CBSE(NCERT) GRADE 12 MATHS 2023 SET1

In the determinant $$\begin{vmatrix} 7 & 5 & 1 \\ 4 & 0 & 6 \\ 5 & 3 & 2 \end{vmatrix},\ M_{23}$ is : (where $M_{ij}$ denotes the minor of element $a_{ij}$)

If $y = \sec(\tan^{-1} x)$, then $\dfrac{dy}{dx}$ at $x = 1$ is equal to :

The value of $k$ for which the function $f$ given by $$f(x)=\begin{cases}2x, & x<5 \\ 2x^2, & x>5 \\ k\cos x, & x=5 \end{cases}$$ is continuous at $x=5$, is :

$\int x^2\cos x^2\sin x\,dx$ equals :

$\int_{4/3}^{4/3} x\sin x\,dx$ equals :

The general solution of the differential equation $y e^x\,dx = e^y\,dy$ is :

The sum of the order and the degree of the differential equation $$x\log x\,\frac{d^2y}{dx^2}+x^3\,\frac{d^4y}{dx^4}=0$$ is :

Let $\vec a$ and $\vec b$ be two unit vectors and $\theta$ is the angle between them. $\vec a+\vec b$ is a unit vector, if :

If $(2\hat i + 6\hat j - 2\hat k)(\hat i + \hat j + \hat k)=0$, then ____ is equal to :

If the position vectors of two points $A$ and $B$ are $\hat i + 2\hat j - 3\hat k$ and $\hat i - 2\hat j + \hat k$ respectively, then the direction cosines of the vector $\overrightarrow{BA}$ are :

The value of ____ for which the lines $\dfrac{x-1}{1}=\dfrac{y-2}{5}=\dfrac{z}{7}$ and $\dfrac{x-3}{1}=\dfrac{y}{2}=\dfrac{z-1}{3}$ are at right angles, is :

Check the injectivity and surjectivity of the function $f:\mathbb N\to\mathbb N$, given by $f(x)=x^3$.

The solution set of the inequation $2x+y\le 5$ is :

The minimum value of $z = 3x + 8y$ subject to the constraints $x\le 20$, $y\le 10$ and $x\ge 0$, $y\ge 0$ is :

Two events $A$ and $B$ will be independent, if :

In a parallelogram ABCD, the sides AB and AD are represented by the vectors $2\hat i-4\hat j+5\hat k$ and $\hat i-2\hat j-3\hat k$ respectively. Find the unit vector parallel to its diagonal AC.

Find the angle between the pair of lines given by $\vec r=\hat i+2\hat j-2\hat k+(\hat i-2\hat j-2\hat k)$ and $\vec r=3\hat i-5\hat j+\hat k+(3\hat i+2\hat j-6\hat k)$.

If $\cos y=x\cos(a+y)$, and $\cos a\ne 1$, prove that $\frac{dy}{dx}=\frac{a\sin y}{a\cos y}$.

Evaluate: $\int \frac{2}{x}\,dx\,x\cos^{-1}x\sin^{-1}e$

Find the projection of the vector $7\hat i-\hat j+8\hat k$ on the vector $\hat i+2\hat j+2\hat k$.

Evaluate: $\int_0^4 dx\,x\tan^{-1}(\log x)$

The radius of an air bubble is increasing at the rate of 0.5 cm/s. At what rate is the surface area of the bubble increasing when the radius is 1.5 cm?

Find: $\int x\cot^{-1}(1)\,dx$

Evaluate : $\int \frac{2}{x}\,dx$

Evaluate : $\int \frac{dx}{x\,\tan^{-1}(\log x)}$

Solve the differential equation $x\,dy-y\,dx=\sqrt{2xy}\,dx$.

From a lot of 10 bulbs which includes 2 defectives, a sample of 2 bulbs is drawn at random without replacement. Find the probability distribution of the number of defective bulbs. Hence, find the mean.

(a) A relation $R$ in the set $A = \{5, 6, 7, 8, 9\}$ is given by $R = \{(x, y) : xy$ is divisible by $2\}$. Write $R$ in roster form and prove that $R$ is an equivalence relation. Also, find the elements related to element $7$. OR (b) Let $A = \{3\}$ and $B = \{1\}$ be two sets. Prove that the function $f : A \to B$ given by $f(x) = 3^2$ is onto. Is the function $f$ one-one? Justify your answer.

Using matrices, solve the following system of linear equations: $x-y+2z=7$; $3x+4y-5z=5$; $2x-y+3z=12$.

Find the area of the region bounded by the curve $y=x$, the line $x=2y+3$ and the $x$-axis, using integration.

Find the foot of the perpendicular drawn from the point $(2,3,8)$ to the line $\frac{x-2}{3}=\frac{y-4}{6}=\frac{z-1}{2}$. Also, find the perpendicular distance of the given line from the given point.

(a) Find the shortest distance between the lines whose vector equations are: $ \vec r = \hat i + 2\hat j - 4\hat k + \lambda(2\hat i + 3\hat j + 6\hat k) $ and $ \vec r = 3\hat i - 3\hat j - 5\hat k + \mu(2\hat i + 3\hat j + 8\hat k) $. OR (b) Find the foot of the perpendicular drawn from the point $(2, 3, 8)$ to the line $\dfrac{x-3}{2} = \dfrac{y-1}{4} = \dfrac{z}{6}$. Also, find the perpendicular distance of the given line from the given point.

Case Study 1 A balloon is being inflated with the help of an air pump, and it remains spherical. Its radius, the surface area and the volume of air in it are all increasing. Based on the above, answer the following questions: (i) Are the quantities: radius, surface area and volume of the spherical balloon changing at the same rate or different rates, when air is filled in it? 1 (ii) Write the expressions for the surface area $(S)$ and the volume $(V)$. 1 (iii) (a) At the instant when the radius of the balloon is $6$ cm and the radius $(r)$ is increasing at the rate of $2$ cm/s, find at what rate the surface area $(S)$ of the balloon is increasing. 2 OR (iii) (b) At the instant when the radius of the balloon is $6$ cm and the radius $(r)$ is increasing at the rate of $2$ cm/s, find at what rate the volume $(V)$ of the spherical balloon is increasing. 2

Case Study 2 A fighter-jet of the enemy is flying along the parabolic path $4y = x^2$. A soldier is located at the point $(0, 5)$ and is aiming to shoot down the jet when it is nearest to him. Based on the above, answer the following questions: (i) Let $(x, y)$ be the position of the jet at any instant. Express the distance between the soldier and the jet as the function $f(x)$. 1 (ii) Taking $S = [f(x)]^2$, find $\dfrac{dS}{dx}$. 1 (iii) (a) What will be the position of the jet when the soldier shoots it down? 2 OR (iii) (b) What will be the distance between the soldier and the jet at the instant when he shoots it down? 2

Case Study 3 Read the following passage and answer the questions given below: There are ten cards numbered $1$ to $10$ and they are placed in a box and then mixed up thoroughly. Then one card is drawn at random from the box. Based on the above, answer the following questions: (i) What is the probability that the number on the drawn card is greater than 4? 2 (ii) If it is known that the number on the drawn card is greater than 4, then what is the probability that it is an even number? 2