CUET Mathematics 2026 22nd May Shift 1 PYQs

$\int \frac{\sqrt{9+(\log x)^2}}{x} dx$ is equal to: (where $C$ is an arbitrary constant and consider $\log_e x = \log x$)

A pair of dice is thrown then probability of getting sum of numbers on dice is '8', is:

$\int_{4}^{10} \frac{\log(x^2)}{\log(x^2)+\log(196-28x+x^2)}dx$ is equal to : (Consider $\log_e x = \log x$)

The differential equation whose order and degree are 3 and 2 respectively is

The corner points of the bounded feasible region determined by $x+3y\le60, x+y\ge10, x\le y, x\ge0, y\ge0$ are A(0, 10), B(5, 5), C(15, 15) and D(0, 20). If the objective function $z=ax+by$ has its maximum value on the line segment CD, then the relation between $a$ and $b$ is:

The value of $\begin{vmatrix} 0 & xy^2 & xz^2 \\ x^2y & 0 & yz^2 \\ x^2z & zy^2 & 0 \end{vmatrix}$ is:

Match List-I with List-II (Consider $\log_e x = \log x$) | List-I | List-II | |---|---| | **Integral** | **Solution: where C is an arbitrary constant** | | (A) $\displaystyle\int \dfrac{f'(x)}{f(x)\log f(x)}\,dx$ | (I) $x\log(\log x) + c$ | | (B) $\displaystyle\int \left(\log(\log x) + \dfrac{1}{\log x}\right) dx$ | (II) $\dfrac{x}{\log x} + C$ | | (C) $\displaystyle\int \dfrac{\log x}{(1+\log x)^2}\,dx$ | (III) $\log\left\vert \log(f(x)) \right\vert + c$ | | (D) $\displaystyle\int \left(\dfrac{1}{\log x} - \dfrac{1}{(\log x)^2}\right) dx$ | (IV) $\dfrac{x}{1+\log x} + c$ | Choose the correct answer from the options given below:

If $A=\begin{bmatrix}4 & 1\\2 & 3\end{bmatrix}$ and $B=\begin{bmatrix}-2 & 3\\1 & 2\end{bmatrix}$ such that $2B-3A+X=0$ then $X$ is

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

Let $A$ be a matrix given by $A=[a_{ij}]_{3\times3}$, $a_{ij}=i+j$, then Match List-I with List-II | List-I | List-II | |---|---| | (A) Number of the elements in $A$ | (I) 3 | | (B) Sum of the diagonal elements | (II) 9 | | (C) $a_{31}+a_{32}-a_{33}$ | (III) -1 | | (D) Cofactor of $a_{13}$ | (IV) 12 | Choose the correct answer from the options given below:

For every square matrix $A$ of order $n\times n$, which of the following statements are correct ? (A) $(adjA)A = A(adjA) = |A|I$, where $I$ is identity matrix of order n. (B) $|kA|=k^n|A|$, k is any scalar. (C) $|adjA|=|A|^n$ (D) $A^2=A$ Choose the correct answer from the options given below:

The function $f:R\to R$ is defined by $f(x)=2x^3+5$, then which of the following statements are correct ? (A) $f(x)$ has no local maximum value. (B) $f(x)$ has no local minimum value. (C) $f(x)$ has both local maximum and local minimum values. (D) $f(x)$ has neither a local maximum value nor a local minimum value Choose the correct answer from the options given below:

If $y=e^{\pi x}$ and $t=x^{\pi}$, then which of following statements is/are TRUE ? (A) $\frac{d^2y}{dt^2}=\frac{y}{t^2}\left(\frac{\pi(x-1)+1}{\pi}\right)x$ (B) $\frac{d^2y}{dt^2}=y\left(\frac{\pi(x-1)+1}{\pi}\right)x$ (C) $\frac{d^2y}{dt^2}=1$ at $x=1$ (D) $\frac{d^2y}{dt^2}=\frac{e^{\pi}}{\pi}$ at $x=1$ Choose the correct answer from the options given below:

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

The interval for which the function $f(x)=\frac{x}{x^2+x+1}$ is an increasing function, is:

The function $f(x)=\log_e(\cos x)$ increases on which of the following intervals for $x\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ ?

Let X be a non empty set and S be the collection of all subsets of X. A relation R in S is defined by $R=\{(A,B): A\subseteq B\}$, where $X\subseteq Y$ means $X$ is proper subset of $Y$. Then R is:

An angle $\theta, 0<\theta<\frac{\pi}{2}$, which increases twice as fast as its sine, is

Area of the region bounded by the curve $y=\sin x, y=\cos x$ and $x-$axis, $\left(0\le x\le\frac{\pi}{2}\right)$ is;

Vector projection of $7\hat i+\hat j+4\hat k$ on the vector $2\hat i+6\hat j-3\hat k$ is

General solution of the differential equation $\frac{dy}{dx}=e^{\frac{x^2}{2}}+xy$ is: (where C is an arbitrary constant)

If the function $f(x)=\begin{cases}\frac{1-\cos kx}{\sin^2 x}, & x\ne0\\\frac{1}{2}, & x=0\end{cases}$ is continuous at $x=0$, then the value of $k$ is

If $adjA=\begin{bmatrix}7 & -3 & 2\\3 & 0 & -3\\-1 & 3 & 1\end{bmatrix}$, then $A^{-1}$ is:

If $\vec a$ and $\vec b$ are two unit vectors and $|\vec a-\vec b|=\sqrt3$, then the value of $|\vec a+\vec b|$ is:

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

Direction cosines of 2 lines are given by the equations $3l+m+5n=0, 6mn-2nl+5lm=0$ then which of the following statements are correct ? (A) Direction ratios of two lines are (1, 2, -1) and (-2, 1, 1). (B) Angle between lines is $\frac{\pi}{3}$. (C) The angle between the lines is $\cos^{-1}\left(-\frac{1}{6}\right)$. (D) Vectors parallel to these lines are $\hat i+2\hat j-\hat k$ and $-2\hat i+\hat j+\hat k$. Choose the correct answer from the options given below:

$\int_{-1}^{1}\log(x+\sqrt{x^2+1})dx$ is equal to

Match List-I with List-II | List-I (Function) | List-II (Principal value of y) | |---|---| | (A) $y=\sin^{-1}\left(-\frac{1}{2}\right)$ | (I) $\frac{3\pi}{4}$ | | (B) $y=\cot^{-1}\left(-\frac{1}{\sqrt3}\right)$ | (II) $-\frac{\pi}{4}$ | | (C) $y=cosec^{-1}(-\sqrt2)$ | (III) $-\frac{\pi}{6}$ | | (D) $y=\cos^{-1}\left(-\frac{1}{\sqrt2}\right)$ | (IV) $\frac{2\pi}{3}$ | Choose the correct answer from the options given below:

Let a function $f(x)=\alpha+(\beta^2+5\beta+6)|x|+\gamma|x|^4$, where $\alpha, \beta$ and $\gamma$ are real number. Then the function $f(x)$ is differentiable at $x=0$ if

If $A=\begin{bmatrix}1 & 1 & 1\\1 & 0 & 3\\1 & -2 & 1\end{bmatrix}$, then Match List-I with List-II | List-I | List-II | |---|---| | (A) $\lvert A\rvert$ | (I) 48 | | (B) $\lvert 2A\rvert$ | (II) 36 | | (C) $\left\lvert\operatorname{adj}(A)\right\rvert$ | (III) 12 | | (D) $2\lvert A\rvert$ | (IV) 6 | Choose the correct answer from the options given below:

If $x=t^3, y=t^4$, then $\frac{d^2y}{dx^2}$ is:

Integrating factor of the differential equation $x(x-1)\frac{dy}{dx}-(x-2)y=x^3(2x-1), x>1$ is:

For a LPP, maximize $z=2x+3y$, subjected to constraints: $x\ge2, x\le7, y\le x, x+y\le10, x\ge0, y\ge0$ Which of the following statements are TRUE ? (A) Vertices of feasible region are (2, 0), (7, 0), (7, 3), (5, 5) and (2, 2). (B) Maximum $z=25$ at point (5, 5). (C) Maximum $z=25$ at infinite numbers of points. (D) The feasible region is unbounded. Choose the correct answer from the options given below:

Let a function $f:\mathbb{R}\to\mathbb{R}$ defined as $f(x)=x-[x]$, (where $\mathbb{R}$ is set of real numbers & $[.]$ denotes greatest integer function) .Then the function:

The linear constraints for which the shaded region in the figure is the solution set, are <img src="https://balti.afterboards.in/oG3DMeIaq2SbT4b" width="400px"/>

Value of $\int_{-2}^{2}|2x+1|dx$ is:

If $\vec a, \vec b$ and $\vec c$ are three vectors, then meaningless expressions are (A) $\vec a.(\vec b\times\vec c)$ (B) $\vec a.(\vec b.\vec c)$ (C) $\vec a\times(\vec b\times\vec c)$ (D) $\vec a\times(\vec b.\vec c)$ Choose the correct answer from the options given below:

If $|\vec a|=2, |\vec b|=7$ and $\vec a\times\vec b=3\hat i+2\hat j+6\hat k$, then angle between $\vec a$ and $\vec b$ is:

Match List-I with List-II Where C is arbitrary constant. | List-I | List-II | |---|---| | (A) $\displaystyle\int \dfrac{dx}{x^2 - a^2} =$ | (I) $\log\left\vert x + \sqrt{x^2+a^2}\right\vert + C$ | | (B) $\displaystyle\int \dfrac{dx}{a^2 - x^2} =$ | (II) $\log\left\vert x + \sqrt{x^2-a^2}\right\vert + C$ | | (C) $\displaystyle\int \dfrac{dx}{\sqrt{x^2 - a^2}} =$ | (III) $\dfrac{1}{2a}\log\left\vert \dfrac{x-a}{x+a}\right\vert + C$ | | (D) $\displaystyle\int \dfrac{dx}{\sqrt{x^2 + a^2}} =$ | (IV) $\dfrac{1}{2a}\log\left\vert \dfrac{a+x}{a-x}\right\vert + C$ | Choose the correct answer from the options given below:

Area bounded by the curves $y=\begin{cases}x+2, & x\ge-1\\-x, & x<-1\end{cases}$, $x=-2, x=3$ and $y=0$ is:

If A is a square matrix such that $A^2=A$ and $I$ is identity matrix of same order, then $(2I+A)^3-19A$ is equal to:

If $A=\begin{bmatrix}3 & -2\\4 & -2\end{bmatrix}$ and $I=\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}$ such that $A^2-A+\lambda I=0$, then value of '$\lambda$' is:

If $A=[a_{ij}]_{3\times3}=\begin{bmatrix}2 & 3 & -1\\1 & 2 & 3\\0 & -1 & 1\end{bmatrix}$ and $B=[b_{ij}]_{3\times2}=\begin{bmatrix}3 & -2\\1 & 4\\1 & 2\end{bmatrix}$ then the value of $a_{12}b_{21}+a_{32}b_{31}$ is

Let height and radius of a right circular cylinder are $h$ and $r$ respectively. If it is open at the top, having a given surface area and greatest volume, then

Let E and F are two independent events such that $P(E)=0.35$ and $P(E\cup F)=0.60$ then which of following statements are TRUE? (A) $P(F)=\frac{5}{13}$ (B) $P(E|\overline{F})=0.35$ (C) $P(E|\overline{F})=0.65$ (D) $P(\overline{E}|\overline{F})=0.65$ Choose the correct answer from the options given below:

If $XA=(I-A)^2$ where $X, A$ and $I$ are $2\times2$ matrices and $A=\begin{bmatrix}1 & 2\\1 & 4\end{bmatrix}$, then $|X|=$

If the lines $\frac{1-x}{2}=\frac{2y}{p}=\frac{z-1}{1}$ and $\frac{3-2x}{p}=\frac{y-1}{2}=\frac{z}{1}$ are parallel then value of 'p' is

A school has to send the report cards of 3 students, but the cleark has not paid attention to match the address on the envelope with the student report card is: The probability that exactly one of the students received his or her own report card is:

Three bags contain a number of red and white balls as follows: Bag I: 3 red balls Bag II: 2 red balls and 1 white ball Bag III: 3 white balls The probability that bag $i$ will be chosen, and a ball is selected from it is $\frac{i}{6}, i=1,2,3$. The probability that a red ball is selected is equal to:

Bag I contains 2 red and 3 blue balls and bag II contains $\alpha$ red and 5 blue balls. One ball is drawn at random from one of the bags and is found to be blue. If the probability that it was drawn from bag II is $\frac{25}{52}$, then $\alpha$ is equal to:

If the matrix $A=\begin{bmatrix}0 & -1 & 3x\\1 & y & -5\\-6 & 5 & 0\end{bmatrix}$ is skew-symmetric, then the value of $(5x-y)$ is

In a kilometer race, A beats B by 30 seconds and B beats C by 15 seconds. If A beats C by 180 m, then Match List-I with List-II | List-I | List-II | |---|---| | (A) Time taken by A to run 1 km | (I) 250 seconds | | (B) Time taken by B to run 1 km | (II) 45 seconds | | (C) Time taken by C to run 1 km | (III) 235 seconds | | (D) A beats C by | (IV) 205 seconds | Choose the correct answer from the options given below:

Sand is pouring from a pipe at the rate of 12 $cm^3$/sec. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm ?

Which of the given statements are true? (A) Coupon rate is the annual interest rate paid by the bond issuer to the bond holder. (B) Bonds can only be issued by government entities. (C) A sinking fund is a fixed term account. (D) If discount rate = coupon rate for a bond, then present value of the bond= face value of the bond Choose the correct answer from the options given below:

Consider the following data: | Year | 1995 | 1996 | 1997 | 1998 | 1999 | 2000 | 2001 | 2002 | |---|---|---|---|---|---|---|---|---| | Sales (in crores) | 6.7 | 5.3 | 4.3 | 6.1 | 5.6 | 7.9 | 5.8 | 6.1 | By using the method of least squares, the trend value for 1997 is :

A manufacturer produces two products, A and B. Both products are processed on two different machines. The available capacity of the first machine is 12 hours and of second machine is 9 hours per day. Each unit of product A requires 3 hours on both machines and each unit of product B requires 2 hours on first machine and 1 hour on second machine. Each unit of product A is sold at ₹7 profit and that of product B at a profit of ₹4. A LPP to maximise the profit is:

Vishal invested ₹20,000 in Tata stocks for 6 years. The value of the investment at the end of each year is given below: | Year 1 | Year 2 | Year 3 | Year 4 | Year 5 | Year 6 | |---|---|---|---|---|---| | ₹20,000 | ₹23,500 | ₹24,000 | ₹24,800 | ₹25,000 | ₹28,000 | The CAGR % of his investment is: [use $(1.4)^{1/6}=1.058$]

The particular solution of the differential equation $\log\left(\frac{dy}{dx}\right)=3x+4y$, given that $y=0$, when $x=0$ is:

The monthly sales of a plywood shop are normally distributed with a standard deviation of ₹900. A statistical study of sales in the last nine months has found a confidence interval for the mean monthly sales with extremes ₹4663 and ₹5839. What were the average sales over the nine-month period?

In a confidence interval, the range of values above and below the sample statistic is called :

If $y=\log_e\left(\frac{x^2}{e^2}\right)$, then $\frac{d^2y}{dx^2}$ is equal to:

The interval in which $f(x)=-2x^3-9x^2-12x+1$ is increasing, is:

Which of the following are true ? (A) If the mean of a normal variate is 12 and standard deviation is 4, then the Z-score of data point 20 is 2. (B) Mean of the numbers obtained on throwing a die having '1' on three faces, '2' on two faces and '5' on one face is 3. (C) If the variance of a Poisson's distribution is 2, then $P(X=0)=\frac{1}{e^2}$. (D) For a binomial distribution, mean = 9 and the standard deviation is $\frac{3}{2}$, then the number of trials is 12. Choose the correct answer from the options given below:

Consider the following hypothesis test $H_0: \mu\le3530$ $H_a: \mu>3530$ A sample of 100 provided a sample mean of $\bar x=3740$ and population standard derivation $S=700$. What is the value of t-statistic ?

Match List-I with List-II | List-I | List-II | |---|---| | (A) Nominal rate of return | (I) $EMI=\frac{Principal+Interest}{Period\ in\ months}$ | | (B) CAGR | (II) $\frac{Original\ value\ of\ the\ asset-Salvage\ value\ of\ the\ asset}{Useful\ life\ of\ the\ asset}$ | | (C) Flat rate method | (III) $\frac{Current\ market\ value\ of\ investment-Original\ investment\ value}{Original\ investment\ value}\times100$ | | (D) Annual depreciation | (IV) $\left(\frac{Investment\ at\ the\ end\ of\ investment\ period}{Investment\ at\ the\ beginning\ of\ the\ period}\right)^{\frac{1}{investment\ period}}-1$ | Choose the correct answer from the options given below:

For the given 9 values, 4,5,5,6,7,8,9,8,10, the three years moving averages are

A vehicle costing ₹9,00,000 has a scrap value of ₹3,60,000. If the annual depreciation charge is ₹60,000, then the useful life of the vehicle using linear depreciation method is:

If matrix $A=\begin{bmatrix}1 & 4 & 5\\3 & 2 & 6\\0 & 1 & 0\end{bmatrix}$, then Match List-I with List-II | List-I | List-II | |---|---| | (A) $\left\vert A \right\vert =$ | (I) $\dfrac{1}{9}$ | | (B) $\left\vert A^{-1} \right\vert =$ | (II) $81$ | | (C) $\left\vert \text{adj } A \right\vert =$ | (III) $9$ | | (D) $\left\vert \text{adj } (\text{adj } A) \right\vert =$ | (IV) $6561$ | Choose the correct answer from the options given below:

Two dice are thrown simultaneously. If X denotes the number of sixes, then the mean of X is :

Which of the following are correct? (A) $A=\begin{bmatrix}-9 & 4 & -3\\-1 & 0 & 4\\2 & 2 & 0\end{bmatrix}$ ; $a_{22}+a_{32}=-9$ where $a_{ij}$ denotes the $(i,j)^{th}$ element of matrix A. (B) $A=\begin{bmatrix}2 & -2\\-2 & 2\end{bmatrix}$; $A^2=pA$, then $p=4$ (C) $\begin{bmatrix}8 & 6\\6 & 16\end{bmatrix}$ is a symmetric matrix (D) $\begin{bmatrix}0 & -5 & 3\\5 & 0 & -7\\-3 & 7 & 0\end{bmatrix}$ is a symmetric matrix Choose the correct answer from the options given below:

A merchant lent out ₹20000 in two parts, one at 8% and other at 10% interest. The yearly average comes out to be 9.2%. Which of the following are correct ? (A) Amount lent at 10% is ₹12000. (B) Amount lent at 8% is ₹12000. (C) Amount lent at 10% is ₹8000. (D) Amount lent at 8% is ₹8000. Choose the correct answer from the options given below:

If $A=\begin{bmatrix}\lambda & 0 & 0\\0 & \lambda & 0\\0 & 0 & \lambda\end{bmatrix}$, $\lambda\ne0$, then $|adjA|$ is equal to

A boat can travel 36 km upstream in 5 hours. If the speed of the stream is 2.4 km/hr, how much time will the boat take to cover a distance of 78 km downstream?

Match List-I with List-II | List-I | List-II | |---|---| | **Inequalities** | **Solution set** | | (A) If $\left\vert 3 - 5x \right\vert < 7$, then $x \in$ | (I) $\left[-\dfrac{5}{2}, \infty\right)$ | | (B) If $3x - 7 > 5x - 1$, then $x \in$ | (II) $(-\infty, -3)$ | | (C) If $\dfrac{x}{3} > \dfrac{x}{2} + 1$, then $x \in$ | (III) $\left(-\dfrac{4}{5}, 2\right)$ | | (D) If $5x + 4 \leq 7x + 9$, then $x \in$ | (IV) $(-\infty, -6)$ | Choose the correct answer from the options given below:

Match List-I with List-II | List-I | List-II | |---|---| | (A) For a random variable X, $E(X)=3$ and $E(X^2)=11$, then $Var(X)$ is | (I) 25 | | (B) For a random variable X, $E(X)=3$ and $E(X^2)=13$, then $Var\left(\frac{X}{2}\right)$ is | (II) 2 | | (C) In a binomial distribution, if $n=20, q=0.75$, then its mean is | (III) 1 | | (D) If in a binomial distribution, mean is 5 and variance is 4, then the number of trials is | (IV) 5 | Choose the correct answer from the options given below:

A steel plant is capable of producing 'x' tonnes per day of low-grade steel and 'y' tonnes per day of high-grade steel, where $y=\frac{32-5x}{10-x}$. If the fixed price of low-grade steel is half that of high-grade steel, then the quantity of low-grade steel that should be produced per day for maximum receipts is

$I=\int_{2}^{8}\frac{\sqrt[3]{x+1}}{\sqrt[3]{x+1}+\sqrt[3]{11-x}}dx$ is equal to :

Jyotsna buys a car for which she makes a down payment of ₹3,50,000 and the balance is to be paid in 3 years by monthly installments of ₹34,000 each. If the financer charges interest at the rate of 12% per annum and uses flat rate method, then the original price of the car is :

What amount is received at the end of every 6 months forever, if ₹72,000 kept in a bank earns 8% per annum compounded half-yearly?

Which of the following holds true? (A) $26\equiv1(mod\ 5)$ (B) $17\equiv3(mod\ 8)$ (C) $-30\equiv-15(mod\ 3)$ (D) $39\equiv12(mod\ 7)$ Choose the correct answer from the options given below:

The matrix X for which $\begin{bmatrix}5 & 4\\1 & 1\end{bmatrix}X=\begin{bmatrix}1 & -2\\1 & 3\end{bmatrix}$ is

The sum and the product of the mean and variance of binomial distribution are 1.8 and 0.8 respectively. The probability of atleast one success is:

A tank is filled by three pipes with uniform flow. The first two pipes operating simultaneously fill the tank in the same time during which the tank is filled by the third pipe alone. The second pipe fills the tank 5 hours faster than the first pipe and 4 hours slower than the third pipe ,then the time required by the first pipe to fill the tank alone is :

If the objective function for a LPP is $Z=3x+4y$ and the corner points for bounded feasible region are (9,0), (4,3), (2,5) and (0,8), then the maximum value of Z occurs at

$I=\int(\log x)^2dx$ is equal to :(Consider $\log_e x=\log x$)