rate of cooling is proportional to the difference between the processor's temperature and the room temperature ($25^\circ\text{C}$). Initially the processor's temperature is $85^\circ\text{C}$. The rate of cooling is defined by the equation $\frac{d}{dt}(T(t))=-k(T(t)-25)$, where $T(t)$ represents the temperature of the processor at time $t$ (in minutes) and $k$ is a constant.

Based on the above information, answer the following question: (ii) How much time will it take for the processor's temperature to reach $40^\circ\text{C}$? Given that $k=0.03$, $\log_e 4 = 1.3863$.

Case Study - 1 Some students are having a misconception while comparing decimals. For example, a student may mention that $78.56 > 78.9$ as $7856 > 789$. In order to assess this concept, a decimal comparison test was administered to the students of class VI through the following question: In the recently held Sports Day in the school, 5 students participated in a javelin throw competition. The distances to which they have thrown the javelin are shown below in the table:

The students were asked to identify who has thrown the javelin the farthest. Based on the test attempted by the students, the teacher concludes that $40\%$ of the students have the misconception in the concept of decimal comparison and the rest do not have the misconception. $80\%$ of the students having misconception answered Bijoy as the correct answer in the paper. $90\%$ of the students who are identified with not having misconception, did not answer Bijoy as their answer.

On the basis of the above information, answer the following question: (i) What is the probability of a student not having misconception but still answers Bijoy in the test?

Case Study - 1 Some students are having a misconception while comparing decimals. For example, a student may mention that $78.56 > 78.9$ as $7856 > 789$. In order to assess this concept, a decimal comparison test was administered to the students of class VI through the following question: In the recently held Sports Day in the school, 5 students participated in a javelin throw competition. The distances to which they have thrown the javelin are shown below in the table:

The students were asked to identify who has thrown the javelin the farthest. Based on the test attempted by the students, the teacher concludes that $40\%$ of the students have the misconception in the concept of decimal comparison and the rest do not have the misconception. $80\%$ of the students having misconception answered Bijoy as the correct answer in the paper. $90\%$ of the students who are identified with not having misconception, did not answer Bijoy as their answer.

On the basis of the above information, answer the following question: (ii) What is the probability that a randomly selected student answers Bijoy as his answer in the test?

Case Study - 2 An engineer is designing a new metro rail network in a city.

Initially, two metro lines, Line A and Line B, each consisting of multiple stations are designed. The track for Line A is represented by $l_1: \frac{x-2}{3} = \frac{y+1}{-2} = \frac{z-3}{4}$ while the track for Line B is represented by $l_2: \frac{x-1}{2} = \frac{y-3}{1} = \frac{z+2}{-3}$.

Based on the above information, answer the following question: (i) Find whether the two metro tracks are parallel.

Case Study - 2 An engineer is designing a new metro rail network in a city.

Initially, two metro lines, Line A and Line B, each consisting of multiple stations are designed. The track for Line A is represented by $l_1: \frac{x-2}{3} = \frac{y+1}{-2} = \frac{z-3}{4}$ while the track for Line B is represented by $l_2: \frac{x-1}{2} = \frac{y-3}{1} = \frac{z+2}{-3}$.

Based on the above information, answer the following question: (ii) Solar panels are to be installed on the rooftop of the metro stations. Determine the equation of the line representing the placement of solar panels on the rooftop of Line A's stations, given that panels are to be positioned parallel to Line A's track ($l_1$) and pass through the point $(1, -2, -3)$.

Find domain of $\sin^{-1}\sqrt{x-1}$.

Simplify $\sin^{-1}\left(\frac{x}{\sqrt{1+x^{2}}}\right)$.