CUET Mathematics 2025 14 May Shift 1 PYQs

If $y = 3e^{2x} + 2e^{3x}$, then $\frac{d^2y}{dx^2} + 6y$ is equal to

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

The interval, on which the function $f(x) = x^2e^{-x}$ is increasing, is equal to

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

If $A = \begin{bmatrix} 0 & 0 & \sqrt{3} \\ 0 & \sqrt{3} & 0 \\ \sqrt{3} & 0 & 0 \end{bmatrix}$, then $|adj A|$ is equal to

The solution of the differential equation $log_e\left(\frac{dy}{dx}\right) = 3x + 4y$ is given by

The probability distribution of a random variable X is given by | X | 0 | 1 | 2 | |---|---|---|---| | P(X) | $1 - 7a^2$ | $\frac{1}{2}a + \frac{1}{4}$ | $a^2$ | If $a > 0$, then $P(0 < x \leq 2)$ is equal to

If the maximum value of the function $f(x) = \frac{\log_ex}{x}$, $x > 0$ occurs at $x = a$, then $a^2f''(a)$ is equal to

$\int_1^4 |x - 2|dx$ is equal to

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

The integral I = $\int \frac{e^{5\log_e x} - e^{4\log_e x}}{e^{3\log_e x} - e^{2\log_e x}} dx$ is equal to

The corner points of the feasible region associated with the LPP: Maximise $Z = px + qy$, $p,q > 0$ subject to $2x + y \leq 10$, $x + 3y \leq 15$, $x, y \geq 0$ are (0, 0), (5, 0), (3, 4) and (0, 5). If optimum value occurs at both (3, 4) and (0, 5), then

Consider the LPP: Minimize $Z = x + 2y$ subject to $2x + y \geq 3$, $x + 2y \geq 6$, $x, y \geq 0$. The optimal feasible solution occurs at

The area (in sq. units) of the region bounded by the parabola $y^2 = 4x$ and the line $x = 1$ is

Which of the following are linear first order differential equations? (A) $\frac{dy}{dx} + P(x)y = Q(x)$ (B) $\frac{dx}{dy} + P(y)x = Q(y)$ (C) $(x - y)\frac{dy}{dx} = x + 2y$ (D) $(1 + x^2)\frac{dy}{dx} + 2xy = 2$ Choose the correct answer from the options given below:

A black and a red die are rolled simultaneously. The probability of obtaining a sum greater than 9, given that the black resulted in a 5 is

If A and B are any two events such that P(B) = P(A and B), then which of the following is correct

Match List-I with List-II $\begin{array}{|l|l|} \hline \rule{0pt}{2.8ex}\text{List-I} & \text{List-II} \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(A) } f(x) = |x| & \text{(I) Not differentiable at } x=-2 \text{ only} \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(B) } f(x) = |x+2| & \text{(II) Not differentiable at } x=0 \text{ only} \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(C) } f(x) = |x^2-4| & \text{(III) Not differentiable at } x=2 \text{ only} \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(D) } f(x) = |x-2| & \text{(IV) Not differentiable at } x=2,-2 \text{ only} \\[1.2ex] \hline \end{array}$ Choose the correct answer from the options given below:

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

The area (in sq. units) of the region bounded by the curve $y = x^5$, the x-axis and the ordinates $x = -1$ and $x = 1$ is equal to

If A and B are invertible matrices then which of the following statement is NOT correct?

Let $\vec{a} = \hat{i} + 4\hat{j} $, $\vec{b} = 4\hat{j} + \hat{k}$ and $\vec{c} = \hat{i} - 2\hat{k}$. If $\vec{d}$ is a vector perpendicular to both $\vec{a}$ and $\vec{b}$ such that $\vec{c} \cdot \vec{d} = 16$, then $|\vec{d}|$ is equal to

If the points $A, B, C$ with position vectors $20\hat{i} + \lambda\hat{j}$, $5\hat{i} - \hat{j}$ and $10\hat{i} - 13\hat{j}$ respectively are collinear, then the value of $\lambda$ is

Consider the differential equation, $x\frac{dy}{dx} = y(\log_e y - \log_e x + 1)$, then which of the following are true? (A) It is a linear differential equation (B) It is a homogenous differential equation (C) Its general solution is $\log_e\left(\frac{y}{x}\right) = Cx$, where C is constant of integration (D) Its general solution is $\log_e\left(\frac{x}{y}\right) = Cy$, where C is constant of integration (E) If $y(1) = 1$, then its particular solution is $y = x$ Choose the correct answer from the options given below:

The rate of change of area of a circle with respect to its circumference when radius is 4cm, is

If A is any event associated with sample space and If $E_1, E_2, E_3$ are mutually exclusive and exhaustive events. Then which of the following are true? (A) $P(A) = P(E_1)P(E_2|A) + P(E_2)P(E_2|A) + P(E_3)P(E_3|A)$ (B) $P(A) = P(A|E_1)P(E_1) + P(A|E_2)P(E_2) + P(A|E_3)P(E_3)$ (C) $P(E_i|A) = \frac{P(A|E_i)P(E_i)}{\sum_{i=1}^3 P(A|E_i)P(E_i)}$, $i = 1,2,3$ (D) $P(A|E_i) = \frac{P(E_i|A)P(E_i)}{\sum_{i=1}^3 P(E_i|A)P(E_i)}$, $i = 1,2,3$ Choose the correct answer from the options given below:

Let $AX = B$ be a system of three linear equations in three variables. Then the system has (A) a unique solutions if $|A| = 0$ (B) a unique solutions if $|A| \neq 0$ (C) no solutions if $|A| = 0$ and (adj A) $B \neq 0$ (D) infinitely many solutions if $|A| = 0$ and (adj A)$B = 0$ Choose the correct answer from the options given below:

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

Match List-I with List-II $\begin{array}{|l|l|} \hline \rule{0pt}{2.8ex}\text{List-I} & \text{List-II} \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(A) The minimum value of } f(x) = (2x - 1)^2 + 3 & \text{(I) } 4 \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(B) The maximum value of } f(x) = -|x + 1| + 4 & \text{(II) } 10 \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(C) The minimum value of } f(x) = \sin(2x) + 6 & \text{(III) } 3 \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(D) The maximum value of } f(x) = -(x - 1)^2 + 10 & \text{(IV) } 5 \\[1.2ex] \hline \end{array}$ Choose the correct answer from the options given below:

Which one of the following set of constraints does the given shaded region represent? <img src="https://balti.afterboards.in/0PPxlbCRpdvsuFO" width="400px"/>

Let $A = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}$ and $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. If $A^T + A = I$, then

The integral I = $\int e^x\left(\frac{x - 1}{3x^2}\right) dx$ is equal to

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

The corner points of the feasible region of the LPP: Minimize $Z = -50x + 20y$ subject to $2x - y \geq -5$, $3x + y \geq 3$, $2x - 3y \leq 12$ and $x, y \geq 0$ are

The function $f(x) = tanx - x$

Let $A = [a_{ij}]_{2 \times 3}$ and $B = [b_{ij}]_{3 \times 2}$, then $|5AB|$ is equal to

The integrating factor of the differential equation $(x log_e x)\frac{dy}{dx} + y = 2log_e x$ is

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

Consider the line $\vec{r} = \hat{i} - 2\hat{j} + 4\hat{k} + \lambda(-\hat{i} + 2\hat{j} - 4\hat{k})$ Match List-I with List-II | List-I | List-II | |---|---| | (A) A point on the given line | (I) $\left(\frac{-1}{\sqrt{21}}, \frac{2}{\sqrt{21}}, \frac{-4}{\sqrt{21}}\right)$ | | (B) direction ratios of the line | (II) $(4, -2, -2)$ | | (C) direction cosines of the line | (III) $(1, -2, 4)$ | | (D) direction ratios of a line perpendicular to given line | (IV) $(-1, 2, -4)$ | Choose the correct answer from the options given below:

Let f: $\mathbb{R} \rightarrow \mathbb{R}$ be defined as $f(x) = 10x$. Then (Where $\mathbb{R}$ is the set of real numbers)

Match List-I with List-II Let A & B are two events such that P(A)=0.8, P(B)=0.5, P(B|A)=0.4 | List-I | List-II | | :--- | :--- | | (A) $P(A \cap B)$ | (I) 0.2 | | (B) $P(A \mid B)$ | (II) 0.32 | | (C) $P(A \cup B)$ | (III) 0.64 | | (D) $P(A')$ | (IV) 0.98 | Choose the correct answer from the options given below:

The shortest distance between the lines $\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ and $\frac{x-2}{4} = \frac{y-4}{6} = \frac{z-5}{8}$ is equal to

If the function $f(x) = \begin{cases} \frac{k\cos x}{\pi - 2x} & : x \neq \frac{\pi}{2} \\ 3 & : x = \frac{\pi}{2} \end{cases}$ is continuous at $x = \frac{\pi}{2}$, then $k$ is equal to

Let $A = \begin{bmatrix} 1 & 2 & 1 \\ -1 & 3 & 2 \\ 2&4&1\end{bmatrix}$ and $M_{ij}$, $A_{ij}$ respectively denote the minor, co-factor of an element $a_{ij}$ of matrix A, then which of the following are true? (A) $M_{22} = -1$ (B) $A_{23} = 0$ (C) $A_{32} = 3$ (D) $M_{23} = 1$ (E) $M_{32} = -3$ Choose the correct answer from the options given below:

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

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

Let $A = \{1, 2, 3\}$. Then, the number of relations containing $(1, 2)$ and $(1, 3)$ which are reflexive and symmetric but not transitive, is

for $|x| < 1$, $sin (tan^{-1}x)$ equal to

If a line makes angles $\alpha, \beta, \gamma$ 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

$\int_{\pi/6}^{\pi/3} \frac{tan x}{tan x + cot x} dx$ is equal to

Match List-I with List-II | List-I | List-II | |---|---| | (A) An observed set of population selected for analysis | (I) Parameter | | (B) A specific characteristic of the population | (II) Hypothesis | | (C) A specific characteristic of the sample | (III) Statistic | | (D) A statement made about a population parameter for testing | (IV) Sample | Choose the correct answer from the options given below:

If $y = a + b(x - 2022)$ is a straight line trend using the least square method for the following data | Year ($x$) | 2020 | 2021 | 2022 | 2023 | 2024 | |---|---|---|---|---|---| | Profit (Rs. '000) ($y$) | 2 | 3 | 4 | 5 | 2 | Then the value of $\frac{a}{b}$ is:

If $e^y = \log x$, then which of the following is true?

Let $e^{\alpha y} + e^{\beta y} + \gamma x^2 + \delta \log|x| + C = 0$, where $C \in \mathbb{R}$ be a particular solution of the differential equation $x(e^{2y} - 1)dy + (x^2 - 1)e^ydx = 0$ and passes through the point $(1, 1)$. The value of $(\alpha + \beta + \gamma + \delta - C)$ is

A sofa set costing ₹ 36000 has a useful life of 10 years. If the annual depreciation is ₹ 3000, then the scrap value by linear method is:

What is the mean of the numbers obtained on throwing a die having written 1 on three faces, 2 on two faces and 5 on 1 face?

Which of the following is NOT a basic requirement of the linear programming problem (LPP)?

Which of the following are correct? (A) Time series analysis does not help to understand the behavior of a variable in the past. (B) Time series predict the future behavior of variable. (C) Time series helps to plan future operations. (D) The main aim of the time series analysis is to derive conclusions after arranging the time series in a systematic manner. Choose the correct answer from the options given below:

A person invested ₹ 10000 in a stock of a company for 6 years. The value of his investment at the end of each year is given in the following table: | 2018 | 2019 | 2020 | 2021 | 2022 | 2023 | |---|---|---|---|---|---| | ₹ 11000 | ₹ 11500 | ₹ 13000 | ₹ 11800 | ₹ 12200 | ₹ 14000 | The compound annual growth rate (CAGR) of his investment is: [Given $(1.4)^{1/6} = 1.058$]

Which of the following are the assumptions underlying the use of t-distribution? (A) The variance of population is known. (B) The samples are drawn from a normally distributed population. (C) Sample standard deviation is an unbiased estimate of the population variance. (D) It depends on a parameter known as degree of freedom. Choose the correct answer from the options given below:

Which of the following is not a component of the time series?

Which of the following statements is incorrect?

The least non-negative remainder when $3^{128}$ is divided by 7 is:

Match List-I with List-II | List-I | List-II | | :--- | :--- | | (Matrix) | (Inverse of the Matrix) | | (A) $\begin{pmatrix} 1 & 7 \\ 4 & -2 \end{pmatrix}$ | (I) $\begin{pmatrix} 2/15 & 1/10 \\ -1/15 & 1/5 \end{pmatrix}$ | | (B) $\begin{pmatrix} 6 & -3 \\ 2 & 4 \end{pmatrix}$ | (II) $\begin{pmatrix} 1/5 & -2/15 \\ -1/10 & 7/30 \end{pmatrix}$ | | (C) $\begin{pmatrix} 5 & 2 \\ -5 & 4 \end{pmatrix}$ | (III) $\begin{pmatrix} 1/15 & 7/30 \\ 2/15 & -1/30 \end{pmatrix}$ | | (D) $\begin{pmatrix} 7 & 4 \\ 3 & 6 \end{pmatrix}$ | (IV) $\begin{pmatrix} 2/15 & -1/15 \\ 1/6 & 1/6 \end{pmatrix}$ | Choose the correct answer from the options given below:

If $X = 11$ and $Y = 3$, then $X \mod Y = (X + aY) \mod Y$ holds

A person wishes to purchase a house for ₹ 39,65,000 with a down payment of ₹ 5,00,000 and balance in equal monthly installments (EMI) for 25 years. If bank charges 6% per annum (compounded monthly), then EMI on reducing balance payment method is: [Given $(1.005)^{300} = 4.465$]

How many minimum number of times must a man toss a fair coin so that the probability of having at least one head is more than 90%?

Which of the following are correct about the Sinking Fund? (A) It is a fixed term account. (B) It is a set-up for a particular upcoming expense. (C) A fixed amount at regular intervals is deposited in the Sinking Fund. (D) It can be used in any emergency. Choose the correct answer from the options given below:

A tub contains 60 litres of milk. From this tub, 6 litres of milk was taken out and replaced with water. This whole process was repeated further two more times. How much milk is there in the tub now?

The slope of the normal to the curve $y = 2x^2$ at $x = 1$ is:

Let $F(z)$ be the cumulative density function of the standard normal variate $z$, then which of the following are correct? (A) $F(z) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^z e^{-\frac{z^2}{2}} dz$, $-\infty < z < \infty$ (B) $F(-z) = 1 - F(z)$ (C) $F(0) = 0$ (D) $F(\infty) = 1$ Choose the correct answer from the options given below:

Which of the following statements are correct in reference to the linear programming problem(LPP): Maximize ${Z} = 5x + 2y$ subject to the following constraints $3x + 5y \leq 15$, $5x + 2y \leq 10$, $x \geq 0, y \geq 0$. (A) The LPP has a unique optimal solution at $(2, 0)$ only. (B) The feasible region is bounded with corner points $(0, 0)$, $(2, 0)$, $(20/19, 45/19)$ and $(0, 3)$. (C) The optimal value is unique, but there are an infinite number of optimal solutions. (D) The feasible region is unbounded. Choose the correct answer from the options given below:

A person can row a boat in still water at the rate of 5 km/hr. It takes him 4 times as long to row upstream of a river as to row downstream to cover same distance in the same river. The speed of flow of the stream is

Which of the following inequalities holds true? (A) $\sqrt{5} + \sqrt{3} > \sqrt{6} + \sqrt{2}$. (B) If $a > b$ and $c < 0$, then $\frac{a}{c} < \frac{b}{c}$. (C) $\frac{1}{x^2} > \frac{1}{x} > 1$, if $0 < x < 1$. (D) If $a$ and $b$ are positive integers and $\frac{a - b}{6.25} = \frac{4}{2.5}$, then $b > a$. Choose the correct answer from the options given below:

For the given 5 values, 15, 18, 21, 27,39; the three year moving averages are:

The probability distribution of the random variable X is given by | X | 0 | 1 | 2 | 3 | |---|---|---|---|---| | P(X) | 0.2 | k | 2k | 2k | The variance of the random variable X is

Let A be a non-singular matrix of order 3 and $|A| = 15$, then $|adj A|$ is equal to

Two runners, Ajay and Vijay complete a 600 m race in 38 seconds and 48 seconds respectively. By how many meters will Ajay defeat Vijay?

An annuity in which the periodic payment begin on a fixed date and continue forever is called

If a 95% confidence interval for a population mean was reported to be 132 to 160 and sample standard deviation $\sigma = 50$, then the size of the sample in the study is: (Given $z_{0.025} = 1.96$)

If $A = \begin{bmatrix} 3 & 7 \\ 4 & -2 \end{bmatrix}$, $X = \begin{bmatrix} \alpha \\ -2 \end{bmatrix}$, $B = \begin{bmatrix} 7 \\ 32 \end{bmatrix}$ and $AX = B$, then the value of the $\alpha$ is

If $P$, $Q$ and $R$ are three singular matrices given by $P = \begin{bmatrix} 2 & 3a \\ 4 & 3 \end{bmatrix}$, $Q = \begin{bmatrix} b & 5 \\ 2a & 6 \end{bmatrix}$ and $R = \begin{bmatrix} a^2 + b^2 - c & 1 - c \\ c + 1 & c \end{bmatrix}$, then the value of $(2a + 6b + 17c)$ is

If $\int \frac{(1 + x \log x)}{xe^{-x}} dx = e^x f(x) + C$, where C is constant of integration, then $f(x)$ is

The original value of an asset minus the accumulated depreciation at a given date is known as

The total cost $C(x)$ in Rupees associated with the production of $x$ units of an item is given by $C(x) = 0.007x^3 + 26x^2 + 15x + 400$. The marginal cost when 10 items are produced is: