CBSE Class 12 Maths 2024 Question Paper 65/4/3 with Solutions

If $\vec{a}$ and $\vec{b}$ are two vectors such that $|\vec{a}|=1$, $|\vec{b}|=2$ and $\vec{a}\cdot \vec{b}=\sqrt{3}$, then the angle between $2\vec{a}$ and $-\vec{b}$ is :

1. (A)$\frac{\pi }{6}$
2. (B)$\frac{\pi }{3}$
3. (C)$\frac{5\pi }{6}$
4. (D)$\frac{11\pi }{6}$

The vectors $\vec{a}=2\hat{i}-\hat{j}+\hat{k}$, $\vec{b}=\hat{i}-3\hat{j}-5\hat{k}$ and $\vec{c}=-3\hat{i}+4\hat{j}+4\hat{k}$ represents the sides of

1. (A)an equilateral triangle
2. (B)an obtuse-angled triangle
3. (C)an isosceles triangle
4. (D)a right-angled triangle

Let $\vec{a}$ be any vector such that $|\vec{a}|=a$. The value of
$|\vec{a}\times \hat{i}{|}^2+|\vec{a}\times \hat{j}{|}^2+|\vec{a}\times \hat{k}{|}^2$ is :

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

If $A=\left[\begin{array}{cc}3& 1\\ -1& 2\end{array}\right]$ and $A^2+7I=kA$, then the value of k is :

1. (A)1
2. (B)2
3. (C)5
4. (D)7

Let $A=\left[\begin{array}{ccc}1& -1& 2\\ 0& 2& -3\\ 3& -2& 4\end{array}\right]$ and $B=\frac{1}{3}\left[\begin{array}{ccc}-2& 0& 1\\ 9& 2& -3\\ 6& 1& \lambda \end{array}\right]$. If AB = I, then the value of $\lambda$ is :

1. (A)$\frac{-9}{4}$
2. (B)$-2$
3. (C)$\frac{-3}{2}$
4. (D)0

Derivative of $x^2$ with respect to $x^3$, is :

1. (A)$\frac{2}{3x}$
2. (B)$\frac{3x}{2}$
3. (C)$\frac{2x}{3}$
4. (D)$6x^5$

The function $f(x)=|x|+|x-2|$ is

1. (A)continuous, but not differentiable at $x=0$ and $x=2$.
2. (B)differentiable but not continuous at $x=0$ and $x=2$.
3. (C)continuous but not differentiable at $x=0$ only.
4. (D)neither continuous nor differentiable at $x=0$ and $x=2$.

The value of ${\int }_0^{\pi }{\tan }^2\left(\frac{\theta }{3}\right)d\theta$ is :

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

The integrating factor of the differential equation $\frac{dy}{dx}+\frac{2}{x}y=0,x\ne 0$ is :

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

The lines $\frac{1-x}{2}=\frac{y-1}{3}=\frac{z}{1}$ and $\frac{2x-3}{2p}=\frac{y}{-1}=\frac{z-4}{7}$ are perpendicular to each other for p equal to :

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

The maximum value of $Z=4x+y$ for a L.P.P. whose feasible region is given below is :

1. (A)50
2. (B)110
3. (C)120
4. (D)170

The probability distribution of a random variable X is :
X: 0, 1, 2, 3, 4
P(X): 0.1, k, 2k, k, 0.1
where k is some unknown constant.
The probability that the random variable X takes the value 2 is :

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

The function $f(x)=kx-\sin x$ is strictly increasing for

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

The Cartesian equation of a line passing through the point with position vector $\vec{a}=\hat{i}-\hat{j}$ and parallel to the line $\vec{r}=\hat{i}+\hat{k}+\mu (2\hat{i}-\hat{j})$, is

1. (A)$\frac{x-2}{1}=\frac{y+1}{0}=\frac{z}{1}$
2. (B)$\frac{x-1}{2}=\frac{y+1}{-1}=\frac{z}{0}$
3. (C)$\frac{x+1}{2}=\frac{y+1}{-1}=\frac{z}{0}$
4. (D)$\frac{x-1}{2}=\frac{y}{-1}=\frac{z-1}{0}$

If $\left[\begin{array}{ccc}a& c& 0\\ b& d& 0\\ 0& 0& 5\end{array}\right]$ is a scalar matrix, then the value of $a+2b+3c+4d$ is :

1. (A)0
2. (B)5
3. (C)10
4. (D)25

Given that $A^{-1}=\frac{1}{7}\left[\begin{array}{cc}2& 1\\ -3& 2\end{array}\right]$, matrix A is :

1. (A)$7\left[\begin{array}{cc}2& -1\\ 3& 2\end{array}\right]$
2. (B)$\left[\begin{array}{cc}2& -1\\ 3& 2\end{array}\right]$
3. (C)$\frac{1}{7}\left[\begin{array}{cc}2& -1\\ 3& 2\end{array}\right]$
4. (D)$\frac{1}{49}\left[\begin{array}{cc}2& -1\\ 3& 2\end{array}\right]$

If $A=\left[\begin{array}{cc}2& 1\\ -4& -2\end{array}\right]$, then the value of $I-A+A^2-A^3+\dots$ is :

1. (A)$\left[\begin{array}{cc}-1& -1\\ 4& 3\end{array}\right]$
2. (B)$\left[\begin{array}{cc}3& 1\\ -4& -1\end{array}\right]$
3. (C)$\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$
4. (D)$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

The integrating factor of the differential equation $(x+2y^2)\frac{dy}{dx}=y(y>0)$ is :

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

Assertion (A) : The relation $R=\{(x,y):(x+y)\text{ is a prime number and }x,y\in N\}$ is not a reflexive relation.
Reason (R) : The number ‘2n’ is composite for all natural numbers n.

1. (A)Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of Assertion (A).
2. (B)Both Assertion (A) and Reason (R) are true and 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) : The corner points of the bounded feasible region of a L.P.P. are shown below. The maximum value of $Z=x+2y$ occurs at infinite points.
Reason (R) : The optimal solution of a LPP having bounded feasible region must occur at corner points.

1. (A)Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of Assertion (A).
2. (B)Both Assertion (A) and Reason (R) are true and 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.

If $y={\cos }^3({\sec }^{2}2t)$, find $\frac{dy}{dt}$.

If $x^y=e^{x-y}$, prove that $\frac{dy}{dx}=\frac{\log x}{(1+\log x{)}^2}$.

The volume of a cube is increasing at the rate of 6 cm³/s. How fast is the surface area of cube increasing, when the length of an edge is 8 cm ?

Show that the function f given by $f(x)=\sin x+\cos x$, is strictly decreasing in the interval $\left(\frac{\pi }{4},\frac{5\pi }{4}\right)$.

Express ${\tan }^{-1}\left(\frac{\cos x}{1-\sin x}\right)$, where $\frac{-\pi }{2}<x<\frac{\pi }{2}$ in the simplest form.

Find the principal value of ${\tan }^{-1}(1)+{\cos }^{-1}\left(-\frac{1}{2}\right)+{\sin }^{-1}\left(-\frac{1}{\sqrt{2}}\right)$.

Find : $\int \frac{2x}{(x^2+1)(x^2-4)}dx$.

Find $\frac{dy}{dx}$, if $y=(\cos x{)}^x+{\cos }^{-1}\sqrt{x}$ is given.

Find the particular solution of the differential equation $\frac{dy}{dx}=y\cot 2x$, given that $y\left(\frac{\pi }{4}\right)=2$.

Find the particular solution of the differential equation
$\left(x{e}^{\frac{y}{x}}+y\right)dx=xdy$, given that $y=1$ when $x=1$.

Find : $\int {\sec }^3\theta d\theta$

A card from a well shuffled deck of 52 playing cards is lost. From the remaining cards of the pack, a card is drawn at random and is found to be a King. Find the probability of the lost card being a King.

A biased die is twice as likely to show an even number as an odd number. If such a die is thrown twice, find the probability distribution of the number of sixes. Also, find the mean of the distribution.

The corner points of the feasible region determined by the system of linear constraints are as shown in the following figure :
(i) If $Z=3x-4y$ be the objective function, then find the maximum value of Z.
(ii) If $Z=px+qy$ where $p,q>0$ be the objective function. Find the condition on p and q so that maximum value of Z occurs at B(4, 10) and C(6, 8).

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

Find : $\int e^x\left[\frac{1}{(1+x^2{)}^{\frac{3}{2}}}+\frac{x}{\sqrt{1+x^2}}\right]dx$

Let $A=R-\{5\}$ and $B=R-\{1\}$. Consider the function $f:A\to B$, defined by $f(x)=\frac{x-3}{x-5}$. Show that f is one-one and onto.

Check whether the relation S in the set of real numbers R defined by $S=\{(a,b):\text{where }a-b+\sqrt{2}\text{ is an irrational number}\}$ is reflexive, symmetric or transitive.

Find the distance between the line $\frac{x}{2}=\frac{2y-6}{4}=\frac{1-z}{-1}$ and another line parallel to it passing through the point $(4,0,-5)$.

If the lines $\frac{x-1}{-3}=\frac{y-2}{2k}=\frac{z-3}{2}$ and $\frac{x-1}{3k}=\frac{y-1}{1}=\frac{z-6}{-7}$ are perpendicular to each other, find the value of k and hence write the vector equation of a line perpendicular to these two lines and passing through the point $(3,-4,7)$.

Find $A^{-1}$, if $A=\left[\begin{array}{ccc}1& 2& 1\\ 2& 3& -1\\ 1& 0& 1\end{array}\right]$. Hence, solve the following system of equations :
$x+2y+z=5$
$2x+3y=1$
$x-y+z=8$

Sketch the graph of $y=x|x|$ and hence find the area bounded by this curve, X-axis and the ordinates $x=-2$ and $x=2$, using integration.