CUET Mathematics 2022 PYQs16 July Shift 2. Free, no login required.

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.

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Let X be the random variable with probability distribution given by the following table.

The value of $P(X \leq 1)$ is:

A random variable X has the following probability distribution:

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?

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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

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

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?

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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

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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: