CBSE Class 12 Maths 2025 Question Paper 65/2/1 with Solutions

The projection vector of vector $\vec{a}$ on vector $\vec{b}$ is

1. (A)$\left(\frac{\vec{a}\cdot \vec{b}}{|\vec{b}{|}^2}\right)\vec{b}$
2. (B)$\frac{\vec{a}\cdot \vec{b}}{|\vec{b}|}$
3. (C)$\frac{\vec{a}\cdot \vec{b}}{|\vec{a}|}$
4. (D)$\left(\frac{\vec{a}\cdot \vec{b}}{|\vec{a}{|}^2}\right)\vec{b}$

The function $f(x)=x^2-4x+6$ is increasing in the interval

1. (A)$(0,2)$
2. (B)$(-\infty ,2]$
3. (C)$[1,2]$
4. (D)$[2,\infty )$

If $f(2a-x)=f(x)$, then ${\int }_0^{2a}f(x)dx$ is

1. (A)${\int }_0^{2a}f\left(\frac{x}{2}\right)dx$
2. (B)${\int }_0^{a}f(x)dx$
3. (C)$2{\int }_a^{0}f(x)dx$
4. (D)$2{\int }_0^{a}f(x)dx$

If $A=\left[\begin{array}{ccc}1& 12& 4y\\ 6x& 5& 2x\\ 8x& 4& 6\end{array}\right]$ is a symmetric matrix, then $(2x+y)$ is

1. (A)$-8$
2. (B)$0$
3. (C)$6$
4. (D)$8$

If $y={\sin }^{-1}x$, $-1\le x\le 0$, then the range of y is

1. (A)$\left(\frac{-\pi }{2},0\right)$
2. (B)$\left[\frac{-\pi }{2},0\right]$
3. (C)$\left[\frac{-\pi }{2},0\right)$
4. (D)$\left(\frac{-\pi }{2},0\right]$

If a line makes angles of $\frac{3\pi }{4},\frac{\pi }{3}$ and $\theta$ with the positive directions of $x$, y and z-axis respectively, then $\theta$ is

1. (A)$\frac{-\pi }{3}$ only
2. (B)$\frac{\pi }{3}$ only
3. (C)$\frac{\pi }{6}$
4. (D)$\pm \frac{\pi }{3}$

If E and F are two events such that $P(E)>0$ and $P(F)\ne 1$, then $P(\bar{E}/\bar{F})$ is

1. (A)$\frac{P(\bar{E})}{P(\bar{F})}$
2. (B)$1-P(\bar{E}/F)$
3. (C)$1-P(E/F)$
4. (D)$\frac{1-P(E\cup F)}{P(\bar{F})}$

Which of the following can be both a symmetric and skew-symmetric matrix ?

1. (A)Unit Matrix
2. (B)Diagonal Matrix
3. (C)Null Matrix
4. (D)Row Matrix

The equation of a line parallel to the vector $3\hat{i}+\hat{j}+2\hat{k}$ and passing through the point $(4,-3,7)$ is :

1. (A)$x=4t+3,y=-3t+1,z=7t+2$
2. (B)$x=3t+4,y=t+3,z=2t+7$
3. (C)$x=3t+4,y=t-3,z=2t+7$
4. (D)$x=3t+4,y=-t+3,z=2t+7$

Four friends Abhay, Bina, Chhaya and Devesh were asked to simplify $4AB+3(AB+BA)-4BA$, where A and B are both matrices of order $2\times 2$. It is known that $A\ne B\ne I$ and $A^{-1}\ne B$.
Their answers are given as :
Abhay : $6AB$
Bina : $7AB-BA$
Chhaya : $8AB$
Devesh : $7BA-AB$
Who answered it correctly ?

1. (A)Abhay
2. (B)Bina
3. (C)Chhaya
4. (D)Devesh

A cylindrical tank of radius 10 cm is being filled with sugar at the rate of $100\pi$ cm${}^3$/s. The rate, at which the height of the sugar inside the tank is increasing, is :

1. (A)0.1 cm/s
2. (B)0.5 cm/s
3. (C)1 cm/s
4. (D)1.1 cm/s

Let $\vec{p}$ and $\vec{q}$ be two unit vectors and $\alpha$ be the angle between them. Then $(\vec{p}+\vec{q})$ will be a unit vector for what value of $\alpha$ ?

1. (A)$\frac{\pi }{4}$
2. (B)$\frac{\pi }{3}$
3. (C)$\frac{\pi }{2}$
4. (D)$\frac{2\pi }{3}$

The line $x=1+5\mu$, $y=-5+\mu$, $z=-6-3\mu$ passes through which of the following point ?

1. (A)$(1,-5,6)$
2. (B)$(1,5,6)$
3. (C)$(1,-5,-6)$
4. (D)$(-1,-5,6)$

If A denotes the set of continuous functions and B denotes set of differentiable functions, then which of the following depicts the correct relation between set A and B ?

1. (A)Figure (A): set A drawn inside set B
2. (B)Figure (B): set B drawn inside set A
3. (C)Figure (C): sets A and B drawn overlapping
4. (D)Figure (D): sets A and B drawn separate (disjoint)

The area of the shaded region (figure) represented by the curves $y=x^2$, $0\le x\le 2$ and y-axis is given by

1. (A)${\int }_0^2{x}^{2}dx$
2. (B)${\int }_0^2\sqrt{y}dy$
3. (C)${\int }_0^4{x}^{2}dx$
4. (D)${\int }_0^4\sqrt{y}dy$

A factory produces two products X and Y. The profit earned by selling X and Y is represented by the objective function $Z=5x+7y$, where $x$ and y are the number of units of X and Y respectively sold. Which of the following statement is correct ?

1. (A)The objective function maximizes the difference of the profit earned from products X and Y.
2. (B)The objective function measures the total production of products X and Y.
3. (C)The objective function maximizes the combined profit earned from selling X and Y.
4. (D)The objective function ensures the company produces more of product X than product Y.

If A and B are square matrices of order m such that $A^2-B^2=(A-B)(A+B)$, then which of the following is always correct ?

1. (A)$A=B$
2. (B)$AB=BA$
3. (C)$A=0$ or $B=0$
4. (D)$A=I$ or $B=I$

If p and q are respectively the order and degree of the differential equation $\frac{d}{dx}{\left(\frac{dy}{dx}\right)}^3=0$, then $(p-q)$ is

1. (A)0
2. (B)1
3. (C)2
4. (D)3

Assertion (A) : $A=\text{diag}[352]$ is a scalar matrix of order $3\times 3$.
Reason (R) : If a diagonal matrix has all non-zero elements equal, it is known as a scalar matrix.

1. (A)Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of the Assertion (A).
2. (B)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. (C)Assertion (A) is true but Reason (R) is false.
4. (D)Assertion (A) is false but Reason (R) is true.

Assertion (A) : Every point of the feasible region of a Linear Programming Problem is an optimal solution.
Reason (R) : The optimal solution for a Linear Programming Problem exists only at one or more corner point(s) of the feasible region.

1. (A)Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of the Assertion (A).
2. (B)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. (C)Assertion (A) is true but Reason (R) is false.
4. (D)Assertion (A) is false but Reason (R) is true.

A vector $\vec{a}$ makes equal angles with all the three axes. If the magnitude of the vector is $5\sqrt{3}$ units, then find $\vec{a}$.

If $\vec{\alpha }$ and $\vec{\beta }$ are position vectors of two points P and Q respectively, then find the position vector of a point R in QP produced such that $QR=\frac{3}{2}QP$.

Evaluate : ${\int }_0^{\frac{\pi }{4}}\sqrt{1+\sin 2x}dx$

Find the values of ‘a’ for which $f(x)=\sin x-ax+b$ is increasing on R.

If $\vec{a}$ and $\vec{b}$ are two non-collinear vectors, then find $x$, such that $\vec{\alpha }=(x-2)\vec{a}+\vec{b}$ and $\vec{\beta }=(3+2x)\vec{a}-2\vec{b}$ are collinear.

If $x=e^{\frac{x}{y}}$, then prove that $\frac{dy}{dx}=\frac{x-y}{x\log x}$.

If $f(x)=\{\begin{array}{ll}2x-3,& -3\le x\le -2\\ x+1,& -2<x\le 0\end{array}$
Check the differentiability of $f(x)$ at $x=-2$.

Solve the differential equation $2(y+3)-xy\frac{dy}{dx}=0$; given $y(1)=-2$.

Solve the following differential equation :
$(1+x^2)\frac{dy}{dx}+2xy=4x^2$.

Let R be a relation defined over N, where N is set of natural numbers, defined as “mRn if and only if m is a multiple of n, m, n $\in$ N.” Find whether R is reflexive, symmetric and transitive or not.

Solve the following linear programming problem graphically :
Minimise $Z=x-5y$
subject to the constraints :
$x-y\ge 0$
$-x+2y\ge 2$
$x\ge 3,y\le 4,y\ge 0$

If $y=\log {\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)}^2$, then show that $x(x+1{)}^2{y}_2+(x+1{)}^2{y}_1=2$.

If $x\sqrt{1+y}+y\sqrt{1+x}=0$, $-1<x<1$, $x\ne y$, then prove that $\frac{dy}{dx}=\frac{-1}{(1+x{)}^2}$.

A die with number 1 to 6 is biased such that $P(2)=\frac{3}{10}$ and probability of other numbers is equal. Find the mean of the number of times number 2 appears on the dice, if the dice is thrown twice.

Two dice are thrown. Defined are the following two events A and B :
$A=\{(x,y):x+y=9\}$, $B=\{(x,y):x\ne 3\}$, where $(x,y)$ denote a point in the sample space.
Check if events A and B are independent or mutually exclusive.

Find : $\int \frac{1}{x}\sqrt{\frac{x+a}{x-a}}dx$.

Using integration, find the area of the region bounded by the line $y=5x+2$, the $x$ – axis and the ordinates $x=-2$ and $x=2$.

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

Find the shortest distance between the lines :
$\frac{x+1}{2}=\frac{y-1}{1}=\frac{z-9}{-3}$ and
$\frac{x-3}{2}=\frac{y+15}{-7}=\frac{z-9}{5}$.

Find the image A′ of the point A(2, 1, 2) in the line $l:\vec{r}=4\hat{i}+2\hat{j}+2\hat{k}+\lambda (\hat{i}-\hat{j}-\hat{k})$. Also, find the equation of line joining AA′. Find the foot of perpendicular from point A on the line l.

Given $A=\left[\begin{array}{ccc}-4& 4& 4\\ -7& 1& 3\\ 5& -3& -1\end{array}\right]$ and $B=\left[\begin{array}{ccc}1& -1& 1\\ 1& -2& -2\\ 2& 1& 3\end{array}\right]$, find AB. Hence, solve the system of linear equations :
$x-y+z=4$
$x-2y-2z=9$
$2x+y+3z=1$

If $A=\left[\begin{array}{ccc}1& 2& 0\\ -2& -1& -2\\ 0& -1& 1\end{array}\right]$, then find $A^{-1}$.
Hence, solve the system of linear equations :
$x-2y=10$
$2x-y-z=8$
$-2y+z=7$