CUET Mathematics 2022 7 Aug Shift 2 PYQs

If A is a square matrix of order 3 and $|A|$ is 2, then value of $|adj(A)|$ is :

Let A and B be square matrices of order 3 and k is a constant. If $|A| \neq 0$ and $|B| \neq 0$, where $|A|$ represents the determinant of A, then which of the following statements are true ? (where A' denotes the transpose of matrix A) (A) $(AB)^{-1} = B^{-1} A^{-1}$ (B) $(A+B)' = A' \times B'$ (C) $(AB)' = A'B'$ (D) $|kA| = k^3 |A|$ (E) $|A'| = |A|$ Choose the correct answer from the options given below :

The number of all possible non-singular matrices of order $2 \times 2$ with each entry 0 or 1 is :

If $y = x^x$, then value of $\frac{dy}{dx}$ at $x = 2$ is :

If $2x + y = 6$ then the maximum value of $x^2 y$ is :

A function $f(x)$ is defined by : $f(x) = \begin{cases} x + 2, & \text{if } x < 0 \\ -x + 2, & \text{if } x > 0 \end{cases}$ Which of the following is true ?

The area of the region bounded by the curve $x = y^2$ and the line $x = 4$ is equal to :

$\int_{0}^{1} x^2 e^{x^3} \, dx$ is equal to :

If $\frac{d}{dx}(f(x)) = 5x^4 - \frac{4}{x^5}$ such that $f(1) = 0$. Then $f(2) - 2f\left(\frac{1}{2}\right)$ is equal to :

If $f(x) = \begin{cases} 1 - 2x, & x \leq 0 \\ 1 + 2x, & x > 0 \end{cases}$, then $\int_{-1}^{1} f(x) \, dx =$

If $y = 500 e^{7x} + 600 e^{-7x}$ and $\frac{d^2 y}{dx^2} = ky$, then the value of k is :

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

Let X be a discrete random variable and probability distribution X is | X | $-1$ | 0 | 1 | |---|---|---|---| | P(X) | $\frac{1}{2}$ | $\frac{1}{5}$ | $\frac{3}{10}$ | Then E(X) is equal to :

In Binomial distribution with parameters $n = 12$ and $p = \frac{1}{3}$, value of $E(X^2) + E(X)$ is :

The solution of LPP max.(z) = $5x + 3y$ subject to $2x + y \leq 6$ $x + y \leq 4$ $x \geq 0, y \geq 0$, is

The modulus function $f : R \to R$, given by $f(x) = |x|$, is :

Let L be the set of all lines in a plane and R be the relation in L defined as $R = \{(l_1, l_2) : l_1 \text{ is perpendicular to } l_2, \text{ where } l_1, l_2 \in L\}$. Choose the correct answer :

Let $A = (a_{ij})$ and $B = (b_{ij})$ are square matrices of same order. (A) The number of possible matrices of order $2 \times 2$ with entries $-1, 0, 1$ is 81. (B) $A + A'$ is skew symmetric matrix (C) $A \cdot A^{-1} = 0$, $|A| \neq 0$ (D) A is skew symmetric matrix if $a_{ij} = -a_{ji}$ for all $i, j$ (E) $(AB)' = A'B'$ Choose the correct answer from the options given below :

Match List - I with List - II. | | List - I | | List - II | |---|---|---|---| | (A) | $A = \begin{bmatrix} 6 & 9 \\ 2 & 3 \end{bmatrix}$, $B = \begin{bmatrix} 2 & 6 & 0 \\ 7 & 9 & 8 \end{bmatrix}$, AB will be | (I) | $\begin{bmatrix} 75 & 117 & 72 \\ 35 & 39 & 24 \end{bmatrix}$ | | (B) | $P = \begin{bmatrix} 8 & 45 & 30 \\ 17 & 19 & 5 \end{bmatrix}$, $Q = \begin{bmatrix} 67 & 72 & 42 \\ 8 & 30 & 19 \end{bmatrix}$, P+Q will be | (II) | $\begin{bmatrix} 75 & 117 & 72 \\ 25 & 39 & 24 \end{bmatrix}$ | | (C) | $\begin{bmatrix} 85 & 42 & 69 \\ 73 & 42 & 50 \end{bmatrix} - \begin{bmatrix} 10 & -75 & -3 \\ 38 & 3 & 26 \end{bmatrix}$ | (III) | $\begin{bmatrix} 75 & 117 & 72 \\ 25 & 49 & 24 \end{bmatrix}$ | | (D) | $2 \begin{bmatrix} 34 & 36 & 30 \\ 12 & 18 & 20 \end{bmatrix} + \begin{bmatrix} 7 & 55 & 12 \\ 1 & 13 & -16 \end{bmatrix}$ | (IV) | $\begin{bmatrix} 75 & 127 & 72 \\ 25 & 49 & 24 \end{bmatrix}$ | Choose the correct answer from the options given below :

$\int_{1}^{5} |x - 2| \, dx =$

The area bounded by $x = \sqrt{9 - y^2}$, $x - y + 3 = 0$ and x-axis is :

$\int \frac{1}{\cos^2 x (1 + \tan x)^3} \, dx =$

$\int \frac{x^2 + 4}{(x^2 + 3)(x^2 + 5)} \, dx = u \tan^{-1}\left(\frac{x}{\sqrt{3}}\right) + v \tan^{-1}\left(\frac{x}{\sqrt{5}}\right) + c$, where c is arbitrary constant, then value of $\frac{1}{u^2} + \frac{1}{v^2}$ is equal to :

The area bounded by the parabola $y^2 = 4ax$ and $x^2 = 4ay$ is :

For $a, b \in R$ and $a < b$, $\int_{a}^{b} \frac{f(x)}{f(x) + f(a + b - x)} \, dx =$

If $y = x^{\sin x}$, then value of $x^{-\sin x} \frac{dy}{dx} - \cos x \log x$ is :

$f(x) = [x]$, where $[\,]$ represents greatest integer function. (A) For $2 \leq x < 3$, $[x] = 3$ (B) For $2 \leq x < 3$, $[x] = 2$ (C) Right hand derivative of f at $x = 2$ is not defined (D) Left hand derivative of f at $x = 2$ is zero (E) $f(x)$ is not differentiable at $x = 2$ Choose the correct answer from the options given below :

If curve represented by differential equation $x \frac{dy}{dx} + y = e^x$ passes through (1, 1), then $y(-1)$ is :

If $U(x) = x + \sqrt{1 + x^2}$, then solution of the differential equation $\frac{dy}{dx} + \sqrt{\frac{1 + y^2}{1 + x^2}} = 0$, is :

The differential equation representing family of curves $y = e^{-2x}(a \cos x + b \sin x)$, where a and b are arbitrary constant, is :

Find the root of perpendicular from origin to the line $\frac{x - 1}{3} = \frac{y - 2}{-2} = \frac{z + 1}{3}$.

The distance of the point $(2, 3, -5)$ from the plane $x + 2y - 2z = 9$ is :

Equation of y-axis in space, in vector form is :

Match List - I with List - II. | | List - I (Two given vector) | | List - II (Projection of $\vec{a}$ on $\vec{b}$) | |---|---|---|---| | (A) | $\vec{a} = \hat{i} - \hat{j}$, $\vec{b} = \hat{i} + \hat{j}$ | (I) | $\frac{2}{\sqrt{5}}$ | | (B) | $\vec{a} = \hat{i} + \hat{j}$, $\vec{b} = 2\hat{i} - \hat{k}$ | (II) | 0 | | (C) | $\vec{a} = \hat{j} + \hat{k}$, $\vec{b} = \hat{i} + \hat{k}$ | (III) | $\sqrt{2}$ | | (D) | $\vec{a} = 2\hat{i} + 3\hat{j}$, $\vec{b} = \hat{i} - \hat{k}$ | (IV) | $\frac{1}{\sqrt{2}}$ | Choose the correct answer from the options given below :

The angle between 2 planes $4x + 8y + z - 8 = 0$ and $y + z - 4 = 0$ is :

The unit vector in the direction of the sum of vectors $\vec{a} = 2\hat{i} + 2\hat{j} - 5\hat{k}$ and $\vec{b} = 2\hat{i} + \hat{j} + 3\hat{k}$ is :

The equation of plane that contains line $\frac{x - 1}{-1} = \frac{y + 1}{2} = \frac{z - 1}{3}$ and also pass through point $(0, 1, 0)$ is :

The corner points of the feasible region determined by inequalities of LPP are $(4, 10)$, $(6, 8)$ and $(6, 5)$. Let $z = 3x + 4y$ be the objective function. Then the sum of maximum value of z and minimum value of z is :

Bag I contains 3 red and 4 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 red. The probability that it was drawn from Bag II is :

In Binomial distribution with parameters $n = 100$ and p, Variance of distribution is maximum when p is equal to :