CUET Mathematics 2025 PYQs30 May Shift 1. Free, no login required.

Which of the following are first order linear differential equations? (A) $\frac{dx}{dy} + P_1(y)x = Q_1(y)$ : $P_1(y)$ and $Q_1(y)$ are functions of y or constant functions (B) $\frac{dy}{dx} + P_2(x)y = Q_2(x)$ : $P_2(x)$ and $Q_2(x)$ are functions of x or constant functions (C) $(x + y)\frac{dy}{dx} = x - 2y$ (D) $(1 + x^2)\frac{dy}{dx} - 2xy = x^2 + 3$ Choose the correct answer from the options given below:

The probability distribution of a random variable X is given by | X | 0 | 1 | 2 | |---|---|---|---| | P(X) | k | 2k | 3k | If k > 0, then $P(0 < X \leq 2)$ is equal to

If A is a square matrix and I is the identity matrix of same order such that $A^2 = I$, then $3(A - I)^3 + 3(A + I)^3 - 15A$ is equal to

The point of local maxima of the function $f(x) = (x - 2)^5(x + 2)^2$ is

If $A = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$ and $B = \begin{bmatrix} 0 & 2 \\ 2 & 0 \end{bmatrix}$, then the matrix AB is equal to

$\int \frac{e^{7\log_e x} - e^{6\log_e x}}{e^{4\log_e x} - e^{3\log_e x}} dx$ is equal to: (Here, c is an arbitrary constant)

The solution of the differential equation $\log_e\left(\frac{dy}{dx}\right) = 5x + 2y$ is given by

$\int_2^5 |x - 3|dx$ equals

The solution set of the linear constraints $x - 2y \geq 0, 2x - y \leq -4, x \geq 0$ and $y \geq 0$ is

The area (in sq. units) of the region bounded by the parabola $y^2 = 8x$ and the line $x = 2$ is

Let $A = [a_{ij}]_{n \times n}$ be a matrix, then match List-I with List-II | List-I | List-II | | --- | --- | | (A) $\vert A\vert = 0$ | (I) $A$ is a symmetric matrix | | (B) $\vert A\vert \neq 0$ | (II) $A$ is a skew-symmetric matrix | | (C) $A^T = A$ | (III) $A$ is a singular matrix | | (D) $A^T = -A$ | (IV) $A$ is a non-singular matrix | Choose the correct answer from the options given below:

If $y = 5e^{2x} + 4e^{3x}$, then $\frac{d^2y}{dx^2}$ equals:

If $A = \begin{bmatrix} 0 & 0 & \sqrt{7} \\ 0 & \sqrt{7} & 0 \\ \sqrt{7} & 0 & 0 \end{bmatrix}$, then $|\text{adj } A|$ is equal to

A car is moving along the curve $y = x^3 + 12$. The point(s) on the curve at which the rate of change of its y-coordinate at a certain time is 3 times the rate of change of its x-coordinate is/are

The corner points of the bounded feasible region associated with the LPP: Maximize $Z=px+qy$, $p,q>0$ are $(0, 0)$, $(3.5, 0)$, $\left(\frac{112}{59}, \frac{135}{59}\right)$ and $(0, 3)$. If the optimum value of Z occurs at both $\left(\frac{112}{59}, \frac{135}{59}\right)$ and $(0, 3)$, then

The area (in sq. units) of the region in the first quadrant bounded by $y = 3\sqrt{1-x^2}$, $x \in [0,1]$ and the x-axis is equal to

Let $f: \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = 100x + 1$, where $\mathbb{R}$ is a set of real numbers, then

$\cos^{-1}\left(\cos\frac{7\pi}{6}\right)$ equals:

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

Let $y = \cos(\sin x^2)$, then the value of $\frac{dy}{dx}$ at $x = \frac{\sqrt{\pi}}{2}$ is equal to

Match List-I with List-II | List-I | List-II | | --- | --- | | Function f(x) | Points of Non-Differentiability | | --- | --- | | (A) $f(x) = \vert x\vert + 1$ | (I) Not differentiable at $x = 3$ only | | (B) $f(x) = \vert x - 3\vert $ | (II) Not differentiable at $x = -3$ only | | (C) $f(x) = \vert x + 3\vert $ | (III) Not differentiable at $x = 3, -3$ only | | (D) $f(x) = \vert x^2 - 9\vert $ | (IV) Not differentiable at $x = 0$ only | Choose the correct answer from the options given below:

The probability that A hits a target is $\frac{1}{5}$ and the probability that B hits it is $\frac{2}{3}$. The probability that the target will be hit if both A and B shoot at it independently is:

The relation R in the set $\{1, 2, 3\}$ given by $R = \{(1, 1), (2, 2), (3, 3), (1, 2), (1, 3), (2, 3)\}$ is:

If a line makes angles $\alpha$, $\beta$ and $\gamma$ with the positive directions of x-axis, y-axis and z-axis respectively, then $\sin^2\alpha + \sin^2\beta + \sin^2\gamma$ is equal to

A die is tossed once. If the random variable X is defined as $X = \begin{cases} 1, & \text{if the die result in an odd number} \\ -1, & \text{if the die result in an even number} \end{cases}$, then the variance of X is

For what value of $\alpha$, the function $f$ defined by $f(x) = \begin{cases} \alpha(x^2 - 2x + 1), & \text{if } x \leq 0 \\ 2x + 1, & \text{if } x > 0 \end{cases}$ is continuous at $x = 0$?

If the points P, Q, R with position vectors $5\hat{i} + \lambda\hat{j}$, $20\hat{i} - \hat{j}$ and $15\hat{i} - 6\hat{j}$ respectively are collinear, then the value of $\lambda$ is

Consider the line $\vec{r} = -2\hat{i} + 3\hat{j} + \hat{k} + \lambda(5\hat{i} - 3\hat{j} - \hat{k})$. Match List-I with List-II | List-I | List-II | |---|---| | (A) A point on the given line | (I) $\left(\frac{5}{\sqrt{35}}, \frac{-3}{\sqrt{35}}, \frac{-1}{\sqrt{35}}\right)$ | | (B) Direction ratios of the given line | (II) (2, 3, 1) | | (C) Direction cosines of the given line | (III) (5, -3, -1) | | (D) Direction ratios of a line perpendicular to given line | (IV) (-2, 3, 1) | Choose the correct answer from the options given below:

The system of equations $x + y - z = 1, 3x + y - 2z = 3, x - y + \lambda z = 1$ has infinite number of solutions if $\lambda$ is equal to

The area of the region $\{(x, y): x^2 + y^2 \leq 1 \leq x + y\}$ is

The rate of change of area of a circle with respect to its circumference when radius in 6 cm, is

Let $A = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$, then $(A^{-1})^T$ equals

Let $A = \begin{bmatrix} 2 & -1 & 3 \\ 1 & 2 & -1 \\ 4 & 1 & 2 \end{bmatrix}$. $M_{ij}$ and $A_{ij}$ respectively denote the minor and cofactor of an element $a_{ij}$ of matrix $A = [a_{ij}]$ (A) $M_{23} = 6$ (B) $A_{22} = -8$ (C) $A_{13} = 7$ (D) $M_{32} = -5$ Choose the correct answer from the options given below:

If $A$ and $B$ are invertible matrices of order $3$ then match List-I with List-II | List-I | List-II | | --- | --- | | (A) $\text{adj}(A)$ | (I) $B^{-1} A^{-1}$ | | (B) $(AB)^{-1}$ | (II) $\vert A\vert ^{-1}$ | | (C) $\vert A^{-1}\vert $ | (III) $\vert A\vert ^2$ | | (D) $\vert \text{adj} A\vert $ | (IV) $\vert A\vert A^{-1}$ | Choose the correct answer from the options given below:

For the differential equation $ydx - (x + 3y^2)dy = 0$, which of the following statements are true? (A) It is a linear differential equation (B) It is a homogenous differential equation (C) Its general solution is $x = 3y^2 + Cy$ : $C$ is an arbitrary constant (D) If $y(0) = 1$, then its particular solution is $x = 3y^2 - 1$ Choose the correct answer from the options given below:

Match List-I with List-II | List-I | List-II | | --- | --- | | (A) The maximum value of $f(x) = \sin(3x) + 6$ | (I) 2 | | (B) The maximum value of $f(x) = -\vert x + 2\vert + 4$ | (II) 5 | | (C) The minimum value of $f(x) = (3x + 1)^2 + 5$ | (III) 7 | | (D) The minimum value of $f(x) = 2 \cos x + 4$ | (IV) 4 | Choose the correct answer from the options given below:

'A' speaks the truth in 80% of the cases while 'B' in 90% of the cases. The probability that they contradict each other in stating the same statement is

The shortest distance between the lines $\vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(2\hat{i} + 3\hat{j} + 4\hat{k})$ and $\vec{r} = (2\hat{i} + 4\hat{j} + 5\hat{k}) + \mu(4\hat{i} + 6\hat{j} + 8\hat{k})$ is equal to

Match List-I with List-II | List-I | List-II | |---|---| | (A) $\int_0^1 \frac{x^2}{1 + x^3} dx$ | (I) 0 | | (B) $\int_0^\pi 3\sin x dx$ | (II) $2\log_e\left(\frac{3}{2}\right)$ | | (C) $\int_{-1}^1 \sin^5 x \cos^6 x dx$ | (III) 6 | | (D) $\int_2^3 \frac{4}{x^2 - 1} dx$ | (IV) $\frac{1}{3}\log_e 2$ | Choose the correct answer from the options given below:

A person can sell a maximum of 20 units of shirts and pants on which a profit of ₹40 is made on each shirt and a profit of ₹30 on each pant. A minimum of 2 shirts are being sold, while pants are sold at least 4 times as many as shirts. Then the maximum profit is:

If A and B are skew-symmetric matrices, then which of the following is not true?

The largest open interval, in which the function $f(x) = \frac{x}{x^2 + 1}$ increases, is

Which one of the following set of constraints represents the shaded region given below? <img src="https://balti.afterboards.in/D5IFnjn6aDlWkeZ" width="300px"/>

$\int \left(\frac{\cos 2x - \cos 2\alpha}{\cos x - \cos \alpha}\right) dx =$ (Given that $c$ is an arbitrary constant)

The product of order and degree of the differential equation $\left(\frac{d^3y}{dx^3}\right)^2 + x^2y\left(\frac{d^2y}{dx^2}\right)^3 = 2x^5$ is:

Match List-I with List-II Let $A$ and $B$ be two events such that $P(A) = 0.2$, $P(B) = 0.4$, $P(B|A) = 0.5$ | List-I | List-II | | --- | --- | | (A) $P(A \cap B)$ | (I) $0.5$ | | (B) $P(A\vert B)$ | (II) $0.8$ | | (C) $P(A \cup B)$ | (III) $0.25$ | | (D) $P(A')$ | (IV) $0.1$ | Choose the correct answer from the options given below:

Let $A = [a_{ij}]_{2 \times 4}$ and $B = [b_{ij}]_{4 \times 2}$, then $|3AB|$ is equal to

If $\hat{i}$, $\hat{j}$ and $\hat{k}$ are unit vectors along the co-ordinate axes OX, OY and OZ respectively, then (A) $\hat{i} \times \hat{j} = \hat{k}$ (B) $\hat{k} \times \hat{i} = -\hat{j}$ (C) $\hat{j} \cdot \hat{j} = 1$ (D) $\hat{j} \cdot \hat{k} = 0$ Choose the correct answer from the options given below:

Let $\vec{a} = 2\hat{i} - \hat{j}, \vec{b} =- 4\hat{j} + k\,\text{and}\,\vec{c} = \hat{i} + 2\hat{k}$. If $\vec{d}$ is a vector perpendicular to both $\vec{a}$ and $\vec{b}$ such that $\vec{c} \cdot \vec{d} = 34$, then $|\vec{d}|$ is equal to

$\int_0^{\pi/2} \frac{\sin^8 x}{\sin^8 x + \cos^8 x} dx$ is equal to

If $A = \begin{bmatrix} a & 1 & -1 \\ 0 & b & 4 \\ 4 & 4 & c \end{bmatrix}$ and $abc = 12$, $b = 4a$, then the value of $|A(adjA)|$ is:

The demand function (in Rs.) for a product is given by $P = 20 - 0.25x$, where P is the price per unit and x is the number of units sold, then the price of one unit, when the revenue is maximized, is:

If $A = \begin{bmatrix} 0 & 1 & 3 \\ 1 & 2 & x \\ 2 & 3 & 1 \end{bmatrix}$ and $A^{-1} = \begin{bmatrix} \frac{1}{2} & -4 & \frac{5}{2} \\ -\frac{1}{2} & 3 & -\frac{3}{2} \\ \frac{1}{2} & y & \frac{1}{2} \end{bmatrix}$, then the value of $8x + 5y$ is:

The differential equation representing the family of curves $y = Ax + \frac{B}{x}$, $x \neq 0$ where A and B are arbitrary constants, is given by

If $xy + \frac{x^2}{y} = x^3y + y$, then $\frac{dy}{dx}$ is equal to

If the corner points of the bounded feasible region of an LPP are (0,2), (3,0), (6,0), (6,8) and (0,5), then the minimum value of objective function F = 4x + 6y occurs at

Match List-I with List-II | List-I | List-II | |---|---| | (A) The chance of observing a specific outcome in an experiment | (I) Parameter | | (B) A method used to estimate the population parameters | (II) Estimation | | (C) A value that describes an entire population | (III) Statistic | | (D) A measurable characteristic of a sample | (IV) Probability distribution | Choose the correct answer from the options given below:

It is currently 9:40 AM. What will be the time in next 610 hours?

Ravi's speed of rowing in still water is 6 km/hr. He rows between two points in a river and return to the same starting point. He took 30 minutes more to cover the distance upstream than downstream. If the speed of the stream is 3 km/hr, then the distance between the two points is:

In a survey for a sample of 300 individuals, 180 persons gave responses 'Yes' and 100 gave responses 'No' and 20 gave "No response". Then point estimate of proposition in the population who responded "Yes" is

A random variable 'X' denotes the number of sixes obtained in three throws of a die. Then, the mean of the distribution is:-

The corner points of the bounded feasible region for an LPP are (0, 20), (3,12), (6,8), and (0,15). The objective function is $Z = \alpha x + \beta y$, where $\alpha, \beta > 0$. If the maximum of Z occurs at the corner points (3,12) and (6,8), then the relationship between $\alpha$ and $\beta$ is:

Consider the following data: | Year (x) | 2010 | 2011 | 2012 | 2013 | 2014 | |---|---|---|---|---|---| | Sale (in crore Rs.) (y) | 9 | 18 | 21 | 29 | 38 | A straight line trend by the method of least square is:

A fair coin is tossed 100 times. The probability of getting head an odd number of times is

Consider $f(x) = \sin(3x) + 4, \forall x \in \mathbb{R}$. Then (A) Maximum value of $f(x)$ is 5 (B) Minimum value of $f(x)$ is 3 (C) Maximum value of $f(x)$ is attained at $x = \frac{\pi}{6}$ (D) Minimum value of $f(x)$ is attained at $x = 0$ Choose the correct answer from the options given below:

A bag contains 6 red balls, 4 green balls and 10 blue balls. Three balls are drawn with replacement. The probability of getting at least 1 green ball is:

Let $A$ be a non singular matrix of order $n \times n$, Then $|\text{adj }(3A)|$ is equal to:

If $A = \begin{bmatrix} 2 & -2 & 1 \\ 0 & 4 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & -2 & 7 \\ 2 & 0 & 6 \end{bmatrix}$ are two matrices such that $3A - 2B + 4C = 0$, then matrix $C$ is equal to:

Assume $P$, $Q$, $R$ and $W$ are matrices of order $3 \times 3$, $a \times 4$, $b \times c$ and $d \times a$ respectively. If $PQ + WR$ is well defined, then the value of $ab + cd$ is:

For the given values 8, 10, 12, 14, 16; the three-year moving averages are:

A piece of machinery is bought for Rs. 50,000. In the first year, it depreciates by 15%, and in each subsequent year, the depreciation rate increases by 5% from the previous year. The value of machinery after 3 years will be:

Increase in the number of patients in the hospitals due to heat stroke is:

A man takes a personal loan worth Rs.3,00,000 at an interest rate of 6% per annum compounded monthly to be repaid by equal monthly installments in 3 years, then the EMI using flat rate method will be:-

In 900 meters race, Ram gives Shyam a start of 150 meters and defeats him by 50 seconds. If the speed of Ram is 4.5 m/sec, then speed of Shyam is

In reference to Inferential Statistics, for 20 degrees of freedom, the least statistical t-value among the given below is:

The effective rate equivalent to a nominal rate of 12% compounded quarterly is: (Given $(1.03)^4=1.1256$)

If $\mathbb{Z}$ and $\mathbb{R}$ denote set of integers and set of real numbers respectively, then match List I with List II. | List-I | List-II | |---|---| | (A) $5x - 3 \leq 3x + 1$, $x \in \mathbb{Z}$ | (I) $x \in (-\infty, -3]$ | | (B) $3x + 17 \leq 2(1 - x)$, $x \in \mathbb{R}$ | (II) $x \in (-\infty, -1)$ | | (C) $13x + 17 \leq 2(1 - x)$, $x \in \mathbb{R}$ | (III) $\{......, -4, -3, .......,0,1\}$ | | (D) $\frac{2x + 3}{5} - 2 > \frac{3(x - 2)}{5}$, $x \in \mathbb{Z}$ | (IV) $\{......, -4, -3, -2\}$ | Choose the correct answer from the options given below:

If an investment value get doubled in 10 years then its compound annual growth rate (CAGR) is: (Given $2^{1/10} = 1.0729$)

A random variable y has the following probability distribution | y | 1 | 2 | 3 | 4 | 5 | |---|---|---|---|---|---| | P(y) | 2k | 3k | k | 4k | 5k | Match List-I with List-II | List-I | List-II | |---|---| | (A) $P(y > 2)$ | (I) $2/5$ | | (B) k | (II) $2/3$ | | (C) $P(y \leq 3)$ | (III) $8/15$ | | (D) $P(2 \leq y \leq 4)$ | (IV) $1/15$ | Choose the correct answer from the options given below:

Two pipes A and B can fill a tank in 5 hours and 6 hours respectively. Pipe C can empty it in 12 hours. If all three pipes are opened together, then the time taken to fill the tank is:

500 g of jaggery syrup has 30% jaggery in it. The quantity of jaggery that should be added to make it 50% of syrup is:

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

Ramesh plans to save some amount required after 10 years for higher studies of his son. He expects the cost of these studies to be Rs.1,00,000. How much should he save at the beginning of each year to accumulate this amount at the end of 10 years, if the interest rate is 12% compounded annually. (Given $(1.12)^{11}=3.477$)

Shown below is the graph of parabola $y^2=x$, the area (in sq. units) of the shaded region is: <img src="https://balti.afterboards.in/OwkXae9o8YL4aIn" width="400px"/>

The scrap value of a machine costing ₹ 75,000 after 4 years of use is ₹ 25000. Using straight line method the annual depreciation of the machine is: