CUET Mathematics 2025 3 June Shift 1 PYQs

If the feasible region of an LPP is bounded and the corresponding objective function is $z = 5x - 9y$, then the objective function attains:

A random variable X has the following probability distribution: | X | 2 | 3 | 4 | 5 | |---|---|---|---|---| | P(X) | 5/k | 7/k | 9/k | 11/k | Then the value of $\frac{k}{4}$ is:

The function $f(x) = 4x^3 - 7x^2$ has point(s) of local minima at

If $A = [a_{ij}]$ is skew symmetric matrix of order 'n', then

$\int \sqrt{1 + \frac{x^2}{9}} dx$ is equal to (Where C is an arbitrary constant)

If $y = \sqrt{ax + b}$ then $y\frac{d^2y}{dx^2} + (\frac{dy}{dx})^2 =$

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

If the points (a, b), (c, d) and (a + c, b + d) are collinear, then

For the function $f(x) = x^x, x > 0$, which of the following are TRUE? (A) $f'(x) = x^x(1 + \log x)$ (B) $x = e$ is the critical point (C) $f$ is increasing in $(\frac{1}{e}, \infty)$ (D) $f$ is increasing in $(0, \infty)$ Choose the *correct* answer from the options given below:

Let A be a matrix such that $A = \begin{bmatrix} 1 & 2 \\ -2 & 3 \end{bmatrix}$. Then which of the following are TRUE? (A) A is non-singular matrix (B) $A^T = A$ (C) A is not invertible matrix (D) A is not skew-symmetric matrix Choose the *correct* answer from the options given below:

The area of the region bounded by the line $y = 2x$ and the x-axis between $x = -2$ and $x = 2$ is

The corner points of a bounded feasible region are (0, 5), (6, 1), (17, 2) and (4, 29). If the maximum value of objective function $z = px + qy$ where $p$ and $q > 0$ occurs at two points (17, 2) and (4, 29), then the relation between $p$ and $q$ is:

If the system of equations $2x + 5y = 7, 6x + \lambda y = 28$ is inconsistent, then

The particular solution of the differential equation $xdy = (2x^2 + 1)dx, x \neq 0$, given that $y = 1$ when $x = 1$ is:

If m and n are respectively the order and degree of the differential equation $(\frac{d^2y}{dx^2})^{2} + (\frac{dy}{dx})^3 + y= 4x$, then the value of $m + n$ is:

The shortest distance between the lines $\vec{r} = \hat{i} + \hat{j} + \lambda(2\hat{i} - \hat{j} + \hat{k})$ and $\vec{r} = 2\hat{i} + \hat{j} - \hat{k} + \mu(4\hat{i} - 2\hat{j} + 2\hat{k})$ is

The function $f(x) = 4 - 3x + 3x^2 - x^3$ is (Here $\mathbb{R}$ is set of real numbers)

Let $A = [a_{ij}]$ be a square matrix of order 3 with $|A| = 2$ and let $C = [c_{ij}]$ where $c_{ij} =$ cofactor of $a_{ij}$ in A. Then $|C|$ is equal to:

Let $\vec{a} = \hat{i} + 4\hat{j} + 2\hat{k}$, $\vec{b} = 3\hat{i} - 2\hat{j} + 7\hat{k}$ and $\vec{c} = 2\hat{i} + \hat{j} + 4\hat{k}$. A vector $\vec{d}$ which is perpendicular to both $\vec{a}$ and $\vec{b}$, and $\vec{c} \cdot \vec{d} = 14$, is:

The range of function $f(x) = 4x^2 + 12x + 7, x \in \mathbb{R}$ is

For a square matrix $A$ of order 3, if $|A| = 2$, then $|adj\ 2A| =$

The sum of the x-coordinates of the corner points of the feasible region for the LPP: Minimize $z = 3x + 2y$ subject to constraints $x + y \leq 14$, $x \geq 4$, $x \leq 8, y \geq 0$ is

If $y = \log_e(\sec e^{x^2})$ then $\frac{dy}{dx} =$

Let $\vec{a} = 3\hat{i} + \hat{j} - 4\hat{k}$ and $\vec{b} = 6\hat{i} + 5\hat{j} - 2\hat{k}$ be two vectors. Then a vector perpendicular to $\vec{a}$ and $\vec{b}$ with magnitude 3 units is

The Cartesian equation of the line passing through the point (1, 2, -1) and parallel to the line $5x - 25 = 14 - 7y = 35z$ is

Area of the region bounded by the curve $y = \sin x$ and x-axis between $x = \frac{\pi}{2}$ and $x = \frac{3\pi}{2}$ is

Consider the differential equation $xdy = (y + 2x^3)dx$. Then which of the following are TRUE? (A) It is a homogeneous differential equation. (B) Product of the order and degree of the differential equation in one. (C) Integrating factor is x. (D) General solution of the differential equation is $y = x^3 + Cx$, where C is an arbitary constant. Choose the *correct* answer from the options given below:

If $A = \begin{bmatrix} 1 & 2 & 3 \\ -4 & -5 & -2 \end{bmatrix}$, $B = \begin{bmatrix} 2 & -3 \\ 4 & -5 \\ 2 & -1 \end{bmatrix}$ and $BA = [b_{ij}]$, then $(b_{23} - b_{31})$ is equal to

The sum of order and degree of the differential equation $(x^2\frac{d^2y}{dx^2})^{3/4} = 5(\frac{dy}{dx})^2 - 3$ is equal to

60% members of a committee favour a certain proposal and 40% members oppose the proposal. A member is selected and let the random variable X = 0 if he opposes and X = 1 if he is in favour. Then the variance of the random variable X is

The value of k for which the system of equations $x + y + z = 1$ $x - ky + z = 1$ $x - y + z = 1$ has more than one solutions is

$\int \frac{dx}{\sqrt{5 - 4x - x^2}}$ is equal to

If A and B are independent events, then which of the following is **not** true?

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

The projection of the vector $2\hat{i} - \hat{j} + 3\hat{k}$ on the vector $3\hat{i} + 2\hat{j} + 6\hat{k}$ is

A vector $\vec{a}$ of magnitude $3\sqrt{2}$ making an angle of $\frac{\pi}{3}$ with $\hat{i}$, $\frac{\pi}{4}$ with $\hat{j}$ and an actue angle $\theta$ with $\hat{k}$, is

The objective function of an LPP is $z = ax + \beta y, (a, \beta > 0)$ in that has to be maximized/minimized subject to constraints $x + y \leq 2$, $x \geq 0$, $y \geq 0$. Then max (z) $-$ min (z) is equal to

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

Match List-I with List-II | List-I | List-II | | --- | --- | | Integral | Value | | --- | --- | | (A) $\int_{-1}^{1} (\vert x\vert + 1) dx$ | (I) 0 | | (B) $\int_{-2}^{2} \vert x + 1\vert dx$ | (II) 2 | | (C) $\int_{-1}^{1} 3\vert x^2\vert dx$ | (III) 5 | | (D) $\int_{-1}^{1} x\vert x\vert dx$ | (IV) 3 | Choose the correct answer from the options given below:

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

If $A = \begin{bmatrix} 1 & 2 \\ 4 & 5 \end{bmatrix}$, then Match **List-I** with **List-II** | List-I | List-II | |---|---| | (A) det (A) | (I) $-\frac{1}{3}$ | | (B) det $(A^{-1})$ | (II) $-12$ | | (C) det (2A) | (III) $-3$ | | (D) det $(3A^T)$ | (IV) $-27$ | Choose the **correct** answer from the options given below:

The area of the smaller region of the circle $x^2 + y^2 = 8$ cut off by the line $x = 2$ is

Let $f(x) = x^2 + \frac{250}{x}$ be any function defined on $\mathbb{R} - \{0\}$, where $\mathbb{R}$ is the set of real numbers. Then which of the following are TRUE? (A) $f'(x) = 2x + \frac{250}{x^2}$ (B) $x = 5$ in the only critical point of $f(x)$ (C) minimum value of $f(x)$ is 75 (D) maximum value of $f(x)$ is 50. Choose the **correct** answer from the options given below:

Let P and Q be any two invertible matrices of the same order. Then Match **List-I** with **List-II** | List-I | List-II | |---|---| | **Matrix** | **Equivalent matrix** | | (A) $(P Q)^{-1}$ | (I) $Q^{-1}P$ | | (B) $(P^{-1}Q)^{-1}$ | (II) $Q P^{-1}$ | | (C) $(P Q^{-1})^{-1}$ | (III) $Q^{-1}P^{-1}$ | | (D) $(P^{-1}Q^{-1})^{-1}$ | (IV) Q P | Choose the **correct** answer from the options given below:

The edge of a cube is increasing at a rate of 7cm/s. The rate of change of area of the cube when edge of the cube is 3cm is:

Match **List-I** with **List-II** | List-I | List-II | | :--- | :--- | | **Function** | **Points of discontinuity** | | (A) $f(x) = \frac{x^2 + 1}{x}$ | (I) $x = 4$ | | (B) $f(x) = \frac{\vert x - 1 \vert}{x - 1}$ | (II) $x = 2$ | | (C) $f(x) = \begin{cases} x - 1, & x < 2 \\ x + 1, & x \ge 2 \end{cases}$ | (III) $x = 0$ | | (D) $f(x) = \frac{1 - x}{(x - 4)}$ | (IV) $x = 1$ | Choose the **correct** answer from the options given below:

If $\frac{d}{dx}[ax^3 + ax^2 + ax + 1] = 9x^2 + 6x + 3$, then $a$ is equal to

The probability that a leap year selected at random will have 53 Mondays is

For the relation $R = \{(a, b): a \leq b\}$ in $\mathbb{R}$, which of the following is correct?

Two persons A and B throw a die alternately till one of them gets a 'three' and wins the game. The probability of A's winning if A starts first is