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

If $\frac{d}{dx}f(x)=2x+\frac{3}{x}$ and $f(1)=1$, then $f(x)$ is

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

Degree of the differential equation $\sin x+\cos \left(\frac{dy}{dx}\right)=y^2$ is

1. (A)2
2. (B)1
3. (C)not defined
4. (D)0

The integrating factor of the differential equation
$(1-y^2)\frac{dx}{dy}+yx=ay$, $(-1<y<1)$ is

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

Unit vector along $\overrightarrow{PQ}$, where coordinates of P and Q respectively are (2, 1, $-1$) and (4, 4, $-7$), is

1. (A)$2\hat{i}+3\hat{j}-6\hat{k}$
2. (B)$-2\hat{i}-3\hat{j}+6\hat{k}$
3. (C)$\frac{-2\hat{i}}{7}-\frac{3\hat{j}}{7}+\frac{6\hat{k}}{7}$
4. (D)$\frac{2\hat{i}}{7}+\frac{3\hat{j}}{7}-\frac{6\hat{k}}{7}$

If in $△ABC$, $\overrightarrow{BA}=2\vec{a}$ and $\overrightarrow{BC}=3\vec{b}$, then $\overrightarrow{AC}$ is

1. (A)$2\vec{a}+3\vec{b}$
2. (B)$2\vec{a}-3\vec{b}$
3. (C)$3\vec{b}-2\vec{a}$
4. (D)$-2\vec{a}-3\vec{b}$

If $|\vec{a}\times \vec{b}|=\sqrt{3}$ and $\vec{a}\cdot \vec{b}=-3$, then angle between $\vec{a}$ and $\vec{b}$ is

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

Equation of line passing through origin and making $30^{\circ }$, $60^{\circ }$ and $90^{\circ }$ with $x$, y, z axes respectively is

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

If A and B are two events such that P(A/B)=2 $\times$ P(B/A) and P(A) + P(B) = $\frac{2}{3}$, then P(B) is equal to

1. (A)$\frac{2}{9}$
2. (B)$\frac{7}{9}$
3. (C)$\frac{4}{9}$
4. (D)$\frac{5}{9}$

Anti-derivative of $\frac{\tan x-1}{\tan x+1}$ with respect to $x$ is :

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)$-\log \left|\sec \left(\frac{\pi }{4}-x\right)\right|+c$

If (a, b), (c, d) and (e, f) are the vertices of $△ABC$ and $\Delta$ denotes the area of $△ABC$, then ${\left|\begin{array}{ccc}a& c& e\\ b& d& f\\ 1& 1& 1\end{array}\right|}^2$ is equal to

1. (A)$2{\Delta }^2$
2. (B)$4{\Delta }^2$
3. (C)$2\Delta$
4. (D)$4\Delta$

The function $f(x)=x|x|$ is

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

If $\tan \left(\frac{x+y}{x-y}\right)=k$, then $\frac{dy}{dx}$ is equal to

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

The objective function Z = a$x$ + by of an LPP has maximum value 42 at (4, 6) and minimum value 19 at (3, 2). Which of the following is true ?

1. (A)a = 9, b = 1
2. (B)a = 5, b = 2
3. (C)a = 3, b = 5
4. (D)a = 5, b = 3

The corner points of the feasible region of a linear programming problem are (0, 4), (8, 0) and $\left(\frac{20}{3},\frac{4}{3}\right)$. If Z = 30$x$ + 24y is the objective function, then (maximum value of Z $-$ minimum value of Z) is equal to

1. (A)40
2. (B)96
3. (C)120
4. (D)136

If A is a 2 $\times$ 3 matrix such that AB and AB${}^{\prime }$ both are defined, then order of the matrix B is

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

If $\left[\begin{array}{cc}2& 0\\ 5& 4\end{array}\right]=P+Q$, where P is a symmetric and Q is a skew symmetric matrix, then Q is equal to

1. (A)$\left[\begin{array}{cc}2& 5/2\\ 5/2& 4\end{array}\right]$
2. (B)$\left[\begin{array}{cc}0& -5/2\\ 5/2& 0\end{array}\right]$
3. (C)$\left[\begin{array}{cc}0& 5/2\\ -5/2& 0\end{array}\right]$
4. (D)$\left[\begin{array}{cc}2& -5/2\\ 5/2& 4\end{array}\right]$

If $\left[\begin{array}{ccc}1& 2& 1\\ 2& 3& 1\\ 3& a& 1\end{array}\right]$ is non-singular matrix and a $\in$ A, then the set A is

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

If $|A|=|kA|$, where A is a square matrix of order 2, then sum of all possible values of k is

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

Assertion (A) : If a line makes angles $\alpha ,\beta ,\gamma$ with positive direction of the coordinate axes, then ${\sin }^2\alpha +{\sin }^2\beta +{\sin }^2\gamma =2$.
Reason (R) : The sum of squares of the direction cosines of a line is 1.

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

Assertion (A) : Maximum value of $({\cos }^{-1}x{)}^2$ is ${\pi }^2$.
Reason (R) : Range of the principal value branch of ${\cos }^{-1}x$ is $\left[\frac{-\pi }{2},\frac{\pi }{2}\right]$.

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

If $\vec{a},\vec{b},\vec{c}$ are three non-zero unequal vectors such that $\vec{a}\cdot \vec{b}=\vec{a}\cdot \vec{c}$, then find the angle between $\vec{a}$ and $\vec{b}-\vec{c}$.

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

Draw the graph of ${\cos }^{-1}x$, where $x\in [-1,0]$. Also, write its range.

If the equation of a line is $x$ = ay + b, z = cy + d, then find the direction ratios of the line and a point on the line.

If $y=\sqrt{ax+b}$, prove that $y\left(\frac{d^{2}y}{d{x}^2}\right)+{\left(\frac{dy}{dx}\right)}^2=0$.

If $f(x)=\{\begin{array}{ll}ax+b& ;0<x\le 1\\ 2x^2-x& ;1<x<2\end{array}$ is a differentiable function in (0, 2), then find the values of a and b.

If the circumference of circle is increasing at the constant rate, prove that rate of change of area of circle is directly proportional to its radius.

Evaluate ${\int }_{\log \sqrt{2}}^{\log \sqrt{3}}\frac{1}{(e^x+e^{-x})(e^x-e^{-x})}dx$

Find the general solution of the differential equation :
$(xy-x^2)dy=y^{2}dx$.

Find the general solution of the differential equation :
$(x^2+1)\frac{dy}{dx}+2xy=\sqrt{x^2+4}$

Two balls are drawn at random one by one with replacement from an urn containing equal number of red balls and green balls. Find the probability distribution of number of red balls. Also, find the mean of the random variable.

A and B throw a die alternately till one of them gets a ‘6’ and wins the game. Find their respective probabilities of wining, if A starts the game first.

Solve the following linear programming problem graphically :
Maximize : Z = $x$ + 2y
subject to constraints : $x+2y\ge 100$,
$2x-y\le 0$,
$2x+y\le 200$,
$x\ge 0$, $y\ge 0$.

Evaluate ${\int }_{-1}^1|x^4-x|dx$.

Find $\int \frac{{\sin }^{-1}x}{(1-x^2{)}^{3/2}}dx$.

Find $\int e^x\left(\frac{1-\sin x}{1-\cos x}\right)dx$

Find the equations of the diagonals of the parallelogram PQRS whose vertices are P(4, 2, $-6$), Q(5, $-3$, 1), R(12, 4, 5) and S(11, 9, $-2$). Use these equations to find the point of intersection of diagonals.

A line $l$ passes through point ($-1$, 3, $-2$) and is perpendicular to both the lines $\frac{x}{1}=\frac{y}{2}=\frac{z}{3}$ and $\frac{x+2}{-3}=\frac{y-1}{2}=\frac{z+1}{5}$. Find the vector equation of the line $l$. Hence, obtain its distance from origin.

Using Integration, find the area of triangle whose vertices are ($-1$, 1), (0, 5) and (3, 2).

A function f: [$-$4, 4] $\to$ [0, 4] is given by $f(x)=\sqrt{16-x^2}$. Show that f is an onto function but not a one-one function. Further, find all possible values of ‘a’ for which $f(a)=\sqrt{7}$.

If $A=\left[\begin{array}{ccc}-3& -2& -4\\ 2& 1& 2\\ 2& 1& 3\end{array}\right]$ and $B=\left[\begin{array}{ccc}1& 2& 0\\ -2& -1& -2\\ 0& -1& 1\end{array}\right]$, then find AB and use it to solve the following system of equations :
$x-2y=3$
$2x-y-z=2$
$-2y+z=3$

If $f(\alpha )=\left[\begin{array}{ccc}\cos \alpha & -\sin \alpha & 0\\ \sin \alpha & \cos \alpha & 0\\ 0& 0& 1\end{array}\right]$, then prove that $f(\alpha )\cdot f(-\beta )=f(\alpha -\beta )$

Recent studies suggest that roughly 12% of the world population is left handed.
Depending upon the parents, the chances of having a left handed child are as follows :
A : When both father and mother are left handed :
Chances of left handed child is 24%.
B : When father is right handed and mother is left handed :
Chances of left handed child is 22%.
C : When father is left handed and mother is right handed :
Chances of left handed child is 17%.
D : When both father and mother are right handed :
Chances of left handed child is 9%.
Assuming that P(A) = P(B) = P(C) = P(D) = $\frac{1}{4}$ and L denotes the event that child is left handed.
Based on the above information, answer the following questions :
(i) Find P(L/C) (1)
(ii) Find P($\overline{L}$/A) (1)
(iii) Find P(A/L) (2)
OR Find the probability that a randomly selected child is left handed given that exactly one of the parents is left handed. (2)

Engine displacement is the measure of the cylinder volume swept by all the pistons of a piston engine. The piston moves inside the cylinder bore
The cylinder bore in the form of circular cylinder open at the top is to be made from a metal sheet of area 75$\pi$ cm${}^2$.
Based on the above information, answer the following questions :
(i) If the radius of cylinder is r cm and height is h cm, then write the volume V of cylinder in terms of radius r. (1)
(ii) Find $\frac{dV}{dr}$. (1)
(iii) Find the radius of cylinder when its volume is maximum. (2)
OR For maximum volume, h > r. State true or false and justify. (2)

The use of electric vehicles will curb air pollution in the long run.
The use of electric vehicles is increasing every year and estimated electric vehicles in use at any time t is given by the function V :
$V(t)=\frac{1}{5}{t}^3-\frac{5}{2}{t}^2+25t-2$
where t represents the time and t = 1, 2, 3.... corresponds to year 2001, 2002, 2003, ....... respectively.
Based on the above information, answer the following questions :
(i) Can the above function be used to estimate number of vehicles in the year 2000 ? Justify. (2)
(ii) Prove that the function V(t) is an increasing function. (2)