CUET Mathematics 2026 PYQs12 May Shift 2. Free, no login required.

The area of region bounded by the curves $4y = x^3, y = 0$ and $x = 4$ in the first quadrant is :

Match **List-I** with **List-II** | List-I (Differential Equation) | List-II (Order and degree) | |---|---| | (A) $\left(\dfrac{d^2y}{dx^2}\right)^2 + e^{\frac{dy}{dx}} = 0$ | (I) order = 2, degree = 1 | | (B) $xy\dfrac{d^2y}{dx^2} + x\left(\dfrac{dy}{dx}\right)^2 - y\dfrac{dy}{dx} = 0$ | (II) order = 2, degree = not defined | | (C) $\left(\dfrac{d^3y}{dx^3}\right)^2 + \log_e\left(\dfrac{dy}{dx}\right) = 4x$ | (III) order = 3, degree = 2 | | (D) $\dfrac{d}{dx}\left(\dfrac{d^2y}{dx^2}\right) + y\dfrac{dy}{dx} = \left(1+\dfrac{dy}{dx}\right)^{1/2}$ | (IV) order = 3, degree = not defined | Choose the **correct** answer from the options given below:

If $x$ is real, then the minimum value of $x^2 - 8x + 25$ is

If $y = \dfrac{\log_e x}{x}$, then $\dfrac{d^2y}{dx^2}$ is :

If $A$ is a $3\times 3$ matrix with $|A|=4$, then $|4A|$ is equal to :

If $A$ is a non-singular matrix of order $n$, then $adj(adj\ A)$ is equal to :

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

If $A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}$, then $A^2 - 5A$ is equal to : (where $I_2$ is identity matrix of order 2)

A die is thrown once then, Match **List-I** with **List-II** | List-I | List-II | |---|---| | (A) Probability of getting odd number | (I) $\dfrac{1}{3}$ | | (B) Probability of getting number more than 4 | (II) $\dfrac{2}{3}$ | | (C) Probability of getting number less than or equal to 4 | (III) $\dfrac{1}{2}$ | | (D) Probability of getting number 2 | (IV) $\dfrac{1}{6}$ | Choose the **correct** answer from the options given below:

The function $f(x) = x^3 + 2x^2 - 1$ is (A) increasing in $\left(\dfrac{-4}{3}, 0\right)$ (B) decreasing in $\left(\dfrac{-4}{3}, 0\right)$ (C) increasing in $\left(-\infty, \dfrac{-4}{3}\right)$ (D) increasing in $(0, \infty)$ Choose the **correct** answer from the options given below:

The solution of the differential equation $\dfrac{dy}{dx} = x^2y^2$ with $y = 1$ at $x = 0$ is

The general solution of the differential equation $\dfrac{dy}{dx} = 1+x+y+xy$ is: (where $C$ is an arbitrary constant) (A) $1+y = x+\dfrac{x^2}{2}+C$ (B) $\log_e|1+y|^2 = x^2+2x+C$ (C) $\log_e|1-y| = x-\dfrac{x^2}{2}+C$ (D) $\log_e|1+y| = x+\dfrac{x^2}{2}+C$ Choose the **correct** answer from the options given below:

$\int\left\{\dfrac{1}{\log_e x} - \dfrac{1}{(\log_e x)^2}\right\}dx$ is equal to : (where $C$ is an arbitrary constant)

If $\displaystyle\int_0^{40} \dfrac{dx}{2x+1} = \log_e k$, then the value of $k$ is :

In the linear programming problem maximize $Z = 4x+3y$ subject to the constraints $x+y \ge 8$, $3x+5y \le 15$, $x\ge0, y\ge0$, the feasible region has

A cylinder with height 'h' and radius of base is 'r' which is open at the top having a given outer surface area, then volume of the cylinder is maximum if

The system of equations in 3 variables $x, y$ and $z$ as $x+y=5$, $y+az=3$, $x+by=8$ has a unique solution if the value of $a$ and $b$ are (A) $a=1, b=1$ (B) $a=0, b=1$ (C) $a \in (-\infty,0)$ and $b\neq1$ (D) $a\neq0$ and $b\in(3,4)$ Choose the **correct** answer from the options given below:

