CUET Mathematics 2025 PYQs2 June Shift 1. Free, no login required.

The solution of the differential equation $ydx + (x - y^2)dy = 0$ is

If $y = \frac{1}{\sqrt[3]{1-x^3}}$ then $\frac{dy}{dx}$ is equal to

The probability distribution of a random variable $x$ is, $P(x) = \frac{k}{2^x}, x = 0, 1, 2, 3$. Then Match List-I with List-II | List-I | List-II | |---|---| | (A) $k$ | (I) $\frac{2}{15}$ | | (B) $P(x = 1)$ | (II) $\frac{1}{5}$ | | (C) $P(1 < x < 3)$ | (III) $\frac{8}{15}$ | | (D) $P(x \geq 2)$ | (IV) $\frac{4}{15}$ | Choose the correct answer from the options given below:

If $\begin{bmatrix}3 & 1\\2 & 1\end{bmatrix}A\begin{bmatrix}2 & 1\\1 & 1\end{bmatrix} = \begin{bmatrix}1 & 1\\0 & 1\end{bmatrix}$, then matrix 'A' is

If the matrix $\begin{bmatrix}3 & 2a & -5\\4 & 0 & b\\-5 & 3 & 10\end{bmatrix}$ is symmetric, then the value of $5a + 2b$ is

If $x$ is real, the minimum value of $x^2 - 8x + 20$ is

The value of $\int_0^1 \log_e\left(\frac{1}{x} - 1\right)dx$ is:

$\int \frac{e^{2x} - e^{-2x}}{e^{2x} + e^{-2x}}dx$ is equal to

Match List-I with List-II | List-I | List-II | |---|---| | (A) The degree of differential equation $\frac{d^3y}{dx^3} = e^{\frac{dx}{dy}}$ | (I) 2 | | (B) The order of differential equation $\left(\frac{dy}{dx}\right)^2 + \frac{d^3y}{dx^3} = 0$ | (II) 4 | | (C) The sum of order and degree of differential equation $\frac{d}{dx}\left(\frac{d^2y}{dx^2}\right) + \left(\frac{dy}{dx}\right)^5 = x$ | (III) not defined | | (D) The number of arbitrary constants in the general solution of a differential equation of order 2 | (IV) 3 | Choose the correct answer from the options given below:

Consider an LPP: Maximise $Z = 50x + 15y$ subjected to constraints $x + y \leq 60$, $5x + y \leq 100$, $x, y \geq 0$. If the maximum value of $Z$ occurs at $x = \alpha$ and $y = \beta$, then the value of $\alpha + \beta$ is

If $A$ is a square matrix of order 3 and $|A| = 5$, then the value of $|-AA^T|$ is

For the function $f(x) = -2x^3 + 3x^2 + 36x - 10$, which of the following is/are true? (A) $f$ is increasing in $(-\infty, -2)$ (B) $f$ is increasing in $(-2, 3)$ (C) $f$ is decreasing in $(-\infty, -2)$ (D) $f$ is decreasing in $(3, \infty)$ Choose the correct answer from the options given below:

If the matrix $\begin{bmatrix}2 & -1 & 3\\ \lambda & 0 & 7\\-1 & 1 & 4\end{bmatrix}$ is not invertible, then value of $\lambda$ is

Linear inequalities corresponding to the shaded feasible region OABCO in the given figure are <img src="https://balti.afterboards.in/XDcoYd2BxodBB7e" width="300px"/>

Area of the region bounded by $y = x^2$ and the line $y = 16$ is

The value of $\lambda$ so that the lines $\frac{1-x}{3} = \frac{7y-14}{2\lambda} = \frac{z-3}{2}$ and $\frac{7-7x}{3\lambda} = \frac{y-5}{1} = \frac{6-z}{5}$ are at right angle, is:

The domain of $y = \cos^{-1}(x^2 - 4)$ is

The solution of the differential equation $\frac{dy}{dx} = \frac{x+y}{x-y}$ is

A problem in mathematics is given to three students whose chances of solving it are 1/2, 1/3, 1/4 respectively. The probability that the problem is solved is

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

Relation R on the set $A = \{1, 2, ..., 15\}$ defined as $R = \{(x, y): y - 4x = 0\}$ is

The feasible region corresponding to the linear constraints of a Linear Programming Problem (LPP) is represented by the shaded region in the given figure. Which of the following is not a constraint to the given LPP? <img src="https://balti.afterboards.in/AaGbmDtdftDHD7T" width="300px"/>

if $A = \begin{bmatrix}1 & 0\\3 & 1\end{bmatrix}$ and $A^4 = \begin{bmatrix}1 & 0\\k & 1\end{bmatrix}$ then value of $k$ is

Consider a closed cylinder of radius $r$ with a fixed surface area. The volume of the cylinder is maximum when its height is

If $A$ is a skew-symmetric matrix of order 5, then $|adjA|$ is equal to

The foot of the perpendicular drawn from the point $(1,6,3)$ to the line $\frac{x}{1} = \frac{y-1}{2} = \frac{z-2}{3}$ is

$\int_0^1 \frac{dx}{\sqrt{1+x} - \sqrt{x}}$ is equal to

If $y = \sin^{-1}x$, then $(1-x^2)\frac{d^2y}{dx^2}$ is equal to

The value of $\int_{-\pi/2}^{\pi/2}(x^5 + x^3\cos x)dx$ is

The function $f(x) = 2\log_e(x-2) - x^2 + 4x + 1, (x > 2)$ is increasing on the interval:

If A (3, 2), B (1, -1) and C (2, 1) are three vertices of a parallelograms ABCD, then its area (in sq.units) is equal to

If the direction ratios of two lines are $a, b, c$ and $(b-c), (c-a), (a-b)$ respectively, then the angle between these lines is:

If $x = -1$ and $x = -2$ are the extreme points of $f(x) = \alpha\log|x| + \beta x^2 + x$ then

The area (in sq.units) of the region bounded by the curve $y = \cos x$ between $x = -\frac{\pi}{2}, x = \frac{\pi}{2}$ and the x-axis is

In a college, 30% students fail in physics, 25% fail in Mathematics and 10% fail in both. One student is chosen at random. The probability that she fails in physics if she has failed in mathematics is

If $x, y$ and $z$ are real number such that $x + y + z = 0$, then value of $\begin{vmatrix}3x & -x+y & -x+z\\x-y & 3y & z-y\\x-z & y-z & 3z\end{vmatrix}$ is

The value of $\lambda$, for which the two vectors $2\hat{i} - \hat{j} + 2\hat{k}$ and $3\vec{i} + \lambda\vec{j} + \hat{k}$ are perpendicular, is:

The area (in sq.units) of the region bounded by the line $2y + x = 8$, the x-axis and the lines $x = 2$ and $x = 4$ is

If $A$ is an invertible matrix of order 3, then Match List-I with List-II | List-I | List-II | | --- | --- | | (A) $\vert \text{adj} A\vert $ | (I) $8\vert A\vert $ | | (B) $\vert A(\text{adj} A)\vert $ | (II) $\vert A\vert ^2$ | | (C) $\vert 2A\vert $ | (III) $\frac{1}{\vert A\vert }$ | | (D) $\vert A^{-1}\vert $ | (IV) $\vert A\vert ^3$ | Choose the correct answer from the options given below:

If $x = a\left(\cos t + \log \tan\frac{t}{2}\right), y = a\sin t$, then value of $\frac{dy}{dx}$ at $t = \frac{\pi}{4}$ is

The area (in sq.units) of a triangle formed by vertices O, A and B where $\vec{OA} = \hat{i} + 2\hat{j} + 3\hat{k}$ and $\vec{OB} = -3\hat{i} - 2\hat{j} + \hat{k}$ is

If the function $f(x) = \begin{cases}\frac{\sin 3x}{x}, & \text{if } x \neq 0\\ \frac{3k}{2}, & \text{if } x = 0\end{cases}$ is continuous at $x = 0$, then the value of $k$ is

A function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = \frac{x}{x^2+1}$ is (where $\mathbb{R}$ is a set of real number)

Which of the following statements are true? (A) If $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$, then $x, y, z$ are called direction ratios of $\vec{r}$. (B) For any two vectors $\vec{a}$ and $\vec{b}$, $\vec{a} + \vec{b} = \vec{b} + \vec{a}$ (C) $\vec{a} \perp \vec{b}$ if and only if $\vec{a} \times \vec{b} = \vec{0}$ (D) Projection of $\vec{b}$ on $\vec{a}$ is $\frac{\vec{a} \cdot \vec{b}}{|\vec{a}|^2}$ Choose the correct answer from the options given below:

The integrating factor of the differential equation $\frac{dy}{dx} = x + xy$ is

An urn contains 5 red and 5 black balls. A ball is drawn at random, its color is noted and is returned to the urn. Moreover, 2 additional balls of the same color are put in the urn and then a ball is drawn at random. The probability that the second drawn ball is red, is:

For any events A and B of a sample space S, which of the following statements are TRUE? (A) $P(S | B) = 1$ (B) $P(A \cap B) = P(A) + P(B) + P(A \cup B)$ (C) $P(\bar{A} | B) = 1 - P(A | B)$ (D) $P(A | B) = \frac{P(A \cap B)}{P(B)}, P(B) \neq 0$ Choose the correct answer from the options given below:

The matrix $A = \begin{bmatrix}0 & 0 & 5\\0 & 5 & 0\\5 & 0 & 0\end{bmatrix}$ is a (A) Diagonal matrix (B) Scalar matrix (C) Square matrix (D) Symmetric matrix Choose the correct answer from the options given below:

Let $\vec{a}$ and $\vec{b}$ are unit vectors. If $\sqrt{3}\vec{a} - \vec{b}$ is a unit vector, then the angle between $\vec{a}$ and $\vec{b}$ is

The corner points of the bounded feasible region for an LLP are: (5, 5), (15, 15), (0, 20) and (0, 10). Let $z = 3x + 9y$ be the objective function. Then the value of $maximum(z) - minimum(z)$ is

If A is a square matrix such that $A^2=A$ and I is the identify matrix of the same order as A then $(I + 2A)^3$-6A is equal to

If $y = e^{\frac{1}{2}\log(1+ \tan^2 x)}$, then $\frac{d^2y}{dx^2}$ is equal to:

The probability distribution of a random discrete variable is given | X | -1 | 0 | 1 | 2 | 3 | |---|---|---|---|---|---| | P(X) | 0.1 | $p$ | 0.3 | $q$ | $r$ | If it is known that P(X=1) is the mean of P(X=0) and P(X=2). Then the value of r is :

The effective rate per annum equivalent to a nominal rate of 8% compounded semi-annually is

The value of $\int_{-a}^a f(x)dx$ where $f(x) = \frac{7^x}{1+7^x}$ is:

The area bounded by the x-axis and the parabola $y = 3x-x^2$ is:

A machine costing Rs 50,000 has a useful life of 4 years.The estimated scrap value is Rs 10,000 . The rate of depreciation per annum is:

A pair of dice is thrown until the sum of numbers appeared is a perfect square or a non-perfect square sum appeared five times in succession. If random variable $X$ denotes the number of non perfect square sums appeared, then $P(X > 0)$ is

If $A = \begin{bmatrix}0 & x^2-6 & -3\\-x & 0 & -8\\x^2-2x & 8 & 0\end{bmatrix}$ is a skew symmetric matrix, then the value(s) of x is/ are - (A) 3 (B) -3 (C) -2 (D) -1 Choose the correct answer from the options given below:

If $A = \begin{bmatrix}1 & 2\\ 0 & 3\end{bmatrix}$ then $|A. adj A|$ is

If CAGR stands for Compound Annual Growth Rate, F.V stands for final value of an investment, P.V stands for present value of an investment and n is the number of years then

If the sum and difference of squares of mean and variance of a Binomial distribution is $\frac{225}{256}$ and $\frac{63}{256}$ respectively, the $P(X \geq 2)$ is:

400 g of apple vinegar has 40% apple juice in it. The amount of apple juice, which should be added to make it 60% in apple vinegar, is:

If $x_1, x_2, x_3, ..., x_n$ are n observations in a sample then which of following is/are TRUE? (A) The mean $\bar{x}$ has n degree of freedom (B) The mean $\bar{x}$ has (n-1) degree of freedom (C) The standard deviation of the sample has (n-1) degree of freedom (D) The standard deviation of the sample has n degree of freedom Choose the correct answer from the options given below:

A and B are two independent events. The probability that both events A and B occur is $\frac{1}{6}$ and the probability that neither of them occur is $\frac{1}{3}$. If P(A) = x, P(B) = y then the value of x+y is.

For the linear programing problem, $Minimize(Z) = 60x + 30y$ subject to: $2x - y \geq -5; 3x + y \geq 3; 2x - 3y \leq 12; x, y \geq 0$ the optimal value of $z$ is

The maximum value of $z$ for the linear programing problem maximize $z = x + y$ subject to the constraints $x + 4y \leq 8, 2x + 3y \leq 12, 3x + y \leq 9, x \geq 0, y \geq 0$ is:

A machine costing ₹ 3,00,000 will have its scrap value of ₹ 50,000. The company at present plans to put ₹ 36,650 per annum at the end of each year in a sinking fund at the rate 5% per annum for the replacement of the machine after its useful life. Suppose the new machine will cost ₹ 4,00,000 at that time, then the useful life (approx.) of the machine is : [Given: $(1.4775)^{1/8} = 1.05$]

Match List-I with List-II | List-I | List-II | |---|---| | (A) A fire in a factory causing production delay for some time is | (I) Secular trend | | (B) Technological progress is | (II) Seasonal trend | | (C) The rise in prices before big festival is example of | (III) Irregular trend | | (D) Rise and fall of share market is | (IV) Cyclic trend | Choose the correct answer from the options given below:

For 95% confidence interval for a population mean reported to be 132 to 142 with standard deviation $\sigma = 17.85$ then the sample size used in this case, is: [Given that: $Z_{0.125} = 1.96$]

A random sample of size 16 has 53 as mean. The sum of the squares of the deviations taken from mean is 150. If the population mean is 56 then the value of t-test statistic is:

The amount of money needed to ensure for a prize of ₹ 5000 at the begining of each year indefinitely if money is worth 5% compounded annually is:

If under pure competition demand and supply functions are given by $p = \sqrt{10 - x}$ and $p = \frac{1}{2}(x-2)$ respectively, where $p$ is price per unit and $x$ is quantity, then the consumer surplus is:

In a kilometer race P runs at 10 m/sec, Q runs at 5 m/sec then which of the following statement(s) is/ are correct ? (A) P wins the race in 100 sec (B) Q wins the race in 100 sec (C) P defeats Q by 500 m (D) Q defeats P by 500 m Choose the correct answer from the options given below:

Solution of the inequality $\frac{2x+3}{4x-5} \geq 0$ is

If maximum value of $f(x) = 2x^3 + 3x^2 - 6ax + 10$ occurs at $x = -3$, then the value of $\alpha$ is ____

Maneesh took a loan of ₹ 9,00,800 from bank at an interest rate of 6% per annum for 10 years. If she has to pay the loan back with the help of equal monthly installments (EMI). Then, the EMI using reduced balance method is (approx): [Given: $(1.005)^{-120}=0.5496$]

Calculate Three Yearly moving averages for the following data | Year | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | |---|---|---|---|---|---|---| | Production (thousand tonnes) | 200 | 220 | 231 | 254 | 202 | 243 |

The unit's digit of $2^{2025}$ is

Which of the given values of $x$ and $y$ make the following pair of matrices equal ? $\begin{bmatrix}2x-1 & 4\\y-1 & 3+2x\end{bmatrix}$ and $\begin{bmatrix}0 & y-2\\5 & 4\end{bmatrix}$

Three pipes A, B & C fill a tank in 3 hours working simultaneously. The pipe C is twice as faster as B and B is twice as fast as A. The time taken by pipe A alone to fill the tank is:

Data available for profit (₹ Thousands) of a company as | Year | 2004 | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | |---|---|---|---|---|---|---|---| | Profit(₹ 000) | 114 | 130 | 126 | 144 | 138 | 156 | 164 | Based on the above data using least square method the trend value for the year '2007' is

The ratio of the speeds of a motor boat and that of the current of water is 26:4. The boat goes certain distance against the current in 6 hrs. The time taken by the boat to come back is____

If $a_{ij}=i+3j$, then the matrix of order 2 with elements as $a_{ij}$ is

$\int e^{(x \log 5)}e^x dx$, is: Where $C$ is the constant of integration.