CUET Mathematics 2026 30th May Shift 1 PYQs

Let $y(x)$ be the solution curve of the differential equation $\frac{dy}{dx} + 2y^2 = 0, y(1) = 1$, then $y\left(\frac{3}{4}\right)$ is equal to:

A box has 5 blue and 4 red balls. One ball is drawn at random and not replaced. Its colour is also not noted, then another ball is drawn at random. The probability of second ball being red is

If E and F are two events such that P(E) = 0.8, P(F) = 0.7 and P(E ∩ F) = 0.6 then $P(\bar{E}/\bar{F})$ is:

The corner points of the bounded feasible region determined by the system of linear constraints of an LPP are (0, 0), (0, 8), (5, 0) and (4, 10). Let $z = 3x - 4y$ be the objective function, then the minimum value of $z$ is:

Let $f(x) = (x-a)^2+(x-b)^2+(x-c)^2$ then $f(x)$ has a minimum value at $x =$

The value (/s) of $x$ satisfying the matrix equation: $x\begin{bmatrix}2x & 2\\3 & x\end{bmatrix}+2\begin{bmatrix}8 & 5x\\4 & 4x\end{bmatrix}=2\begin{bmatrix}(x^2+8) & 24\\10 & 6x\end{bmatrix}$ is/are:

The function $f(x) = -x^3 + 12x^2 - 36x + 21$ is A. Increasing on $(-\infty, 2)$ B. Decreasing on $(-\infty, 2)$ C. Increasing on $(2, 6)$ D. Decreasing on $(6, \infty)$ Choose the correct answer from the options given below:

The area of the region lying in the first quadrant and bounded by $y=8x^2, x=0, y=1$ and $y=4$ is:

Let A, B be square matrices of order $3\times3$. Then $|A.adj(A).B|$ is

Match the LIST-I with LIST-II | | LIST-I<br>Differential equation | | LIST-II<br>Sum of the order and the degree of the differential equation | |---|---|---|---| | A. | $\dfrac{d}{dx}\left(\dfrac{dy}{dx}\right) = 1$ | I. | $2$ | | B. | $4 + \left(\dfrac{dy}{dx}\right)^4 = 7\left(\dfrac{d^2y}{dx^2}\right)^3$ | II. | $3$ | | C. | $x^3\left(\dfrac{d^2y}{dx^2}\right)^2 + x\left(\dfrac{dy}{dx}\right)^4 + 1 = 0$ | III. | $5$ | | D. | $\dfrac{dy}{dx} = x^4 e^{-3y}$ | IV. | $4$ | Choose the correct answer from the options given below:

If A and B are two symmetric matrices of the same order, then

Which of the following is/are correct? A. $\int e^x\left[\frac{1}{(x-2)}-\frac{1}{(x-2)^2}\right]dx=\frac{e^x}{(x-2)}+C$, where C is an arbitrary constant. B. $\int e^x\left[\frac{1}{(x-2)^3}-\frac{3}{(x-2)^4}\right]dx=\frac{e^x}{(x-2)^2}+C$, where C is an arbitrary constant. C. $\int e^x\left[\frac{x-4}{(x-2)^3}\right]dx=\frac{e^x}{(x-2)^2}+C$, where C is an arbitrary constant. D. $\int e^x\left[\frac{1}{x+2}-\frac{3}{(x+2)^2}\right]dx=\frac{e^x}{(x+2)^2}+C$, where C is an arbitrary constant. Choose the correct answer from the options given below:

Let $f(x)=\begin{cases}\frac{1-\cos4x}{x^2}, & \text{if } x<0\\2\lambda, & \text{if } x=0\\\frac{\sqrt{x}}{\sqrt{16+\sqrt{x}}-4}, & \text{if } x>0\end{cases}$ such that $f$ continuous at $x=0$. Then, the value of $\lambda$ is equal to:

$\int e^x\left(\log x+\frac{1}{x^2}\right)dx$ is equal to: (consider $\log x=\log_e x$) [C is an arbitrary constant]

Match the LIST-I with LIST-II | | LIST-I<br>(Matrix) | | LIST-II<br>(Type of matrix) | |---|---|---|---| | A. | $A = [a_{ij}]_{m \times m}$ where $a_{ij} = 0\ \forall\, i \neq j$ | I. | Scalar matrix | | B. | $A = [a_{ij}]_{m \times 1}$ | II. | Diagonal matrix | | C. | $A = [a_{ij}]_{1 \times n}$ | III. | Column matrix | | D. | $A = [a_{ij}]_{m \times m}$ where $a_{ij} = 0\ \forall\, i \neq j$ and $a_{ij} = k\ \forall\, i = j, k \neq 0$ | IV. | Row matrix | Choose the correct answer from the options given below:

In reference to a linear programming problem (LPP), which of the following statements are true? A. The optimal value of the objective function is attained at the points given by intersections of inequation with the axes only. B. The objective function of a LPP is a linear function to be optimized. C. Every LPP admits an optimal solution. D. The region represented by the inequation system, $x, y\geq0$, $x+2y\leq4, 3x+y\geq3, 4x+3y\geq6$ is bounded in the first quadrant. Choose the correct answer from the options given below:

The area bounded by the y-axis, $y=\cos x$ and $y=\sin x$, when $0\leq x\leq\frac{\pi}{4}$, is

Let $A=\begin{bmatrix}1 & \sin\theta & 1\\-\sin\theta & 1 & \sin\theta\\-1 & -\sin\theta & 1\end{bmatrix}$, where $\theta\in[0,2\pi]$ then which of the following statements are correct? A. $|A|=2+2\sin^2\theta$ B. Maximum value of $|A|$ is 1. C. Maximum value of $|A|$ is –1. D. $|A|\in[2,4]$ Choose the correct answer from the options given below:

The value of the determinant $\Delta=\begin{vmatrix}\cos\alpha\cos\beta & \cos\alpha\sin\beta & -\sin\alpha\\-\sin\beta & \cos\beta & 0\\\sin\alpha\cos\beta & \sin\alpha\sin\beta & \cos\alpha\end{vmatrix}$ is:

For a LPP, the objective function is $z=4x+3y$ and the feasible region determined by a set of linear constraints is shown in the graph as a shaded portion. Which one of the following statement is true ? <img src="https://balti.afterboards.in/MlHK4OHrR07hALc" width="400px"/>

Which of the following statements are correct? A. The unit vector in the direction of the vector $\vec{a}=\hat{i}+\hat{j}+2\hat{k}$ is $\frac{1}{\sqrt{6}}\hat{i}+\frac{1}{\sqrt{6}}\hat{j}+\frac{2}{\sqrt{6}}\hat{k}$. B. A vector in the direction of the vector $5\hat{i}-\hat{j}+2\hat{k}$ which has magnitude 8 units is $40\hat{i}-8\hat{j}+16\hat{k}$. C. The vector joining the points P(2, 3, 0) and Q(–1, –2, –4) directed from P to Q is $-3\hat{i}-5\hat{j}-4\hat{k}$. D. The position vector of the mid-point of the vector joining the points P(2, 3, 4) and Q(4, 1, –2) is $3\hat{i}+2\hat{j}-\hat{k}$. Choose the correct answer from the options given below:

If $A=\begin{bmatrix}2 & 3\\1 & 2\end{bmatrix}, I=\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}$ and $A^2=\alpha A+\beta I$, for some constant $\alpha$ and $\beta$ then the value of $\alpha$ and $\beta$ respectively:

Integrating factor of the differential equation $x\log x\frac{dy}{dx}+y=2\log x$ is equal to: (Consider $\log x=\log_e x$)

The relation R in the set of real numbers, defined as R = {(a, b): 1 + ab > 0}, is:

Which of the following statements are correct ? A. If $\vec{a}$ and $\vec{b}$ are two adjacent sides of a triangle, then the area of the triangle is $\frac{1}{2}|\vec{a}\times\vec{b}|$ B. If $\vec{a}$ and $\vec{b}$ are two adjacent sides of a triangle, then the area of the triangle is $\frac{1}{2}|\vec{a}.\vec{b}|$ C. If $\vec{a}$ and $\vec{b}$ are two adjacent sides of a parallelogram, then its area is $|\vec{a}\times\vec{b}|$ D. If $\vec{a}$ and $\vec{b}$ are two adjacent sides of a parallelogram, then its area is $|\vec{a}.\vec{b}|$ Choose the correct answer from the options given below:

The signum function $f:R\to R$, defined by $f(x)=\begin{cases}-1, & x<0\\0, & x=0\\1, & x>0\end{cases}$ is: (Where R is set of real numbers)

If A and B are two independent events such that P(A) = 0.5, P(B) = 0.3 then Match the LIST-I with LIST-II | LIST-I | LIST-II | |---|---| | A. $P(A/B)$ | I. 0.35 | | B. $P(B/A)$ | II. 0.5 | | C. $P(A\cap B)$ | III. 0.3 | | D. $P(\bar{A}\cap\bar{B})$ | IV. 0.15 | Choose the correct answer from the options given below:

A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimeters of gas per second. The rate at which the radius of the balloon increases when the radius is 15 cm, is:

If the function $f(x)=\begin{cases}\frac{x^3-1}{x-1}, & x<1\\a, & x=1\\\frac{b\sin(x-1)}{x-1}, & x>1\end{cases}$ is continuous at $x=1$, then which of the following statements are correct ? A. $a=b$ B. $a+b$ is a multiple of 4 C. $2a-b$ is a multiple of 3 D. $a+4b$ is divisible by 5 Choose the correct answer from the options given below:

$\int\frac{\tan^5\sqrt{x}.\sec^2\sqrt{x}}{\sqrt{x}}dx$ is equal to:

Given a line $L:\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$, then Match the LIST-I with LIST-II | LIST-I | LIST-II | |---|---| | A. The foot of perpendicular from point (1, 6, 3) on line L | I. $2\sqrt{13}$ | | B. Perpendicular distance from point (1, 6, 3) to line L | II. (1, 0, 7) | | C. Image of point (1, 6, 3) with respect to line L | III. (1, 3, 5) | | D. Distance between points (1, 6, 3) and (1, 0, 7) | IV. $\sqrt{13}$ | Choose the correct answer from the options given below:

The value of $\int_{-\pi/3}^{\pi/3}\frac{1}{1+e^{\tan x}}dx$ is:

In a sphere of radius $r$, a right circular cone of height $h$ having a maximum curved surface area is inscribed. The expression for the curved surface area of the cone is

Match the LIST-I with LIST-II | | LIST-I<br>Number of the elements in a matrix | | LIST-II<br>Number of matrices of possible order | |---|---|---|---| | A. | $17$ | I. | $8$ | | B. | $8$ | II. | $2$ | | C. | $12$ | III. | $6$ | | D. | $24$ | IV. | $4$ | Choose the correct answer from the options given below:

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

A bag contains 17 tickets numbered from 1 to 17. A ticket is drawn, and then another ticket is drawn without replacing the first one. Then the probability that both the tickets show at least one even number, is

The area bounded by an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and the lines $x=0, x=ae$ and the $x$-axis in the first quadrant, where $e$ is the eccentricity of the ellipse, is:

The shortest distance between lines $L_1:\frac{x+3}{-4}=\frac{6-y}{-3}=\frac{z}{2}$ and $L_2:\frac{x+2}{-4}=\frac{y}{1}=\frac{z-7}{1}$ is:

The value of $\int_{-\pi/2}^{\pi/2}(\sin|x|+\cos|x|)dx$ equals:

The general solution of the differential equation $\frac{dy}{dx}-\frac{y}{x}+\text{cosec}\left(\frac{y}{x}\right)=0$ is (Consider $\log x=\log_e x$)

if $x=\sqrt{a^{\tan^{-1}t}}$ and $y=\sqrt{a^{\cot^{-1}t}}$ then which one of the following is true?

The vectors from origin to the points A and B are $\vec{a}=2\hat{i}-3\hat{j}+2\hat{k}$ and $\vec{b}=2\hat{i}+3\hat{j}+\hat{k}$ respectively, then the area (in Sq. unit) of triangle OAB is:

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

Probabilities of solving a specific mathematical problem independently by three students are $\frac{1}{2},\frac{1}{3}$ and $\frac{1}{4}$. If all the three students try to solve the problem independently, the probability that the problem is solved, is:

The Cartesian equation of a line is $6x-2=3y+1=2z-2$. Then its vector from is:

The function $f(x)=\cot^{-1}x+x$ increases in the interval:

The value of $\begin{vmatrix}1 & \log_ba\\\log_ab & 1\end{vmatrix}$ is:

Let $L_1$ and $L_2$ lines such that $L_1:\vec{r}=\hat{i}+\hat{j}+\lambda(2\hat{i}-\hat{j}+\hat{k})$, $L_2:\vec{r}=2\hat{i}+\hat{j}-\hat{k}+\mu(4\hat{i}-2\hat{j}+2\hat{k})$ and $\theta$ be the acute angle between $L_1$ and $L_2$, then Match the LIST-I with LIST-II | | LIST-I | | LIST-II | |---|---|---|---| | A. | $\cos\theta$ | I. | $\dfrac{\sqrt{66}}{6}$ | | B. | $\sin\theta$ | II. | $3$ | | C. | the shortest distance (in units) between $L_1$ and $L_2$ | III. | $1$ | | D. | (sum of squares of direction cosines of $L_1$) $+2$ | IV. | $0$ | Choose the correct answer from the options given below:

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

A company manufactured a product through 4 units. A, B, C and D. The probability that a product is manufactured are $\frac{3}{10},\frac{1}{5},\frac{1}{10}$ and $\frac{2}{5}$ by units A, B, C and D respectively. The probability that it will be defective are $\frac{1}{4},\frac{1}{3}$ and $\frac{1}{12}$ if it is produced by A, B, C respectively and there is no defective from unit D. A product is chosen, it is found to be defective. Match the LIST-I with LIST-II | LIST-I | LIST-II | |---|---| | A. The Probability that the defective product is from A | I. 4/9 | | B. The Probability that the defective product is from B | II. 0 | | C. Probability of defective product | III. 1/2 | | D. Probability that the defective product is from D | IV. 3/20 | Choose the correct answer from the options given below: