CUET Mathematics 2025 13 May Shift 2 PYQs

The general solution of the differential equation $\frac{dy}{dx} = xy + x + y + 1$ is

If $A = \begin{bmatrix} a & 4 & -5 \\ d & b & -6 \\ 5 & e & c \end{bmatrix}$ is a skew symmetric matrix, then value of $a + b + c + d + e$ is equal to

A random variable X has the following probability distribution | X | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |---|---|---|---|---|---|---|---|---|---| | P(X) | a | 3a | 5a | 7a | 9a | 11a | 13a | 15a | 17a | Then the values of 'a' and P(0 < X < 5) respectively are

$\int \left(\frac{1}{log_e t} - \frac{1}{(log_e t)^2}\right) dt$ is equal to

If $A = \begin{bmatrix} 1 & 2 & 1 \\ 2 & 3 & 1 \\ 0 & 0 & 1 \end{bmatrix}$ then $|adj(3A^T)|^2$ is equal to

$\int_0^2 x(2-x)^n dx$ is equal to

The objective function of an LPP is $z = ax + by$. If the maximum value of the objective function is 180, which occurs at two points (15,15) and (0,20), then which one of the following is true?

If $A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ then $A^{-1}$ is equal to

Match List-I with List-II | List-I | List-II | |---|---| | **Differential equation** | **Degree** | | (A) $\frac{d^2y}{dx^2} + \sqrt{\frac{dy}{dx}} - y = 0$ | (I) 6 | | (B) $\sqrt{\frac{d^3y}{dx^3}} - \sqrt[12]{\frac{d^2y}{dx^2}} = 0$ | (II) Not defined | | (C) $\left(\frac{d^2y}{dx^2}\right)^2 + \frac{dy}{dx} + e^{\frac{dx}{dx}} = x^2$ | (III) 3 | | (D) $\sqrt[3]{\frac{dy}{dx}} - \frac{d^2y}{dx^2} = e^x$ | (IV) 2 | Choose the correct answer from the options given below:

If P, Q and R are matrices of order 2x3, 3x5 and 5x3 respectively. Then which of the following are valid? (A) P Q R (B) P R Q (C) Q R (D) R Q (E) P R Choose the correct answer from the options given below:

The function $f(x) = \frac{x}{2} + \frac{2}{x}, x \neq 0$ is increasing on (A) $(-\infty, -2)$ (B) $(-2, 2)$ (C) $(2, \infty)$ (D) $(-1, 1)$ Choose the correct answer from the options given below:

The absolute maximum value of the function $f(x) = 4x - \frac{1}{2}x^2$ in the interval $\left[-2, \frac{9}{2}\right]$ is

The area (in sq. units) of the region bounded by the lines $y = 2x + 3$, the x – axis and the ordinates $x = -2$ and $x = 2$ is equal to

Let $e^y(x+1) = 1$. Then which of the following are TRUE? (A) $\frac{d^2y}{dx^2} = -\frac{1}{(x+1)^2}$ (B) $\frac{d^2y}{dx^2} = \left(\frac{dy}{dx}\right)^2$ (C) $\left.\frac{d^2y}{dx^2}\right|_{x=0} = -1$ (D) $\left.\frac{d^2y}{dx^2}\right|_{x=0} = 1$ (E) $\left.\frac{d^2y}{dx^2}\right|_{x=1} = \frac{1}{4}$ Choose the correct answer from the options given below:

If the corner points of the bounded feasible region of an LPP with objective function Maximize $z = 2x + 3y$ are (0,0), (1,2) and (1,1), then its optimal value is

Let $f(x)=\begin{cases}\dfrac{|x|}{x},&x\ne0\\1,&x=0\end{cases}$ and $g(x)=\begin{cases}x\sin\left(\dfrac{1}{x}\right),&x\ne0\\0,&x=0\end{cases}$ Then at the origin, which one of the following is true?

Let $A = \begin{bmatrix} 2 & -3 & 4 \\ 0 & 1 & 5 \\ -4 & 2 & 3 \end{bmatrix}$ and $a_{ij}$ be any element of matrix A, i, j ∈ {1,2,3}, then which of the following are TRUE? (A) Minor of $a_{23} = 16$ (B) Minor of $a_{23} = -8$ (C) Cofactor of $a_{23} = -16$ (D) Cofactor of $a_{23} = 8$ (E) Cofactor of $a_{13} = 4$ Choose the correct answer from the options given below:

If A and B are independent events such that $P(A|B) = \frac{1}{3}$ and $P(B) = \frac{1}{2}$, then the value of $P(A \cap B)$ is equal to

Match List-I with List-II Let $f: A \rightarrow B$ be a function given by $f(x) = x^2$ | List-I | List-II | |---|---| | **Domain and Co-domain** | **Kind** | | (A) $A = \mathbb{R}$ and $B = \mathbb{R}$ | (I) $f$ is both one-one and onto | | (B) $A = \mathbb{R}$ and $B = [0, \infty]$ | (II) $f$ is one-one but not onto | | (C) $A = B = [0, \infty]$ | (III) $f$ is not one-one but onto | | (D) $A = [0, \infty]$ and $B = \mathbb{R}$ | (IV) $f$ is neither one-one nor onto | Choose the correct answer from the options given below:

The shortest distance between the lines $\vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(2\hat{i} + 3\hat{j} + 4\hat{k})$ and $\vec{r} = (2\hat{i} + 4\hat{j} + 5\hat{k}) + \mu(3\hat{i} + 4\hat{j} + 5\hat{k})$ is equal to

The feasible region of the linear programming problem is represented below: <img src="https://balti.afterboards.in/2p1sfNGWOxx6GJZ" width="400px"/> The constraints of this LPP are

Let $\begin{vmatrix} x & 2 \\ 18 & x \end{vmatrix} = \begin{vmatrix} -4 & -2 \\ -8 & -4 \end{vmatrix}$. Then (A) $x = -4$ (B) $x = -6$ (C) $x = 4$ (D) $x = 6$ Choose the correct answer from the options given below:

$\int_0^1 tan^{-1}\left(\frac{2x-1}{1+x-x^2}\right)dx$ is equal to

The probability of not getting 53 Sundays in a leap year is

If $x = \frac{1-t}{1+t}$ and $y = \frac{3t}{1+t}$, then $\frac{d^2y}{dx^2}$ is equal to

Match List-I with List-II | List-I | List-II | | --- | --- | | (A) Point of minima of $f(x) = \vert x+1\vert $ | (I) 1 | | (B) Minimum value of $f(x) = \vert x\vert $ | (II) -1 | | (C) Maximum value of $f(x) = 1 - x^2$ | (III) 2 | | (D) Minimum value of $f(x) = 2 + \sin^2 x$ | (IV) 0 | Choose the correct answer from the options given below:

The area (in sq. units) of the region bounded by the curve $y = \sin x, -2\pi \leq x \leq 2\pi$ and $x - axis$ is equal to

The length of line segment joining the points with position vectors $2\hat{i} - 2\hat{j} + 3\hat{k}$ and $5\hat{i} + 2\hat{j} + 3\hat{k}$ is

The optimal value of the objective function of the LPP, Minimize $Z = 3x - 2y$ subject to constraints $x + y \ge 10$, $3x + 5y \ge 15$, $x \ge 0$, $y \ge 0$, is equal to:

Match List-I with List-II | List-I | List-II | |---|---| | (A) Angle between $\hat{i} - \hat{j}$ and $\hat{i} + \hat{j}$ | (I) $\pi$ | | (B) Angle between $\hat{i} - \hat{j} + \hat{k}$ and $-\hat{i} + \hat{j} - \hat{k}$ | (II) $\frac{3\pi}{4}$ | | (C) Angle between $\hat{i} + \hat{j}$ and $-\hat{i}$ | (III) $\frac{\pi}{4}$ | | (D) Angle between $\hat{i} + \hat{k}$ and $\hat{k}$ | (IV) $\frac{\pi}{2}$ | Choose the correct answer from the options given below:

Let L be the set of all lines in a plane and R be the relation on set L defined by $R = \{(L_1, L_2): L_1 \perp L_2\}$ Then R is (A) an equivalence Relation (B) a symmetric Relation (C) not a transitive Relation (D) a reflexive Relation Choose the correct answer from the options given below:

The area (in sq. units) of the region $\{(x,y): 3x^2 \leq y \leq |x|\}$ is equal to

If $y = \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right), 0 < x < 1$, then $\frac{dy}{dx}$ is equal to

A vector of magnitude 8 units in the direction perpendicular to both the vectors $\hat{i} + \hat{j} + \hat{k}$ and $2\hat{i} + \hat{k}$ is

$\int_{-\frac{5}{2}}^{\frac{5}{2}} |x| dx$ is equal to

If A and B are square matrices of order 3 such that $|A| = 3$ and $|B| = -1$, then $|3AB|$ is equal to

The sum of two positive numbers is 60. If the sum of their squares in minimum, then the absolute value of the difference of their cubes is

Which of the following statements is (are) true? (A) $B^T AB$ is a skew-symmetric matrix if A is a symmetric matrix (B) $B^T AB$ is a symmetric matrix if A is a symmetric matrix (C) $B^T AB$ is a symmetric matrix if A is a skew-symmetric matrix (D) $B^T AB$ is a skew-symmetric matrix if B is a skew-symmetric matrix (E) $B^T AB$ is a symmetric matrix if B is a symmetric matrix Choose the correct answer from the options given below:

If $f(x) = x^3 e^{-x}$, then the value of $f''(1)$ is equal to

For what value of k, the following system of equations has infinitely many solutions? $x + 2y = 5, 3x + ky = 15$

If $A$ is a square matrix and $I$ is an identity matrix of same order such that $A^2 = A$, then $(2I + A)^2 - 5A$ is

If the lines $\frac{1-x}{3} = \frac{3y-6}{k} = \frac{3-z}{-2}$ and $\frac{1-x}{2k} = \frac{y-5}{3} = \frac{6-z}{5}$ are perpendicular to each other, then $k$ is equal to

If A and B are two events such that $P(A|B) = P(B|A)$, and $A \cap B \neq \phi$ then

$\int \left(\frac{\cos x - \sin x}{1 + \sin 2x}\right) dx$ is equal to

The integrating factor of the differential equation, $x^2 \frac{dy}{dx} + xy = log_e x$ is equal to

Match List-I with List-II | List-I | List-II | |---|---| | (A) $\sin^{-1}(-1)$ | (I) $\frac{5\pi}{6}$ | | (B) $\cot^{-1}(-1)$ | (II) $\frac{-\pi}{2}$ | | (C) $\sec^{-1}\left(\frac{-2}{\sqrt{3}}\right)$ | (III) $\frac{\pi}{4}$ | | (D) $\tan^{-1}(1)$ | (IV) $\frac{3\pi}{4}$ | Choose the correct answer from the options given below:

A man is known to speak truth 3 out of 4 times. He throws a die and reports that it is four. The probability that it is actually four is

The general solution of differential equation $\frac{dy}{dx} = e^{x+y}$ is

Let $\vec{a}, \vec{b}$ be two vectors such that $|\vec{a}| = 2, |\vec{b}| = 3, \vec{a} \cdot \vec{b} = 4$, then $|\vec{a} - \vec{b}|$ is equal to

The direction cosines of a line equally inclined with the co-ordinate axes are

Arrange the following as per statistical inferences: (A) Data Analysis (B) Population (C) Making Inferences (D) Data Collection Choose the correct answer from the options given below:

In a box containing 100 bulbs, 10 are defective. The probability that out of a sample of 5 bulbs, none is defective is

For the system $\begin{bmatrix} 2 & -3 \\ -4 & 6 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 5 \\ -10 \end{bmatrix}$ which of the following statements are correct? (A) The system has no solution. (B) The system is consistent. (C) It has infinitely many solutions. (D) It has a unique solution. Choose the correct answer from the options given below:

If A is an invertible matrix of order 3 and the determinant of A is 9, then the determinant of $A^{-1}$ is:

The non-negative remainder when $7^{30}$ is divided by 5 is

At 6% converted quarterly, the present value of a perpetuity of ₹ 900 payable at the end of each quarter is:

If $a > b$ and $c < 0$, then which of the following is NOT correct? (A) $ac < bc$ (B) $a + c < b + c$ (C) $a - c < b - c$ (D) $ac > bc$ Choose the correct answer from the options given below:

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

For the given five values 15,24,18,33,42, the three years moving averages are

If the matrix $\begin{bmatrix} 0 & 7 & -12 \\ -7 & 0 & -5 \\ 2a & 5 & 3b \end{bmatrix}$ is skew-symmetric, then the value of $(4a + 3b)$ is:

Match List-I with List-II | List-I | List-II | |---|---| | **(Matrix)** | **(Determinant)** | | (A) $\begin{bmatrix} 1 & 7 \\ -3 & 5 \end{bmatrix}$ | (I) 24 | | (B) $\begin{bmatrix} -2 & 5 \\ -3 & -3 \end{bmatrix}$ | (II) 32 | | (C) $\begin{bmatrix} -12 & 8 \\ -16 & 8 \end{bmatrix}$ | (III) 21 | | (D) $\begin{bmatrix} 15 & 9 \\ -21 & -11 \end{bmatrix}$ | (IV) 26 | Choose the correct answer from the options given below:

If $r_{eff}$ = effective rate of interest, $r$ = nominal rate of interest and $m$ = number of conversion periods per year, the relationship between nominal rate and effective rate of interest is:

When the two independent small samples of sizes $n_1$ and $n_2$ with means $\bar{x}_1$ and $\bar{x}_2$ respectively are drawn from populations with identical population variances, the test-statistic is computed as

If the integral $I = \int \left[log_e(log_e x)^2 + \frac{a}{log_e x}\right] dx = x log_e(log_e x)^2 + C$, where C is constant of integration. Then the value of $a$ is:

The speed of a motorboat in still water and that of the current of water is in a ratio of 27:5. The boat goes along the current from point A to point B in 3 hours 40 minutes. How much time will it take to come back from B to A?

Let $P, I$ and $n$ be the principal of the loan, the total interest on the principal and number of months in the loan period respectively, then the EMI by Flat Rate Method is:

The interval in which the function $g(x) = x^2 e^{-x}$ is increasing is:

A shopkeeper has 300 Kg millet, a part of which he sells at 10 % profit. The remaining quantity of millet was of poor quality, and he sold it at 5 % loss. In the whole transaction, he made a profit of 7 %. The quantity of the millet sold at 5 % loss is:

Which of the following statements about the Sinking Fund are correct? (A) It is a fund established by a business entity by setting aside revenue over a period of time to fund a future capital expense. (B) It is a fund established by a business entity by setting aside revenue over a period of time to fund a future repayment of a long-term debt. (C) It is set up for any purpose that it may serve. (D) It is a fund that is accumulated for the purpose of paying off a financial obligation at some future designated date. Choose the correct answer from the options given below:

If $x^2 - y^2 = 1$, then which of the following is correct? (A) $(x^2 - 1)\left(\frac{dy}{dx}\right)^2 = x^2$ (B) $(x^2 - 1)\left(\frac{d^2y}{dx^2}\right)^2 = x^2$ (C) $(x^2 - 1)^3\left(\frac{d^2y}{dx^2}\right)^2 = x^2$ (D) $(x^2 - 1)^3\left(\frac{d^2y}{dx^2}\right)^2 = 1$ Choose the correct answer from the options given below:

On which of the following components, the pattern and behavior of the data in any time series is based? (A). Secular trend component (B). Seasonal component (C). Cyclical component (D). Regular component Choose the correct answer from the options given below:

Which of the following is incorrect about the Linear Programming Problem (LPP)?

Which of the following statements are NOT correct about Standard Normal Distribution? (A) The probability curve of the Standard Normal Distribution is a bell-shaped curve. (B) The Standard Normal variate (Z) score describes the position of each data point in terms of its distance from the mean, when measured in standard deviation units. (C) The Z-score is negative if the data point lies above the mean, and positive if it lies below the mean. (D) There is a 95.45 % probability of randomly selecting a score between $\mu - \sigma$ and $\mu + \sigma$, when $\sigma$ is standard deviation and $\mu$ is mean. Choose the correct answer from the options given below:

Two pipes can fill a cistern in 8 and 12 hours respectively. The pipes are opened simultaneously, and it takes 12 minutes more to fill the cistern due to leakage. If the cistern is full, what will be the time taken by the leakage to empty it?

A manufacturing unit makes two models, 'classic' and 'supreme' of the scooter. Each piece of the classic model requires 9 labour hours for assembling and 1 labour hour for finishing. Each piece of supreme model requires 12 labour hours for assembling and 3 labour hour for finishing. For assembling and finishing, the maximum labour hours available are 180 and 30 respectively. The company makes a profit of ₹ 10000 on each piece of the classic model and ₹ 15000 on each piece of the supreme model. Which of the following options describes the given Linear Programming Problem (LPP) to maximize the profit Z (Max Z) (where x and y are the number of pieces of the classic model and the supreme model respectively)?

Let a pair of dice be thrown and the random variable X be the sum of the numbers that appear on the two dice. Match List-I with List-II | List-I | List-II | |---|---| | **X** | **Probability, P(X)** | | (A) 4 | (I) $\frac{1}{6}$ | | (B) 5 | (II) $\frac{5}{36}$ | | (C) 6 | (III) $\frac{1}{12}$ | | (D) 7 | (IV) $\frac{1}{9}$ | Choose the correct answer from the options given below:

The general solution of the differential equation $e^x dy + (y e^x + 2x)dx = 0$ is

A random variable X has the following probability distribution: | X | 0 | 1 | 2 | 3 | |---|---|---|---|---| | P(X) | 0.1 | 0.2 | 0.3 | 0.4 | The variance of the X will be:

When data of the variable is collected at distinct time intervals for a specified period of time, it is called

Let A, B, C, D and E be matrices of order $2 \times n, 3 \times k, 2\times p, n \times 3$ and $p \times k$ respectively. Choose the correct statement(s) from the following? (A) EB + DB will be defined if $k = 3, p = n$. (B) EB + DB will be defined if $k = 2, p = 3$. (C) If n = p = 2, then the order of the matrix $5A^2 - 3C$ is $2 \times 2$. (D) If n = p, then the order of the matrix $5A^2 - 3C$ is $p \times k$. Choose the correct answer from the options given below:

The number of tangents to the curve $xy - 3y + 2 = 0$ having slope 2 is:

A coin is tossed twice and outcomes are recorded. If the random variable X represents the number of heads in the experiment, then the expectation of X will be:

Let $A = [a_{ij}]$ be a square matrix of order 2 with elements either 0 or 1. Then the difference between the possible number of singular and non-singular matrices is

Which of the following statements are correct about the Compound Annual Growth Rate (CAGR)? (A) It can be used to compare historical returns on different investment portfolios. (B) It helps smooth returns when growth rates are expected to be volatile and inconsistent. (C) It is unable to track the performance of various business measures of one or multiple companies alongside one another. (D) It can be used to calculate the average growth of a single investment. Choose the correct answer from the options given below:

On 1st April 2024, person 'X' purchased a machinery costing ₹ 65000 and spent ₹ 10000 on its installation. The estimated effective life of the machinery is 5 years with a scrap value of ₹ 10000. The annual depreciation using the straight-line method with the accounting year ending on 31st March 2025 is: