CUET Mathematics 2025 22 May Shift 2 PYQs

The largest interval, in which the function $f(x) = x^3 + 2x^2 - 1$ is increasing, is:

$\int \frac{\sin 2x \, dx}{\sqrt{9 - \cos^4 x}}$ equals

Match List-I with List-II | List-I | List-II | |---|---| | **Differential Equation** | **Order and Degree** | | (A) $\left(\frac{d^2y}{dx^2}\right)^2 = e^x\left(\frac{dy}{dx}\right)^4 + 1 = 0$ | (I) order = 1 and degree = 2 | | (B) $\left(\frac{dy}{dx}\right)^2 + xy = 0$ | (II) order = 2 and degree = 1 | | (C) $\left(1 + \frac{dy}{dx}\right)^{3/2} = 4\left(\frac{d^2y}{dx^2}\right)^2$ | (III) order = 2 and degree = 2 | | (D) $\sqrt\frac{d^2y}{dx^2} + 1 = \frac{dy}{dx}$ | (IV) order = 2 and degree = 4 | Choose the correct answer from the options given below:

The area (in sq. units) of the triangle whose vertices are $(0, 0)$, $(a, 0)$, $(0, b)$, is equal to

If $A = \begin{bmatrix} 2 & 3 \\ 5 & -2 \end{bmatrix}$ be such that $A^{-1} = KA$, then the value of K is:

The probability distribution of a random variable x is given below. | x | 0 | 1 | 2 | 3 | |---|---|---|---|---| | P(x) | k/3 | k/2 | k/4 | k/7 | Then the value of k is

Let $A = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} 4 & -6 \\ -2 & 4 \end{bmatrix}$ (A) $\det(A^T) = 1$ (B) $AB = I$, where $I$ is the identity matrix of order 2. (C) $A^{-1} = \begin{bmatrix} 2 & -3 \\ -1 & 2 \end{bmatrix}$ (D) adj $(B) = \begin{bmatrix} 4 & 2 \\ 6 & 4 \end{bmatrix}$ Choose the correct answer from the options given below:

Area (in sq. units) of the region bounded by the curve $y^2 = 4x$, $y$-axis and the line $y = 3$ is

The solution set of inequality $3x + 5y < 4$ is

For LPP: Maximize $z = 2x + 3y$ subject to the constraints $x + y \geq 2$, $x + 2y \geq 3$, $x \geq 0$, $y \geq 0$, which of the following graph represents the feasible region of the above LPP as shaded portion?

$\int_0^2 (|x| + |x - 2|) dx =$

The maximum value of $\left(\frac{1}{x}\right)^x$ for $x > 0$ is

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

The solution of the differential equation $\frac{dy}{dx} = (1 + x^2)(1 + y^2)$ is (Here C is an arbitrary constant)

If $A = \begin{bmatrix} x & 3 \\ 2 & 4 \end{bmatrix}$, $B = \begin{bmatrix} 2 & 3 \\ y & 3 \end{bmatrix}$ and $C = \begin{bmatrix} z & 1 \\ 8 & 2 \end{bmatrix}$ are singular matrices then: (A) $x > y$ (B) $y > z$ (C) $z > x$ (D) $x \neq y \neq z$ Choose the correct answer from the options given below:

If A and B are independent events and $P(A) = \frac{1}{2}$ $P(B) = \frac{1}{3}$, then Match List-I with List-II | List-I | List-II | |---|---| | (A) $P(A \cap B)$ | (I) $\frac{1}{2}$ | | (B) $P(\bar{A})P(B) + P(A)P(\bar{B})$ | (II) $\frac{1}{3}$ | | (C) $P(A \mid B) + P(B \mid A)$ | (III) $\frac{1}{6}$ | | (D) $P(A \cap \bar{B})$ | (IV) $\frac{5}{6}$ | Choose the correct answer from the options given below:

