CUET Mathematics 2025 PYQs21 May Shift 1. Free, no login required.

Match List-I with List-II | List-I | List-II | | --- | --- | | Differential Equation | General solution | | --- | --- | | (A) $\dfrac{dy}{dx} = \dfrac{y}{x}; x \neq 0$ | (I) $y = cx; c \text{ is an arbitrary constant}$ | | (B) $x dx - y dy = 0; y \neq 0, x \neq 0$ | (II) $x^2 - y^2 = c; c \text{ is an arbitrary constant}$ | | (C) $\dfrac{(x^2 - 1)}{y^2 + 1}\dfrac{dx}{dy} = 1$ | (III) $2x + 3y = c; c \text{ is an arbitrary constant}$ | | (D) $2 dx + 3 dy = 0$ | (IV) $(x^3-y^3) = c + 3(x+y); c \text{ is an arbitrary constant}$ | Choose the correct answer from the options given below:

If $A = \begin{bmatrix} 3 & -2 \\ 4 & -2 \end{bmatrix}$, $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ and $A² = KA - 2I$, then the value of K is

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

Let $A = [a_{ij}]_{3×2}$ and $B = [b_{ij}]_{3×4}$ be two matrices. Then the order of the matrix $(A^T . B)^T$ is:

If the objective function $z = px + qy$ has its maximum value at the points (2, 1) and (0, 6), then the relationship between p and q is:

For $x > e$, $\int \frac{dx}{x - \sqrt{x}}$ is equal to

The area (in square units) of the region enclosed between the lines $x + y = 2$, $x = 0$, $x = 3$ and $x$-axis is equal to

If the random variable X has the following probability distribution: | X | 0 | 1 | 2 | Otherwise | |---|---|---|---|---| | P(X) | K | 3K | 5K | 0 | then K is equal to

Match List-I with List-II | List-I | List-II | | --- | --- | | Definite integral | Value | | --- | --- | | (A) $\int_{1}^{e} \frac{\log x}{x} dx$ | (I) 4 | | (B) $\int_{-2}^{2} x^3(1 - x^2) dx$ | (II) $\frac{1}{2}$ | | (C) $\int_{1}^{2} x \, dx$ | (III) 0 | | (D) $\int_{-2}^{2} \vert x\vert dx$ | (IV) $\frac{3}{2}$ | Choose the correct answer from the options given below:

Let $M_{ij}$ and $A_{ij}$ denote respectively minors and co-factors of the element in the $i^{th}$ row and $j^{th}$ column of the matrix $A = \begin{bmatrix} 1 & 2 & -1 \\ 3 & 2 & 3 \\ 4 & -1 & 0 \end{bmatrix}$. Then: (A) $M_{32} = 6$ (B) $M_{23} = 9$ (C) $A_{32} = -6$ (D) $A_{23} = -9$ Choose the correct answer from the options given below:

The inverse of the matrix $A = \begin{bmatrix} \frac{1}{2} & 0 & 0 \\ 0 & \frac{1}{4} & 0 \\ 0 & 0 & \frac{1}{8} \end{bmatrix}$ is

The real valued function $f(x) = x^{15} + 5x^9 + 10$ is increasing for___________.

The corner points of the bounded feasible region determined by the system of linear constraints are $(0, 0), (5, 0), (6, 5), (6, 8), (4, 10), (0,8)$. Let $z = 3x - 4y$ be the objective function. Then the minimum value of the objective function z occurs at

The number of arbitrary constants in the general solution of a differential equation with degree 1 and order 3, is

If $y = ax^2 + bx$ has minima at $x = 2$ and the minimum value is -12, then which of the following are correct? (A) $a = 3$ (B) $a = -3$ (C) $b = 12$ (D) $b = -12$ Choose the correct answer from the options given below:

A can solve 90% problems and B can solve 70% problems of the book. A problem is selected at random from the book. The probability that the problem is solved, is equal to

Match List-I with List-II | List-I | List-II | |---|---| | (Inverse Trigonometric Function) | (Principal Value) | | (A) $\sin^{-1}(-\frac{1}{2})$ | (I) ${\pi}/{6}$ | | (B) $\cos^{-1}(-\frac{1}{2})$ | (II) $-{\pi}/{6}$ | | (C) $\tan^{-1}(-\sqrt{3})$ | (III) ${2\pi}/{3}$ | | (D) $\cot^{-1}(\sqrt{3})$ | (IV) $-{\pi}/{3}$ | Choose the correct answer from the options given below:

The value of $\int \left\{ \frac{1}{\log_e x} - \frac{1}{(\log_e x)^2} \right\} dx$ is

The value of k for which the lines $\frac{2x - 3}{4} = \frac{3 - y}{k} = \frac{z - 2}{-2}$ and $\frac{x - 2}{1} = \frac{y}{4} = \frac{5 - z}{3}$ are perpendicular to each other is:

If $\begin{vmatrix} 1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c \end{vmatrix}$, where a, b and c are non zero constants, then the value of $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$ is

Match List-I with List-II | List-I | List-II | |---|---| | (A) The vectors $\lambda\hat{i}$ $ + \hat{j} + 2\hat{k}$ and $\vec{i} + \lambda\hat{j} + \hat{k}$ are perpendicular if λ is equal to | (I) 1 | | (B) The vectors $3\hat{i} + 6\hat{j} - \hat{k}$ and $2\hat{i} + 4\hat{j} - \lambda\hat{k}$ are collinear if λ is equal to | (II) -1 | | (C) The number of vectors of unit-length which are perpendicular to both the vectors $\vec{a}$ = $\hat{i} + \hat{j} + 2\hat{k}$ and $\vec{b}$ = $3\hat{i} - \hat{j} + 5\hat{k}$ is | (III) 2/3 | | (D) If $\lvert \vec{a} \rvert = 1$ and $\vec{a} + \vec{b} = \vec{0}$, then $\lvert \vec{b} \rvert$ is equal to | (IV) 2 | Choose the correct answer from the options given below:

The area (in square units) bounded by the curve $y = \cos x$ between $x = 0$ and $x = 2\pi$ in first quadrant is equal to:

Match List-I with List-II | List-I | List-II | |---|---| | (A) Maximum value of $f(x) = \sin^2 x - \cos^2 x$, $\forall x \in (\pi, 2\pi)$ is | (I) 0 | | (B) Minimum value of $f(x) = \sin x \cos x$ | (II) 1 | | (C) Point of Minima of $f(x) = x^x$ $(x > 0)$ | (III) $-\frac{1}{2}$ | | (D) Maximum value of $f(x) = -x^{2026}$ | (IV) $\frac{1}{e}$ | Choose the correct answer from the options given below:

If $\vec{a}, \vec{b}, \vec{c}$ are vectors such that $\vec{a} + \vec{b} + \vec{c} = \vec{0}$ and $|\vec{a}| = 1, |\vec{b}| = 2, |\vec{c}| = 5$, then the expression $\vec{a}\cdot\vec{b} + \vec{b}\cdot\vec{c} + \vec{c}\cdot\vec{a}$ equals

If $\vec{a}$, $\vec{b}$ and $\sqrt{3}\vec{a} - \vec{b}$ are three unit vectors, then the angle between $\vec{a}$ and $\vec{b}$ is:

If the matrix $A = \begin{bmatrix} 1 & 3 \\ 2 & 1 \end{bmatrix}$, then the value of det $(A^2 - 2A)$ is equal to

If $x\sqrt{1 + y} + y\sqrt{1 + x} = 0$, where $|x| < 1, |y| < 1$ and $x ≠ y$, then

A integrating factor of the differential equation $\frac{dy}{dx} + \frac{y}{x} = \frac{1}{x^2}$, $(x > 0)$ is equal to

The value of $\int_0^{\pi/2} \log_e \left(\frac{5 + 2 \sin x}{5 + 2 \cos x}\right) dx$ is

If R be a relation on the set of integers Z, given by R = {(a, b) : (a - b) is a multiple of 3}, then R is:

Let N be set of natural numbers, then the function $f: \mathbb{N} \rightarrow \mathbb{N}$, defined by $f(x) = \begin{cases} \frac{n+1}{2} & \text{, if } n \text{ is odd} \\ \frac{n}{2} & \text{, if } n \text{ is even} \end{cases}$, is

Match List-I with List-II (Given that $c$ is an arbitrary constant) | List-I | List-II | | --- | --- | | (A) $\int \frac{dx}{\sqrt{a^2 - x^2}} =$ | (I) $\log_e \vert x + \sqrt{x^2 - a^2}\vert + c$ | | (B) $\int \sqrt{a^2 - x^2} dx =$ | (II) $\sin^{-1} \frac{x}{a} + c$ | | (C) $\int \sqrt{x^2 - a^2} dx =$ | (III) $\frac{x}{2} \sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{-1} \frac{x}{a} + c$ | | (D) $\int \frac{dx}{\sqrt{x^2 - a^2}} =$ | (IV) $\frac{x}{2} \sqrt{x^2 - a^2} - \frac{a^2}{2} \log_e \vert x + \sqrt{x^2 - a^2}\vert + c$ | Choose the correct answer from the options given below:

If a line makes angles $\alpha$, $\beta$ and $\gamma$ with the positive directions of coordinate axes respectively, then $\cos 2\alpha + \cos 2\beta + \cos 2\gamma$ is equal to

Let A be any square matrix of order n, then which of the following are true? (A) $| \text{adj } A| = |A|^{n-1}$ (B) $|A^{-1}| = \frac{1}{|A|}$ (C) $| \text{adj } A| = |A|^n$ (D) $(A^T)^{-1} = (A^{-1})^T$ Choose the correct answer from the options given below:

If A and B are two independent events, then which of the following is/are true? (A) $P(A \cap B) = 0$ (B) $P(A \cup B) = 1 - P(A')P(B')$ (C) $P(A \cup B) = P(A)P(B)$ (D) $P(A \cap B) = P(A)P(B)$ Choose the correct answer from the options given below:

Let the matrix $A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}$. Then which of the following are true? (A) adj $A = \begin{bmatrix} 4 & -3 \\ -1 & 2 \end{bmatrix}$ (B) det $(A) = 5$ (C) det (adjA) = 25 (D) If $A^3 = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, then $a + b = c + d$ Choose the correct answer from the options given below:

The greatest integer function $f(x) = [x]$ is differentiable for all values of

The interval in which the function $f(x) = 2x^3 + 3x^2 - 12x + 1$ is strictly increasing, is

The general solution of the differential equation $x(1 + y^2)dx + y(1 + x^2)dy = 0$ is

The corner points of the bounded feasible region determined by the system of linear constraints are (0, 0), (5, 0), (3, 4) and (0, 5). Let $Z = px + qy$ where $p, q > 0$. Condition on p and q so that the maximum of Z occurs at both (5, 0) and (3, 4) is

A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be heart then the probability of the missing card to be a heart is:

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

For some constant 'k', if the system of linear equations $2x - y + 3z = 1$ $x - 2y + z = 3$ $kx + y - z = 0$ has a unique solution, then

The point on the curve $y^2 = 8x$ for which the abscissa and ordinate change at the same rate, is

If $f(x) = \begin{cases} \frac{\tan(\frac{\pi}{4} - x)}{\cot 2x} & , x ≠ \frac{\pi}{4} \\ 2K + 1 & , x = \frac{\pi}{4} \end{cases}$ is continuous at $x = \frac{\pi}{4}$, then the value of K is equal to

If $\begin{bmatrix} x+y & 4 \\ 1+z & y \end{bmatrix} = \begin{bmatrix} 2 & 4 \\ 5 & 6 \end{bmatrix}$, then

The acute angle between the lines $\vec{r} = (4\hat{i} - \hat{j}) + \lambda(2\hat{i} + \hat{j} - 3\hat{k})$ and $\frac{x-1}{1} = \frac{y+1}{-3} = \frac{z-2}{2}$ is

Let $|\vec{a}| = 5, |\vec{b}| = 2$ and $\vec{a}\cdot\vec{b} = 6$, then the value of $|\vec{a} \times \vec{b}|$ is equal to

Let E and F be two events such that $P(E) = \frac{1}{3}$, $P(F) = \frac{1}{4}$ and $P(E \cap F) = \frac{1}{5}$. Then the value of $P(F|E)$ is equal to

Consider a L.P.P, Maximize $Z = 3x + 5y$ subject to constraints $2x + 6y ≤ 6$, $x - y ≥ 0$, $x ≥ 0$, $y ≥ 0$. Then which of the following are true? (A) The feasible region of L.P.P is bounded region (B) The corner points of the feasible region are (0, 0), (2, 2), (0, 1) (C) Maximum value of Z is 9 (D) Point (1, 3) lies in the feasible region Choose the correct answer from the options given below:

There are 50 telephone lines in an exchange. The probability that any one of them will be busy is 0.1, then the probability that all the lines are busy?

The system of equations $x + ky = 0$ $3x + 5y = 0$ has infinitely many solutions, then k is equal to

The differential equation representing the curve $y = e^{2x}(a + bx)$, where a, b are arbitrary constants is

Anshu takes a personal loan of ₹10,00,000 at the rate of 12% per annum for 2 years, then the EMI by using flat rate method is

If A is a square matrix of order 3 such that $A( {adj } A) = \begin{bmatrix} -2 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & -2 \end{bmatrix}$, then $|A|$ is equal to

If $(t_1, y_1), (t_2, y_2), (t_3, y_3), ..., (t_n, y_n)$ denote the time series and $y_t$ are the trend values of the variable y, then $\sum(y - y_t)$, the sum of the deviations of y from their corresponding trend value is equal to:

A mobile phone costing ₹50000 has a useful life of 5 years. If the annual depreciation is ₹5000, then by using a linear method, its scrap value is

What sum of money is needed to invest now, so as to get ₹5000 at the beginning of every month forever, if the money is worth 6% per annum compounded monthly?

If $\begin{bmatrix} 1 & 3 & 9 \\ 1 & x & x^2 \\ 4 & 6 & 9 \end{bmatrix}$ is singular matrix, where $x \in \mathbb{N}$ (where N set of natural number), then x is equal to

If a random variable y follows Poisson's distribution such that $P(X = 2) = 9 P(X = 4) + 90 P(X = 6)$, then sum of the mean and variance of X is

The rise and fall of share market is an example of

The point on the curve $\frac{x^2}{4} + \frac{y^2}{9} = 1$ at which the tangent to the curve is parallel to the x-axis is

If $\begin{bmatrix} x-y & t \\ 2x-y & w \end{bmatrix} = \begin{bmatrix} -1 & 2 \\ 0 & 1 \end{bmatrix}$, then $2x + y + 3t + w$ is equal to

The least positive integer x satisfying $29 \equiv x \pmod{5}$ is

If for the following data: | Year (x) | 2000 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 | |---|---|---|---|---|---|---|---| | Production(in tonnes) (y) | 40 | 45 | 46 | 42 | 47 | 50 | 46 | the equation of the straight line trend is $y = 45.143 + 1.036(x - 2003)$, then the trend value for the year 2004 is

Consider the following hypothesis test, $H_0: \mu = 15$ $H_a: \mu ≠ 15$ A sample of 50 provided a sample mean of 14.15. If the sample standard deviation is 3, then the value of the test statistic t0-test is

The mean of the number of heads in a simultaneous toss of three coins is

The demand for a certain product is represented by the function $p = 300 + 25x - x^2$ (in rupees), where x is the number of units demanded and p is the price per unit, then the marginal revenue when 15 units are sold, is

A die is thrown 6 times. If getting an odd number is a success, then the probability of at least 5 successes is

A pipe A can fill a tank in 4 hours. There are two outlet pipes, B and C, from the tank which can empty it in 8 and 10 hours respectively. If all the three pipes are opened simultaneously, how long will it take to fill the tank?

In a game of 100 points, A can give B 20 points and C 28 points. How many points can B give C?

In a linear programming problem(LPP), the maximum value of the objective function $Z = 2x + 5y$ subjected to the constraints: $2x + 3y ≤ 6$ $2x + y ≤ 4$ $x, y ≥ 0$ is

Ram can row a boat in still water at the rate of 6 km/hr. He finds that it takes him twice as long to row upstream of a river as to row downstream of the same river, then the speed of the current of the river is

Which of the following are correct about the linear programming problem(LPP)? (A) The objective function is always linear. (B) A corner point of a feasible region is a point in the region which is the intersection of two boundary lines. (C) The common region determined by all the linear constraints of a LPP is called the feasible region. (D) If an LPP admits optimal solution at two points then its optimal values occurs at an infinite number of points. Choose the correct answer from the options given below:

Which of the following statements are correct? (A) In the sinking fund a fixed amount at regular intervals is deposited. (B) Sinking fund is a long-term account which can be closed any time. (C) In a saving account, any amount, any time can be deposited. (D) Sinking fund can be used only for the purpose it was created. Choose the correct answer from the options given below:

The demand function for a commodity is $p = 35 - 2x - x^2$, then the consumer's surplus at equilibrium price $p_0 = 20$ is

Which of the following is correct about the compound annual growth rate(CAGR)?

If $f(x) = x^2 - 4x + 13, x \in \mathbb{R}$, then which of the following are correct? (A) $x = 2$ is a stationary point of $f(x)$. (B) $f(x)$ is increasing function on $(2, \infty)$ (C) $f(x)$ have maxima at $x = 2$ (D) $f(2) = 9$ Choose the correct answer from the options given below:

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

Which of the following are correct? (A) The probability curve in t-distribution is symmetric about the line t=0. (B) t-axis is an asymptote of the curve. (C) The variable t of t-distribution ranges from $-\infty$ to $\infty$ (D) As the number of degrees of freedom increases, the t-distribution curve moves closer to the binomial distribution. Choose the correct answer from the options given below:

For the given five values 18, 25, 20, 36, 49, the three years moving averages are

If water is mixed with milk to gain 20% by selling the mixture at cost price. Then ratio of water to milk in mixture, is:

If we reject the null hypothesis when it is true, we might be making

The number of years required for a sum of money to get tripple at the effective rate of 4% is : (Given $(1.04)^{28} = 3)$

The solution set of the inequation $|2x - 3| ≤ 4$ is