CUET Mathematics 2025 22 May Shift 1 PYQs

The function $f(x) = x^2e^{-2x}$ increases on

The minimum value of $Z = 2x + y$ subjected to $x + y \geq 10, 2x + 3y \leq 26, x, y \geq 0$ is

If A is a square matrix and I is an identity matrix of same order such that $A^2 = A$, then $(I + A)^3 - 8I$ is equal to

$\int \frac{x}{(x-1)(x-2)} dx$ is equal to ( where $C$ is a constant of integration)

If $A = \begin{bmatrix} a & a & a \\ o & a & a \\ o & o & a \end{bmatrix}$, then $|adj A|$ is equal to

The function $f(x) = \frac{-3}{4}x^4 - 8x^3 - \frac{45}{2}x^2 + 163$ has a local maxima at

Let A be a non-singular square matrix of order 3 and $|adj A| = 5$ then $|A|$ is equal to

Let $x = t^2, y = t^3$. Then $\frac{d^2y}{dx^2}$ is equal to

Match **List-I** with **List-II** | List-I | List-II | |---|---| | **Differential Equation** | **Degree** | | (A) $xy\frac{d^2y}{dx^2} + x\left(\frac{dy}{dx}\right)^2 - y\frac{dy}{dx} = 0$ | (I) 3 | | (B) $\frac{d^2y}{dx^2} + \log\left(\frac{dy}{dx}\right) = 0$ | (II) 1 | | (C) $\left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^3 + \frac{dy}{dx} + 1 = 0$ | (III) not defined | | (D) $2x^2\left(\frac{d^2y}{dx^2}\right)^3 - 5\left(\frac{dy}{dx}\right)^3 + y = 0$ | (IV) 2 | 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

Let X denotes the number of doublets obtained in 3 throws of a pair of dice. Match **List-I** with **List-II** | List-I | List-II | |---|---| | (A) $P(X = 0)$ | (I) $\frac{1}{216}$ | | (B) $P(X = 1)$ | (II) $\frac{15}{216}$ | | (C) $P(X = 2)$ | (III) $\frac{75}{216}$ | | (D) $P(X = 3)$ | (IV) $\frac{125}{216}$ | Choose the correct answer from the options given below:

The corner points of the bounded feasible region determined by the system of linear inequalities are $(0,0)$, $(4,0)$, $(2,4)$ and $(0,5)$. If maximum value of $z = ax + by$, where $a,b > 0$, occurs at both $(2,4)$ and $(4,0)$ then

$\int_0^1 x e^x dx$ is equal to

If a matrix has 8 elements then the possible order(s) it may have (A) $8 \times 1$ (B) $5 \times 3$ (C) $6 \times 2$ (D) $2 \times 4$ Choose the correct answer from the options given below:

The solution of the differential equation $\frac{dr}{dt} = -rt, r(0) = r_0$ is

The corner points of the feasible region determined by a system of linear constraints are $(0, 0)$, $(0, 40)$, $(20, 40)$, $(60, 20)$, $(60, 0)$. If the objective function is $z = 4x + 3y$, then which one of the following is true?

$\int \frac{dx}{2\sin^2 x + 5\cos^2 x}$ is equal to

If a line makes angle $\pi/3$ and $\pi/4$ with the positive directions of x-axis and y-axis respectively, then the acute angle made by the line with positive direction of z-axis is

Let $A = \begin{bmatrix} 152 & 105 & 3 \\ 149 & 25 & 35 \\ 2 & 1 & 0 \end{bmatrix}$. If $A_{ij}$ denotes the co-factor of an element $a_{ij}$ of the matrix A, then the value of $a_{11}A_{21} + a_{12}A_{22} + a_{13}A_{23}$ is equal to

Two persons A and B throw a die alternately till one of them gets a six and wins the game. If A begins, then the probabilities of winning of A and B respectively are

A and B are two sets such that $n(A) = 5$ and $n(B) = 7$. The number of one-one functions from A to B is

The number of equivalence relation on the set $\{1, 2, 3\}$ containing $(1, 2)$ and $(2, 1)$ is

Match **List-I** with **List-II** | List-I | List-II | |---|---| | (A) Equations of line through $(5, -4, 6)$ with direction ratios $3, 7, 2$ | (I) $\frac{x+3}{5} = \frac{y+7}{-4} = \frac{z+2}{6}$ | | (B) Equations of line through $(3, 7, 2)$ with direction ratios $5, -4, 6$ | (II) $\frac{x-3}{5} = \frac{y-7}{-4} = \frac{z-2}{6}$ | | (C) Equations of line through $(-5, 4, -6)$ with direction ratios $3, 7, 2$ | (III) $\frac{x-5}{3} = \frac{y+4}{7} = \frac{z-6}{2}$ | | (D) Equations of line through $(-3, -7, -2)$ with direction ratios $5, -4, 6$ | (IV) $\frac{x+5}{3} = \frac{y-4}{7} = \frac{z+6}{2}$ | Choose the correct answer from the options given below:

Let $\vec{a} = \hat{i} + \hat{j} + \hat{k}$ and $\vec{b} = \hat{i} + \hat{2j} + 3\hat{k}$ then a unit vector perpendicular to both vectors $(\vec{a} + \vec{b})$ and $(\vec{a} - \vec{b})$ is equal to

The area (in sq. units) bounded by the curve $y = \cos x$ and x-axis between $x = 0$ and $x = \frac{3\pi}{2}$ is

The rate of change of volume of a sphere with respect to its surface area, when radius is 4 cm, is equal to

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

Consider the LPP: Max $Z = 5x + 3y$ subject to $3x + 5y \leq 15, 5x + 2y \leq 10, x \geq 0, y \geq 0$ Match **List-I** with **List-II** | List-I | List-II | |---|---| | (A) Objective function | (I) $3x + 5y \geq 15$ | | (B) One constraint | (II) $x, y \geq 0$ | | (C) Non-negative restrictions | (III) $Z = 5x + 3y$ | | (D) Point $(1, 2)$ does not lie in the region | (IV) $3x + 5y \leq 15$ | Choose the correct answer from the options given below:

If A is a skew-symmetric matrix, then which of the following statements is **NOT** true? (A) A is singular if order of A is odd (B) A is non-singular (C) $A^{2025}$ is a skew-symmetric matrix (D) $A^{2025}$ is a symmetric matrix (E) all diagonal elements of A are zeros Choose the correct answer from the options given below:

Which of the following functions $f(x)$ are differentiable at $x = 0$? (A) $|x|$ (B) $|x - 1|$ (C) $[x]$, where $[t]$ denotes the greatest integer $\leq t$ (D) $|x + 1|$ (E) $x^2$ Choose the correct answer from the options given below:

Which of the following statements is/are true? (A) The vector sum of the three sides of a triangle in order is $\vec{0}$ (B) The magnitude $(r)$, direction ratios $(a, b, c)$ and direction cosines $(l, m, n)$ of any vector $\vec{r} = a\hat{i} + b\hat{j} + c\hat{k}$ are related as $l = \frac{a}{r}, m = \frac{b}{r}, n = \frac{c}{r}$ (C) If θ is the angle between two vectors $\vec{a}$ and $\vec{b}$, then their cross product is given as $\vec{a} \times \vec{b} = |\vec{a}||\vec{b}|\sin \theta$ (D) The cross product of two vectors is commutative Choose the correct answer from the options given below:

If $y = \sin^{-1} x + \sin^{-1} \sqrt{1-x^2}, x \in (-1, 0)$, then $\frac{dy}{dx}$ is equal to

The solution of the differential equation $\frac{dy}{dx} - \frac{y}{x} = 2\log_e x$

Let A be a square matrix of order n, then which of the following are TRUE? (A) $|adj A| = |A|^{n-1}$ (B) $|A. adj A| = |A|^n$ (C) $A. (adj A) = |A|$ (D) $|KA| = K|A|$ (E) $|A^{-1}| = \frac{1}{|A|}, |A| \neq 0$ Choose the correct answer from the options given below:

If the lines $\frac{1-x}{3} = \frac{y-2}{2\lambda} = \frac{z-3}{2}$ and $\frac{x-1}{3\lambda} = \frac{y-1}{1} = \frac{6-z}{5}$ are perpendicular, then $\lambda$ is equal to

If $\begin{bmatrix} 1 & 2 & 1\end{bmatrix}$ $\begin{bmatrix}1 & 2 & 0 \\ 2 & 0 & 1 \\ 1 & 0 & 2 \end{bmatrix} \begin{bmatrix} 0 \\ 2 \\ x\end{bmatrix} = 0, $then value of x is

The value(s) of $K$, for which the system of linear equations $2x + y + z = 1, x + Ky - z = \frac{3}{2}$ and $3y - 5z = 9$ does not possess a unique solution is

If $f(a-x) = f(x)$, then $\int_0^a xf(x)dx$ is equal to

Let $[x]$ denote the greatest integer $\leq t$ and $a \mathbb{Z} = [ax: x \in \mathbb{Z}, a \in \mathbb{R}]$ (where $\mathbb{Z}$ is set of integer and $\mathbb{R}$ is set of real number). The set of points of discontinuity of the function $f(x) = [2x]$ is given by

The value of $\tan^2(\sec^{-1} 2) + \cot^2(\cosec^{-1} 3)$ is equal to

Consider two independent events A and B such that $P(A) = 0.3$, $P(B) = 0.6$. Match **List-I** with **List-II** | List-I | List-II | |---|---| | (A) $P(A$ and $B)$ | (I) 0.28 | | (B) $P(A$ and not $B)$ | (II) 0.18 | | (C) $P(A$ or $B)$ | (III) 0.12 | | (D) $P$(neither A nor B) | (IV) 0.72 | Choose the correct answer from the options given below:

Let $\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$ and $\vec{b} = -2\hat{i} + 3\hat{j} - 4\hat{k}$, then which of the following statements are correct? (A) $|\vec{a}| = \sqrt{14}$ (B) $|\vec{b}| = 29$ (C) $\vec{a} \cdot \vec{b} = 8$ (D) Angle between $\vec{a}$ and $\vec{b}$ is $\cos^{-1}\left(\frac{-8}{\sqrt{406}}\right)$ Choose the correct answer from the options given below:

The minimum value of the function $f(x) = 3\sin x - 4\cos x, x \in [-4\pi, 4\pi]$ is equal to

If E and F are independent events associated with an experiment, then which one of the following statements is correct?

Which of the following statements is/are true? (A) $(\tan^{-1} y - x)dy = (1 + y^2)dx$ is a differential equation where variables are separable. (B) $(1 + x^2)dy + 2xydy = \cot x \ dx (x \neq 0)$ is a first order linear differential equation. (C) $(4x + 6y + 5)dy - (3y + 2x + 4)dx = 0$ is not a homogeneous differential equation. (D) $(xy)dx - (x + y^2)dy = 0$ is a homogeneous differential equation. Choose the correct answer from the options given below:

Probability that a man speaks truth is $\frac{3}{4}$. He throws a die and reports that it is a six. The probability that it is actually a six is

The area (in sq. units) of the region bounded by the curve $y = \sqrt{16-x^2}$ and x-axis is

The function $f(x) = \log_e(\sin x), x \in (0, \pi)$ is (A) strictly increasing on $\left(0, \frac{\pi}{2}\right)$ (B) strictly decreasing on $\left(0, \frac{\pi}{2}\right)$ (C) strictly increasing on $\left(\frac{\pi}{2}, \pi\right)$ (D) strictly decreasing on $\left(\frac{\pi}{2}, \pi\right)$ (E) strictly increasing on $(0, \pi)$ Choose the correct answer from the options given below:

If $A = \begin{bmatrix} 0 & 1 & -3 \\ -1 & 0 & 5 \\ 3 & -5 & 0 \end{bmatrix}$, then the value of $|A^{2025}|$ is

$\int \left(\frac{1}{\log_e x} - \frac{1}{(\log_e x)^2}\right)dx$ is equal to