CBSE Class 12 Maths 2023 Question Paper 65/2/3 with Solutions

If the angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi }{3}$ and $|\vec{a}\times \vec{b}|=3\sqrt{3}$, then the value of $\vec{a}\cdot \vec{b}$ is

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

The position vectors of three consecutive vertices of a parallelogram ABCD are A $(4\hat{i}+2\hat{j}-6\hat{k})$, B $(5\hat{i}-3\hat{j}+\hat{k})$ and C $(12\hat{i}+4\hat{j}+5\hat{k})$. The position vector of D is given by

1. (A)$-3\hat{i}-5\hat{j}-10\hat{k}$
2. (B)$21\hat{i}+3\hat{j}$
3. (C)$11\hat{i}+9\hat{j}-2\hat{k}$
4. (D)$-11\hat{i}-9\hat{j}+2\hat{k}$

If for two events A and B, P(A $-$ B) = $\frac{1}{5}$ and P(A) = $\frac{3}{5}$, then $P\left(\frac{B}{A}\right)$ is equal to

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

If ${\int }_0^{2\pi }{\cos }^{2}xdx=k{\int }_0^{\pi /2}{\cos }^{2}xdx$, then the value of k is

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

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

Number of symmetric matrices of order 3 $\times$ 3 with each entry 1 or $-$ 1 is

1. (A)512
2. (B)64
3. (C)8
4. (D)4

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

If $\frac{d}{dx}[f(x)]=ax+b$ and $f(0)=0$, then $f(x)$ is equal to

1. (A)a + b
2. (B)$\frac{a{x}^2}{2}+bx$
3. (C)$\frac{a{x}^2}{2}+bx+c$
4. (D)b

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)$\frac{2\hat{i}}{7}+\frac{3\hat{j}}{7}-\frac{6\hat{k}}{7}$

Equation of a line passing through point (1, 2, 3) and equally inclined to the coordinate axis, is

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

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.

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.

If points A, B and C have position vectors $2\hat{i}$, $\hat{j}$ and $2\hat{k}$ respectively, then show that $△ABC$ is an isosceles triangle.

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 equal sides of an isosceles triangle with fixed base 10 cm are increasing at the rate of 4 cm/sec, how fast is the area of triangle increasing at an instant when all sides become equal ?

Find the coordinates of points on line $\frac{x}{1}=\frac{y-1}{2}=\frac{z+1}{2}$ which are at a distance of $\sqrt{11}$ units from origin.

If $y=\sqrt{ax+b}$, then 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.

Solve the following Linear Programming problem graphically :
Maximize : Z = 3$x$ + 3.5y
subject to constraints : $x+2y\ge 240$,
$3x+1.5y\ge 270$,
$1.5x+2y\le 310$,
$x\ge 0$, $y\ge 0$.

Find $\int \frac{x+2}{\sqrt{x^2-4x-5}}dx$.

Evaluate ${\int }_{-a}^{a}f(x)dx$, where $f(x)=\frac{9^x}{1+9^x}$.

Find $\int e^{{\cot }^{-1}x}\left(\frac{1-x+x^2}{1+x^2}\right)dx$.

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 numbers are selected from first six even natural numbers at random without replacement. If X denotes the greater of two numbers selected, find the probability distribution of X.

A fair coin and an unbiased die are tossed. Let A be the event, “Head appears on the coin” and B be the event, “3 comes on the die”. Find whether A and B are independent events or not.

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

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

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.

Find the area of the smaller region bounded by the curves $\frac{x^2}{25}+\frac{y^2}{16}=1$ and $\frac{x}{5}+\frac{y}{4}=1$, using integration.

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)