CUET Mathematics 2025 2 June Shift 1 PYQs

The solution of the differential equation $ydx + (x - y^2)dy = 0$ is

If $y = \frac{1}{\sqrt[3]{1-x^3}}$ then $\frac{dy}{dx}$ is equal to

The probability distribution of a random variable $x$ is, $P(x) = \frac{k}{2^x}, x = 0, 1, 2, 3$. Then Match List-I with List-II | List-I | List-II | |---|---| | (A) $k$ | (I) $\frac{2}{15}$ | | (B) $P(x = 1)$ | (II) $\frac{1}{5}$ | | (C) $P(1 < x < 3)$ | (III) $\frac{8}{15}$ | | (D) $P(x \geq 2)$ | (IV) $\frac{4}{15}$ | Choose the correct answer from the options given below:

If $\begin{bmatrix}3 & 1\\2 & 1\end{bmatrix}A\begin{bmatrix}2 & 1\\1 & 1\end{bmatrix} = \begin{bmatrix}1 & 1\\0 & 1\end{bmatrix}$, then matrix 'A' is

If the matrix $\begin{bmatrix}3 & 2a & -5\\4 & 0 & b\\-5 & 3 & 10\end{bmatrix}$ is symmetric, then the value of $5a + 2b$ is

If $x$ is real, the minimum value of $x^2 - 8x + 20$ is

The value of $\int_0^1 \log_e\left(\frac{1}{x} - 1\right)dx$ is:

$\int \frac{e^{2x} - e^{-2x}}{e^{2x} + e^{-2x}}dx$ is equal to

Match List-I with List-II | List-I | List-II | |---|---| | (A) The degree of differential equation $\frac{d^3y}{dx^3} = e^{\frac{dx}{dy}}$ | (I) 2 | | (B) The order of differential equation $\left(\frac{dy}{dx}\right)^2 + \frac{d^3y}{dx^3} = 0$ | (II) 4 | | (C) The sum of order and degree of differential equation $\frac{d}{dx}\left(\frac{d^2y}{dx^2}\right) + \left(\frac{dy}{dx}\right)^5 = x$ | (III) not defined | | (D) The number of arbitrary constants in the general solution of a differential equation of order 2 | (IV) 3 | Choose the correct answer from the options given below:

Consider an LPP: Maximise $Z = 50x + 15y$ subjected to constraints $x + y \leq 60$, $5x + y \leq 100$, $x, y \geq 0$. If the maximum value of $Z$ occurs at $x = \alpha$ and $y = \beta$, then the value of $\alpha + \beta$ is

If $A$ is a square matrix of order 3 and $|A| = 5$, then the value of $|-AA^T|$ is

For the function $f(x) = -2x^3 + 3x^2 + 36x - 10$, which of the following is/are true? (A) $f$ is increasing in $(-\infty, -2)$ (B) $f$ is increasing in $(-2, 3)$ (C) $f$ is decreasing in $(-\infty, -2)$ (D) $f$ is decreasing in $(3, \infty)$ Choose the correct answer from the options given below:

If the matrix $\begin{bmatrix}2 & -1 & 3\\ \lambda & 0 & 7\\-1 & 1 & 4\end{bmatrix}$ is not invertible, then value of $\lambda$ is

Linear inequalities corresponding to the shaded feasible region OABCO in the given figure are <img src="https://balti.afterboards.in/XDcoYd2BxodBB7e" width="300px"/>

Area of the region bounded by $y = x^2$ and the line $y = 16$ is

The value of $\lambda$ so that the lines $\frac{1-x}{3} = \frac{7y-14}{2\lambda} = \frac{z-3}{2}$ and $\frac{7-7x}{3\lambda} = \frac{y-5}{1} = \frac{6-z}{5}$ are at right angle, is:

The domain of $y = \cos^{-1}(x^2 - 4)$ is

The solution of the differential equation $\frac{dy}{dx} = \frac{x+y}{x-y}$ is

A problem in mathematics is given to three students whose chances of solving it are 1/2, 1/3, 1/4 respectively. The probability that the problem is solved is

$\int_{-1}^1 \frac{x^3 + |x| + 1}{x^2 + 2|x| + 1}dx$ is equal to

Relation R on the set $A = \{1, 2, ..., 15\}$ defined as $R = \{(x, y): y - 4x = 0\}$ is

The feasible region corresponding to the linear constraints of a Linear Programming Problem (LPP) is represented by the shaded region in the given figure. Which of the following is not a constraint to the given LPP? <img src="https://balti.afterboards.in/AaGbmDtdftDHD7T" width="300px"/>

if $A = \begin{bmatrix}1 & 0\\3 & 1\end{bmatrix}$ and $A^4 = \begin{bmatrix}1 & 0\\k & 1\end{bmatrix}$ then value of $k$ is

Consider a closed cylinder of radius $r$ with a fixed surface area. The volume of the cylinder is maximum when its height is

If $A$ is a skew-symmetric matrix of order 5, then $|adjA|$ is equal to

The foot of the perpendicular drawn from the point $(1,6,3)$ to the line $\frac{x}{1} = \frac{y-1}{2} = \frac{z-2}{3}$ is

$\int_0^1 \frac{dx}{\sqrt{1+x} - \sqrt{x}}$ is equal to

If $y = \sin^{-1}x$, then $(1-x^2)\frac{d^2y}{dx^2}$ is equal to

The value of $\int_{-\pi/2}^{\pi/2}(x^5 + x^3\cos x)dx$ is

The function $f(x) = 2\log_e(x-2) - x^2 + 4x + 1, (x > 2)$ is increasing on the interval:

If A (3, 2), B (1, -1) and C (2, 1) are three vertices of a parallelograms ABCD, then its area (in sq.units) is equal to

If the direction ratios of two lines are $a, b, c$ and $(b-c), (c-a), (a-b)$ respectively, then the angle between these lines is:

If $x = -1$ and $x = -2$ are the extreme points of $f(x) = \alpha\log|x| + \beta x^2 + x$ then

The area (in sq.units) of the region bounded by the curve $y = \cos x$ between $x = -\frac{\pi}{2}, x = \frac{\pi}{2}$ and the x-axis is

In a college, 30% students fail in physics, 25% fail in Mathematics and 10% fail in both. One student is chosen at random. The probability that she fails in physics if she has failed in mathematics is

If $x, y$ and $z$ are real number such that $x + y + z = 0$, then value of $\begin{vmatrix}3x & -x+y & -x+z\\x-y & 3y & z-y\\x-z & y-z & 3z\end{vmatrix}$ is

The value of $\lambda$, for which the two vectors $2\hat{i} - \hat{j} + 2\hat{k}$ and $3\vec{i} + \lambda\vec{j} + \hat{k}$ are perpendicular, is:

The area (in sq.units) of the region bounded by the line $2y + x = 8$, the x-axis and the lines $x = 2$ and $x = 4$ is

If $A$ is an invertible matrix of order 3, then Match List-I with List-II | List-I | List-II | | --- | --- | | (A) $\vert \text{adj} A\vert $ | (I) $8\vert A\vert $ | | (B) $\vert A(\text{adj} A)\vert $ | (II) $\vert A\vert ^2$ | | (C) $\vert 2A\vert $ | (III) $\frac{1}{\vert A\vert }$ | | (D) $\vert A^{-1}\vert $ | (IV) $\vert A\vert ^3$ | Choose the correct answer from the options given below:

If $x = a\left(\cos t + \log \tan\frac{t}{2}\right), y = a\sin t$, then value of $\frac{dy}{dx}$ at $t = \frac{\pi}{4}$ is

The area (in sq.units) of a triangle formed by vertices O, A and B where $\vec{OA} = \hat{i} + 2\hat{j} + 3\hat{k}$ and $\vec{OB} = -3\hat{i} - 2\hat{j} + \hat{k}$ is

If the function $f(x) = \begin{cases}\frac{\sin 3x}{x}, & \text{if } x \neq 0\\ \frac{3k}{2}, & \text{if } x = 0\end{cases}$ is continuous at $x = 0$, then the value of $k$ is

A function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = \frac{x}{x^2+1}$ is (where $\mathbb{R}$ is a set of real number)

Which of the following statements are true? (A) If $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$, then $x, y, z$ are called direction ratios of $\vec{r}$. (B) For any two vectors $\vec{a}$ and $\vec{b}$, $\vec{a} + \vec{b} = \vec{b} + \vec{a}$ (C) $\vec{a} \perp \vec{b}$ if and only if $\vec{a} \times \vec{b} = \vec{0}$ (D) Projection of $\vec{b}$ on $\vec{a}$ is $\frac{\vec{a} \cdot \vec{b}}{|\vec{a}|^2}$ Choose the correct answer from the options given below:

The integrating factor of the differential equation $\frac{dy}{dx} = x + xy$ is

An urn contains 5 red and 5 black balls. A ball is drawn at random, its color is noted and is returned to the urn. Moreover, 2 additional balls of the same color are put in the urn and then a ball is drawn at random. The probability that the second drawn ball is red, is:

For any events A and B of a sample space S, which of the following statements are TRUE? (A) $P(S | B) = 1$ (B) $P(A \cap B) = P(A) + P(B) + P(A \cup B)$ (C) $P(\bar{A} | B) = 1 - P(A | B)$ (D) $P(A | B) = \frac{P(A \cap B)}{P(B)}, P(B) \neq 0$ Choose the correct answer from the options given below:

The matrix $A = \begin{bmatrix}0 & 0 & 5\\0 & 5 & 0\\5 & 0 & 0\end{bmatrix}$ is a (A) Diagonal matrix (B) Scalar matrix (C) Square matrix (D) Symmetric matrix Choose the correct answer from the options given below:

Let $\vec{a}$ and $\vec{b}$ are unit vectors. If $\sqrt{3}\vec{a} - \vec{b}$ is a unit vector, then the angle between $\vec{a}$ and $\vec{b}$ is

The corner points of the bounded feasible region for an LLP are: (5, 5), (15, 15), (0, 20) and (0, 10). Let $z = 3x + 9y$ be the objective function. Then the value of $maximum(z) - minimum(z)$ is