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: