CUET Mathematics 2025 14 May Shift 2 PYQs

The region represented by the constraints $x \geq 0, y \geq 0$ of an LPP is

If $A = \begin{bmatrix} 2 & 1 & -1 \\ 0 & 1 & 2 \\ 2 & -1 & \lambda \end{bmatrix}$ is a singular matrix, then the value of $\lambda$ is

Let the random variable X represent the positive difference between the number of heads and the number of tails obtained when a coin is tossed 6 times. Then probability $P(X \leq 3)$ is equal to

Match **List-I** with **List-II** The function $f(x) = 2x^3 - 15x^2 + 36x + 5$ for $x \in [2,5]$ has | List-I | List-II | |---|---| | (A) absolute maximum value | (I) 5 | | (B) absolute minimum value | (II) 60 | | (C) point of absolute maxima | (III) 3 | | (D) point of absolute minima | (IV) 32 | Choose the **correct** answer from the options given below:

For $x \in \mathbb{R} - \{-1,0,1\}$, $\int \frac{1}{x - x^5}dx$ is equal to

Let A be any square matrix of order 3 and $B = \begin{bmatrix} 0 & -4 & 2 \\ 4 & 0 & 3 \\ -2 & -3 & 0 \end{bmatrix}$. Then the matrix $ABA^T$ is a

If $y = \frac{1}{1+x^{b-a}+x^{c-a}} + \frac{1}{1+x^{c-b}+x^{a-b}} + \frac{1}{1+x^{a-c}+x^{b-c}}$ then $\frac{d^2y}{dx^2}$ is

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

$\int_{-1}^{1}(x^7 + x^5 + x^3 + x + 1)dx$ is equal to

The greatest possible value of '$a$' such that the function $f(x) = x^2 + a x + 1$ is always decreasing in the interval [1, 2] is:

If $A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 0 & 2 \\ x & 1 & 1 \end{bmatrix}$ and $A^{-1} = \frac{1}{4}\begin{bmatrix} -2 & 0 & y \\ 5 & -2 & -1 \\ 1 & 2 & -1 \end{bmatrix}$, then values of x and y, are:

Let the matrix $A = [a_{ij}]_{3\times3}$ be defined by $a_{ij} = \begin{cases} 2i + 3j, & i < j \\ 5, & i = j \\ 3i - 2j, & i > j \end{cases}$ The number of elements in the matrix A which are greater than 7, is:

Particular solution of the differential equation $x(1 + y^2)dx - y(1 + x^2)dy = 0$, given $y = 0$ when $x = 1$, is

The maximum value of the objective function $Z = 8x + 2y$ of an LPP subject to constraints $2x + y \leq 3, 2x + 3y \leq 6, x \geq 0, y \geq 0$ is:

The area of the region (in square units) bounded by $x=1, x=2$ and the curve $y^2 = 4x$ in the first quadrant is

An urn I contains 3 white and 4 blue balls, while urn II contains 5 white and 6 blue balls. One ball is drawn at random from one of the urns and it is found to be white. The probability that it was drawn from urn II is

The values of $\lambda$ for which the system of equation $x + 2y + z = 14, - x + y + z = 10, x + \lambda y + z = 2$ has unique solution is

General solution of the differential equation $\frac{dy}{dx} = e^{\frac{x^2}{2}} + xy$ is

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

If a person A speaks the truth in 80% cases and the person B speaks the truth in 75% cases, then the probability that they contradict each other in a statement is

If A and B are two invertible matrices, then which of the following statements are correct? (A) $|A^{-1}| = |A|^{-1}$ (B) $adjA = |A|A^{-1}$ (C) $(AB)^{-1} = A^{-1}B^{-1}$ (D) $(A + B)^{-1} = A^{-1} + B^{-1}$ Choose the **correct** answer from the options given below:

If $\vec{a} = 2\hat{i} - 3\hat{j} + \hat{k}$ and $\vec{b} = 2\hat{i} + \hat{j} - \hat{k}$, then which of the following statements is/are correct? (A) $\vec{a}$ and $\vec{b}$ are collinear (B) $\vec{a}$ and $\vec{b}$ are perpendicular (C) Angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{4}$ (D) $|\vec{a} + \vec{b}| = 2\sqrt{5}$ Choose the **correct** answer from the options given below:

Match List-I with List-II | List-I | List-II | | --- | --- | | Function | Derivative | | --- | --- | | (A) $y = \sin^{-1} x + \sin^{-1} \sqrt{1 - x^2}; \vert x\vert < 1$ | (I) $\frac{dy}{dx} = \frac{1}{2y-1}$ | | (B) $y = \sqrt{x + y}, x+y > 0 \text{ and } y \neq \frac{1}{2}$ | (II) $\frac{dy}{dx} = 10^x \log_e 10$ | | (C) $y = \log_{10} x, x > 0$ | (III) $\frac{dy}{dx} = 0$ | | (D) $y = 10^x$ | (IV) $\frac{dy}{dx} = \frac{1}{x \log_e 10}$ | Choose the correct answer from the options given below:

If $\left|\begin{matrix} x & 8 \\ 4 & x \end{matrix}\right| = \left|\begin{matrix} 6 & 2 \\ 18 & 6 \end{matrix}\right|$, then $x$ is/are equal to

If A is a square matrix of order $3 \times 3$ and $|A| = 4$. The value of $|(adjA).A|$ is

The function $f(x) = \frac{x - 2}{x + 1}, x \neq -1$ is increasing when (Where $\mathbb{R}$ is a set of real numbers)

The minimum value of the objective function $z = x + 2y$ of an L.P.P. subject to constraints $2x + y \geq 3, \frac {x} {2} + 2y \geq 6, x \geq 0, y \geq 0$ is:

For the differential equation $(x + y)dy + (x - y)dx = 0$, which of the following is/are correct? (A) Differential equation is homogeneous (B) Order of differential equation is 1 (C) Integrating factor of differential equation is $e^x$ (D) Degree of the equation is not defined Choose the **correct** answer from the options given below:

Match **List-I** with **List-II** Consider two vectors $\vec{a} = \hat{i} + 2\hat{j} - \hat{k}$ and $\vec{b} = -3\hat{i} - 6\hat{j} + 3\hat{k}$, then | List-I | List-II | |---|---| | (A) Angle between $\vec{a}$ and $\vec{b}$ is | (I) $\cos^{-1}\left(\frac{1}{\sqrt{6}}\right)$ | | (B) Angle between $\vec{a}$ and $x$-axis is | (II) $\cos^{-1}\left(\frac{2}{\sqrt{6}}\right)$ | | (C) Angle between $\vec{b}$ and $x$-axis is | (III) $\pi$ | | (D) Angle between $\vec{a}$ and $y$-axis is | (IV) $\cos^{-1}\left(-\frac{1}{\sqrt{6}}\right)$ | Choose the **correct** answer from the options given below:

Let a relation R = {(a, b) : a is a factor of b, a, b $\in$ N}. Then, R is ______.

The angle between the lines $l_1: \frac{x + 1}{1} = \frac{2 - y}{2} = \frac{z - 1}{1}$ and $l_2: \frac{x - 1}{4} = \frac{2y - 4}{6} = \frac{z - 1}{2}$ is

Match **List-I** with **List-II**. Here [x] denotes the greatest integer function $\begin{array}{|l|l|} \hline \rule{0pt}{2.8ex}\text{List-I} & \text{List-II} \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(A) } f(x) = [x] & \text{(I) is continuous everywhere but not differentiable at } x=-1 \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(B) } f(x) = |x-1| & \text{(II) is continuous everywhere except at all integral values} \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(C) } f(x) = e^{|x|} & \text{(III) is continuous everywhere but not differentiable at } x=1 \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(D) } f(x) = |x+1| & \text{(IV) is continuous everywhere but not differentiable at } x=0 \\[1.2ex] \hline \end{array}$ Choose the **correct** answer from the options given below:

If $\vec{a}$ is a unit vector perpendicular to both the vectors $\vec{b} = \hat{j} + \hat{2k}$ and $\vec{c} = \hat{i} + 2\hat{j}$, then $\hat{a}$ is equal to

If $A = \begin{bmatrix} x & -3 & 4 \\ 3 & y & -5\\-4&z&0 \end{bmatrix}$ is a Skew-Symmetric matrix and $adj \ A = [a_{ij}]_{3 \times3}$, then $a_{11} + a_{22} + a_{33}$ is equal to

Match **List-I** with **List-II** | List-I | List-II | |---|---| | **Lines** | **Direction Ratios** | | (A) $\frac{x - 1}{2} = \frac{2 - y}{1} = z$ | (I) 1, 3, -1 | | (B) $\frac{2x - 1}{2} = \frac{y + 1}{3} = \frac{1 - z}{1}$ | (II) 2, -2, 0 | | (C) $\frac{x + 1}{2} = \frac{3 - y}{2}, z = 2$ | (III) 2, -1, 1 | | (D) $\frac{2x - 3}{4} = \frac{1 - 2y}{2} = \frac{z}{5}$ | (IV) 2, -1, 5 | Choose the **correct** answer from the options given below:

$\int_{-\pi}^{\pi} \frac{e^{\sin x}}{e^{\sin x} + e^{-\sin x}}dx$ is equal to

Consider the region bounded by the lines $y - 1 = x, x = -2, x = 3$ and $x$ - axis. Then (A) The area of the bounded region is given by $\int_{-2}^{3}(x + 1)dx$ (B) The numerical value of the area is $\frac{15}{2}$ sq. units (C) The numerical value of the area is 8 sq. units (D) The numerical value of the area is $\frac{17}{2}$ sq. units Choose the **correct** answer from the options given below:

Consider the function $f(x) = \sin x$ in the interval $[\pi, 2\pi]$ then which of the following statements are correct? (A) $x = \frac{3\pi}{2}$ is its stationary point. (B) Its maximum value is 1 (C) Its minimum value is -1 (D) It attains its maximum value at $\pi$ and $2\pi$ Choose the **correct** answer from the options given below:

The radius of spherical balloon is decreasing at the rate of 0.1cm/sec, the rate at which its volume is decreasing, when its radius is 0.5cm is

For $x \in [-1,1]$, if $4\sin^{-1}x + \cos^{-1}x = \pi$ then $x$ is equal to

For an LPP: Maximize $z = 3x + 9y$, $x \geq 0, y \geq 0$, the feasible region OAB is shown in the figure, then the other constraints are <img src="https://balti.afterboards.in/HfoMd9Ve3v1q1ag" width="400px"/>

If $A = \begin{bmatrix} 2 & -3 & 4 \\ -3 & 5 & x \\ 4 & 3 & 0 \end{bmatrix}$ is a symmetric matrix and $B = \begin{bmatrix} 0 & 2 & -10 \\ -2 & z & 6 \\ y & -6 & 0 \end{bmatrix}$ is a skew-symmetric matrix, then the value of $(xy + yz + zx)$ is

The function $f: [0, \infty) \rightarrow \mathbb{R}$ defined by, $f(x) = 2x^2 + 3$, is

The maximum value of the objective function $Z = 2x + y$ of an LPP, subject to the constraints $x \leq 6, y \leq 2, x - y \leq 0$, $x \geq 0, y \geq 0$ is

Consider a line $\vec{r} = (\hat{i} + 4\hat{j}) + \lambda(2\hat{i} - 2\hat{j} + 3\hat{k})$, then which of the following statements are correct? (A) it passes through point (9, -4, 12) (B) it passes through point (1, 4, -1) (C) its direction cosine's are $\frac{2}{\sqrt{17}}, \frac{-2}{\sqrt{17}}, \frac{3}{\sqrt{17}}$ (D) its Cartesian equation is $\frac{x - 1}{2} = \frac{y - 4}{-2} = \frac{z}{3}$ Choose the **correct** answer from the options given below:

If $y = x\sin y$, then $\frac{dy}{dx}$ is:

If $I_n = \int_{0}^{\pi/4} \tan^n x dx$ then $I_{2024} + I_{2026}$ is equal to:

Let $\vec{a} = \hat{i} + \hat{j}$, $\vec{b} = \hat{i} - \hat{j}$ and $\vec{c} = \hat{i} + \hat{j} + \hat{k}$. If $\hat{m}$ is a unit vector perpendicular to both $\vec{a}$ and $\vec{b}$, then $|\vec{c}.\hat{m}|$ is equal to

A problem in Mathematics is given to two students X and Y whose chances of solving it are $\frac{1}{3}$ and $\frac{1}{4}$ respectively. The probability that only X solves the problem, is:

Area of the region bounded by the curve $y^2 = 4x$, $y$-axis and the line $y = 3$ is equal to

If we take 8 identical slips of paper and write the number 0 on one of them, the number 1 on three of the slips, the number 2 on three of the slips and the number 3 on one of the slips. These slips are folded, put in a box and roughly mixed. One slip is drawn at random from the box. If X is the random variable denoting the number written on the drawn slip, the variance of X is:

If the matrix $\begin{bmatrix} 0 & 1 & 4x\ \\ -1 & 0 & -5 \\ 2 & 5 & y \end{bmatrix}$ is skew-symmetric, then

A person has an initial investment of ₹ 25000 in an investment plan. After 2. years it has grown ₹ 30000, then the rate of return on his investment is

If $x^2 - y^2 = t - \frac{1}{t}$, and $x^4 + y^4 = t^2 + \frac{1}{t^2}$, then which of the following is correct?

In what ratio should a shopkeeper mix two types of rice, one costing ₹20 per kg and another costing ₹40 per kg to get a rice variety costing ₹ 28 per kg?

A runs 3 times as fast as B. If A gives B a start of 30 meters, how far must the goal on the race course be so that A and B reach it at the same time?

The normal distribution curve is symmetrical about [$\mu$ = mean, $\sigma$= standard deviation]

For the given 5 values 24, 18, 33, 42, 24; the 3-year moving averages are:

Due to which of the following, the irregular variations in a time series are caused: (A) Floods (B) Rise in prices before festivals (C) A fire in a factory (D) Epidemics Choose the **correct** answer from the options given below:

If the objective function $Z = px + qy, p > 0, q > 0$ of a linear programming problem attains its optimal value at the points (4, 7) and (5, 5) and $pq = 50$ then

If $A = \begin{bmatrix} 4 & 5 \\ 2 & 1 \end{bmatrix}$ and $I$ is an identity matrix of order 2, then $A - 3I$ equals

For the system of equations AX = B, which of the following is correct?

Which of the following are the components of a time series? (A) Cyclic component (B) Regular component (C) Seasonal component (D) Economic component Choose the **correct** answer from the options given below:

Which of the following are NOT correct about "Sinking Fund"? (A) It does not have any specific purpose. (B) It can be used in any emergency. (C) Any amount, any time can be deposited in it. (D) It is set up for a particular upcoming expense. Choose the **correct** answer from the options given below:

If $\int \frac{dx}{(x-1)^3/^4. (x+2)^5/^4} = a[1 - g(x)]^b + c$, where $c$ is a constant of integration, then which of the following are true? (A) $a = \frac{2}{3}$ (B) $\beta = \frac{3}{4}$ (C) $3\alpha + 4\beta = 5$ (D) $g(x) = \frac{3}{(x+2)}$ Choose the **correct** answer from the options given below:

In a game, a person is paid Rs. 2 if he gets all heads or all tails when three coins are tossed, and he will pay Rs. 2 if either one or two heads show. What can he expect to win on an average per game?

If an investment of Rs. 12000 becomes Rs. 72000 in 4 years, then the compound annual growth rate is:

A person can row a boat at 5 km/hr in still water. If the speed of water current in a river is 1 km/hr, and it takes him 1 hour to row to a place and come back, how far off is the place?

A machine costing Rs. 25000 has a useful life of 4 years. The estimated scrap value is Rs. 5000. The annual depreciation by linear method is

Which of the following inequalities are NOT correct? (A) If $a > 1, b > 1,$ then $\log_b a + \log_a b \leq 2$ (B) For any real number $x, (9^x + 9^{1-x}) \geq 9$ (C) If $a,b,c$ are non-zero real numbers of the same sign, then $\left(\frac{a}{b} + \frac{b}{c} + \frac{c}{a}\right) \leq 3$ (D) If $a,b,c$ are three distinct real numbers, then $(a + b)(b + c)(c + a) \geq 8abc$ Choose the **correct** answer from the options given below:

The value of the definite integral $I = \int_{1}^{2} \frac{1}{x(1 + x^2)}dx$ is:

The number of all possible matrices of order $2 \times 3$ with entries -1 or 1 is

A specific characteristic of a sample is known as a

The present value of a sequence of payments of Rs. 2000 made at the end of every 6 months and continuing forever, if money is worth 8% per annum compounded semi-annually, is:

The general solution of the differential equation $\frac{dy}{dx} = e^{x-y} + x^2e^{-y}$ is equal to:

Mr. X wishes to purchase a house for ₹ 14,51,400 from a bank and decided to repay the loan by equal monthly installments (EMI) in 10 years. If bank charges interest at 9 % per annum compounded monthly, then the EMI is: [Given that $(1.0075)^{120} = 2.4514]$

If A is a square matrix of order 3 such that $|A| = 3$, then $|adj(adj A)|$ is equal to:

Three pipes A, B and C can fill a tank in 12 hours, 15 hours and 20 hours respectively. If A is open all the time and B and C are open for one hour each alternately, in how many hours will the tank be full?

Which of the following are NOT correct regarding the equation of tangent and normal to the curve $y = \frac{x-11}{(x-2)(x-3)}$ at the point, where it cuts the $x$-axis? (A) The point of contact is (11, 0). (B) The equation of tangent is $x - 72y - 11 = 0$ (C) The equation of normal is $72x + y - 11 = 0$ (D) The slope of the tangent at the given point of contact is $\frac{1}{88}$ Choose the **correct** answer from the options given below:

Consider the linear programming problem(LPP): *Minimize* $Z = x + y$ $x + 2y \leq 4,$ $3x + y \geq 3,$ $4x + 3y \geq 6,$ $x, y \geq 0.$ Which of the following is correct for the above linear programming problem (LPP): (A) The LPP has a bounded feasible region. (B) The LPP has a unique optimal solution. (C) The optimal value of the LPP exists at the point (3/2, 0) (D) The corner points of the feasible region are (3/2, 0), (3/5, 6/5), (2/5, 6/5) and (4, 0) Choose the **correct** answer from the options given below:

The function $f(x) = kx^3 + 6kx^2 + 18x + 17$ is increasing on $\mathbb{R}$(set of real numbers) if:

A lot of 50 watches is known to have 10 defective watches. If 8 watches are selected one by one with a replacement at random, then the probability that there will be at least one defective watch is:

The least non-negative remainder when $2^{101}$ is divided by 5 is:

The total cost $c(x)$ associated with the production of $x$ units of an item is given by $c(x) = 0.001x^3 + 0.06x^2 + 20x + 500$. The marginal cost when 10 units are produced is:

The simple random sample consists of six observations: 5, 8, 10, 7, 10, 14. The point estimate of the population mean is: