CUET Mathematics 2022 10 Aug Shift 1 PYQs

If $A$ is square matrix of order 3 and $A \cdot (Adj.(A)) = 10I$, then the value of $\frac{1}{25}|Adj.(A)|$ is

Let $A$ and $B$ be two non-singular, square matrices of same order, and A. $(AB)^{-1} = B^{-1} \cdot A^{-1}$ B. $(A+B)^{-1} = B^{-1} + A^{-1}$ C. $adj. A = |A| \cdot A^{-1}$ D. $det(A^{-1}) = [det A]^{-1}$ Choose the correct answer from the options given below

If $\begin{vmatrix} -1 & a & a^2 \\ -1 & b & b^2 \\ -1 & c & c^2 \end{vmatrix}^2 = \lambda$ and $a - b = 1$, $b - c = 2$ and $c - a = 3$, then the value of $\lambda$ is

If $\begin{bmatrix} 2x-1 & -3 & 6 \\ 3 & 3y-2 & 4 \\ -6 & -4 & 4z-3 \end{bmatrix}$ is skew symmetric matrix, then $xyz$ is equal to

The solution of differential equation $\sqrt{x+1} - \sqrt{x-1}\frac{dy}{dx} = 0$ is

The point(s) on the curve $\frac{x^2}{9} + \frac{y^2}{64} = 1$, at which the tangents are parallel to x-axis are

A die is tossed four times. The probability of getting an odd number at least once, is

A manufacturer of electronic circuit has a stock of 200 resistors, 120 transistors and 150 capacitors and is required to produce two types of circuits A and B. Type A requires 20 resistors, 10 transistors and 10 capacitors. Type B requires 10 resistors, 20 transistors and 30 capacitors. If the profit on type A circuit is Rs. 50 and that on type B circuit is Rs. 60, identify the constraints for this LPP, if it was assumed that x circuit B of type A and y circuits of type B was produced by the manufacturer. A. $x + 2y \geq 15$ B. $2x + y \leq 20$ C. $x + 2y \leq 12$ D. $x, y \leq 0$ Choose the correct answer from the options given below

An energy DRONE is flying along the curve $y = x^2 + 7$. A soldier is placed at $(3, 7)$. The nearest distance of the DRONE from soldier's position is

The line $y = x$, partition the area of the circle $(x-1)^2 + y^2 = 1$, into two segments. The area of the major segment is

The maximum value of $x^{-x}$ is

If $\int (x + \sqrt{x^2 - 1})^2 \, dx = \alpha \cdot x + \beta x^3 + \gamma (x^2 - 1)^{\frac{3}{2}} + C$, where $C$ is arbitrary constant, then the value of $3(\alpha + \beta + \gamma)$ is

The probability distribution of a random variable $X$ is | x | 0 | 1 | 2 | 3 | | --- | --- | --- | --- | --- | | P(X = x) | $\frac{1}{4}$ | $\frac{1}{8}$ | $\frac{1}{8}$ | $\frac{1}{2}$ | The variance of $X$ is

$\int_0^1 \frac{dx}{x^2 + x + 1}$

If the order and degree of the differential equation $\sqrt{\frac{d^2y}{dx^2}} = \left(1 + \frac{dy}{dx}\right)^{\frac{1}{3}}$ are $a$ and $b$ respectively, then the value of $a^2 + b^2$ is

The integrating factor of differential equation $x\frac{dy}{dx} + 2y = x^2 \log x$ is

General solution of the differential equation $\frac{dy}{dx} + \frac{\sqrt{1 - y^2}}{\sqrt{1 - x^2}} = 0$ is A. $\tan^{-1} x + \tan^{-1} y = C$ B. $\sin^{-1} x - \cos^{-1} y = C$ C. $x\sqrt{1 - y^2} - y\sqrt{1 - x^2} = C$ D. $\sin^{-1} x + \sin^{-1} y = C$ E. $\cos^{-1} x + \cos^{-1} y = C$ (where C is arbitrary constant) Choose the correct answer from the options given below

The absolute maximum value of $y = x^3 - 3x + 2$, $0 \leq x \leq 2$, is

Match List I with List II | List - I | List - II | |---|---| | A. $\int_{-\pi/2}^{\pi/2} \sin^7 x \, dx$ | I. $\frac{\pi}{2}$ | | B. $\int_{-\pi/2}^{\pi/2} \sin^2 x \, dx$ | II. $\frac{\pi}{4}$ | | C. $\int_0^{\pi/2} \frac{1}{1 + \tan x} \, dx$ | III. 0 | | D. $\int_0^{\pi} \lvert \cos x \rvert \, dx$ | IV. 2 |

The portion of the area enclosed by the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, that lies in the first quadrant is

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

$\int \frac{\sin(\tan^{-1} x)}{1 + x^2} dx =$

The distance of the plane $\vec{r} \cdot \left(\frac{2}{7}\hat{i} + \frac{3}{7}\hat{j} - \frac{6}{7}\hat{k}\right) = 2$ from the origin is

If $|\vec{a}| = 8$, $|\vec{b}| = 3$ and $|\vec{a} \times \vec{b}| = 12$, then the value of $\vec{a} \cdot \vec{b}$ is

The direction ratios of the line $\frac{1-x}{3} = \frac{7y - 14}{2} = \frac{z-3}{2}$ are

The vertices of a closed convex polygon representing the feasible region of the LPP with, objective function $z = 5x + 3y$ are $(0, 0)$, $(3, 1)$, $(1, 3)$ and $(0, 2)$. The maximum value of $z$ is

$\int \frac{x^2 + 1}{x^4 + 1} dx =$

$\int_0^{4\pi} \frac{x}{1 + |\cos x|} dx =$

A man is known to speak truth 4 out of 5 times. He throws a die and reports that five appears. Then the probability that actual five appears on the dice is

Five numbers taken out from numbers 1-30 and arrange them in ascending order. The probability that the third number will be 20 is

Bag I contains 4 red and 5 black balls, while another Bag II contains 5 red and 6 black balls. One ball is drawn at random from one of the bags and it is found to be black. Then the probability that it was drawn from Bag II, is

A card is picked at random from a pack of 52 playing cards. If the picked card is a queen, then probability of card to be of spade type also, is

The differential equation representing family of curves $y = ae^{mx} + be^{nx}$, where $a$ and $b$ are arbitrary constants, is

The interval in which the function, $f(x) = 7 - 4x - x^2$ is strictly increasing is

The area (in square units) of minor segment of the circle $x^2 + y^2 = 25$ cut off by the line $x = \frac{5}{2}$ is

If $\int_0^{\pi/2} \sqrt{\tan x} \, dx = \frac{\lambda}{\sqrt{2}}$, then the value of $\lambda$ is

A. A relation $R$ on a set $A$ is called an equivalence relation, if it is reflexive, symmetric and transitive. B. The function $f : R \to R$ defined by $f(x) = e^x$ is not one-one. C. The one-one function is also known as injective function. D. The onto function is also known as subjective function. E. A function $f : X \to Y$ is said to be many-one, if two or more than two elements in set $X$ have the different image in set $Y$. Choose the correct answer from the option given below:

If $|\vec{a}| = 3|\vec{b}|$, $|\vec{b}| = 2$ and angle between $\vec{a}$ and $\vec{b}$ is $60^\circ$, then $|\vec{a} - \vec{b}|$ is equal to:

If $A = \begin{bmatrix} 1 & -1 & 1 \\ 1 & -2 & -2 \\ 2 & 1 & 3 \end{bmatrix}$ and square matrix $B$ satisfy $AB = 8I$, then the value of $|B|$ is:

The angle between the vectors $\hat{i} - \hat{j}$ and $\hat{j} - \hat{k}$ is: