CUET Mathematics 2025 PYQs22 May Shift 2. Free, no login required.

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

$\int \frac{\sin 2x \, dx}{\sqrt{9 - \cos^4 x}}$ equals

Match List-I with List-II | List-I | List-II | |---|---| | **Differential Equation** | **Order and Degree** | | (A) $\left(\frac{d^2y}{dx^2}\right)^2 = e^x\left(\frac{dy}{dx}\right)^4 + 1 = 0$ | (I) order = 1 and degree = 2 | | (B) $\left(\frac{dy}{dx}\right)^2 + xy = 0$ | (II) order = 2 and degree = 1 | | (C) $\left(1 + \frac{dy}{dx}\right)^{3/2} = 4\left(\frac{d^2y}{dx^2}\right)^2$ | (III) order = 2 and degree = 2 | | (D) $\sqrt\frac{d^2y}{dx^2} + 1 = \frac{dy}{dx}$ | (IV) order = 2 and degree = 4 | Choose the correct answer from the options given below:

The area (in sq. units) of the triangle whose vertices are $(0, 0)$, $(a, 0)$, $(0, b)$, is equal to

If $A = \begin{bmatrix} 2 & 3 \\ 5 & -2 \end{bmatrix}$ be such that $A^{-1} = KA$, then the value of K is:

The probability distribution of a random variable x is given below. | x | 0 | 1 | 2 | 3 | |---|---|---|---|---| | P(x) | k/3 | k/2 | k/4 | k/7 | Then the value of k is

Let $A = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} 4 & -6 \\ -2 & 4 \end{bmatrix}$ (A) $\det(A^T) = 1$ (B) $AB = I$, where $I$ is the identity matrix of order 2. (C) $A^{-1} = \begin{bmatrix} 2 & -3 \\ -1 & 2 \end{bmatrix}$ (D) adj $(B) = \begin{bmatrix} 4 & 2 \\ 6 & 4 \end{bmatrix}$ Choose the correct answer from the options given below:

Area (in sq. units) of the region bounded by the curve $y^2 = 4x$, $y$-axis and the line $y = 3$ is

The solution set of inequality $3x + 5y < 4$ is

For LPP: Maximize $z = 2x + 3y$ subject to the constraints $x + y \geq 2$, $x + 2y \geq 3$, $x \geq 0$, $y \geq 0$, which of the following graph represents the feasible region of the above LPP as shaded portion?

$\int_0^2 (|x| + |x - 2|) dx =$

The maximum value of $\left(\frac{1}{x}\right)^x$ for $x > 0$ is

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

The solution of the differential equation $\frac{dy}{dx} = (1 + x^2)(1 + y^2)$ is (Here C is an arbitrary constant)

If $A = \begin{bmatrix} x & 3 \\ 2 & 4 \end{bmatrix}$, $B = \begin{bmatrix} 2 & 3 \\ y & 3 \end{bmatrix}$ and $C = \begin{bmatrix} z & 1 \\ 8 & 2 \end{bmatrix}$ are singular matrices then: (A) $x > y$ (B) $y > z$ (C) $z > x$ (D) $x \neq y \neq z$ Choose the correct answer from the options given below:

If A and B are independent events and $P(A) = \frac{1}{2}$ $P(B) = \frac{1}{3}$, then Match List-I with List-II | List-I | List-II | |---|---| | (A) $P(A \cap B)$ | (I) $\frac{1}{2}$ | | (B) $P(\bar{A})P(B) + P(A)P(\bar{B})$ | (II) $\frac{1}{3}$ | | (C) $P(A \mid B) + P(B \mid A)$ | (III) $\frac{1}{6}$ | | (D) $P(A \cap \bar{B})$ | (IV) $\frac{5}{6}$ | Choose the correct answer from the options given below:

If the function $f(x) = \begin{cases} ax + 2, & x \leq 1 \\ x^2 + 3x + b, & x > 1 \end{cases}$ is differentiable at $x = 1$, then the value of $(2a + b)$ is

If $\int_0^1 \frac{e^x}{1 + x} dx = m$, then the value of $\int_0^1 \frac{e^x}{(1 + x)^2} dx$ is:

Let A and B be 3×3 matrices such that $A \neq B$. If $A^3 = B^3$ and $A^2B = B^2A$, then the determinant of $A^2 + B^2$ is:

A boat 10 m high floating at a uniform speed of 13 meters per minute(m/min) away from a lamp post 15 m high. Then the rate at which the length of shadow of the boat increases is:

$\int \frac{dx}{x^3\sqrt{(1 + x^4)}} =$

Match List-I with List-II | List-I | List-II | | --- | --- | | (A) A square matrix $P$ is said to be non-singular if | (I) $\vert P\vert = 0$ | | (B) A square matrix $P$ is said to be singular if | (II) $P P^T$ is symmetric | | (C) If a matrix $P$ is both symmetric and skew-symmetric, then | (III) $\vert P\vert \neq 0$ | | (D) If $P$ is a square matrix, then | (IV) $P$ is a null matrix | Choose the correct answer from the options given below:

If $y = \log_e\left(\frac{e^2}{x^2}\right)$ for $x \neq 0$, then $\frac{d^2y}{dx^2}$ equals

The maximum value of the determinant of the matrix $\begin{bmatrix} 1 & 1 & 1 \\ 1 & 1+\sin x & 1 \\ 1+\cos x & 1 & 1 \end{bmatrix}$ is: (where $x$ is real)

Consider the function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = \frac{x}{x^2+1}$ then

If $A^{-1}$ exists for the matrix $A = \begin{bmatrix} 1 & \lambda & -1 \\ -1 & 1 & 0 \\ \lambda & 1 & 1 \end{bmatrix}$ then

The particular solution of the differential equation $\left[x \sin^2\left(\frac{y}{x}\right) - y\right]dx + xdy = 0$, $y = \frac{\pi}{4}$ when $x = 1$ is

Match List-I with List-II | List-I | List-II | |---|---| | **Differential Equation** | **Integrating Factor** | | (A) $\frac{dy}{dx} + 2xy = 1$ | (I) $x$ | | (B) $x\frac{dy}{dx} + 2xy = 1$ | (II) $e^{2x}$ | | (C) $x\frac{dy}{dx} + y = 1$ | (III) $x^2$ | | (D) $x\frac{dy}{dx} + 2y = 2$ | (IV) $e^{x^2}$ | Choose the correct answer from the options given below:

let $\vec{a}$ be a non-zero vector of magnitude '$a$' and $\lambda$ is a non-zero scalar, then $\lambda\vec{a}$ is a unit vector if

If the function $f(x) = 2x^2 - kx + 7$, is increasing on $[1,2]$, then $k$ lies in the interval

The value of $\cot\left(\cos^{-1}\frac{7}{25}\right)$ is

The area of the smaller region bounded by the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$ and the straight line $3x + 4y = 12$ is:

If $\vec{a} = 2\hat{i} + m\hat{j} - n\hat{k}$ and $\vec{b} = l\hat{i} - 3\hat{j} + 4\hat{k}$ such that $\vec{a} = 2\vec{b}$ then the value of $14{l} + m + n$ is:

If $P(A) = \frac{3}{10}$, $P(B) = \frac{2}{5}$ and $P(A \cup B) = \frac{3}{5}$ then the value of $P(B|A) + P(A|B)$ is:

Match List-I with List-II Where $\mathbb{R}$ is set of real numbers | List-I | List-II | |---|---| | (A) $\sin x$ is continuous on: | (I) $\mathbb{R} - \{0\}$ | | (B) $ \tan x$ is continuous on: | (II) $\mathbb{R}$ | | (C) $\cot x$ is continuous on: | (III) $\mathbb{R} - \{n\pi: n \in \mathbb{Z}\}$ | | (D) $x^{-n}, n \in \mathbb{N}$ is continuous on: | (IV) $\mathbb{R} - \left\{(2n + 1)\frac{\pi}{2}: n \in \mathbb{Z}\right\}$ | Choose the correct answer from the options given below:

Bag I contains 3 black and 2 white balls. Bag II contains 2 black and 4 white balls. A bag is selected at random and then a ball is drawn from it. The probability that the ball drawn is black is:

If A and B are symmetric matrices of order 3 x 3 then the matrix $2AB - BA$ is:

Acute angle between the lines $\frac{x}{3} = \frac{y}{4} = \frac{z}{5}$ and $\frac{x-1}{4} = \frac{y+1}{-3} = \frac{z+10}{5}$ is:

Consider the LPP: Maximize $z = 5x + 3y$ subject to $3x + 5y \leq 15$, $5x + 2y \leq 10$, $x,y \geq 0$. The optimal feasible solution occurs at

If $\vec{a}$, $\vec{b}$, $\vec{c}$ are unit vectors such that $\vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c} = 0$, and the angle between $\vec{b}$ and $\vec{c}$ is $\frac{\pi}{6}$, then

The co-ordinates of the point where the line $\frac{x+3}{3} = \frac{y-1}{-1} = \frac{z-5}{-5}$ cuts $yz$-plane are:

If the minimum value of the objective function $Z = ax + by$ of an LPP occurs at two points $(3, 5)$ and $(5, 3)$, then

If $P\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} = \begin{bmatrix} -7 & -8 & -9 \\ 2 & 4 & 6 \end{bmatrix}$, then matrix P is equal to:

Match List-I with List-II | List-I | List-II | |---|---| | **Equation of line** | **Information** | | (A) $\vec{r} = (3\hat{i} - 2\hat{j} + \hat{k}) + \lambda(\hat{j} - 2\hat{k})$ | (I) Direction ratios are 2, 4, -1 | | (B) $\frac{2-x}{1} = \frac{2y+1}{4}$, $z = 2$ | (II) Perpendicular to $2\hat{i} - \hat{j} + \hat{k}$ | | (C) $\frac{x}{1} = \frac{y-3}{2} = \frac{3-4z}{2}$ | (III) Passing through the point $(3, -2, 1)$ | | (D) $\vec{r} = (3\hat{i} + 2\hat{j} + \hat{k}) + \lambda(2\hat{i} + \hat{j} - 3\hat{k})$ | (IV) Direction ratios are -1, 2, 0 | Choose the correct answer from the options given below:

Area of the bounded region between the curve $y = |x - 2|$ and the line $y = 2$ is:

A relation R on the set $A = \{1, 2, 3, \ldots, 13, 14\}$ defined as $R = \{(x,y): 3x - y = 0\}$ is

Two cards are drawn successively with replacement from a well-shuffled deck of 52 cards. The probability distribution of number of aces is given by:

If $a$, $b$ and $c$ are distinct prime numbers then the value of $\begin{vmatrix} a-b & b-c & c-a \\ b-c & c-a & a-b \\ c-a & a-b & b-c \end{vmatrix}$ is equal to

For any vector $\vec{a}$, the value of $|\vec{a} \times \hat{i}|^2 + |\vec{a} \times \hat{j}|^2 + |\vec{a} \times \hat{k}|^2$ is equal to:

Match List-I with List-II (where $c$ is an arbitrary constant) | List-I | List-II | | --- | --- | | (A) $\int \tan x \, dx$ | (I) $\log\vert \sec x + \tan x\vert + c$ | | (B) $\int \cot x \, dx$ | (II) $\log\vert \sec x\vert + c$ | | (C) $\int \sec x \, dx$ | (III) $\log\vert \sin x\vert + c$ | | (D) $\int \cosec x \, dx$ | (IV) $\log\vert \cosec x - \cot x\vert + c$ | Choose the correct answer from the options given below:

Siyaram has 100 kg apples, he sells a part of it at 8% profit and rest of it at 18% profit. The overall profit he earns is 14%. The quantity he sold at 18% profit is:

Ajesh has set up a sinking fund in order to have ₹ 10,00,000 after 10 years for his son's education. The amount should be set aside at the end of every 6 months into an account paying 5% per annum compounded half yearly is: [given $(1.025)^{20} = 1.6386$]

The graph given below represents which of the following function? <img src="https://balti.afterboards.in/B69GKLFdaZzveBZ" width="300px"/>

If it is 7:00 pm currently in the clock, what will the clock show (in am or pm) after 674 hours?

A machine costing ₹ 36000 has an effective life of 5 years with scrap value of ₹ 5000 following a linear method of depreciation. Which of the following statements are **correct**? (A) The value of the machine after 1 year is ₹ 31000 (B) The value of the machine after 2 years is ₹ 23600 (C) The value of the machine after 3 years is ₹ 18400 (D) The value of the machine after 4 years is ₹ 11200 Choose the **correct** answer from the options given below:

A boat covers a distance of 51 km downstream in 3 hours and takes $7\frac{2}{7}$ hours to cover the same distance upstream. If the boat's speed in still water is 12 km/hrs, then the speed of the stream is:

If $\begin{bmatrix} ab & cd \\ a+c & b+d \end{bmatrix} = \begin{bmatrix} 2 & -3 \\ 4 & 1 \end{bmatrix}$ where $a$, $b$, $c$, $d$ are integers, then which of the following are true? (A) $a + d = 0$ (B) $b + d = 3$ (C) $b + d = 1$ (D) $c + d = 2$ Choose the **correct** answer from the options given below:

Match List-I with List-II | List-I | List-II | | --- | --- | | Matrix Product | Order of resultant matrix | | --- | --- | | (A) $[a_{ij}]_{2 \times 3} \times [b_{ij}]_{3 \times 4}$ | (I) $2 \times 4$ | | (B) $[a_{ij}]_{2 \times 1} \times [b_{ij}]_{1 \times 3}$ | (II) Not possible | | (C) $[a_{ij}]_{3 \times 2} \times [b_{ij}]_{3 \times 2}$ | (III) $3 \times 3$ | | (D) $[a_{ij}]_{3 \times 3} \times [b_{ij}]_{3 \times 3}$ | (IV) $2 \times 3$ | Choose the correct answer from the options given below:

Match List-I with List-II | List-I | List-II | |---|---| | **Example** | **Time-series component** | | (A) Labour strike | (I) Secular-trend | | (B) Continuous decline in death rate | (II) Seasonal | | (C) Rise in prices before Diwali | (III) Cyclical | | (D) Rise and fall of the share-market | (IV) Irregular | Choose the **correct** answer from the options given below:

At what rate of interest will the present value of a perpetuity of ₹ 500 payable at the end of every 6 months be ₹ 20,000?

From the below-mentioned graph of shaded feasible region of a linear programming problem (LPP) with objective function $z = 1.50x + 1.00y$; the maximum value of $z$ will be: <img src="https://balti.afterboards.in/TZVsJcDnM85gOPS" width="300px"/>

Which of the following statements are true? (A) The function $f(x) = \frac{x^4}{4} - \frac{4}{3}x^3 + \frac{x^2}{2} + 6x$ has 3 critical points. (B) The function $f(x) = |x| + 3$ has no minimum value. (C) A local maximum value is always the absolute maximum value. (D) $f(x) = x^2$ has minima at $x=0$. Choose the **correct** answer from the options given below:

Rajesh calculated 95% confidence level. What does he mean by that? (A) He can be 95% confident that his sample will include the population parameter. (B) He can be 95% confused that his sample will include the population parameter. (C) He can be 5% confident that his sample will not include the population parameter. (D) He can be 5% confident that his sample will include the population parameter. Choose the **correct** answer from the options given below:

The solution of the system of equations $2x + \frac{1}{2}y - z = 1$, $2y = 3$, $x + 2z = 4$ is:

If $\int \frac{2x - 5}{(2x - 3)^3} e^{2x} dx = \frac{\lambda e^{2x}}{(2x - 3)^2} + C$, where $C$ is an arbitrary constant then the value of $\lambda$ is

The rate of change of the area of a circle with respect to its radius $r$, when $r = 3$cm, is:

Which of the following are correct about t-test statistics? (A) The mean of t-test distribution is 0. (B) It depends on the degrees of freedom. (C) It depends on population standard deviation. (D) With more degrees of freedom, it closely resembles standard normal distribution. Choose the **correct** answer from the options given below:

The curve $x = y^2$ and $xy = k$ cut orthogonally, then $k^2$ is equal to:

The area bounded by $y = 3x + 1$, $x = 0$, $y = 0$ and $x = a$ is 8 Sq.units. Then value of $a$ (where $a > 0$) is

A 95% confidence interval for a population mean was reported to be 152 to 160. If sample standard deviation $\sigma=15$, then sample size used in this study is (Given $Z_{0.025}=1.96$)

If three year moving averages for five data items given by 16, 18, 20, 10, 21 are $x$, $y$ and $z$ respectively, then:

Suppose X has Poisson distribution such that $3 P(X=1) = 2 P(X=2)$ then $P(X>0)$ is:

Match List-I with List-II | List-I | List-II | |---|---| | (A) Marginal average cost if cost function $C(x) = \frac{50}{\sqrt{x}}$ | (I) $50\sqrt{x}$ | | (B) Marginal average cost if cost function $C(x) = 50\sqrt{x}$ | (II) $-\frac{75}{x^2\sqrt{x}}$ | | (C) Revenue function if demand function $P=\frac{50}{\sqrt{x}}$ | (III) $\frac{-25}{x\sqrt{x}}$ | | (D) Marginal revenue if demand function $P=50\sqrt{x}$ | (IV) $75\sqrt{x}$ | Choose the **correct** answer from the options given below:

The value of $\begin{vmatrix} 7! & 8! & 9! \\ 8! & 9! & 10! \\ 9! & 10! & 11! \end{vmatrix}$ is:

The declared rate of return compounded semi annually equivalent to the effective rate of return 10.25% per annum is:

The values of '$a$' and '$b$' if the equation of straight line trend by least square method is given by $y = a + bx$ such that $\sum x = 0$, $\sum y = 84$, $\sum xy = 108$, $\sum x^2 = 70$ for 6 observation at are:

For a Binomial distribution, B(n,p), where p+q=1, the sum and product of mean and variance are 8 and 12 respectively, when the value of n is:

A die is thrown 4 times and getting 3 is considered a success. The probability of 2 successes is:

The unit's digit of $13^{13}$ is:

Rahul invested ₹ 20000 in a mutual fund in year 2018. If the value of mutual fund increased to ₹ 32000 in year 2023. Then the compound annual growth rate of his investment is: $[given \, that(1.6)^{1/5} = 1.098]$

Consider the matrices $A = \begin{bmatrix} 9 & 0 & 0 \\ 0 & 16 & 0 \\ 0 & 0 & 25 \end{bmatrix}$ and $B = \begin{bmatrix} \frac{1}{5} & 0 & 0 \\ 0 & \frac{1}{4} & 0 \\ 0 & 0 & \frac{1}{3} \end{bmatrix}$. The value of $|(AB)^{-1}|$ is

Anush takes a loan of ₹ 150000 @ 16% annual interest for 5 years. His EMI (Equally Monthly Installment) on monthly basis under flat rate system is:

The random variable X can take values 0, 1, 2. If $P(X=0)=P(X=1)=\alpha$, and $E(X^2)=E(X)$, then which of the following are correct? (A) $E(X) = 2-3\alpha$ (B) $E(X^2) = 4+7\alpha$ (C) $\alpha = \frac{1}{2}$ (D) $\alpha = \frac{1}{5}$ Choose the **correct** answer from the options given below:

Two pipes P and Q can fill a tank in 26 minutes and 52 minutes respectively. Both the pipes are opened together for some time and then pipe P is closed. If the tank is filled in 26 minutes, then after how many minutes pipe P is closed?

The maximum value of the objective function $z = 10x + 15y$ of an L.P.P. subjected to the constraints $2x + 4y \leq 8$, $3x + y \leq 6$, $-x - y \geq -4$, $x \geq 0$, $y \geq 0$ is: