CUET Mathematics 2025 15 May Shift 1 PYQs

Let $A = [a_{ij}]_{3 \times 3}$ such that $|A| = -5$. Then the value of $\det(5A)$ is equal to

If $f(x) = a \log_e|x| + bx^2 + x$ has critical points at $x = -2$ and $x = 1$, then

Let x denote the number of doublets in three throws of a pair of dice with the following probability distribution. | x | 0 | 1 | 2 | 3 | |---|---|---|---|---| | P(x) | $\frac{25}{72}k$ | $\frac{15}{72}k$ | $\frac{3}{72}k$ | $\frac{1}{360}k$ | If value of k is equal to $\frac{m}{n} \cdot \gcd(m,n) = 1$, then $m + n$ is equal to

The number of all possible matrices of order $2 \times 3$ with each entry 0 or 1 is

Match List-I with List-II | List-I | List-II | | --- | --- | | (A) The value of $\int_{0}^{4} \vert x\vert \, dx$ is | (I) 3 | | (B) The value of $\int_{-2}^{2} \vert x\vert \, dx$ is | (II) -1 | | (C) The value of $\int_{0}^{3} [x] \, dx$ is | (III) 8 | | (D) The value of $\int_{-1}^{1} [x] \, dx$ is | (IV) 4 | Choose the correct answer from the options given below:

The area of the region bounded by parabola $x^2 = 4y$, straight line $x = 2$ and $x$-axis, is

If A is a square matrix such that $A^2 = A$ and I is the identity matrix of same order as A, then the matrix $(2I+A)^3 - 19A - 3I$ is equal to

If the optimal value of the objective function $z = px + y$ of an L.P.P occurs at two corner points (2, 11) and (4, 5) of its bounded feasible region, then its optimal value is

The function $f(x) = x^2 - 4x + 6$ is (A) Strictly decreasing on $(-\infty, 2) \cup (2, \infty)$ (B) Strictly increasing on $(2, \infty)$ (C) Strictly increasing on $(-\infty, \infty)$ (D) Strictly decreasing on $(-\infty, 2)$ Choose the correct answer from the options given below:

The general solution of the differential equation $(1 + e^x)dy + ye^x dx = 0$, where $y > 0$, is

Match List-I with List-II | List-I | List-II | |---|---| | (Differential equation) | (Order and Degree) | | (A) $\frac{d^3y}{dx^3} + y^2 + e^{dy/dx} = 0$ | (I) order = 3, degree = 1 | | (B) $\left(\frac{d^2y}{dx^2}\right)^3 + \left(\frac{dy}{dx}\right)^2 + \frac{dy}{dx} + 1 = 0$ | (II) order = 3, degree not defined | | (C) $2x^2\frac{d^2y}{dx^2} - 3\left(\frac{dy}{dx}\right)^2 + y = 0$ | (III) order = 2, degree = 3 | | (D) $\frac{d^3y}{dx^3} + 2\left(\frac{dy}{dx}\right)^2 + \frac{dy}{dx} = 0$ | (IV) order = 2, degree = 1 | Choose the correct answer from the options given below:

If $x = 4t$ and $y = \frac{4}{t}$, then $\frac{d^2y}{dx^2}$ is

If A is an invertible matrix of order 2, then $\det(( {adj } A)^{-1})$ is equal to

The constraints of the given shaded feasible region below of an L.P.P., for non-negative variable constraints $x$ and $y$ are <img src="https://balti.afterboards.in/IECuq9aGxsHDLdW" width="300px"/>

For $x \neq -1$, if $\int \frac{xe^x dx}{(1+x)^2} = \frac{ae^x}{(1+x)^b} + c$, where a, b are fixed numbers and c is the integration constant, then $a + b$ is equal to

For $x \in \left(0, \frac{\pi}{2}\right)$, $\int \frac{\sin x + \cos x}{\sqrt{\sin 2x}} dx$ is equal to

Consider the lines $l_1: \frac{x-1}{0} = \frac{y-1}{1} = \frac{2-z}{1}$ and $l_2: \frac{x}{2} = \frac{y}{0} = \frac{2z-1}{4}$, then which of the following are correct? (A) Direction Ratio's of $l_1 = <0, 1, 1>$ (B) Direction Ratio's of $l_2 = <2, 0, 2>$ (C) Angle between $l_1$ and $l_2 =$ $\frac{\pi}{3}$ (D) Angle between $l_1$ and $l_2 = $ $\frac{2\pi}{3}$ Choose the correct answer from the options given below:

For $|x| < 1$, if $x = \cos\left(\frac{1}{a}\log y\right)$, then

A linear programming problem is as follows: Minimize $z = 2x + 3y$ Subject to the constraints $x \ge 3, x \le 9, y \ge 0, x - y \ge 0, x + y \le 14$. The feasible region has 5 corner points including

The straight line $\frac{x+3}{3} = \frac{y+2}{4} = \frac{z+1}{0}$ is

If $\int e^x\left(\frac{x-1}{(x+1)^3}\right)dx = \frac{Ae^x}{(x+1)^B} + C$, where C is constant of integration then which of the following are correct? (A) $A = -1$ (B) $A = 1$ (C) $B = 3$ (D) $B = 2$ Choose the correct answer from the options given below:

The area bounded by the curve $y = 4 + 3x - x^2$ and $x$-axis is equal to

Which of the following functions are increasing on $x \in \left(0, \frac{\pi}{2}\right)$? (A) $f(x) = \sin x$ (B) $f(x) = \cos x$ (C) $f(x) = \tan x$ (D) $f(x) = \cos 3x$ Choose the correct answer from the options given below:

Match List-I with List-II if $P(A) = \frac{3}{7}$, $P(B) = \frac{4}{7}$ and $P(A \cup B) = \frac{5}{7}$ | List-I | List-II | | :--- | :--- | | (A) $P(A \cap B)$ | (I) $\dfrac{2}{3}$ | | (B) $P(A \mid B)$ | (II) $\dfrac{5}{7}$ | | (C) $P(B \mid A)$ | (III) $\dfrac{1}{2}$ | | (D) $P(A' \cup B')$ | (IV) $\dfrac{2}{7}$ | Choose the correct answer from the options given below:

If $\vec{a} = 2\hat{i} - \hat{j} + 3\hat{k}$ and $\vec{b} = 2\hat{i} + 2\hat{j} + \hat{k}$, then Match List-I with List-II | List-I | List-II | |---|---| | (A) Projection of $\vec{a}$ on $\vec{b}$ is | (I) $-7\hat{i} + 4\hat{j} + 6\hat{k}$ | | (B) $\vec{a} \times \vec{b}$ is | (II) $\frac{1}{\sqrt{101}}(-7\hat{i} + 4\hat{j} + 6\hat{k})$ | | (C) unit vector along $\vec{a} + \vec{b}$ is | (III) $\frac{5}{3}$ | | (D) Unit vector perpendicular to both $\vec{a}$ & $\vec{b}$ is | (IV) $\frac{1}{\sqrt{33}}(4\hat{i} + \hat{j} + 4\hat{k})$ | Choose the correct answer from the options given below:

If $\vec{a}, \vec{b}$ and $\vec{c}$ are three vectors such that $\vec{a} + \vec{b} + \vec{c} = \vec{0}$, $|\vec{a}| = 7$, $|\vec{b}| = 3$ and $|\vec{c}| = 5$, then angle between $\vec{b}$ and $\vec{c}$ is

If $\alpha, \beta$ and $\gamma$ are angle of inclinations of a line with x, y and z axes respectively, then the value of $2(\cos 2\alpha + \cos 2\beta + \cos 2\gamma)$ is

If $A = \begin{bmatrix} 0 & l & -3 \\ -2 & 0 & 1 \\ m & -1 & 0 \end{bmatrix}$ is a skew symmetric matrix, then

The maximum value of $f(x) = \frac{1}{4x^2 + 2x + 1}$ is

If the function defined by $f(x) = \begin{cases} \ kx^2 + 1, & \text{if } x \le 1 \\ 2 , & \text{if } x > 1 \end{cases}$ is continuous at $x = 1$, then k is equal to

Match List-I with List-II An urn contains 4 white and 3 red balls. In a random draw of three balls, the probability of | List-I | List-II | |---|---| | (A) No red ball is | (I) $\frac{12}{35}$ | | (B) Only 1 red ball is | (II) $\frac{1}{35}$ | | (C) Exactly 2 red balls is | (III) $\frac{4}{35}$ | | (D) no white ball is | (IV) $\frac{18}{35}$ | Choose the correct answer from the options given below:

The area of a triangle whose vertices are $(x_1, y_1), (x_2, y_2)$ and $(x_3, y_3)$ is given by the absolute value of

Which of the following functions from $\mathbb{Z}$ to $\mathbb{Z}$ is a bijective function? (where $\mathbb{Z}$ is set of integers)

The position vector of a point which divides the line joining the points with position vectors $(\vec{a} - 2\vec{b})$ and $(2\vec{a} + \vec{b})$ externally in the ratio 2:1, is

If a computer code is correctly programmed, it gives 90% acceptable results. But if it is not correctly programmed, it gives only 40% acceptable results. From previous experience, it is observed that only 80% of codes are correctly programmed. If after a certain programming, the code gives 2 acceptable results, then the approximate probability that the code is correctly programmed is

For $x \in \left(0, \frac{\pi}{2}\right)$, $\int \frac{1}{\sin^2 x + \sin 2x} dx$ is equal to

If $\vec{p}$ and $\vec{q}$ are two unit vectors such that $|\vec{p} + \vec{q}| = \sqrt{2}$, then which of the following are correct? (A) $|\vec{p}| = |\vec{q}| = 1$ (B) $\vec{p}$ and $\vec{q}$ are orthogonal vectors (C) $\vec{p}$ and $\vec{q}$ are collinear vectors (D) $(4\vec{p} - \vec{q}).(2\vec{p} + \vec{q}) = 7$ Choose the correct answer from the options given below:

The maximum value of the objective function $z = 2x + 3y$ of an L.P.P. subjected to the constraints $x - y \le 1$, $x + y \le 3$, $x, y \ge 0$ is

If $A = \begin{bmatrix} -2 \\ -1 \\ -4 \end{bmatrix}$, $B = [-1 \quad 2 \quad 3]$, then the value of $A'B'$ is

Let A= {1, 2, 3}, then the possible equivalence relations on A are: (A) {(1, 1), (2, 2) , (3, 3)} (B) {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)} (C) {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1, 3)} (D) {(1, 1), (2, 2), (3, 3), (1, 3), (3, 1)} Choose the correct answer from the options given below:

If A is a square matrix, then $(A^T - A)$ is-

Consider the differential equation $\frac{dy}{dx} + y \tan x = \sec x$, then which of the following statements are correct? (A) It is homogeneous (B) It has $\sec x$ as its integrating factor (C) It's general solution is $y \sec x = \tan x + c$, where c is arbitary constant. (D) It's degree is not defined Choose the correct answer from the options given below:

The value of $\begin{vmatrix} 1 & x & y \\ 1 & x+y & y \\ 1 & x & x+y \end{vmatrix}$ is

The system of equations $x + y + z = 7$ $x + 2y + 3z = 5$ $x + 3y + \lambda z = \mu$ has a unique solution, if

The area (in sq. units) bounded by the parabola $y^2 = 16x$ and its latus rectum is

If $2P(A) = P(B) = \frac{5}{13}$ and $P(A|B) = \frac{2}{5}$, then $P(A \cap B)$ is

For the principal value branch, the value of $\sin\left(\frac{\pi}{2} - \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right)$ is

The function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = \begin{cases} x^2, & x \ge 1 \\ x, & x < 1 \end{cases}$ is

For $y \neq 0$, the particular solution of the differential equation $2ye^{x/y}dx + (y - 2xe^{x/y})dy = 0$ at the point (1, 1) is

If the area of an equilateral triangle is increasing at the rate of $4\sqrt{3}$ cm²/sec, then the rate of increase of its perimeter when the side is 4cm, is

In a game of billiards, A can give B, 15 points in 60 and A can give C, 20 points in 60. How many points can B give C in a game of 90?

Let the corner points of the bounded feasible region of the linear programming problem (LPP) $Z = ax+by$ be: (0, 0), (2, 0), (20/19, 45/19) and (0, 3). If the optimal value of Z occurs at both points (2, 0) and (20/19, 45/19), then the relation between a and b is:

Which of the following are correct? (A) The function $f(x) = 3x+12$ is increasing on R. (B) The function $f(x) = e^{2x}$ is decreasing on R. (C) The function $f(x) = x^2-x-1$ is neither increasing nor decreasing on (-1, 1). (D) The function $f(x) = x^3-3x^2+4x$ is increasing on R. Choose the correct answer from the options given below:

Which of the following is not the specification of the Sinking Fund?

Match List-I with List-II | List-I | List-II | | --- | --- | | (A) If $A$ is a non-singular matrix of order $n$, then $\vert A(\text{adj} A)\vert $ is equal to | (I) $\vert A\vert ^{n-1}$ | | (B) If $A$ is a non-singular matrix of order $n$, then $\vert \text{adj}(\text{adj} A)\vert $ is equal to | (II) $\vert A\vert ^{n-2} A$ | | (C) If $A$ is a non-singular matrix of order $n$, then $\text{adj}(\text{adj} A)$ is equal to | (III) $\vert A\vert ^n$ | | (D) If $A$ is a non-singular matrix of order $n$, then $\vert (\text{adj} A)\vert $ is equal to | (IV) $\vert A\vert ^{(n-1)^2}$ | Choose the correct answer from the options given below:

Mr. X invested Rs. 4,00,000 in shares for 5 years. The value of this investment was Rs. 4,50,000 at the end of the second year, Rs. 490000 at the end of the third year and on maturity, the final value stood at Rs. 6,00,000. The compound annual growth rate of this investment is: [Given that: $(1.5)^{1/5} = 1.084]$

If $I = \int \frac{x^4 + x^2 + 1}{x^2 - x + 1} dx = \alpha x + \beta x^2 + \gamma x^3 + \delta$, $\delta$ is constant of integration, then $(\alpha + 2\beta + 3\gamma)$ equals

Let $y(x) = a(x + 1) \log(x + 1) + bx + 5$ be the solution of the differential equation e$^\frac{dy}{dx} = x + 1_{;}y(0) = 5$, then the value of $(a + b)$ is:

Which of the following is not correct about the Central Limit Theorem?

Under which of the following conditions the Poisson distribution is the limiting case of, binomial distribution: (A) The number of trials is indefinitely large. (B) The probability of success for each trial is indefinitely small. (C) The product of the number of trials and the probability of success for each trial is finite. (D) The probability of success for each trial is indefinitely large. Choose the correct answer from the options given below:

| Year (t) | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | |---|---|---|---|---|---|---|---| | Sales (in Rs. crores) (y) | 6 | 8 | 9 | 11 | 13 | 17 | 20 | In reference to the above data, which of the following statements are correct? (where $x = t - 2013$) (A) If the equation of the straight line trend is $y = 12 + 2.29 x$, then the trend value for the year 2017 is 21.16. (B) If the equation of the straight line trend is $y = 12 + 2.29 x$, then the trend value for the year 2013 is 11. (C) If the equation of the straight line trend is $y = 12 + 2.29 x$, then the trend value for the year 2015 is 17. (D) If $(t_1, y_2),(t_2, y_2),(t_3, y_3),......,(t_n, y_n)$ denote the time series and $y_t$ are the trend values of the variable y, then $\sum(y - y_t) = 0$. Choose the correct answer from the options given below:

If $X$ is a normal variate with mean 12 and standard deviation 4, then $P[X \ge 20]$ is: [Given that: $P[0 \le Z \le 2] = 0.4772]$

Which of the following are correct? (A) If A and B are symmetric matrices such that AB = BA, then AB is symmetric. (B) If A and B are symmetric matrices of the same order, then (A+B) is a symmetric matrix. (C) If A and B are symmetric matrices of the same order, then (AB-BA) is a symmetric matix. (D) If A and B are symmetric matrices of the same order, then (AB+BA) is a skew symmetric matrix Choose the correct answer from the options given below:

A person has purchased a home for Rs.10,00,000 with down payment of Rs 2,00,000. He amortize the balance at 9% per annum compounded monthly for 25 years then the equal monthly installment (EMI) is: [Given that: $\frac{(1.0075)^{300} - 1}{(.0075)(1.0075)^{300}} = 119.1616]$

A motorbike costing Rs. 1,25,000 has a scrap value of Rs. 25,000. If the annual depreciation charge is Rs. 12,500, then the useful life of the bike is(by using linear method):

Let $\begin{vmatrix} 3 & y \\ x & 1 \end{vmatrix} = \begin{vmatrix} 3 & 2 \\ 4 & 1 \end{vmatrix}$ and $x, y$ are natural numbers, then the number of solutions for the system is:

The inequality $\frac{5x - 2}{3} - \frac{7x - 3}{5} > \frac{x}{4}$ holds when

Pure milk costs Rs. 75 per litre. A milkman adds water to 28 liters of pure milk and sells the mixture at Rs. 60 per litre. How many liters of water does he add?

For predicting the straight line trend in the sales of cars (in thousands) on the basis of 5 consecutive years' data, the company makes use of a 3-year moving averages method. If the sales of the cars for respective years are 15, 24, 18, 33 and 42 respectively, then which of the following averages will not be computed?

A boat takes 1 hour 30 minutes less to travel 36 km downstream than to travel the same distance upstream. If the speed of the boat in still water is 10 km/hr, then the speed of the stream is:

Let $A = [a_{ij}]$ be a square matrix of order 3 with each entry either 0 or 1, then the number of all such possible matrices is:

A tank has two outlet pipes, A and B, which together take 6 hours to empty a full tank when they are opened simultaneously. The tank was initially half-full and both the outlets were opened, after an hour an inlet pipe was also opened. If the inlet alone can fill the empty tank in 4 hours, how much time will it now take to fill the tank completely?

At what rate of interest will the present value of a perpetuity of Rs. 1000 payable at the end of every six months be Rs. 20000?

For the linear programming problem (LPP), $Maximize Z = 7x + 9y$, subject to constraints, $x - y \le -1, -x + y \le 0, x, y \ge 0.$ Which of the following is correct?

If the function $f(x) = 2x^3 + 9x^2 + 12x-1$ is given,then $f(x)$ have

In a game, a man wins Rs. 5 for getting a number greater than 4 and loses Rs. 1 otherwise, when a fair dice is thrown. The man decided to throw a die thrice but to quit as and when he gets a number greater than 4. The expected value of the amount(in Rs.) he wins or loses is:

Which of the following are correct about equated monthly installments (EMI)? (A) The EMI depends on principal borrowed, rate of interest and tenure of the loan. (B) It is a fixed amount made by borrower to the lender every month. (C) The interest remains constant for every EMI in reducing balance method. (D) As we pay off our loan, the outstanding principal amount decreases with every EMI in reducing balance method. Choose the correct answer from the options given below:

Let us suppose that two independent random samples of sizes $n_1$ and $n_2$ has been drawn from the same normal population then degree of freedom of statistic t-distribution is:

The value of the integral $I = \int_0^1 \frac{1}{\sqrt{x^2 + 2x + 3}} dx$ is:

The level of production where the revenue from sales is equal to the cost of production and marketing is known as

Let A be a square matrix of order 2 such that $\begin{bmatrix} 2 & 1 \\ 3 & 2 \end{bmatrix} A \begin{bmatrix} -3 & 2 \\ 5 & -3 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, then A is:

A 95 % confidence interval states that the population mean is greater than 152 and less than 160. If $\sigma = 15$ and $Z_{0.025} = 1.96$, then what sample size was used in the study?

The probability of a man hitting a target is 1/2. How many times must he fire so that the probability of hitting the target at least once is more than 90%?

The remainder when $2^{340}$ is divided by 341 is:

The rise in prices before festivals is an example of