If the function $f(x) = \begin{cases} ax + 2, & x \leq 1 \\ x^2 + 3x + b, & x > 1 \end{cases}$ is differentiable at $x = 1$, then the value of $(2a + b)$ is

If $\int_0^1 \frac{e^x}{1 + x} dx = m$, then the value of $\int_0^1 \frac{e^x}{(1 + x)^2} dx$ is:

Let A and B be 3×3 matrices such that $A \neq B$. If $A^3 = B^3$ and $A^2B = B^2A$, then the determinant of $A^2 + B^2$ is:

A boat 10 m high floating at a uniform speed of 13 meters per minute(m/min) away from a lamp post 15 m high. Then the rate at which the length of shadow of the boat increases is:

$\int \frac{dx}{x^3\sqrt{(1 + x^4)}} =$

Match List-I with List-II | List-I | List-II | | --- | --- | | (A) A square matrix $P$ is said to be non-singular if | (I) $\vert P\vert = 0$ | | (B) A square matrix $P$ is said to be singular if | (II) $P P^T$ is symmetric | | (C) If a matrix $P$ is both symmetric and skew-symmetric, then | (III) $\vert P\vert \neq 0$ | | (D) If $P$ is a square matrix, then | (IV) $P$ is a null matrix | Choose the correct answer from the options given below:

If $y = \log_e\left(\frac{e^2}{x^2}\right)$ for $x \neq 0$, then $\frac{d^2y}{dx^2}$ equals

The maximum value of the determinant of the matrix $\begin{bmatrix} 1 & 1 & 1 \\ 1 & 1+\sin x & 1 \\ 1+\cos x & 1 & 1 \end{bmatrix}$ is: (where $x$ is real)

Consider the function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = \frac{x}{x^2+1}$ then

If $A^{-1}$ exists for the matrix $A = \begin{bmatrix} 1 & \lambda & -1 \\ -1 & 1 & 0 \\ \lambda & 1 & 1 \end{bmatrix}$ then

The particular solution of the differential equation $\left[x \sin^2\left(\frac{y}{x}\right) - y\right]dx + xdy = 0$, $y = \frac{\pi}{4}$ when $x = 1$ is

Match List-I with List-II | List-I | List-II | |---|---| | **Differential Equation** | **Integrating Factor** | | (A) $\frac{dy}{dx} + 2xy = 1$ | (I) $x$ | | (B) $x\frac{dy}{dx} + 2xy = 1$ | (II) $e^{2x}$ | | (C) $x\frac{dy}{dx} + y = 1$ | (III) $x^2$ | | (D) $x\frac{dy}{dx} + 2y = 2$ | (IV) $e^{x^2}$ | Choose the correct answer from the options given below:

let $\vec{a}$ be a non-zero vector of magnitude '$a$' and $\lambda$ is a non-zero scalar, then $\lambda\vec{a}$ is a unit vector if

If the function $f(x) = 2x^2 - kx + 7$, is increasing on $[1,2]$, then $k$ lies in the interval

The value of $\cot\left(\cos^{-1}\frac{7}{25}\right)$ is

The area of the smaller region bounded by the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$ and the straight line $3x + 4y = 12$ is:

If $\vec{a} = 2\hat{i} + m\hat{j} - n\hat{k}$ and $\vec{b} = l\hat{i} - 3\hat{j} + 4\hat{k}$ such that $\vec{a} = 2\vec{b}$ then the value of $14{l} + m + n$ is:

If $P(A) = \frac{3}{10}$, $P(B) = \frac{2}{5}$ and $P(A \cup B) = \frac{3}{5}$ then the value of $P(B|A) + P(A|B)$ is:

Match List-I with List-II Where $\mathbb{R}$ is set of real numbers | List-I | List-II | |---|---| | (A) $\sin x$ is continuous on: | (I) $\mathbb{R} - \{0\}$ | | (B) $ \tan x$ is continuous on: | (II) $\mathbb{R}$ | | (C) $\cot x$ is continuous on: | (III) $\mathbb{R} - \{n\pi: n \in \mathbb{Z}\}$ | | (D) $x^{-n}, n \in \mathbb{N}$ is continuous on: | (IV) $\mathbb{R} - \left\{(2n + 1)\frac{\pi}{2}: n \in \mathbb{Z}\right\}$ | Choose the correct answer from the options given below:

Bag I contains 3 black and 2 white balls. Bag II contains 2 black and 4 white balls. A bag is selected at random and then a ball is drawn from it. The probability that the ball drawn is black is:

If A and B are symmetric matrices of order 3 x 3 then the matrix $2AB - BA$ is:

Acute angle between the lines $\frac{x}{3} = \frac{y}{4} = \frac{z}{5}$ and $\frac{x-1}{4} = \frac{y+1}{-3} = \frac{z+10}{5}$ is:

Consider the LPP: Maximize $z = 5x + 3y$ subject to $3x + 5y \leq 15$, $5x + 2y \leq 10$, $x,y \geq 0$. The optimal feasible solution occurs at

If $\vec{a}$, $\vec{b}$, $\vec{c}$ are unit vectors such that $\vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c} = 0$, and the angle between $\vec{b}$ and $\vec{c}$ is $\frac{\pi}{6}$, then

The co-ordinates of the point where the line $\frac{x+3}{3} = \frac{y-1}{-1} = \frac{z-5}{-5}$ cuts $yz$-plane are:

If the minimum value of the objective function $Z = ax + by$ of an LPP occurs at two points $(3, 5)$ and $(5, 3)$, then

If $P\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} = \begin{bmatrix} -7 & -8 & -9 \\ 2 & 4 & 6 \end{bmatrix}$, then matrix P is equal to:

Match List-I with List-II | List-I | List-II | |---|---| | **Equation of line** | **Information** | | (A) $\vec{r} = (3\hat{i} - 2\hat{j} + \hat{k}) + \lambda(\hat{j} - 2\hat{k})$ | (I) Direction ratios are 2, 4, -1 | | (B) $\frac{2-x}{1} = \frac{2y+1}{4}$, $z = 2$ | (II) Perpendicular to $2\hat{i} - \hat{j} + \hat{k}$ | | (C) $\frac{x}{1} = \frac{y-3}{2} = \frac{3-4z}{2}$ | (III) Passing through the point $(3, -2, 1)$ | | (D) $\vec{r} = (3\hat{i} + 2\hat{j} + \hat{k}) + \lambda(2\hat{i} + \hat{j} - 3\hat{k})$ | (IV) Direction ratios are -1, 2, 0 | Choose the correct answer from the options given below:

Area of the bounded region between the curve $y = |x - 2|$ and the line $y = 2$ is:

A relation R on the set $A = \{1, 2, 3, \ldots, 13, 14\}$ defined as $R = \{(x,y): 3x - y = 0\}$ is

Two cards are drawn successively with replacement from a well-shuffled deck of 52 cards. The probability distribution of number of aces is given by:

If $a$, $b$ and $c$ are distinct prime numbers then the value of $\begin{vmatrix} a-b & b-c & c-a \\ b-c & c-a & a-b \\ c-a & a-b & b-c \end{vmatrix}$ is equal to

For any vector $\vec{a}$, the value of $|\vec{a} \times \hat{i}|^2 + |\vec{a} \times \hat{j}|^2 + |\vec{a} \times \hat{k}|^2$ is equal to:

Match List-I with List-II (where $c$ is an arbitrary constant) | List-I | List-II | | --- | --- | | (A) $\int \tan x \, dx$ | (I) $\log\vert \sec x + \tan x\vert + c$ | | (B) $\int \cot x \, dx$ | (II) $\log\vert \sec x\vert + c$ | | (C) $\int \sec x \, dx$ | (III) $\log\vert \sin x\vert + c$ | | (D) $\int \cosec x \, dx$ | (IV) $\log\vert \cosec x - \cot x\vert + c$ | Choose the correct answer from the options given below: