CBSE Class 12 Maths 2023 Question Paper 65/3/3 with Solutions

If the angle between the vectors $\vec{a}$ and $\vec{b}$ is $\frac{\pi }{4}$ and $|\vec{a}\times \vec{b}|=1$, then $\vec{a}\cdot \vec{b}$ is equal to

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

$\vec{a}$ and $\vec{b}$ are two non-zero vectors such that the projection of $\vec{a}$ on $\vec{b}$ is 0. The angle between $\vec{a}$ and $\vec{b}$ is :

1. (A)$\frac{\pi }{2}$
2. (B)$\pi$
3. (C)$\frac{\pi }{4}$
4. (D)0

In $\Delta$ ABC, $\overrightarrow{AB}=\hat{i}+\hat{j}+2\hat{k}$ and $\overrightarrow{AC}=3\hat{i}-\hat{j}+4\hat{k}$. If D is mid-point of BC, then vector $\overrightarrow{AD}$ is equal to :

1. (A)$4\hat{i}+6\hat{k}$
2. (B)$2\hat{i}-2\hat{j}+2\hat{k}$
3. (C)$\hat{i}-\hat{j}+\hat{k}$
4. (D)$2\hat{i}+3\hat{k}$

The equation of a line passing through point (2, – 1, 0) and parallel to the line $\frac{x}{1}=\frac{y-1}{2}=\frac{2-z}{2}$ is :

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

X and Y are independent events such that $P(X\cap \bar{Y})=\frac{2}{5}$ and $P(X)=\frac{3}{5}$. Then P(Y) is equal to :

1. (A)$\frac{2}{3}$
2. (B)$\frac{2}{5}$
3. (C)$\frac{1}{3}$
4. (D)$\frac{1}{5}$

The value of k for which function $f(x)=\{\begin{array}{ll}{x}^2,& x\ge 0\\ kx,& x<0\end{array}$ is differentiable at x = 0 is :

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

If $y=\frac{\cos x-\sin x}{\cos x+\sin x}$, then $\frac{dy}{dx}$ is :

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

The number of feasible solutions of the linear programming problem given as
Maximize $z=15x+30y$ subject to constraints :
$3x+y\le 12$, $x+2y\le 10$, $x\ge 0$, $y\ge 0$ is

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

The feasible region of a linear programming problem is shown in the figure below :
Which of the following are the possible constraints ?

1. (A)$x+2y\ge 4,x+y\le 3,x\ge 0,y\ge 0$
2. (B)$x+2y\le 4,x+y\le 3,x\ge 0,y\ge 0$
3. (C)$x+2y\ge 4,x+y\ge 3,x\ge 0,y\ge 0$
4. (D)$x+2y\ge 4,x+y\ge 3,x\le 0,y\le 0$

A and B are square matrices of same order. If $(A+B{)}^2=A^2+B^2$, then :

1. (A)AB = BA
2. (B)AB = – BA
3. (C)AB = O
4. (D)BA = O

If $A\cdot (\text{adj }A)=\left[\begin{array}{ccc}3& 0& 0\\ 0& 3& 0\\ 0& 0& 3\end{array}\right]$, then the value of $|A|+|\text{adj }A|$ is equal to :

1. (A)12
2. (B)9
3. (C)3
4. (D)27

A and B are skew-symmetric matrices of same order. AB is symmetric, if :

1. (A)AB = O
2. (B)AB = – BA
3. (C)AB = BA
4. (D)BA = O

For what value of $x\in \left[0,\frac{\pi }{2}\right]$, is $A+A^{\prime }=\sqrt{3}I$, where $A=\left[\begin{array}{cc}\cos x& \sin x\\ -\sin x& \cos x\end{array}\right]$ ?

1. (A)$\frac{\pi }{3}$
2. (B)$\frac{\pi }{6}$
3. (C)0
4. (D)$\frac{\pi }{2}$

Let A be the area of a triangle having vertices $(x_1,y_1)$, $(x_2,y_2)$ and $(x_3,y_3)$. Which of the following is correct ?

1. (A)$\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|=\pm A$
2. (B)$\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|=\pm 2A$
3. (C)$\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|=\pm \frac{A}{2}$
4. (D)${\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|}^2=A^2$

$\int 2^{x+2}dx$ is equal to :

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

$\int e^{-x}\left(\frac{x+1}{x^2}\right)dx$ is equal to :

1. (A)$\frac{e^{-x}}{x}+C$
2. (B)$\frac{e^x}{x}+C$
3. (C)$\frac{e^x}{x^2}+C$
4. (D)$-\frac{e^{-x}}{x}+C$

The value of ${\int }_0^{\pi /2}\log \tan xdx$ is :

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

What is the product of the order and degree of the differential equation $\frac{d^{2}y}{d{x}^2}\sin y+{\left(\frac{dy}{dx}\right)}^3\cos y=\sqrt{y}$ ?

1. (A)3
2. (B)2
3. (C)6
4. (D)not defined

Assertion (A) : Range of $[{\sin }^{-1}x+2{\cos }^{-1}x]$ is $[0,\pi ]$.
Reason (R) : Principal value branch of ${\sin }^{-1}x$ has range $\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$.

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) : A line through the points (4, 7, 8) and (2, 3, 4) is parallel to a line through the points $(-1,-2,1)$ and (1, 2, 5).
Reason (R) : Lines $\vec{r}={\vec{a}}_1+\lambda {\vec{b}}_1$ and $\vec{r}={\vec{a}}_2+\mu {\vec{b}}_2$ are parallel if ${\vec{b}}_1\cdot {\vec{b}}_2=0$.

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.

If $y=x^{\frac{1}{x}}$, then find $\frac{dy}{dx}$ at x = 1.

If $x=a\sin 2t$, $y=a(\cos 2t+\log \tan t)$, then find $\frac{dy}{dx}$.

If $\vec{r}=3\hat{i}-2\hat{j}+6\hat{k}$, find the value of $(\vec{r}\times \hat{j})\cdot (\vec{r}\times \hat{k})-12$.

Find the direction cosines of the line whose Cartesian equations are
$5x-3=15y+7=3-10z$.

Find the points on the curve $6y=x^3+2$ at which ordinate is changing 8 times as fast as abscissa.

Evaluate : $3{\sin }^{-1}\left(\frac{1}{\sqrt{2}}\right)+2{\cos }^{-1}\left(\frac{\sqrt{3}}{2}\right)+{\cos }^{-1}(0)$

Draw the graph of $f(x)={\sin }^{-1}x$, $x\in \left[-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right]$. Also, write range of f(x).

A pair of dice is thrown simultaneously. If X denotes the absolute difference of numbers obtained on the pair of dice, then find the probability distribution of X.

There are two coins. One of them is a biased coin such that P (head) : P (tail) is 1 : 3 and the other coin is a fair coin. A coin is selected at random and tossed once. If the coin showed head, then find the probability that it is a biased coin.

Find the general solution of the differential equation :
$\frac{d}{dx}(x{y}^2)=2y(1+x^2)$

Solve the following differential equation :
$x{e}^{\frac{y}{x}}-y+x\frac{dy}{dx}=0$

Evaluate :
${\int }_0^{\pi /4}\log (1+\tan x)dx$

Find :
$\int \frac{\cos x}{\sin 3x}dx$

Find :
$\int x^2\log (x^2+1)dx$

Find :
${\int }_1^4\frac{1}{\sqrt{2x+1}-\sqrt{2x-1}}dx$

Solve the following linear programming problem graphically :
Minimize $z=x+2y$
subject to the constraints
$2x+y\ge 3$,
$x+2y\ge 6$,
$x\ge 0$,
$y\ge 0$.

Find the value of b so that the lines $\frac{x-1}{2}=\frac{y-b}{3}=\frac{z-3}{4}$ and $\frac{x-4}{5}=\frac{y-1}{2}=z$ are intersecting lines. Also, find the point of intersection of these given lines.

Find the equations of all the sides of the parallelogram ABCD whose vertices are A(4, 7, 8), B(2, 3, 4), C(– 1, – 2, 1) and D(1, 2, 5). Also, find the coordinates of the foot of the perpendicular from A to CD.

Check whether a function $f:\mathbb{R}\to \left[-\frac{1}{2},\frac{1}{2}\right]$ defined as $f(x)=\frac{x}{1+x^2}$ is one-one and onto or not.

The area of the region bounded by the line y = mx (m > 0), the curve $x^2+y^2=4$ and the x-axis in the first quadrant is $\frac{\pi }{2}$ units. Using integration, find the value of m.

If $A=\left[\begin{array}{ccc}1& 0& 2\\ 0& 2& 1\\ 2& 0& 3\end{array}\right]$, then show that $A^3-6A^2+7A+2I=O$.