CUET Mathematics 2022 16 July Shift 2 PYQs

The number of all possible matrices of order 3 x 3 with each entry belonging to the set {0, 1} is:

The values of $x$ and $y$ in the equation $2\begin{bmatrix} x & 1 \\ 4 & -3 \end{bmatrix} + 3\begin{bmatrix} -2 & 1 \\ 2 & y-2 \end{bmatrix} = \begin{bmatrix} 2 & 5 \\ 14 & 3 \end{bmatrix}$ are respectively:

The matrix $\begin{bmatrix} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \end{bmatrix}$ is: A. a square matrix B. a scalar matrix C. a diagonal matrix D. an identity matrix Which of the above statements are true? Choose the correct answer from the options given below:

$\begin{vmatrix} a+b & 1 & 0 \\ a^2-b^2 & a-b & 1 \\ a^3+b^3 & a^2+b^2+ab & a^2-b^2 \end{vmatrix} =$

The equation of the normal to the curve $y = x - \frac{1}{x}$ at $(1,0)$ is:

The minimum value of $x^2 - 8x + 17$ on the set $\mathbb{R}$ of all real numbers is:

The value of $\int_{-3}^{3} \log_e\left(\frac{4+x}{4-x}\right) dx$ is

$\int \frac{x^2+1}{(x+1)^2} e^x \, dx$ is equal to

Integrating factor of the differential equation $\frac{dy}{dx} - \frac{1}{x} y = 1$ is:

The order and the degree of the differential equation $\frac{d^2y}{dx^2} + x\left(\frac{dy}{dx}\right)^2 = 2x^2 \log\left(\frac{d^2y}{dx^2}\right)$ are respectively:

Match List I with List II. | List I | List II | |---|---| | A. $\frac{d^2y}{dx^2} + \frac{dy}{dx} = 0$ | I. order 3, degree 1 | | B. $\left(\frac{d^2y}{dx^2}\right)^2 = 0$ | II. order 2, degree 2 | | C. $\frac{d^3y}{dx^3} + \frac{d^2y}{dx^2} + y = 0$ | III. order 2, degree 1 | | D. $\sin\left(\frac{dy}{dx}\right) + 5y = 0$ | IV. order 1, degree is not defined | Choose the correct answer from the options given below:

Let X be the random variable with probability distribution given by the following table. | X | 0 | 1 | 2 | 3 | |---|---|---|---|---| | P(X = x) | $\frac{1}{8}$ | k | $\frac{3}{8}$ | $\frac{1}{8}$ | The value of $P(X \leq 1)$ is:

A random variable X has the following probability distribution: | x | 1 | 2 | 3 | 4 | |---|---|---|---|---| | p(x) | 2k | 4k | 3k | k | The value of E(X) is:

The smaller area enclosed by the curve $y = |x|$ and the circle $(x-a)^2 + y^2 = a^2$ is:

$\int \frac{x}{(x-1)(x-2)} dx =$ (where c is an arbitrary constant)

If $f:\mathbb{R} \to [-5, \infty)$ is defined as $f(x) = x^2 - 5$, then the function f is A. one-one B. many-one C. onto D. into Which of the above statements are true? Choose the correct answer from the options given below:

A relation R = {(1,1),(2,2),(3,3),(1,2),(2,1),(1,3),(2,3)} on A = {1,2,3} will be an equivalence relation, if we delete: Choose the correct answer from the options given below:

Match List I with List II | List I | List II | |---|---| | A. If $f(x) = 2x$ and $g(x) = \frac{x^2}{2} + 1$, then $\frac{g(x)}{f(x)}$ is | I. discontinuous at exactly three points. | | B. The function $f(x) = \frac{4-x^2}{4x-x^3}$ is | II. continuous everywhere | | C. The function $f(x) = \lvert x \rvert + \lvert x-1 \rvert$ is | III. discontinuous at $x = 0$. | | D. The function $f(x) = \lvert \sin x \rvert$ is | IV. continuous at $x = 0$ and $x = 1$ | Choose the correct answer from the options given below:

Which of the following statements are true? (A) $f:\mathbb{R} \to \mathbb{R}$ given by $f(x) = 3x$ is one-one onto. (B) $f:\mathbb{R} \to \mathbb{R}$ given by $f(x) = x^4$ is one-one and onto. (C) $f:\mathbb{Z} \to \mathbb{Z}$ given by $f(x) = x^2$ is neither one-one nor onto. (D) $f:\mathbb{R} \to \mathbb{R}$ given by $f(x) = |x|$ is neither one-one nor onto. (Where $\mathbb{R}$ is the set of all real numbers and $\mathbb{Z}$ is the set of all integers) Choose the correct answer from the options given below:

Match List I with List II | List I (Functions) | List II (Principal value branches) | |---|---| | A. $f(x) = \cos^{-1} x$ | I. $[0, \pi]$ | | B. $f(x) = \tan^{-1} x$ | II. $[0, \pi] - \left\{\frac{\pi}{2}\right\}$ | | C. $f(x) = \text{cosec}^{-1} x$ | III. $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] - \{0\}$ | | D. $f(x) = \sec^{-1} x$ | IV. $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ | Choose the correct answer from the options given below:

The graph of $\sin^{-1} x$ is represented by:

The value of $\text{cosec}^{-1}(-2) - 2\sec^{-1}(-2)$ is equal to:

If P and Q are symmetric matrices of same order, then $(PQ - QP)$ is

If P is matrix of order $m \times n$ and Q is a matrix such that PQ' and Q'P are both computable, then the order of matrix Q is

If A and B are square matrices of order 3 such that $|A| = 2$, $|B| = 3$, and $|2A \cdot \text{adj}(3(\text{adj}B))| = 2^\alpha \cdot 3^\beta$, then value of $\alpha + \beta$ is:

If $A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 1 & 4 \\ 3 & 2 & 1 \end{bmatrix}$, then which of the following is the value of $(\text{adj } A)^{-1}$

Read the following statements carefully: A. Determinant is a square matrix B. If A be any given square matrix of order n, then $A(\text{adj}A) = (\text{adj}A)A = |A|I$. C. If A and B are nonsingular matrices of the same order, then AB and BA are also nonsingular matrices of the same order. D. If A is a nonsingular matrix, then its inverse does not exist Which of the above statements are true? Choose the correct answer from the options given below:

The function $f(x) = |x - 1|$ is

The interval in which the function given by $f(x) = x^2 e^{-x}$ is strictly increasing is:

The line $y = mx$ ($m > 0$) partitions the area of the circle $x^2 + y^2 = a^2$ ($a > 0$) in the ratio:

If $\begin{vmatrix} (a-x)^2 & (a-y)^2 & (a-z)^2 \\ (b-x)^2 & (b-y)^2 & (b-z)^2 \\ (c-x)^2 & (c-y)^2 & (c-z)^2 \end{vmatrix} = \lambda(a-b)(b-c)(c-a) \cdot (x-y)(y-z)(z-x)$ then the value of $\lambda$ is:

The value of $\int_{\pi/2}^{\pi} \frac{1}{1 + \cot x} dx$ is equal to:

The number of arbitrary constants in the general solution of a differential equation of fourth order is:

Match List I with List II | List I | List II | |---|---| | A. $\int \frac{dx}{x^2 - a^2}$ is equal to | I. $\frac{x}{2}\sqrt{x^2 + a^2} + \frac{a^2}{2}\log\left\lvert x + \sqrt{x^2 + a^2}\right\rvert + C$ | | B. $\int \frac{dx}{\sqrt{x^2 + a^2}}$ is equal to | II. $\frac{1}{2a}\log\left\lvert \frac{x-a}{x+a}\right\rvert + C$ | | C. $\int \sqrt{a^2 - x^2} \, dx$ is equal to | III. $\frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2}\sin^{-1}\frac{x}{a} + C$ | | D. $\int \sqrt{a^2 + x^2} \, dx$ is equal to | IV. $\log\left\lvert x + \sqrt{x^2 + a^2}\right\rvert + C$ | Choose the correct answer from the options given below:

If $\vec{a} = \hat{i} - \hat{j} + \hat{k}$, $\vec{b} = 2\hat{i} + \hat{j} - 3\hat{k}$, $\vec{c} = 2\hat{i} - \hat{j} + 7\hat{k}$ and $\vec{a} \times (\vec{b} \times \vec{c}) = \lambda \vec{b} + \mu \vec{c}$ (When $\lambda$, $\mu$ are scalars), then the value of $\lambda + \mu$ is:

If $|\vec{a}| = |\vec{b}| = |\vec{a} + \vec{b}| = 1$, then $|\vec{a} - \vec{b}|$ is equal to:

If $\theta$ is the acute angle between two unit vectors $\vec{a}$ and $\vec{b}$, then $\cos\frac{\theta}{2} =$

The two lines given by $\vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \mu(\hat{i} - \hat{j} + \hat{k})$ and $\vec{r} = (2\hat{i} - \hat{j} - \hat{k}) + \mu(-\hat{i} + \hat{j} - \hat{k})$ A. are perpendicular B. are parallel. C. have shortest distance 0. D. have shortest distance $\sqrt{26}$. E. have shortest distance $\sqrt{78}$. Which of the above statements are true? Choose the correct answer from the options given below:

The correct order of steps from A to E, for finding value of p, so that the lines $\frac{1-x}{3} = \frac{7y-14}{2p} = \frac{z-3}{2}$ and $\frac{7-7x}{3p} = \frac{y-5}{1} = \frac{6-z}{5}$ are at right angle is: A. $p = \frac{70}{11}$ B. $(-3) \times \left(\frac{-3p}{7}\right) + 1 \times \left(\frac{2p}{7}\right) + 2 \times (-5) = 0$ C. $\frac{x-1}{-3} = \frac{y-2}{2\frac{p}{7}} = \frac{z-3}{2}$, $\frac{x-1}{-\frac{3p}{7}} = \frac{y-5}{1} = \frac{z-6}{-5}$ D. $\frac{9p}{7} + \frac{2p}{7} - 10 = 0$ E. $\frac{11p}{7} = 10$ Choose the correct answer from the options given below:

Image of origin with respect to plane $x + y + z = 3$ is: