CUET Mathematics 2025 29 May Shift 2 PYQs

Match List-I with List-II: | List-I | List-II | | --- | --- | | **Differential Equations** | **Degree/Order** | | (A) Degree of the differential equation $\frac{d^3y}{dx^3} + 2 \log x.y = 0$ | (I) 3 | | (B) Order of the differential equation $\frac{d^4y}{dx^4} + \left(\frac{dy}{dx}\right)^4 + xy = 0$ | (II) 2 | | (C) Degree of the differential equation $\left(\frac{d^4y}{dx^4}\right)^2 + \left(\frac{dy}{dx}\right)^3 + x^2y = 0$ | (III) 1 | | (D) Order of the differential equation $\frac{d^3y}{dx^3} + y\left(\frac{dy}{dx}\right)^3 = 0$ | (IV) 4 |

The value of $\int_{-1}^{1}|x|dx$ is

If $A = [a_{ij}]$ is a square matrix of order 2 such that $a_{ij} = \begin{cases} 2, & \text{when } i \neq j \\ 0, & \text{when } i = j \end{cases}$, then det $(A^2)$ is:

Suppose that A, B and C are matrices of order $m \times n$, $n \times 5$ and $5 \times q$ respectively. The restriction on $m$, $n$ and $q$ so that AB-BC is defined are

Match List-I with List-II | List-I | List-II | | --- | --- | | Function f(x) | Interval for increasing/decreasing of function f(x) | | --- | --- | | (A) $f(x) = x\vert x\vert $ | (I) Decreases on $(0, \infty)$ | | (B) $f(x) = x^2 + 2x - 5$ | (II) Increases on $(3, \infty)$ | | (C) $f(x) = x^2 - 6x + 9$ | (III) Decreases on $(-\infty, -1)$ | | (D) $f(x) = -x^2$ | (IV) Increases on $(-\infty, \infty)$ | Choose the correct answer from the options given below:

If $A = \begin{bmatrix} -1 & 2 & 3x \\ 2y & 4 & -1 \\ 6 & -1 & 0 \end{bmatrix}$ is a symmetric matrix, then the value of $2x - y$ is:

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

$\int (e^{x\log a} + e^{a\log x}) dx$ is equal to (where $a > 1$)

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

If $xy = e^{(x-y)}$, then $\frac{dy}{dx}$ is equal to:

If $f(x) = 2x^3 - 15x^2 + 36x + 1$, $x \in [1, 5]$, then the absolute minimum value of $f(x)$ is:

The particular solution of the differential equation $\frac{dy}{dx} = 8yx$ when $y = 1$ at $x = 0$

The maximum value of a LPP $z = 3x + 4y$ subject to the constraints: $x + y \leq 6$, $x \geq 0$, $y \geq 0$ is:

The system of equation $2x + \lambda y = 8$, $\lambda x + 8y = 3$ has a unique solution if the value of $\lambda$ is (are):

If $z = 5x + 8y$ is the objective function of a LPP and (0, 0), (3, 1), (2, 4), (0, 3), (5, 0) are corner points of the bounded feasible region, then the maximum value of the objective function is

The direction cosines of a line which makes equal angles with co-ordinate axes are:

Match List-I with List-II | List-I | List-II | | --- | --- | | Integral | Solution: C is an arbitrary constant | | --- | --- | | (A) $\int \frac{dx}{x^2 + 25}$ | (I) $\frac{1}{10} \log \left\vert \frac{5 + x}{5 - x} \right\vert + C$ | | (B) $\int \frac{dx}{x^2 - 25}$ | (II) $\log \vert x + \sqrt{x^2 - 25}\vert + C$ | | (C) $\int \frac{dx}{25 - x^2}$ | (III) $\frac{1}{5} \tan^{-1} \left( \frac{x}{5} \right) + C$ | | (D) $\int \frac{dx}{\sqrt{x^2 - 25}}$ | (IV) $\frac{1}{10} \log \left\vert \frac{x - 5}{x + 5} \right\vert + C$ | Choose the correct answer from the options given below:

$$\frac{d^2}{dx^2} \left\{ \det \begin{bmatrix} x^3 & x \\ 2 & e^x \end{bmatrix} \right\}$$ equals

Let the equation of lines be as $L_1: \vec{r_1} = \vec{a_1} + \lambda\vec{b_1}$ and $L_2: \vec{r_2} = \vec{a_2} + \lambda\vec{b_2}$ such that $\vec{a_1} - \vec{a_2} = 2\hat{i} + 4\hat{j} + 4\hat{k}$ and $\vec{b_1} \times \vec{b_2} = 8\hat{i} - 4\hat{k}$. Then the shortest distance between $L_1$ and $L_2$ is

Match List-I with List-II Let $\theta$ be the angle between the vectors $\vec{a}$ and $\vec{b}$. | List-I | List-II | | --- | --- | | (A) $\vec{a} \cdot \vec{b}$ | (I) $\dfrac{\vec{a} \cdot \vec{b}}{\vert \vec{b}\vert ^2} \vec{b}$ | | (B) $\vec{a} \times \vec{b}$ | (II) $\vec{a} \cdot \vec{b} = 0$ | | (C) Projection vector of $\vec{a}$ on $\vec{b}$ ($\ne{0}$) | (III) $\vert \vec{a}\vert \vert \vec{b}\vert \sin \theta \, \hat{n}$ where $\hat{n}$ is a unit vector perpendicular to both $\vec{a}$ and $\vec{b}$ | | (D) $\vec{a}$ and $\vec{b}$ are orthogonal vectors | (IV) $\vert \vec{a}\vert \vert \vec{b}\vert \cos \theta$ |

Assume that R is a relation on the set Z of integers and it is given by $(x, y) \in R \Leftrightarrow |x - y| \leq 1$. Then, R is

If $x = a\sin 2t(1 + \cos 2t)$ and $y = b\cos 2t(1 - \cos 2t)$, then $(\frac{dy}{dx})_{\text{at } x=\frac{\pi}{4}}$ is equal to

If $\vec{a}$ and $\vec{b}$ are two non-zero orthogonal vectors, then $|\vec{a} + \vec{b}|$ is equal to

The probabilities of occurrance of two events A and B are 0.45 and 0.20 respectively. The probability of their simultaneous occurrence is 0.06. The probability that neither A nor B occurs is

The rate of change of volume of a sphere with respect to its surface area, when the radius is 6cm is:

The function $f(x) = x^2 - x + 1$ is

Let $f(x) = 4x^3 - 18x^2 + 27x - 5$, $x \in R$. Then which of the following statements are TRUE? (A) $f''(x) = 24x - 36$ (B) f has local maxima at $x = \frac{3}{2}$ but no minima (C) f has neither maxima nor minima (D) f has both maxima and minima Choose the correct answer from the options given below:

If the area of a triangle whose vertices are (-1, 3), (1, -5) and (k, 2) where $k > 0$ is 30 sq. units, then the value of k is

The area of the region bounded by the curves $y = x^2 + 2$, $y = x$, $x = 0$ and $x = 2$ is

For the differential equation $x\frac{dy}{dx} + 3y = x^2\log_e x$, which of the following statements are TRUE? (A) Product of order and degree is 1 (B) Integrating factor is $x^3$ (C) Integrating factor is $3x$ (D) General solution is $y = \frac{x^3}{36}(6\log_e|x| - 1) + Cx^{-3}$, C is an arbitrary constant. Choose the correct answer from the options given below:

If the system of equation $x - y + z = 4$ $x - 2y - 2z = 9$ $2x + y + \lambda z = 1$ has a unique solution, then

If $A = \begin{bmatrix} 1 & 2 \\ 4 & -3 \end{bmatrix}$ and $f(x) = 2x^2 - 4x + 5$, the $f(A)$ is equal to

A vector of magnitude 9, which is perpendicular to both the vectors $(4\hat{i} - \hat{j} + 8\hat{k})$ and $(-\hat{j} + \hat{k})$ is

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

$\int \frac{dx}{(1+5\sin^2 x)}$ is equal to

The area (in sq. units) of the bigger portion of region enclosed by the curves $4x^2 + 9y^2 = 36$ and $2x + 3y = 6$ is

The corner points of the feasible region with the constraints $x + y \leq 30$, $x + y \geq 15$, $y \leq 20$, $x \leq 15$ and $x$, $y \geq 0$ are

Let A and B be two events. Then which of the following statements are TRUE? (A) $P(B|A) = \frac{P(A \cap B)}{P(A)}$, provided $P(A) \neq 0$ (B) $P(B') = 1 + P(B)$ (C) $P(A \cup B) = P(A) + P(B) + P(A \cap B)$ (D) $P(A \cap B) = P(A).P(B)$ If A and B are independent events Choose the correct answer from the options given below:

Match List-I with List-II If the random variable x has the following distribution: | x | 0 | 1 | 2 | otherwise | |---|---|---|---|-----------| | P(x) | k | k | 2k | 0 | | List-I | List-II | |---|---| | (A) k | (I) $\frac{3}{4}$ | | (B) P(x ≥ 2) | (II) $\frac{1}{4}$ | | (C) P(x ≤ 2) | (III) $\frac{1}{2}$ | | (D) P(0 < x ≤ 2) | (IV) 1 | Choose the correct answer from the options given below:

If $f(x) = |x| + |x - 5|$, then which of the following statements are TRUE? (A) f is a continuous function every where (B) f is a continuous function except $x = 5$ and $x = 0$ (C) f is a continuous function except $x = 0$ but not differentiable at $x = 5$ (D) f is a continuous function everywhere but not differentiable at $x = 0$ and $x = 5$ Choose the correct answer from the options given below:

The value of $\cos(2\cos^{-1}x + \sin^{-1}x)$ at $x = \frac{1}{5}$ is

If $x \neq y \neq z$ then $\begin{vmatrix} 1 & x & x^2 \\ 1 & y & y^2 \\ 1 & z & z^2 \end{vmatrix}$ is equal to

If $\int_0^a 3x^2dx = 8$, then the value of $a$ is:

A bag contain 8 blue and 12 green balls. Two balls are drawn in succession without replacement. The probability that first is blue and second is green is

The function $f: [-1, 1] \rightarrow R$ (set of real numbers) given by $f(x) = \frac{x}{x+3}$ is

The number of arbitrary constants in the general solution of a differential equation of order 4 and degree 1 is

If $A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}$ then $A^2 - 5A$ is equal to (where I is identity matrix of order 2)

If $\vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c}$, $\vec{a} \times \vec{b} = \vec{a} \times \vec{c}$ and $\vec{a} \neq {0}$, then the vector $\vec{b}$ in equal to.

Which of the following statement are correct? (A) $A = [a_{ij}]_{n \times n}$ is a diagonal matrix if $a_{ij} = 0$ when $i = j$ (B) A square matrix $A = [a_{ij}]$ is called a symmetric matrix if $a_{ij} = a_{ji}$ for all $i, j$ (C) A square matrix $A = [a_{ij}]$ is called a skew-symmetric matrix if $a_{ij} = -a_{ji}$ for all $i, j$ (D) For every square matrix $A$, there exist an identity matrix of the same order such that $IA = AI = I$ Choose the correct answer from the options given below:

The feasible region of a LPP is bounded. The corresponding objective function is Z= 6x - 7y. Then objective function attains: