CBSE Class 12 Maths 2023 Question Paper 65/1/2 with Solutions

$\int \frac{1+\tan x}{1-\tan x}dx$ is equal to :

1. (A)${\sec }^2\left(\frac{\pi }{4}+x\right)+C$
2. (B)${\sec }^2\left(\frac{\pi }{4}-x\right)+C$
3. (C)$\log \left|\sec \left(\frac{\pi }{4}+x\right)\right|+C$
4. (D)$\log \left|\sec \left(\frac{\pi }{4}-x\right)\right|+C$

${\int }_0^{\frac{\pi }{6}}{\sec }^2\left(x-\frac{\pi }{6}\right)dx$ is equal to :

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

The sum of the order and the degree of the differential equation $\frac{d^{2}y}{d{x}^2}+{\left(\frac{dy}{dx}\right)}^3=\sin y$ is :

1. (A)5
2. (B)2
3. (C)3
4. (D)4

The value of p for which the vectors $2\hat{i}+p\hat{j}+\hat{k}$ and $-4\hat{i}-6\hat{j}+26\hat{k}$ are perpendicular to each other, is :

1. (A)3
2. (B)$-3$
3. (C)$-\frac{17}{3}$
4. (D)$\frac{17}{3}$

If the vector $\hat{i}-b\hat{j}+\hat{k}$ is equally inclined to the coordinate axes, then the value of b is :

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

If $\vec{a}+\vec{b}=\hat{i}$ and $\vec{a}=2\hat{i}-2\hat{j}+2\hat{k}$, then $|\vec{b}|$ equals :

1. (A)$\sqrt{14}$
2. (B)3
3. (C)$\sqrt{12}$
4. (D)$\sqrt{17}$

Direction cosines of a line perpendicular to both x-axis and z-axis are :

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

If $P\left(\frac{A}{B}\right)=0.3$, $P(A)=0.4$ and $P(B)=0.8$, then $P\left(\frac{B}{A}\right)$ is equal to :

1. (A)0.6
2. (B)0.3
3. (C)0.06
4. (D)0.4

For what value of k may the function $f(x)=\{\begin{array}{ll}k(3x^2-5x),& x\le 0\\ \cos x,& x>0\end{array}$ become continuous ?

1. (A)0
2. (B)1
3. (C)$-\frac{1}{2}$
4. (D)No value

If $A=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$ and $(3I+4A)(3I-4A)=x^{2}I$, then the value(s) x is/are :

1. (A)$\pm \sqrt{7}$
2. (B)0
3. (C)$\pm 5$
4. (D)25

The general solution of the differential equation $xdy-(1+x^2)dx=dx$ is :

1. (A)$y=2x+\frac{x^3}{3}+C$
2. (B)$y=2\log x+\frac{x^3}{3}+C$
3. (C)$y=\frac{x^2}{2}+C$
4. (D)$y=2\log x+\frac{x^2}{2}+C$

If $f(x)=a(x-\cos x)$ is strictly decreasing in $\mathbb{R}$, then ‘a’ belongs to

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

The corner points of the feasible region in the graphical representation of a linear programming problem are (2, 72), (15, 20) and (40, 15). If $z=18x+9y$ be the objective function, then :

1. (A)z is maximum at (2, 72), minimum at (15, 20)
2. (B)z is maximum at (15, 20), minimum at (40, 15)
3. (C)z is maximum at (40, 15), minimum at (15, 20)
4. (D)z is maximum at (40, 15), minimum at (2, 72)

The number of corner points of the feasible region determined by the constraints $x-y\ge 0$, $2y\le x+2$, $x\ge 0$, $y\ge 0$ is :

1. (A)2
2. (B)3
3. (C)4
4. (D)5

If for a square matrix A, $A^2-3A+I=O$ and $A^{-1}=xA+yI$, then the value of $x+y$ is :

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

If $\left|\frac{A^{-1}}{2}\right|=\frac{1}{k|A|}$, where A is a $3\times 3$ matrix, then the value of k is :

1. (A)$\frac{1}{8}$
2. (B)8
3. (C)2
4. (D)$\frac{1}{2}$

Let A be a $3\times 3$ matrix such that $|\text{adj }A|=64$. Then $|A|$ is equal to :

1. (A)8 only
2. (B)$-8$ only
3. (C)64
4. (D)8 or $-8$

If $A=\left[\begin{array}{cc}3& 4\\ 5& 2\end{array}\right]$ and $2A+B$ is a null matrix, then B is equal to :

1. (A)$\left[\begin{array}{cc}6& 8\\ 10& 4\end{array}\right]$
2. (B)$\left[\begin{array}{cc}-6& -8\\ -10& -4\end{array}\right]$
3. (C)$\left[\begin{array}{cc}5& 8\\ 10& 3\end{array}\right]$
4. (D)$\left[\begin{array}{cc}-5& -8\\ -10& -3\end{array}\right]$

Assertion (A) : Equation of a line passing through the points (1, 2, 3) and $(3,-1,3)$ is $\frac{x-3}{2}=\frac{y+1}{3}=\frac{z-3}{0}$.
Reason (R) : Equation of a line passing through points $(x_1,y_1,z_1)$, $(x_2,y_2,z_2)$ is given by $\frac{x-x_1}{x_2-x_1}=\frac{y-y_1}{y_2-y_1}=\frac{z-z_1}{z_2-z_1}$.

1. (A)Both Assertion (A) and Reason (R) are true and 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 and Reason (R) is false.
4. (D)Assertion (A) is false and Reason (R) is true.

Assertion (A) : The number of onto functions from a set P containing 5 elements to a set Q containing 2 elements is 30.
Reason (R) : Number of onto functions from a set containing m elements to a set containing n elements is $n^m$.

1. (A)Both Assertion (A) and Reason (R) are true and 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 and Reason (R) is false.
4. (D)Assertion (A) is false and Reason (R) is true.

Position vectors of the points A, B and C as shown in the figure below are $\vec{a}$, $\vec{b}$ and $\vec{c}$ respectively.
If $\overrightarrow{AC}=\frac{5}{4}\overrightarrow{AB}$, express $\vec{c}$ in terms of $\vec{a}$ and $\vec{b}$.

Check whether the lines given by equations $x=2\lambda +2$, $y=7\lambda +1$, $z=-3\lambda -3$ and $x=-\mu -2$, $y=2\mu +8$, $z=4\mu +5$ are perpendicular to each other or not.

If $y=(x+\sqrt{x^2-1}{)}^2$, then show that $(x^2-1){\left(\frac{dy}{dx}\right)}^2=4y^2$.

Find the sub-intervals in which $f(x)=\log (2+x)-\frac{x}{2+x}$, $x>-2$ is increasing or decreasing.

A function $f:A\to B$ defined as $f(x)=2x$ is both one-one and onto. If $A=\{1,2,3,4\}$, then find the set B.

Evaluate :
${\sin }^{-1}\left(\sin \frac{3\pi }{4}\right)+{\cos }^{-1}\left(\cos \frac{3\pi }{4}\right)+{\tan }^{-1}(1)$

For two non-zero vectors $\vec{a}$ and $\vec{b}$, if $|\vec{a}-\vec{b}|=|\vec{a}+\vec{b}|$, then find the angle between $\vec{a}$ and $\vec{b}$.

Find the general solution of the differential equation :
$\frac{dx}{dy}=\frac{e^{x/y}\left(\frac{x}{y}-1\right)}{1+e^{x/y}}$.

Find the particular solution of the differential equation $\frac{dy}{dx}+\cot x\cdot y={\cos }^{2}x$, given that when $x=\frac{\pi }{2}$, $y=0$.

Solve the following linear programming problem graphically :
Maximize $P=100x+5y$
subject to the constraints
$x+y\le 300$,
$3x+y\le 600$,
$y\le x+200$,
$x,y\ge 0$.

The probability distribution of a random variable X is given below :
X: 1, 2, 3
P(X): $\frac{k}{2}$, $\frac{k}{3}$, $\frac{k}{6}$
(i) Find the value of k.
(ii) Find $P(1\le X<3)$.
(iii) Find E(X), the mean of X.

A and B are independent events such that $P(A\cap \bar{B})=\frac{1}{4}$ and $P(\bar{A}\cap B)=\frac{1}{6}$. Find P(A) and P(B).

Evaluate :
${\int }_0^{\frac{\pi }{2}}{e}^x\sin xdx$

Find :
$\int \frac{1}{\cos (x-a)\cos (x-b)}dx$

Evaluate :
${\int }_0^{\frac{\pi }{2}}|\sin x-\cos x|dx$

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

Find the vector and the Cartesian equations of a line passing through the point $(1,2,-4)$ and parallel to the line joining the points $A(3,3,-5)$ and $B(1,0,-11)$. Hence, find the distance between the two lines.

Find the equations of the line passing through the points A(1, 2, 3) and B(3, 5, 9). Hence, find the coordinates of the points on this line which are at a distance of 14 units from point B.

Find the area of the region $\{(x,y):x^2+y^2\le 1\le x+y\}$, using integration.

A relation R is defined on a set of real numbers $\mathbb{R}$ as
$R=\{(x,y):x\cdot y\text{ is an irrational number}\}$.
Check whether R is reflexive, symmetric and transitive or not.

If $A=\left[\begin{array}{ccc}1& 2& -2\\ -1& 3& 0\\ 0& -2& 1\end{array}\right]$ and $B^{-1}=\left[\begin{array}{ccc}3& -1& 1\\ -15& 6& -5\\ 5& -2& 2\end{array}\right]$, find $(AB{)}^{-1}$.

Solve the following system of equations by matrix method :
$x+2y+3z=6$
$2x-y+z=2$
$3x+2y-2z=3$

A tank, as shown in the figure below, formed using a combination of a cylinder and a cone, offers better drainage as compared to a flat bottomed tank.
A tap is connected to such a tank whose conical part is full of water. Water is dripping out from a tap at the bottom at the uniform rate of 2 cm³/s. The semi-vertical angle of the conical tank is $45^{\circ }$.
On the basis of given information, answer the following questions :
(i) Find the volume of water in the tank in terms of its radius r. (1)
(ii) Find rate of change of radius at an instant when $r=2\sqrt{2}$ cm. (1)
(iii) (a) Find the rate at which the wet surface of the conical tank is decreasing at an instant when radius $r=2\sqrt{2}$ cm. (2)
OR (iii) (b) Find the rate of change of height ‘h’ at an instant when slant height is 4 cm. (2)

There are different types of Yoga which involve the usage of different poses of Yoga Asanas, Meditation and Pranayam as shown in the figure below :
The Venn diagram below represents the probabilities of three different types of Yoga, A, B and C performed by the people of a society. Further, it is given that probability of a member performing type C Yoga is 0.44.
On the basis of the above information, answer the following questions :
(i) Find the value of x. (1)
(ii) Find the value of y. (1)
(iii) (a) Find $P\left(\frac{C}{B}\right)$. (2)
OR (iii) (b) Find the probability that a randomly selected person of the society does Yoga of type A or B but not C. (2)

The equation of the path traced by a roller-coaster is given by the polynomial $f(x)=a(x+9)(x+1)(x-3)$. If the roller-coaster crosses y-axis at a point $(0,-1)$, answer the following :
(i) Find the value of ‘a’. (2)
(ii) Find $f^{\prime \prime }(x)$ at $x=1$. (2)