CUET Mathematics 2025 3 June Shift 2 PYQs

Which of the following functions has a local minima at $x = 0$? (A) $f(x) = x^3$ (B) $f(x) = |x|$ (C) $f(x) = x^2$ (D) $f(x) = x^{-2}$ Choose the correct answer from the options given below:

If A be a square matrix of order 3 such that $|A| = 2$, then $|adj(2A)|$ is equal to

Which of the following terms are associated with a linear programming problem? (A) Constraints (B) Independent events (C) Feasible region (D) Objective function Choose the correct answer from the options given below:

If A is an invertible matrix, then which of the following statement(s) is/are TRUE? (A) $|A^{-1}| = |A|$ (B) $(A^{-1})^{-1} = A$ (C) $A^{-1} = \frac{adj A}{|A|}$ (D) $(A^T)^{-1} = (A^{-1})^T$ Choose the correct answer from the options given below:

If A and B are symmetric matrices, then AB - BA is

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

The general solution of the differential equation $\frac{dy}{dx} = -4xy^2$ is given by

If $x = at^2, y = 2at$; then $\frac{d^2y}{dx^2}$ is equal to

Assume A, B and C are matrices of order $m \times n$, $n \times 3$ and $3 \times q$ respectively. The restrictions on $_{m,n}$ and $_q$ so that $AB + BC$ is defined are

The area (in sq. units) of the region bounded by $y = -1, y = 2, x = y^3$ and $x = 0$ is equal to

Function $f(x) = x^x, x > 0$ decreases on the interval

Let X denotes the number of heads in a simultaneous toss of three coins, then $P(0 < X < 3)$ is

If $z = 3x + 4y$ be the objective function of a of a linear programming problem (LPP) and (3, 1), (2, 4), (0, 4), (5, 0) be corner points of the bounded feasible region. Then the maximum value of objective function is

Match List-I with List-II | List-I | List-II | |------------|-------------| | (A) Degree of this differential equation $\frac{d^4y}{dx^4} + 2\log_e\left(\frac{d^3y}{dx^3}\right) = 0$ | (I) 1 | | (B) Order of this differential equation $e^{\left(\frac{dy}{dx}\right)^3} + 3y\left(\frac{d^2y}{dx^2}\right)^3 = 0$ | (II) 4 | | (C) Degree of $\frac{d^4y}{dx^4} + \left(\frac{dy}{dx}\right)^2 = 0$ | (III) not defined | | (D) Order of the differential equation $2\frac{d^4y}{dx^4} + \left(\frac{d^2y}{dx^2}\right)^5 = 0$ | (IV) 2 | Choose the correct answer from the options given below:

$\int_0^8 (x^{\frac{2}{3}} + 1) dx$ is equal to

If the area of a triangle whose vertices are $(-2, 4)$, $(2, -6)$ and $(k, 4)$, $(k > 0)$ is 35 squnits, then the value of k is

Let $f(x)=\begin{cases} |x|+3 & \text{if } x\le -3 \\ -2x & \text{if } -3<x<3 \\ 6x+2 & \text{if } x\ge 3 \end{cases}$ Then, which of the following is true?

The region represented by the system of inequalities $x, y \geq 0, y \leq 6, x + y \leq 3$

$\int e^{2x}(\sin x + \frac{1}{2}\cos x) dx$ is equal to

Three students A, B and C can respectively solve 50%, 25% and 20% of the problems in a book. A particular problem is selected at random from the book. The probability that at least one of them will solve the problem is

The direction ratios of the line perpendicular to the lines $\dfrac{x-5}{2} = \dfrac{y+11}{-3} = \dfrac{z+3}{1}$and $\dfrac{x-7}{1} = \dfrac{y+2}{2} = \dfrac{z-4}{-2}$ are proportional to:

If $\vec{a}$ is any vector, then $|\vec{a} \times \hat{i}|^2 + |\vec{a} \times \hat{j}|^2 + |\vec{a} \times \hat{k}|^2$ is equal to

If $\begin{vmatrix} 2x & 5 \\ 8 & x \end{vmatrix} = \begin{vmatrix} 3 & 0 \\ 4 & -8 \end{vmatrix}$, then value(s) of x is/are

The vector equation of line passing through $(-1, 3, -2)$ and perpendicular to the lines $\frac{x+4}{1} = \frac{y}{2} = \frac{z-3}{3}$ and $\frac{x+2}{-3} = \frac{y+5}{2} = \frac{z-6}{5}$ is

For the differential equation $x\frac{dy}{dx} + 2y = x^2\log_e x$ (A) Integrating factor is $2x$ (B) Integrating factor is $x^2$ (C) General Solution is $y = \frac{x^2}{16}(4\log_e|x| - 1) + Cx^{-2}$ Where C is an arbitrary constant. (D) General Solution is $y = \frac{x^4}{16}(4\log_e|x| - 1) + C$ Where C is an arbitrary constant. Choose the correct answer from the options given below:

The area (in sq. units) of the region enclosed by the curve $9x^2 + 4y^2 = 36$ is

If $\vec{a}$ and $\vec{b}$ are two vectors such that $|\vec{a}| = 10, |\vec{b}| = 2$ and $\vec{a} \cdot \vec{b} = 12$, then $|\vec{a} \times \vec{b}|$ is equal to

The angle at which the line, $\frac{x-1}{0} = \frac{2-y}{-1} = \frac{2z-3}{-2}$ is inclined with the positive direction of z-axis is

The probabilities of occurrence of two events E and F are 0.25 and 0.50 respectively. The probability of their simultaneous occurrence is 0.14. The probability that neither E nor F occurs is

The relation R on the set of real numbers defined by $R = \{(a, b): a \leq b^2\}$ is

Let $f(x) = x^3 - 6x^2 + 9x - 8$ be a function, then which of the following statements are TRUE? (A) $f'(x) = 3(x - 1)(x - 3)$ (B) The critical points of the function are $x = 1$ and $x = 3$ (C) $x = 1$ is the point of local minimum (D) The local maximum value is $-4$ Choose the correct answer from the options given below:

If $x^m y^n = (x + y)^{m+n}$, then $\frac{d^2y}{dx^2}$ is equal to:

The number of arbitrary constants in the particular solution of a differential equation of order 4 and degree 3 is

If A and B are two square matrices of same order such that $AB = A$ and $BA = B$, then the value of $A^{2024} + B^{2024}$ is equal to

The area (in sq.units) of region bounded by $y^2 = 9x$, $x = 2$, $x = 4$ and the $x$-axis in the first quadrant is

Let $\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}, \vec{b} = -\hat{i} + 2\hat{j} + \hat{k}, \vec{c} = 3\hat{i} + \hat{j}$ be three vectors. If $(\vec{a} + \lambda\vec{b})$ is perpendicular to $\vec{c}$, then the value of $\lambda$ is

The sides of an equilateral triangle are increasing at the rate of 5 cm/sec. The rate at which the area increases when the side is 20 cm, is

The function $f: [-1, 1] \rightarrow R$ is given by $f(x) = \frac{x}{x + 2}$

The corner points of the feasible region of a LPP with the constraints $x + 2y \leq 40$, $3x + y \geq 30$, $4x + 3y \geq 60$, $x, y \geq 0$ are

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

Let $f(x) = \log_e(\sin x), x \in (0, \pi)$, then which of the following statements is/are TRUE? (A) $f(x)$ is increasing on $(0, \pi/2)$ (B) $f(x)$ is decreasing on $(\pi/2, \pi)$ (C) $f(x)$ is increasing on $(0, \pi)$ (D) $f(x)$ is decreasing on $(0, \pi)$ Choose the correct answer from the options given below:

If $\int \sqrt\frac{1-x}{{1+x}} dx = a\sqrt{1-x^2} + \beta \sin^{-1}x + C$, Where C is an arbitrary constant, then which of the following are TRUE? (A) $\alpha = 1$ (B) $\alpha = -1$ (C) $\beta = 1$ (D) $\beta = -1$ Choose the correct answer from the options given below:

Which of the following statement is/are correct? (A) A square matrix $A = [a_{ij}]$ is called a symmetric matrix if $a_{ij} = a_{ji}$ for all $i, j$ (B) $A = [a_{ij}]_{m \times m}$ is a diagonal matrix if $a_{ij} = 0$ when $i = j$ (C) A square matrix $A = [a_{ij}]$ is called a skew symmetric matrix, if $a_{ij} = -a_{ji}$ for all $i, j$ (D) The multiplication of diagonal matrices of same order is commutative Choose the correct answer from the options given below:

A bag contains 4 red and 6 black balls. Two balls are drawn in succession without replacement. The probability that the first is red and the second is black is

If $\sin y = x \cos(a + y)$, then $\frac{dy}{dx}$ is equal to

The probability distribution of a random variable X is | X | 0 | 1 | 2 | 3 | 4 | |---|---|---|---|---|---| | P(X) | 0.2 | k | k | 2k | k | Match List-I with List-II | List-I | List-II | |---|---| | (A) value of k | (I) $\frac{16}{25}$ | | (B) $P(x \geq 2)$ | (II) $\frac{9}{25}$ | | (C) $P(x = 3)$ | (III) $\frac{4}{25}$ | | (D) $P(x < 2)$ | (IV) $\frac{8}{25}$ | Choose the correct answer from the options given below:

Let $\theta$ be the angle between two vectors $\vec{a}$ and $\vec{b}$. Then match List-I with List-II | List-I | List-II | | --- | --- | | (A) $\sin \theta$ | (I) $\dfrac{\vec{a} \cdot \vec{b}}{\vert \vec{a}\vert \vert \vec{b}\vert }$ | | (B) $\cos \theta$ | (II) $\vert \vec{a} \times \vec{b}\vert $ | | (C) Area of the parallelogram with adjacent sides represented by $\vec{a}$ and $\vec{b}$ | (III) $\dfrac{\vec{a} \cdot \vec{b}}{\vert \vec{a}\vert }$ | | (D) Projection of $\vec{a}$ on $\vec{b}$ | (IV) $\dfrac{\vert \vec{a} \times \vec{b}\vert }{\vert \vec{a}\vert \vert \vec{b}\vert }$ | Choose the correct answer from the options given below:

If $A = \begin{bmatrix} 1 & 5 \\ 7 & 12 \end{bmatrix}, B = \begin{bmatrix} 9 & 1 \\ 7 & 8 \end{bmatrix}$ and C are three matrices such that $3A + 5B + 2C = 0$, then the matrix C is equal to

Match List-I with List-II | List-I | List-II | | --- | --- | | (A) $\int \dfrac{dx}{x^2 - 16}$ | (I) $\dfrac{1}{8} \log \left\vert \dfrac{4 + x}{4 - x} \right\vert + c$, Where C is an arbitrary constant, | | (B) $\int \dfrac{dx}{x^2 + 16}$ | (II) $\log \left\vert x + \sqrt{x^2 - 16} \right\vert + c$, Where C is an arbitrary constant, | | (C) $\int \dfrac{dx}{16 - x^2}$ | (III) $\dfrac{1}{8} \log \left\vert \dfrac{x - 4}{x + 4} \right\vert + c$, Where C is an arbitrary constant, | | (D) $\int \dfrac{dx}{\sqrt{x^2 - 16}}$ | (IV) $\dfrac{1}{4} \tan^{-1} \left( \dfrac{x}{4} \right) + c$, Where C is an arbitrary constant, | Choose the correct answer from the options given below:

The system of equations $x + y + z = 4$ $x + 2y + 3z = 12$ $x + 3y + \lambda z = \mu$ has a unique solution if