CUET Mathematics 2025 27 May Shift 1 PYQs

Two cards are drawn simultaneously at random from a well shuffled pack of 52 Cards. Let X be the random variable which denotes number of kings in the draw. Then the probability distribution of X is

The feasible region represented by the constraints: $x + 2y \geq 100$, $2x - y \leq 0$, $2x + y \leq 200$, $x \geq 0$, $y \geq 0$ of an LPP is: <img src="https://balti.afterboards.in/uiW8X6FYeOPBw6q" width="300px"/>

The value of $\int \frac{x^5}{\sqrt{1 + x^3}} dx$ is

If A and B are symmetric matrices of the same order, then which of the following are true? (A) AB - BA is a skew symmetric matrix (B) AB is a symmetric matrix (C) AB is a scalar matrix (D) AB + BA is a symmetric matrix Choose the correct answer from the options given below:

Let $A = [a_{ij}]$ is given by $A = \begin{bmatrix} 1 & -1 & 2 \\ 3 & 4 & -5 \\ 2 & -1 & 3 \end{bmatrix}$. Then the matrix $B = [b_{ij}]$, where $b_{ij}$ = Minor of $a_{ij}$ is:

If $f(x) = \begin{vmatrix} 0 & x-1 & x-2 \\ x+1 & 0 & x-3 \\ x+2 & x+3 & 0 \end{vmatrix}$, then the value of $f(0)$ is equal to:

The interval on which the function $f(x) = x^3 + 2x^2 - 1$ is decreasing, is

If A and B are invertible matrices of the same order, then $(AB)^{-1}$ is equal to

The value of $\int_{-1}^{1} |x^3 - x| dx$ is

The area of the region bounded by the parabola $y^2 = x$ and the straight line $2y = x$ is

If $y = (x+1)(x^2+1)(x^4+1)(x^8+1)$ then $\frac{dy}{dx}$ at $x = -1$ is

For the function $f(x) = e^{-2x}(2-x)^2$, the point of local maxima is:

The particular solution of the differential equation $\frac{dy}{dx} + \frac{3y}{x} = 0$, $y(1) = 1$ is

The corner points of the bounded feasible region determined by the system of linear constraints are (15,0), (40,0), (4,18) and (6, 12). If objective function is Z = 30x + 20y, then the sum of the maximum and the minimum values of Z is

Which one of the following equations is a homogeneous differential equation?

The function $f: \mathbb{R} \rightarrow [-1, 1]$ defined by $f(x) = \cos x$ is:

A coin is tossed and a die is thrown. The probability that the outcome will be a tail on the coin or a number greater than 3 on the die is

If $c_{ij}$ denotes the cofactor of element $a_{ij}$ of the matrix $A = \begin{bmatrix} 1 & 2 & -1 \\ 0 & -3 & 2 \\ 4 & 2 & 3 \end{bmatrix}$ then the value of $c_{21} \cdot c_{33}$ is

$\int \frac{e^{2x} - 1}{e^{2x} + 1} dx =$

If A and B are independent events, then which of the following is/are true? (A) $\bar{A}$ and B are independent events (B) $P(A \cap B) = 0$ (C) $\bar{A}$ and $\bar{B}$ are independent events (D) $P(A \cap B) = P(A) + P(B)$ Choose the correct answer from the options given below:

For $x \in \mathbb{R}$, if $f(x) = -(x-1)^2 + 2$, then (A) $f$ is an increasing function on $(-\infty, 1]$ (B) $f$ has no critical points (C) $f$ has a maximum value at $x = 1$ (D) $f$ has a minimum value at $x = 1$ Choose the correct answer from the options given below:

If $P(A) = \frac{3}{5}$, $P(B) = \frac{1}{2}$ and $P(A \cap B) = \frac{1}{4}$, then $P(\overline{A} | \overline{B})$ is

Cosine of the acute angle between the lines $\frac{x-3}{2} = \frac{y-2}{1} = \frac{z-5}{2}$ and $\frac{x-1}{6} = \frac{y-3}{-3} = \frac{z+6}{2}$ is

The domain of the function $y = \sin^{-1}(x-1) + \cos^{-1}\sqrt{x-1}$ is:

Match List-I with List-II | List-I | List-II | | --- | --- | | (A) If vector $\vec{a}$ and $\vec{b}$ are such that $\vec{a} = \lambda \vec{b}$ and $\vert \vec{a}\vert = \vert \vec{b}\vert $, then | (I) $\vec{a}$ and $\vec{b}$ are orthogonal | | (B) Projection vector of $\vec{a}$ on $\vec{b}$ | (II) $[0, 12]$ | | (C) $\vec{a}$ and $\vec{b}$ are non-zero vectors such that $\vert \vec{a} + \vec{b}\vert = \vert \vec{a} - \vec{b}\vert $, then | (III) $\vec{a} = \pm \vec{b}$ | | (D) If $\vert \vec{a}\vert = 4, -3 \le \lambda \le 2$, then the range of $\vert \lambda \vec{a}\vert $ | (IV) $(\dfrac{\vec{a} \cdot \vec{b}}{\vert \vec{b}\vert ^2}) \vec{b}$ | Choose the correct answer from the options given below:

The following system of equations: $x + y - z = 7$ $4x + \lambda y - \lambda z = 3$ $3x + 2y - 4z = 5$ does not possess a solution if the value of $\lambda$ is:

A balloon which always remains spherical, has a variable diameter $\frac{3}{2}(5x+7)$. Then the rate of change of its volume with respect to x is

The corner points of the bounded feasible region determined by a set of constraints in an LPP are $P(0, 5)$, $Q(3, 5)$, $R(5, 0)$ and $S(4, 1)$. If the objective function is $z = ax + 2by$, where, $a, b > 0$, then the condition on $a$ and $b$ such that the maximum value of $z$ occurs at $Q$ and $S$ is

If $\vec{a}$ is a unit vector and $(\vec{x} - \vec{a}) \cdot (\vec{x} + \vec{a}) = 15$, then the value of $|\vec{x}|$ is:

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

Match List-I with List-II | List-I | List-II | |---|---| | (A) $\tan^{-1}\frac{2}{11} + \tan^{-1}\frac{7}{24}$ | (I) $\frac{3\pi}{4}$ | | (B) $\tan^{-1}2 + \tan^{-1}3$ | (II) $\pi$ | | (C) $\tan^{-1}1 + \tan^{-1}2 + \tan^{-1}3$ | (III) $\tan^{-1}\frac{1}{2}$ | | (D) $\tan^{-1}\frac{1}{7} + \tan^{-1}\frac{1}{13}$ | (IV) $\tan^{-1}\frac{2}{9}$ | Choose the correct answer from the options given below:

Match List-I with List-II | List-I | List-II | |---|---| | (A) The number of possible matrices of order 3x3 with each entry 1 or 0 | (I) $2^4$ | | (B) The number of possible matrices of order 2x3 with each entry 1 or 0 | (II) $2^9$ | | (C) The number of possible matrices of order 2x3 with each entry 0,1,2 | (III) $2^6$ | | (D) The number of possible matrices of order 2x2 with each entry 1 or 0 | (IV) $3^6$ | Choose the correct answer from the options given below:

If A is a square matrix such that $A^2 = A$ and I is the identity matrix of same order as A, then the value of $(A-2I)^2 - (2A + I)^2 + 11A$ is:

$\int e^{-x}(\cot x + \cosec^2 x)dx =$

If $x = a\sec^3 \theta$, $y = a \tan^3 \theta$, then $\frac{d^2y}{dx^2}$ equals.

The area of the region bounded by the curves $y = x$ and $y = x^3$ is:

Which of the following statements are true? (A) The vector joining the points P(2, 3, 0) and Q(-1,-2,-4) directed from P to Q is $\vec{PQ} = -3\hat{i} - 5\hat{j} - 4\hat{k}$ (B) Projection of a vector $\vec{a}$ on other vector $\vec{b}$ is $\frac{\vec{a}.\vec{b}}{|\vec{a}|}$ (C) If $\vec{a} = \hat{i} - 2\hat{j} + \hat{k}$ and $\vec{b} = -2\hat{i} + 4\hat{j} + 5\hat{k}$ then $\vec{a} + \vec{b} = -\hat{i} + 2\hat{j} + 6\hat{k}$ (D) If $\theta$ is the angle between $\vec{a}$ and $\vec{b}$ then $\cos \theta = \frac{\vec{a}.\vec{b}}{|\vec{a}||\vec{b}|}$ Choose the correct answer from the options given below:

Let $L_1$ and $L_2$ be two lines, represented as, $L_1: \vec{r} = \hat{i} + \hat{j} + \lambda(2\hat{i} - \hat{j} + \hat{k})$ and $L_2: \vec{r} = 2\hat{i} + \hat{j} - \hat{k} + \mu(4\hat{i} - 2\hat{j} + 2\hat{k})$, where $\lambda$ and $\mu$ are scalars. Then which of the following are true? (A) $L_1$ is perpendicular to $L_2$. (B) $L_1$ is parallel to $L_2$. (C) $L_1$ passes through the point (1, 1, 0) (D) $L_2$ passes through the point (2, 1, -1) Choose the correct answer from the options given below:

For the LPP: minimize $z = 6x + 3y$ subject to the constraints $4x + y \geq 80$ $x + 5y \geq 115$ $3x + 2y \leq 150$ $x \geq 0, y \geq 0$ then the minimum value of z is

The solution of the differential equation $(x^2 + xy)dy = (x^2 + y^2)dx$ is

If $y = \sqrt{\sin x + \sqrt{\sin x + \sqrt{\sin x + \cdots + \infty}}}$ then $\frac{dy}{dx}$ equals to

If $f(x) = \begin{cases} ax - 1 & {if } x \ > 1\\ \ 2x + 1 & {if } x < 1 \end{cases}$ is continuous at $x = 1$, then $a$ equals

If $A$ and $B$ square matrices of order 3 such that $|A| = -1$, $|B| = 5$, then the value of $|2AB|$ is:

Let box I contains 3 black and 4 white balls, box II contains 2 black and 2 white balls, box III contains 4 black and 3 white balls. A box is selected at random and then a ball is randomly drawn from the selected box. If the color of the ball is black then the probability that the ball is drawn from box III, is:

Match List-I with List-II | List-I | List-II | |---|---| | (A) Line : $x = 2y + 1 = z - 1$ | (I) Crosses $xz$ plane at (1, 0, 1) | | (B) Line : $x + 1 = 2y + 1 = z$ | (II) Crosses $xz$ plane at (0, 0, 1) | | (C) Line : $x - 1 = 2y = z + 1$ | (III) Crosses $xz$ plane at (1, 0, -1) | | (D) Line : $x - 1 = 2y = z - 1$ | (IV) Crosses $xz$ plane at (1, 0, 2) | Choose the correct answer from the options given below:

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

The projection of the vector $5\hat{i} + \hat{j} - 3\hat{k}$ on the vector $\hat{i} + 2\hat{j} - 3\hat{k}$ is

If $\begin{vmatrix} p-a & 0 & c-r \\ 0 & q-b & c-r \\ a & b & r \end{vmatrix} = 0$, then the value of $\dfrac{p}{p-a} + \dfrac{q}{q-b} + \dfrac{r}{r-c}$ is

If $f(x) = \sin x - \cos x$, $x \in [0, 2\pi]$ then (A) $f(x)$ is increasing in $(0, \frac{3\pi}{4})$ (B) $f(x)$ is decreasing in $(0, \frac{3\pi}{4})$ (C) $f(x)$ is decreasing in $(\frac{3\pi}{4}, \frac{7\pi}{4})$ (D) $f(x)$ is decreasing in $(\frac{7\pi}{4}, 2\pi)$ Choose the correct answer from the options given below:

Match List-I with List-II | List-I | List-II | |---|---| | (A) The degree of the differential equation $\left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^2 = x\sin\left(\frac{dy}{dx}\right)$ | (I) 4 | | (B) The degree of differential equation $\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^{1/4} + x^{1/5} = 0$ | (II) 1 | | (C) The degree of differential equation $\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^3 + 6y^5 = 0$ | (III) Not defined | | (D) The degree of differential equation $1 + \left(\frac{dy}{dx}\right)^4 = 7\left(\frac{d^2y}{dx^2}\right)^3$ | (IV) 3 | Choose the correct answer from the options given below:

Let $A$ be a non-singular square matrix of order $n$, then Match List-I with List-II | List-I | List-II | | --- | --- | | (A) $A(\text{adj} A)$ | (I) $\frac{1}{\vert A\vert }$ | | (B) $\vert \text{adj} A\vert $ | (II) $\vert A\vert ^n$ | | (C) $\vert A^{-1}\vert $ | (III) $\vert A\vert I$ | | (D) $\vert A(\text{adj} A)\vert $ | (IV) $\vert A\vert ^{n-1}$ | Choose the correct answer from the options given below:

Mr. Jayesh plans to save amount for higher studies of his daughter, required after 10 years. How much amount should he save at the beginning of each year to accumulate Rs.1,00,000 at the end of 10 years. If rate of interest is 12% compounded annually? [Given $(1.12)^{11} = 3.5$]

Consider the following hypothesis test: $H_0: \mu \leq 3432$ $H_a: \mu > 3432$ A sample of 96 provided a sample mean $\bar{x} = 3648$ and sample standard deviation $s=802$ then the degree of freedom of t-distribution is:

Mohini purchases a house worth Rs. 50 lakhs and makes a down payment of Rs. 11.2 lakhs. She pays the remaining amount on monthly EMI using a reducing balance method. The bank charges 6% per annum compounded monthly for a tenure of 25 years. Her EMI is: [Given: $(1.005)^{-300} \approx 0.224$]

The marginal cost of production of x units of a commodity is $56 + \frac{3}{2}x$. It is known that fixed costs are Rs.115. Then the total cost of producing 50 units is:-

In a survey question for a sample of 250 individuals, 120 persons gave response 'yes', 80 persons gave response 'no' and 50 gave 'no response'. The point estimate of the proportion in the population who responded 'yes' is:

A runs 9 times slower than B. If B gives A a start of 80 meters, how far must be the wining post on the tracks so that A and B reach there at the same time?

The five month moving averages for the following data | Month ($r^{th}$) | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 | 51 | |---|---|---|---|---|---|---|---|---|---| | Actual Demand | 105 | 106 | 110 | 110 | 114 | 121 | 130 | 128 | 137 | is:

If the area above x-axis, bounded by the curves $y = 3^{\beta x}$, $x = 0$ and $x = 3$ is $\frac{26}{\log_e 3}$, then the value of $\beta$ is:

Match List-I with List-II | List-I | List-II | |---|---| | (A) Corner point of a feasible region | (I) The line segment joining any two arbitrary points of the region always lies entirely within the region | | (B) Bounded feasible region | (II) can not be enclosed within a circle | | (C) Unbounded feasible region | (III) can be enclosed within a circle | | (D) Convex region | (IV) Is a point of intersection of two boundary lines in the feasible region | Choose the correct answer from the options given below:

For the objective function $Z = 3x + 5y$ subject to constraints $x + 3y \geq 3$, $x + y \geq 2$, $x \geq 0$, $y \geq 0$:

A swimmer whose speed in swimming pool is 5 km/h, swims between two points in a river and returns back to starting point. He took 20 minutes more to cover the distance upstream than to cover downstream. If the speed of stream is 2 km/h, then the distance between two points is

If $x, y \in \mathbb{R}$ then match List-I with List-II | List-I | List-II | | --- | --- | | (A) $\vert x\vert < \vert y\vert $ | (I) iff $x^2 > y^2$ | | (B) $\vert x\vert > \vert y\vert $ | (II) iff $x^2 \le y^2$ | | (C) $\vert x\vert \le \vert y\vert $ | (III) iff $x^2 < y^2$ | | (D) $\vert x\vert \ge \vert y\vert $ | (IV) iff $x^2 \ge y^2$ | Choose the correct answer from the options given below:

As per the graph given below: <img src="https://balti.afterboards.in/04rh2eS7VMjqGKY" width="200px"/> Match List-I with List-II | List-I | List-II | |---|---| | (Function/Area/point) | (Representation) | | (A) Consumers Surplus | (I) y=g(x) | | (B) Supply function | (II) v | | (C) Demand function | (III) s | | (D) Equilibrium point | (IV) y=f(x) | Choose the correct answer from the options given below:

If matrix $A = \begin{bmatrix} x & 2 & 3 \\ a & y & -5 \\ b & c & 0 \end{bmatrix}$ is a skew-symmetric matrix, then (A) $x + y + c = 5$ (B) $c = 5$ (C) $a + b + c = 0$ (D) $a + b - c = 10$ Choose the correct answer from the options given below:

Let X be a random variable. Let E (X) and Var (X) denote the mean and the variance of X respectively. Then match List-I with List-II | List-I | List-II | |---|---| | (A) If Var (X) = $a$, then Var (2X + 3) is | (I) 11$a$ | | (B) If E (X) = $a$, then E (2X) is | (II) 6$a$ | | (C) If Var (X) = $a$, then Var(3X - $a$) + Var ($\sqrt{2}x + \beta$) is | (III) 4$a$ | | (D) If E (X) = $\frac{5a}{12}$, then E (12X + $a$) is | (IV) 2$a$ | Choose the correct answer from the options given below:

The water and milk in two vessels are in the ratio:1:1 and 3:8 respectively. In what ratio, the mixtures in the vessels be mixed to obtain a new mixture containing water and milk in the ratio 4:7?

If an asset costs Rs. 50,000 with an estimated useful life of 6 years and a scrap value of Rs. 5000. Then by using a linear depreciation method, the annual depreciation of the asset will be:

The largest open interval in which the function $f(x) = 4x^3 - 5x^2 - 8x + 12$ increases, is:

Consider the following hypothesis test. $H_0: \mu \leq 12$ $H_a: \mu > 12$ If a sample of 25 is taken with sample mean 15 and a sample standard deviation of 6, then the value of t-test statistic is:

It is given that 3% of items manufactured by an industry are defective. The probability that a packet of 250 items contains one defective item is: [Given: $e^{-7.5} \approx 0.000553$]

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

If $525 \equiv (10 + K)( {mod } 7)$ where $K \in \mathbb{N}$, then the least value of $K$ is:

For independent events $A_1, A_2, A_3, ..., A_n$ if $P(A_i) = \frac{1}{i+1}$, $i = 1, 2, 3, ..., n$, then the probability that none of the events occur is:

Match List-I with List-II | List-I | List-II | |---|---| | (Example) | (Components of Time Series) | | (A) Lockouts and strikes | (I) Secular trend | | (B) Rise and fall of share market | (II) Seasonal trend | | (C) Continuous decline in death rate | (III) Irregular trend | | (D) The rise in prices before Diwali | (IV) Cyclical trend | Choose the correct answer from the options given below:

A fair coin is tossed a fixed number of times. If the probability of getting 11 heads is equal to the probability of getting 13 heads, then the probability of getting 2 heads is:

Rs. 2,50,000 cash is equivalent to a perpetuity of Rs.7500 payable at the end of each quarter. Then the rate of interest is

Two pipes can fill a tank in 6 minutes and 12 minutes respectively, and a third pipe can empty the tank at the rate of 18 liters per minute. If all the pipes working together can fill the empty tank in 5 minutes, the capacity of the tank is:

Ram had invested Rs. 15,000 in a mutual fund and the value of the investment at the time of redemption was Rs. 25,000. If the compound annual growth rate (CAGR) is 8.88%, then the number of years for which Ram has invested the amount is: [Given: $\log 1.089 \approx 0.0370$ and $\log 1.667 \approx 0.2220$]

A cylindrical drum of radius 7 cm and height 2 m is being kept in a vertical position filled with milk. If the milk is leaking at 14 cm³/sec from its lower base, then the rate of decrease in the level of milk is: [Take $\pi = \frac{22}{7}$]

The value of $\int_{-5}^{5} |x + 3| dx$ is

If $\begin{bmatrix} -1 & 1 & 0 \\ a & b & 1 \\ 1 & 2 & 1 \end{bmatrix}$ is a singular matrix, then the relation between $a$ and $b$ is:

For the system of linear equations $x + y + z = 5000$ $6x + 7y + 8z = 35800$ $6x + 7y - 8z = 7000$ the values of x, y and z are:

Consider the following data | Year (x) | 2014 | 2016 | 2018 | 2020 | 2022 | 2024 | |---|---|---|---|---|---|---| | Profit (in Rs. Thousand) (y) | 7 | 9 | 10 | 12 | 14 | 14 | Then for the above data the equation of straight line trend by method of least square is given by:

A person has set up a sinking fund in order to have Rs. 10,00,000 after 10 years for his child education. The amount should put bi-annually into account paying 5% per annum compounded semi-annually is: [Given $(1.025)^{20} = 1.6386$]