CBSE Class 12 Maths 2023 Question Paper 65/4/2 with Solutions

If $\left[\begin{array}{ccc}1& 1& 1\\ 0& 1& 1\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}6\\ 3\\ 2\end{array}\right]$, then the value of $(2x+y-z)$ is :

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

If a matrix $A=\left[\begin{array}{ccc}1& 2& 3\end{array}\right]$, then the matrix AA′ (where A′ is the transpose of A) is :

1. (A)14
2. (B)$\left[\begin{array}{ccc}1& 0& 0\\ 0& 2& 0\\ 0& 0& 3\end{array}\right]$
3. (C)$\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 1\\ 3& 1& 2\end{array}\right]$
4. (D)$\left[\begin{array}{c}14\end{array}\right]$

If $x\left[\begin{array}{c}1\\ 2\end{array}\right]+y\left[\begin{array}{c}2\\ 5\end{array}\right]=\left[\begin{array}{c}4\\ 9\end{array}\right]$, then :

1. (A)$x=1$, $y=2$
2. (B)$x=2$, $y=1$
3. (C)$x=1$, $y=-1$
4. (D)$x=3$, $y=2$

If A is a square matrix and $A^2=A$, then $(I+A{)}^2-3A$ is equal to :

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

The value of the determinant $\left|\begin{array}{ccc}2& 7& 1\\ 1& 1& 1\\ 10& 8& 1\end{array}\right|$ is :

1. (A)47
2. (B)− 79
3. (C)49
4. (D)− 51

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

1. (A)continuous and differentiable everywhere.
2. (B)continuous and differentiable nowhere.
3. (C)continuous everywhere, but differentiable everywhere except at $x=0$.
4. (D)continuous everywhere, but differentiable nowhere.

If $y=\log (\sin e^x)$, then $\frac{dy}{dx}$ is :

1. (A)$\cot e^x$
2. (B)$\text{cosec }{e}^x$
3. (C)$e^x\cot e^x$
4. (D)$e^x\text{cosec }{e}^x$

$\int e^{5\log x}dx$ is equal to :

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

${\int }_0^4(e^{2x}+x)dx$ is equal to :

1. (A)$\frac{15+e^8}{2}$
2. (B)$\frac{16-e^8}{2}$
3. (C)$\frac{e^8-15}{2}$
4. (D)$\frac{-e^8-15}{2}$

A unit vector along the vector $4\hat{i}-3\hat{k}$ is :

1. (A)$\frac{1}{7}(4\hat{i}-3\hat{k})$
2. (B)$\frac{1}{5}(4\hat{i}-3\hat{k})$
3. (C)$\frac{1}{\sqrt{7}}(4\hat{i}-3\hat{k})$
4. (D)$\frac{1}{\sqrt{5}}(4\hat{i}-3\hat{k})$

If $\theta$ is the angle between two vectors $\vec{a}$ and $\vec{b}$, then $\vec{a}\cdot \vec{b}\ge 0$ only when :

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

The integrating factor for solving the differential equation $x\frac{dy}{dx}-y=2x^2$ is :

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

The number of solutions of the differential equation $\frac{dy}{dx}=\frac{y+1}{x-1}$, when $y(1)=2$, is :

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

Distance of the point (p, q, r) from y-axis is :

1. (A)q
2. (B)|q|
3. (C)|q| + |r|
4. (D)$\sqrt{p^2+r^2}$

If the direction cosines of a line are $\left(\frac{1}{a},\frac{1}{a},\frac{1}{a}\right)$, then :

1. (A)$0<a<1$
2. (B)$a>2$
3. (C)$a>0$
4. (D)$a=\pm \sqrt{3}$

For two events A and B, if P(A) = 0.4, P(B) = 0.8 and P(B/A) = 0.6, then $P(A\cup B)$ is :

1. (A)0.24
2. (B)0.3
3. (C)0.48
4. (D)0.96

Which of the following points satisfies both the inequations $2x+y\le 10$ and $x+2y\ge 8$ ?

1. (A)(− 2, 4)
2. (B)(3, 2)
3. (C)(− 5, 6)
4. (D)(4, 2)

The solution set of the inequation $3x+5y<7$ is :

1. (A)whole xy-plane except the points lying on the line $3x+5y=7$.
2. (B)whole xy-plane along with the points lying on the line $3x+5y=7$.
3. (C)open half plane containing the origin except the points of line $3x+5y=7$.
4. (D)open half plane not containing the origin.

Assertion (A) : All trigonometric functions have their inverses over their respective domains.
Reason (R) : The inverse of ${\tan }^{-1}x$ exists for some $x\in \mathbb{R}$.

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 lines $\vec{r}=\overrightarrow{a_1}+\lambda \overrightarrow{b_1}$ and $\vec{r}=\overrightarrow{a_2}+\mu \overrightarrow{b_2}$ are perpendicular, when $\overrightarrow{b_1}\cdot \overrightarrow{b_2}=0$.
Reason (R) : The angle $\theta$ between the lines $\vec{r}=\overrightarrow{a_1}+\lambda \overrightarrow{b_1}$ and $\vec{r}=\overrightarrow{a_2}+\mu \overrightarrow{b_2}$ is given by $\cos \theta =\frac{\overrightarrow{b_1}\cdot \overrightarrow{b_2}}{|\overrightarrow{b_1}||\overrightarrow{b_2}|}$

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.

Find the interval in which the function $f(x)=2x^3-3x$ is strictly increasing.

Find the vector equation of the line passing through the point (2, 1, 3) and perpendicular to both the lines
$\frac{x-1}{1}=\frac{y-2}{2}=\frac{z-3}{3}$; $\frac{x}{-3}=\frac{y}{2}=\frac{z}{5}$.

The equations of a line are $5x-3=15y+7=3-10z$. Write the direction cosines of the line and find the coordinates of a point through which it passes.

Find the domain of $y={\sin }^{-1}(x^2-4)$.

Evaluate :
${\cos }^{-1}\left[\cos \left(-\frac{7\pi }{3}\right)\right]$

If $(x^2+y^2{)}^2=xy$, then find $\frac{dy}{dx}$.

If $\vec{a}=4\hat{i}-\hat{j}+\hat{k}$ and $\vec{b}=2\hat{i}-2\hat{j}+\hat{k}$, then find a unit vector along the vector $\vec{a}\times \vec{b}$.

Find :
$\int \frac{x^2}{x^2+6x+12}dx$

Two fair dice are thrown simultaneously. If X denotes the number of sixes, find the mean of X.

Find the particular solution of the differential equation
$\frac{dy}{dx}=\frac{x+y}{x}$, $y(1)=0$.

Find the general solution of the differential equation
$e^x\tan ydx+(1-e^x){\sec }^{2}ydy=0$.

Evaluate :
${\int }_{\pi /4}^{\pi /2}{e}^{2x}\left(\frac{1-\sin 2x}{1-\cos 2x}\right)dx$

Evaluate :
${\int }_{-2}^2\frac{x^2}{1+5^x}dx$

Find :
$\int \frac{e^x}{\sqrt{5-4e^x-e^{2x}}}dx$

Evaluate :
${\int }_0^{\pi /2}\sqrt{\sin x}{\cos }^{5}xdx$

Solve the following linear programming problem graphically :
Maximise $z=-3x-5y$
subject to the constraints
$-2x+y\le 4$,
$x+y\ge 3$,
$x-2y\le 2$,
$x\ge 0$, $y\ge 0$.

Using integration, find the area of the region bounded by the circle $x^2+y^2=16$, line $y=x$ and y-axis, but lying in the 1st quadrant.

Show that the following lines do not intersect each other :
$\frac{x-1}{3}=\frac{y+1}{2}=\frac{z-1}{5}$; $\frac{x+2}{4}=\frac{y-1}{3}=\frac{z+1}{-2}$

Find the angle between the lines
$2x=3y=-z$ and $6x=-y=-4z$.

If N denotes the set of all natural numbers and R is the relation on $N\times N$ defined by (a, b) R (c, d), if $ad(b+c)=bc(a+d)$. Show that R is an equivalence relation.

Let $f:\mathbb{R}-\left\{-\frac{4}{3}\right\}\to \mathbb{R}$ be a function defined as $f(x)=\frac{4x}{3x+4}$. Show that f is a one-one function. Also, check whether f is an onto function or not.

Find the inverse of the matrix $A=\left[\begin{array}{ccc}1& -1& 2\\ 0& 2& -3\\ 3& -2& 4\end{array}\right]$. Using the inverse, $A^{-1}$, solve the system of linear equations
$x-y+2z=1$; $2y-3z=1$; $3x-2y+4z=3$.

A building contractor undertakes a job to construct 4 flats on a plot along with parking area. Due to strike the probability of many construction workers not being present for the job is 0.65. The probability that many are not present and still the work gets completed on time is 0.35. The probability that work will be completed on time when all workers are present is 0.80.
Let : $E_1$ : represent the event when many workers were not present for the job;
$E_2$ : represent the event when all workers were present; and
E : represent completing the construction work on time.
Based on the above information, answer the following questions :
(i) What is the probability that all the workers are present for the job ? (1)
(ii) What is the probability that construction will be completed on time ? (1)
(iii) What is the probability that many workers are not present given that the construction work is completed on time ? (2)
OR (iii) What is the probability that all workers were present given that the construction job was completed on time ? (2)

Let f(x) be a real valued function. Then its
Left Hand Derivative (L.H.D.) : $L{f}^{\prime }(a)={\lim }_{h\to 0}\frac{f(a-h)-f(a)}{-h}$
Right Hand Derivative (R.H.D.) : $R{f}^{\prime }(a)={\lim }_{h\to 0}\frac{f(a+h)-f(a)}{h}$
Also, a function f(x) is said to be differentiable at x = a if its L.H.D. and R.H.D. at x = a exist and both are equal.
For the function $f(x)=\{\begin{array}{ll}|x-3|,& x\ge 1\\ \frac{x^2}{4}-\frac{3x}{2}+\frac{13}{4},& x<1\end{array}$
answer the following questions :
(i) What is R.H.D. of f(x) at x = 1 ? (1)
(ii) What is L.H.D. of f(x) at x = 1 ? (1)
(iii) Check if the function f(x) is differentiable at x = 1. (2)
OR (iii) Find $f^{\prime }(2)$ and $f^{\prime }(-1)$. (2)

Sooraj’s father wants to construct a rectangular garden using a brick wall on one side of the garden and wire fencing for the other three sides as shown in the figure. He has 200 metres of fencing wire.
Based on the above information, answer the following questions :
(i) Let ‘x’ metres denote the length of the side of the garden perpendicular to the brick wall and ‘y’ metres denote the length of the side parallel to the brick wall. Determine the relation representing the total length of fencing wire and also write A(x), the area of the garden. (2)
(ii) Determine the maximum value of A(x). (2)