CUET Mathematics 2025 15 May Shift 1 PYQs

Let $A = [a_{ij}]_{3 \times 3}$ such that $|A| = -5$. Then the value of $\det(5A)$ is equal to

If $f(x) = a \log_e|x| + bx^2 + x$ has critical points at $x = -2$ and $x = 1$, then

Let x denote the number of doublets in three throws of a pair of dice with the following probability distribution. | x | 0 | 1 | 2 | 3 | |---|---|---|---|---| | P(x) | $\frac{25}{72}k$ | $\frac{15}{72}k$ | $\frac{3}{72}k$ | $\frac{1}{360}k$ | If value of k is equal to $\frac{m}{n} \cdot \gcd(m,n) = 1$, then $m + n$ is equal to

The number of all possible matrices of order $2 \times 3$ with each entry 0 or 1 is

Match List-I with List-II | List-I | List-II | | --- | --- | | (A) The value of $\int_{0}^{4} \vert x\vert \, dx$ is | (I) 3 | | (B) The value of $\int_{-2}^{2} \vert x\vert \, dx$ is | (II) -1 | | (C) The value of $\int_{0}^{3} [x] \, dx$ is | (III) 8 | | (D) The value of $\int_{-1}^{1} [x] \, dx$ is | (IV) 4 | Choose the correct answer from the options given below:

The area of the region bounded by parabola $x^2 = 4y$, straight line $x = 2$ and $x$-axis, is

If A is a square matrix such that $A^2 = A$ and I is the identity matrix of same order as A, then the matrix $(2I+A)^3 - 19A - 3I$ is equal to

If the optimal value of the objective function $z = px + y$ of an L.P.P occurs at two corner points (2, 11) and (4, 5) of its bounded feasible region, then its optimal value is

The function $f(x) = x^2 - 4x + 6$ is (A) Strictly decreasing on $(-\infty, 2) \cup (2, \infty)$ (B) Strictly increasing on $(2, \infty)$ (C) Strictly increasing on $(-\infty, \infty)$ (D) Strictly decreasing on $(-\infty, 2)$ Choose the correct answer from the options given below:

The general solution of the differential equation $(1 + e^x)dy + ye^x dx = 0$, where $y > 0$, is

Match List-I with List-II | List-I | List-II | |---|---| | (Differential equation) | (Order and Degree) | | (A) $\frac{d^3y}{dx^3} + y^2 + e^{dy/dx} = 0$ | (I) order = 3, degree = 1 | | (B) $\left(\frac{d^2y}{dx^2}\right)^3 + \left(\frac{dy}{dx}\right)^2 + \frac{dy}{dx} + 1 = 0$ | (II) order = 3, degree not defined | | (C) $2x^2\frac{d^2y}{dx^2} - 3\left(\frac{dy}{dx}\right)^2 + y = 0$ | (III) order = 2, degree = 3 | | (D) $\frac{d^3y}{dx^3} + 2\left(\frac{dy}{dx}\right)^2 + \frac{dy}{dx} = 0$ | (IV) order = 2, degree = 1 | Choose the correct answer from the options given below:

If $x = 4t$ and $y = \frac{4}{t}$, then $\frac{d^2y}{dx^2}$ is

If A is an invertible matrix of order 2, then $\det(( {adj } A)^{-1})$ is equal to

The constraints of the given shaded feasible region below of an L.P.P., for non-negative variable constraints $x$ and $y$ are <img src="https://balti.afterboards.in/IECuq9aGxsHDLdW" width="300px"/>

For $x \neq -1$, if $\int \frac{xe^x dx}{(1+x)^2} = \frac{ae^x}{(1+x)^b} + c$, where a, b are fixed numbers and c is the integration constant, then $a + b$ is equal to

For $x \in \left(0, \frac{\pi}{2}\right)$, $\int \frac{\sin x + \cos x}{\sqrt{\sin 2x}} dx$ is equal to

Consider the lines $l_1: \frac{x-1}{0} = \frac{y-1}{1} = \frac{2-z}{1}$ and $l_2: \frac{x}{2} = \frac{y}{0} = \frac{2z-1}{4}$, then which of the following are correct? (A) Direction Ratio's of $l_1 = <0, 1, 1>$ (B) Direction Ratio's of $l_2 = <2, 0, 2>$ (C) Angle between $l_1$ and $l_2 =$ $\frac{\pi}{3}$ (D) Angle between $l_1$ and $l_2 = $ $\frac{2\pi}{3}$ Choose the correct answer from the options given below:

For $|x| < 1$, if $x = \cos\left(\frac{1}{a}\log y\right)$, then

A linear programming problem is as follows: Minimize $z = 2x + 3y$ Subject to the constraints $x \ge 3, x \le 9, y \ge 0, x - y \ge 0, x + y \le 14$. The feasible region has 5 corner points including

The straight line $\frac{x+3}{3} = \frac{y+2}{4} = \frac{z+1}{0}$ is

If $\int e^x\left(\frac{x-1}{(x+1)^3}\right)dx = \frac{Ae^x}{(x+1)^B} + C$, where C is constant of integration then which of the following are correct? (A) $A = -1$ (B) $A = 1$ (C) $B = 3$ (D) $B = 2$ Choose the correct answer from the options given below:

The area bounded by the curve $y = 4 + 3x - x^2$ and $x$-axis is equal to

Which of the following functions are increasing on $x \in \left(0, \frac{\pi}{2}\right)$? (A) $f(x) = \sin x$ (B) $f(x) = \cos x$ (C) $f(x) = \tan x$ (D) $f(x) = \cos 3x$ Choose the correct answer from the options given below:

Match List-I with List-II if $P(A) = \frac{3}{7}$, $P(B) = \frac{4}{7}$ and $P(A \cup B) = \frac{5}{7}$ | List-I | List-II | | :--- | :--- | | (A) $P(A \cap B)$ | (I) $\dfrac{2}{3}$ | | (B) $P(A \mid B)$ | (II) $\dfrac{5}{7}$ | | (C) $P(B \mid A)$ | (III) $\dfrac{1}{2}$ | | (D) $P(A' \cup B')$ | (IV) $\dfrac{2}{7}$ | Choose the correct answer from the options given below:

If $\vec{a} = 2\hat{i} - \hat{j} + 3\hat{k}$ and $\vec{b} = 2\hat{i} + 2\hat{j} + \hat{k}$, then Match List-I with List-II | List-I | List-II | |---|---| | (A) Projection of $\vec{a}$ on $\vec{b}$ is | (I) $-7\hat{i} + 4\hat{j} + 6\hat{k}$ | | (B) $\vec{a} \times \vec{b}$ is | (II) $\frac{1}{\sqrt{101}}(-7\hat{i} + 4\hat{j} + 6\hat{k})$ | | (C) unit vector along $\vec{a} + \vec{b}$ is | (III) $\frac{5}{3}$ | | (D) Unit vector perpendicular to both $\vec{a}$ & $\vec{b}$ is | (IV) $\frac{1}{\sqrt{33}}(4\hat{i} + \hat{j} + 4\hat{k})$ | Choose the correct answer from the options given below:

If $\vec{a}, \vec{b}$ and $\vec{c}$ are three vectors such that $\vec{a} + \vec{b} + \vec{c} = \vec{0}$, $|\vec{a}| = 7$, $|\vec{b}| = 3$ and $|\vec{c}| = 5$, then angle between $\vec{b}$ and $\vec{c}$ is

If $\alpha, \beta$ and $\gamma$ are angle of inclinations of a line with x, y and z axes respectively, then the value of $2(\cos 2\alpha + \cos 2\beta + \cos 2\gamma)$ is

If $A = \begin{bmatrix} 0 & l & -3 \\ -2 & 0 & 1 \\ m & -1 & 0 \end{bmatrix}$ is a skew symmetric matrix, then

The maximum value of $f(x) = \frac{1}{4x^2 + 2x + 1}$ is

If the function defined by $f(x) = \begin{cases} \ kx^2 + 1, & \text{if } x \le 1 \\ 2 , & \text{if } x > 1 \end{cases}$ is continuous at $x = 1$, then k is equal to

Match List-I with List-II An urn contains 4 white and 3 red balls. In a random draw of three balls, the probability of | List-I | List-II | |---|---| | (A) No red ball is | (I) $\frac{12}{35}$ | | (B) Only 1 red ball is | (II) $\frac{1}{35}$ | | (C) Exactly 2 red balls is | (III) $\frac{4}{35}$ | | (D) no white ball is | (IV) $\frac{18}{35}$ | Choose the correct answer from the options given below:

The area of a triangle whose vertices are $(x_1, y_1), (x_2, y_2)$ and $(x_3, y_3)$ is given by the absolute value of

Which of the following functions from $\mathbb{Z}$ to $\mathbb{Z}$ is a bijective function? (where $\mathbb{Z}$ is set of integers)

The position vector of a point which divides the line joining the points with position vectors $(\vec{a} - 2\vec{b})$ and $(2\vec{a} + \vec{b})$ externally in the ratio 2:1, is

If a computer code is correctly programmed, it gives 90% acceptable results. But if it is not correctly programmed, it gives only 40% acceptable results. From previous experience, it is observed that only 80% of codes are correctly programmed. If after a certain programming, the code gives 2 acceptable results, then the approximate probability that the code is correctly programmed is

For $x \in \left(0, \frac{\pi}{2}\right)$, $\int \frac{1}{\sin^2 x + \sin 2x} dx$ is equal to

If $\vec{p}$ and $\vec{q}$ are two unit vectors such that $|\vec{p} + \vec{q}| = \sqrt{2}$, then which of the following are correct? (A) $|\vec{p}| = |\vec{q}| = 1$ (B) $\vec{p}$ and $\vec{q}$ are orthogonal vectors (C) $\vec{p}$ and $\vec{q}$ are collinear vectors (D) $(4\vec{p} - \vec{q}).(2\vec{p} + \vec{q}) = 7$ Choose the correct answer from the options given below:

The maximum value of the objective function $z = 2x + 3y$ of an L.P.P. subjected to the constraints $x - y \le 1$, $x + y \le 3$, $x, y \ge 0$ is

If $A = \begin{bmatrix} -2 \\ -1 \\ -4 \end{bmatrix}$, $B = [-1 \quad 2 \quad 3]$, then the value of $A'B'$ is

Let A= {1, 2, 3}, then the possible equivalence relations on A are: (A) {(1, 1), (2, 2) , (3, 3)} (B) {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)} (C) {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1, 3)} (D) {(1, 1), (2, 2), (3, 3), (1, 3), (3, 1)} Choose the correct answer from the options given below:

If A is a square matrix, then $(A^T - A)$ is-

Consider the differential equation $\frac{dy}{dx} + y \tan x = \sec x$, then which of the following statements are correct? (A) It is homogeneous (B) It has $\sec x$ as its integrating factor (C) It's general solution is $y \sec x = \tan x + c$, where c is arbitary constant. (D) It's degree is not defined Choose the correct answer from the options given below:

The value of $\begin{vmatrix} 1 & x & y \\ 1 & x+y & y \\ 1 & x & x+y \end{vmatrix}$ is

The system of equations $x + y + z = 7$ $x + 2y + 3z = 5$ $x + 3y + \lambda z = \mu$ has a unique solution, if

The area (in sq. units) bounded by the parabola $y^2 = 16x$ and its latus rectum is

If $2P(A) = P(B) = \frac{5}{13}$ and $P(A|B) = \frac{2}{5}$, then $P(A \cap B)$ is

For the principal value branch, the value of $\sin\left(\frac{\pi}{2} - \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right)$ is

The function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = \begin{cases} x^2, & x \ge 1 \\ x, & x < 1 \end{cases}$ is

For $y \neq 0$, the particular solution of the differential equation $2ye^{x/y}dx + (y - 2xe^{x/y})dy = 0$ at the point (1, 1) is

If the area of an equilateral triangle is increasing at the rate of $4\sqrt{3}$ cm²/sec, then the rate of increase of its perimeter when the side is 4cm, is