CBSE(NCERT) GRADE 12 MATHS 2024 SET1

If $A = \begin{bmatrix} x\cos x & x\sin x \\ x\sin x & -x\cos x \end{bmatrix}$, then the value of $x$, for which $A$ is an identity matrix, is

If the matrix $A = \begin{bmatrix} 0 & 3-b \\ 3 & 0 & a \\ 7 & -5 & 0 \end{bmatrix}$ is a skew-symmetric matrix, then the values of ‘a’ and ‘b’ are :

If $\dfrac{3x^2 - x}{4 - x^2 + x} = \dfrac{31}{2 - 6}$, then the value of $x$ is :

If $\begin{bmatrix} 7 & 9 \\ 14 & 8 \end{bmatrix} = \begin{bmatrix} 1 & 3 \\ 2 & 1 \end{bmatrix} X$, then matrix $X$ is :

The value of $k$, for which $f(x) = \begin{cases} \dfrac{\sin x}{x^3}, & x \ne -\dfrac{\pi}{3} \\ k, & x = -\dfrac{\pi}{3} \end{cases}$ is continuous at $x = -\dfrac{\pi}{3}$, is :

Let the vectors $\vec a$ and $\vec b$ be such that $|\vec a| = 3$ and $|\vec b| = \dfrac{3}{2}$, then $\vec a \times \vec b$ is a unit vector, if the angle between $\vec a$ and $\vec b$ is :

If $\vec a = 2\hat i - 2\hat j + \hat k$, $\vec b = \hat i + 2\hat j - 3\hat k$ and $\vec c = 2\hat i - \hat j + 4\hat k$, then the projection of $(\vec c - \vec b)$ along $\vec a$ is :

The angle between the lines $\dfrac{x-1}{3} = \dfrac{y-2}{5} = \dfrac{z-4}{1}$ and $\dfrac{x-3}{1} = \dfrac{y-7}{2} = \dfrac{z-5}{3}$ is :

The Cartesian equations of a line are given as $6x - 2 = 3y + 1 = 2z - 2$. The direction ratios of the line are :

The solution set of the inequation $2x + 3y < 6$ is :

The maximum value of the objective function $z = 3x + 5y$ subject to the constraints $x \ge 0$, $y \ge 0$ and $4x + 3y \le 12$ is :

If the points $A(3, -2)$, $B(k, 2)$ and $C(8, 8)$ are collinear, then the value of $k$ is :

If $\vec a$, $\vec b$ and $\vec c$ are unit vectors such that $\vec a + \vec b + \vec c = \vec 0$, then $(\vec a \cdot \vec b + \vec b \cdot \vec c + \vec c \cdot \vec a)$ is equal to :

$\int \dfrac{x\cos x\sin x}{2x\cos^2 x}\,dx$ is equal to :

A differential equation $\dfrac{dy}{dx} = 1 - x + y - xy$ has solution :

The solution of the differential equation $\dfrac{dy}{dx} = 1 - x + y - xy$ is :

The degree of the differential equation $x^3 \dfrac{d^2y}{dx^2} + y^4 \dfrac{dy}{dx} + y^5 = 0$ is :

The integrating factor of the differential equation $\dfrac{dy}{dx} + y \tan x = 2x + x^2 \tan x$ is :

The probabilities of A, B and C solving a problem are $\tfrac{1}{3}$, $\tfrac{1}{5}$ and $\tfrac{1}{6}$ respectively. The probability that the problem is solved, is :

Assertion (A): If the side of a square is increasing at the rate of $0.2$ cm/s, then the rate of increase of its perimeter is $0.8$ cm/s. Reason (R): Perimeter of a square = 4 (side).

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

If $(\cos x)^y=(\cos y)^x$, find $\dfrac{dy}{dx}$.

Find the intervals on which the function $f(x)=10-6x-2x^2$ is (a) strictly increasing (b) strictly decreasing.

Show that of all rectangles inscribed in a given circle, the square has the maximum area.

Find: $$\int \sec^3(1+3x)\cot(1+3x)\,dx$$

If $x=a\cos\theta$ and $y=b\sin\theta$, then prove that $$\frac{d^2y}{dx^2}=-\frac{a^4}{b^3y^4}.$$

Find: $$\int \frac{x^2+2x}{(x+1)^3}\,dx$$

Find: $$\int x^5 e^{x-4}\,dx$$

Solve the differential equation $\dfrac{dy}{dx}+2y\tan x=\sin x$, given that $y=0$ when $x=\dfrac{\pi}{3}$.

The corner points of the feasible region determined by the system of linear constraints are A$(0,40)$, B$(20,40)$, C$(60,20)$ and D$(60,0)$. The objective function of the L.P.P. is $z=4x+3y$. Find the point of the feasible region at which the value of objective function is maximum and the point at which the value is minimum. Hence, find the maximum and the minimum values.

Find the probability distribution of the number of doublets in three throws of a pair of dice.

(a) A card is randomly drawn from a well-shuffled pack of 52 playing cards. Events A and B are defined as under: A: Getting a card of diamond B: Getting a queen Determine whether the events A and B are independent or not.

Let $A=\mathbb{R}-\{4\}$ and $B=\mathbb{R}-\{1\}$ and let $f:A\to B$ be defined by $f(x)=\dfrac{4x-3}{x-3}$ for all $x\in A$. Show that $f$ is one-one and onto.

(a) Let $A = \{x \mid x \in \mathbb{Z}, 0 \le x \le 12\}$. Show that the relation $R = \{(a, b) : a, b \in A, (a-b)$ is divisible by $4\}$ is an equivalence relation. Find the set of elements related to $2$.

Using matrices, solve the following system of linear equations: $$3x+4y+2z=8;\ 2y-3z=3;\ x-2y+6z=-2$$

Using matrices, solve the following system of linear equations: $3x + 4y + 2z = 8$; $2y - 3z = 3$; $x - 2y + 6z = -2$

Using integration, find the area enclosed by $y=x^2$, $x=-1$, $x=1$ and the $x$-axis.

Using integration, find the area of the region bounded by the curve $y = x^2$, $x = -1$, $x = 1$ and the x-axis.

(a) Write the vector equations of the following lines and hence find the shortest distance between them: $\frac{x-1}{2} = \frac{y-1}{1} = \frac{z-6}{1}$ and $\frac{x-3}{1} = \frac{y+5}{-2} = \frac{z-7}{1}$

(b) Find the length and the coordinates of the foot of the perpendicular drawn from the point $P(5, 9, 3)$ to the line $\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-4}{3}$. Also, find the coordinates of the image of the point $P$ in the given line.

Case Study – 1 The relation between the height of the plant ($y$ in cm) with respect to exposure to sunlight is governed by the relation $y = 4x - \frac{1}{2}x^2$, where $x$ is the number of days it is exposed to sunlight. Based on the above, answer the following questions: (i) Find the rate of growth of the plant with respect to sunlight. 1 (ii) What is the number of days it will take for the plant to grow to the maximum height? 2 (iii) What is the maximum height of the plant? 1

Case Study – 2 Two clubs P and Q played a cricket match in which each club selected one team. The players of clubs P and Q were placed in the vectors $\overrightarrow{AB}$ and $\overrightarrow{CD}$ respectively, where the points $A$, $B$, $C$ and $D$ are given as: $A(3, 4, 0)$, $B(5, 3, 3)$, $C(6, -4, 1)$ and $D(13, -5, -4)$. Based on the above, answer the following questions: (i) Write the direction cosines of $\overrightarrow{AB}$. 1 (ii) Write a unit vector in the direction of $\overrightarrow{CD}$. 1 (iii) (a) Find the angle between vectors $\overrightarrow{AB}$ and $\overrightarrow{CD}$. 2 OR (iii) (b) Write a vector perpendicular to both $\overrightarrow{AB}$ and $\overrightarrow{CD}$. 2

A cricket match is organised between two clubs P and Q for which a team from each club is chosen. Remaining players of club P and club Q are respectively sitting along the lines AB and CD, where the points are A(3, 4, 0), B(5, 3, 3), C(6, – 4, 1) and D(13, – 5, – 4). Based on the above, answer the following questions : (i) Write the direction ratios of vector $\vec{AB}$. (ii) Write a unit vector in the direction of $\vec{CD}$. (iii) (a) Find the angle between vectors $\vec{AB}$ and $\vec{CD}$. OR (iii) (b) Write a vector perpendicular to both $\vec{AB}$ and $\vec{CD}$.

A coach is training 3 players. He observes that player A can hit a target 4 times in 5 shots, player B can hit 3 times in 4 shots and player C can hit 2 times in 3 shots. Based on the above, answer the following questions : (i) Find the probability that all three players miss the target. (ii) Find the probability that all of them hit the target. (iii) (a) Find the probability that only one of them hits the target. OR (iii) (b) Find the probability that exactly two of them hit the target.