The general solution of the differential equation $ydx + (x-y^3)dy = 0$ is :

The area of the region bounded by the curves $x^2=4y$ and $x=4y$ is

If $\vec{a}.\vec{b} = \vec{a}.\vec{c}$ and $\vec{a}\times\vec{b} = \vec{a}\times\vec{c}$, $\vec{a}\neq\vec{0}$, then

Let $A$ be set of all real numbers and $R$ be a relation on $A$ defined by $R=\{(a,b): 1+ab>0\}$, then relation $R$ is

A bag contains 4 balls. Let E be an event in which two balls are drawn at random and found to be white, then Match **List-I** with **List-II** | List-I | List-II | |---|---| | (A) Probability that all balls are white | (I) $\dfrac{1}{6}$ | | (B) Probability of drawing 2 white ball at random given that there are 3 white ball | (II) $\dfrac{3}{5}$ | | (C) P(E) | (III) $\dfrac{1}{2}$ | | (D) Probability of drawing 2 white ball at random given that there is only 2 white ball | (IV) $\dfrac{5}{9}$ | Choose the **correct** answer from the options given below:

Let $\vec{a}=\hat{i}-\hat{j}$, $\vec{b}=3\hat{j}-\hat{k}$ and $\vec{c}=7\hat{i}-\hat{k}$. Then a vector $\vec{d}$ which is perpendicular to both $\vec{a}$ and $\vec{b}$ and $\vec{c}.\vec{d}=1$ is

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

If the points $A(-1,3,2)$, $B(-4,2,-2)$ and $C(5,5,\lambda)$ are collinear, then value of $\lambda$ is :

Two vectors $\vec{a}$ and $\vec{b}$ are inclined at angle $\theta=60°$ such that $|\vec{a}|=1$, $|\vec{b}|=2$, then $\left|(\vec{a}+3\vec{b})\times(3\vec{a}-\vec{b})\right|^2$ is equal to

The feasible region for an LPP is shown in the given figure. Let $Z=3x+4y$ be the objective function. Then which of the following is/are TRUE? <img src="https://balti.afterboards.in/keY7zvrtj9K8Dlg" width="400px"/> (A) The minimum value of Z is 156 (B) The minimum value of Z is 152 (C) The maximum value of Z is 196 (D) Neither minimum nor maximum value of Z exists Choose the **correct** answer from the options given below:

If angle between the lines $\dfrac{x-1}{\propto} = \dfrac{y+2}{-5} = \dfrac{z+\frac{7}{3}}{\beta}$ and $x=z, y=0$ is $\dfrac{\pi}{4}$, then

The values of $x, y, z$ and $w$ such that $\begin{bmatrix} x-y & 2z+w \\ 2x-y & 2x+w \end{bmatrix} = \begin{bmatrix} 5 & 3 \\ 12 & 15 \end{bmatrix}$ are given by

Match **List-I** with **List-II** | List-I (Functions) | List-II (Range, principal value) | |---|---| | (A) $\sin^{-1}x$ | (I) $[0,\pi]-\left\{\dfrac{\pi}{2}\right\}$ | | (B) $\tan^{-1}x$ | (II) $\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]-\{0\}$ | | (C) $cosec^{-1}x$ | (III) $\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]$ | | (D) $\sec^{-1}x$ | (IV) $\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$ | Choose the **correct** answer from the options given below:

From a lot of 10 items containing 3 defectives, a sample of 4 items is drawn at random. Let X denotes the number of defective items in the sample. If the sample is drawn randomly, then Match **List-I** with **List-II** (P(X) denotes the probability of occurence of X) | List-I | List-II | |---|---| | (A) P (X = 0) | (I) $\dfrac{^3C_3\times{}^7C_1}{^{10}C_4}$ | | (B) P (X = 1) | (II) $\dfrac{^7C_4\times{}^3C_0}{^{10}C_4}$ | | (C) P (X = 2) | (III) $\dfrac{^3C_1\times{}^7C_3}{^{10}C_4}$ | | (D) P (X = 3) | (IV) $\dfrac{^3C_2\times{}^7C_2}{^{10}C_4}$ | Choose the **correct** answer from the options given below:

Let $A=R-\{2\}$ and $B=R-\{1\}$ and a function $f:A\to B$ defined by $f(x)=\dfrac{x-1}{x-2}\ \forall x\in A$, then the function $f(x)$ is :

If $x^2+y^2 = t+\dfrac{1}{t}$ and $x^4+y^4=t^2+\dfrac{1}{t^2}$, then $\dfrac{dy}{dx}$ is equal to

If $[2\ \ x\ \ 3]\begin{bmatrix}-1 & 0 & -1\\ -1 & 1 & 0\\ 0 & 1 & 1\end{bmatrix}\begin{bmatrix}x\\0\\-1\end{bmatrix}=[-4]$, then the value(/s) of $x$ is

For a given line L: $\dfrac{x}{2}=\dfrac{y+1}{3}=\dfrac{-z-3}{-5}$, which of the following is/are True ? (A) The direction ratio of line L is (2, 3, -5) (B) The perpendicular distance of the point (1, 1, 3) on the line L is $\sqrt{3}$ (C) The foot of perpendicular from the point (1, 1, 3) on line L is (2, 2, 1) (D) The image of point (1, 1, 3) with respect to line L is (3, 3, 1) Choose the **correct** answer from the options given below:

If $x=a(\theta+\sin\theta)$, $y=a(1-\cos\theta)$, then $\dfrac{d^2y}{dx^2}=$

The linear constraints for which the shaded region in the figure are: <img src="https://balti.afterboards.in/v0QsIyGIrwouZMF" width="500px"/>

The position vectors of the points A and B are $\vec{a}=2\hat{i}-3\hat{j}+2\hat{k}$ and $\vec{b}=2\hat{i}+3\hat{j}+\hat{k}$ respectively, then area of triangle OAB is:

The maximum value of $f(x)=\dfrac{1}{3(x^2-x+1)}$ is :

The area of region $\{(x,y): 9x^2+16y^2\le144, x\ge0, y\ge0\}$ is-

If $A=[a_{ij}]_{3\times3}$ and $a_{ij}=2i-j$, then Match **List-I** with **List-II** | List-I | List-II | |---|---| | (A) $a_{21}+a_{32}$ | (I) 0 | | (B) $\lvert A\rvert$ | (II) 19 | | (C) Number of the elements in A | (III) 7 | | (D) $a_{22}.a_{31}+a_{21}.a_{33}$ | (IV) 9 | Choose the **correct** answer from the options given below:

For the function $f(x)=ax^3-9ax^2+9x+3, x\in R$, then which of the following statements are True? (A) $f'(x)=3[ax^2-6ax+3]$ (B) It is increasing for $a\in\left[0,\dfrac{1}{3}\right]\ \forall x\in R$ (C) It is increasing for $a\in(-\infty,0)\ \forall x\in R$ (D) It is increasing for $\forall a\in R$ Choose the **correct** answer from the options given below:

Suppose that A and B are independent events with $P(A)=\dfrac{1}{7}$ and $P(B)=\dfrac{3}{7}$, then $P(A\cap B)$ is

Which of the following differential equations has $y=x$ as one of the particular solution?

If $\begin{vmatrix}1 & 1 & 1\\ c & b-x & a\\ b & a & c-x\end{vmatrix}=0$, then $x$ is equal to

Function $f(x)=[x]$, greatest integer function is (A) Continuous at $x=2.5$ (B) Continuous at $x=1$ (C) Differentiable at $x=2$ (D) Differentiable at $x=-1.2$ Choose the **correct** answer from the options given below:

If $\begin{vmatrix}3 & x\\ x & 1\end{vmatrix}=\begin{vmatrix}3 & 2\\ 4 & 1\end{vmatrix}$, then value of $x$ is/are (A) $x=2$ (B) $x=-2$ (C) $x=2\sqrt2$ (D) $x=-2\sqrt2$ Choose the **correct** answer from the options given below:

If $\displaystyle\int\dfrac{x^3\,dx}{\sqrt{1+x^2}} = a(1+x^2)^{3/2}+b\sqrt{1+x^2}+c$, where 'C' is an arbitrary constant, then the value of $a$ and $b$ are

Given that the events A and B are mutually exclusive such that $P(A)=\dfrac{1}{2}$, $P(A\cup B)=\dfrac{2}{3}$ and $P(B)=p$, then the value of $p$ is:

The value of $\displaystyle\int_{-1}^{1}|x^4-x|\,dx$ is equal to:

A person can row a motor boat in still water at the rate of 12 km/hr. It takes him thrice as long to row upstream of a river as to row downstream of the same river. The speed of the stream is :

Match the components of time series with the examples associated to them. Match **List-I** with **List-II** | List-I | List-II | |---|---| | (A) Secular Trend | (I) Monthly sale of soft drink increases in summer | | (B) Seasonal variations | (II) Rise and fall of the share market | | (C) Cyclic Variations | (III) Data regarding national income | | (D) Irregular Variations | (IV) Sudden fall in crop yield due to floods | Choose the **correct** answer from the options given below:

The matrix which is both symmetric and skew-symmetric, is a:

Which of the following approaches is not associated to valuation of bonds ?

The total revenue received from the sale of $x$ units of a product is given by $R(x)=24x+5x^2+3$. The marginal revenue when $x=10$ is,

If a container contains 't' units of a particular liquid. Out of which 's' units are taken out and replaced by water. Then after $p-1$ operations, the quantity of pure liquid left in the container is given by:-

Let $Z=ux+vy$, $u,v>0$ be the objective function of a linear programming problem. Then the condition on $u$ and $v$ so that the maximum of $Z$ occurs at both points $(3,4)$ and $(0,5)$ is :

If $y=7^x\log(5x^3)$, then $\dfrac{dy}{dx}$ is equal to: (consider $\log_e x = \log x$)

A perpetual bond generates an annual return of ₹1,50,000 payable at the end by each year. The present value of this perpetuity if the money is worth 7% per annum will be

If A is an invertible matrix of order 3 and $|A|=7$, then the value of $|adj\ A|$ is:

If $\begin{vmatrix}4x & 9\\ -4 & -5\end{vmatrix}=\begin{vmatrix}4 & 8\\ 2 & 3\end{vmatrix}$, then the value of $x$ is :

If $X$ is a Poisson variate such that $4P(X=1)=3P(X=2)$, then $P(X=0)=$

If $\begin{bmatrix}-2 & 0\\ x & 4\end{bmatrix}+3\begin{bmatrix}y & 0\\ 2 & -1\end{bmatrix}=I$, where $I$ is $2\times2$ unit matrix, then the value of $x$ and $y$ are:

If $x=2at^5$ and $y=at^{10}$, then $\dfrac{dy}{dx}=$

For the values 22, 25, 28, 31, 40, 43, the three yearly moving averages are

A ring is dropped into a quiet pond and waves move in circles at a speed of 2 cm/second. At the instant, when the radius of the circular wave is 12 cm, how fast is the enclosed area increasing?

Match the rational functions with the correct form of the partial fractions. [ $p,q,r,a,b,c,A,B,C$ are arbitrary constants] Match **List-I** with **List-II** | List-I (Rational Function) | List-II (Partial Fraction) | |---|---| | (A) $\dfrac{px^2+qx+r}{(x-a)(x^2+bx+c)}$ | (I) $\dfrac{A}{x-a}+\dfrac{B}{(x-a)^2}+\dfrac{C}{x-b}$ | | (B) $\dfrac{px^2+qx+r}{(x-a)^2(x-b)}, a\neq b$ | (II) $\dfrac{A}{x-a}+\dfrac{B}{(x-a)^2}$ | | (C) $\dfrac{px^2+qx+r}{(x-a)(x-b)(x-c)}, a\neq b\neq c$ | (III) $\dfrac{A}{x-a}+\dfrac{Bx+C}{x^2+bx+c}$ | | (D) $\dfrac{px+q}{(x-a)^2}$ | (IV) $\dfrac{A}{x-a}+\dfrac{B}{x-b}+\dfrac{C}{x-c}$ | Choose the **correct** answer from the options given below:

The probability distribution of a random variable $X$ is given as $P(X=x)=\begin{cases}kx^3 & \text{for } x=1,2,3\\ 3kx & \text{for } x=4,5\\ 0 & \text{otherwise}\end{cases}$, where $k$ is a constant. The value of $P(X\ge4)$ is

Match **List-I** with **List-II** | List-I | List-II | |---|---| | (A) If an inlet pipe can fill a tank in $x$ hours, but due to a leakage in the bottom it is filled in $y$ hours, then time (in hours) taken to empty the tank due to leakage | (I) $\dfrac{yz+zx-xy}{xyz}$ | | (B) If two inlet pipes take $x$ & $y$ hours respectively to fill the tank and the third outlet pipe takes $z$ hours to empty the tank. If all three of them are opened together, then the net part of the tank filled in 1 hour | (II) $\dfrac{1}{x}$ | | (C) If a pipe can fill the tank completely in $x$ hours, then the part of the tank filled in 1 hour | (III) $\dfrac{xy}{y-x}$ | | (D) If two inlet pipes can fill a tank in $x$ & $y$ hours. If both pipes are opened at the same time, then net part of the tank filled in 1 hour | (IV) $\dfrac{x+y}{xy}$ | Choose the **correct** answer from the options given below:

If the sample size ($n$) is 32 in testing of hypothesis by t-test, then the degree of freedom ($\gamma$) is equal to :

A function $f(x)$ is defined as: $f(x) = x^3-6x^2+9x+15$ For the function $f(x)$, which of the following statements are correct ? (A) The critical points are $x=1$ and $x=3$. (B) $x=1$ is a point of local minimum. (C) The local maximum value is 19. (D) $x=3$ is a point of local minimum. Choose the **correct** answer from the options given below:

A car whose cost is ₹9,00,000 will depreciate to a scrap value of ₹1,20,000 in 12 years. Which of the following statements are correct ? (A) The annual depreciation is ₹65,000 (B) The book value of the car at the end of the fourth year is ₹6,40,000 (C) The book value of the car at the end of the sixth year is ₹5,00,000 (D) The book value of the car at the end of the fifth year is ₹5,75,000 Choose the **correct** answer from the options given below:

The following data are from a simple random sample: 3, 6, 8, 15, 5, 11. What is the point estimate of the population mean?

The last digit (unit digit) of $12^{25}$ is

The solution set of the inequation $|8x-7|\le\dfrac{2}{3}$ is

A runs four times as fast as B. If A gives B a start of 90 meters, how far must the goal on the race course be so that A and B reach it at the same time ?

Match **List-I** with **List-II** | List-I (Types of Probability Sampling) | List-II (Characteristic) | |---|---| | (A) Simple random sampling | (I) Every member of the population is arranged in a particular order. The first member is selected at random and then the remaining members are selected at regular intervals. | | (B) Systematic sampling | (II) The population is divided into subgroups and each subgroup has similar characteristics of the whole population. | | (C) Stratified sampling | (III) Every member of the population has an equal chance of being selected. | | (D) Cluster Sampling | (IV) The population is divided into subgroups based on the relevant characteristics | Choose the **correct** answer from the options given below:

If a coin is tossed 8 times and getting a head is considered as success, then which of the following statements are correct ? (A) The probability of getting 4 heads is $\dfrac{35}{128}$ (B) The mean of the number of successes is 4. (C) The variance of the number of successes is 2. (D) The probability of getting 6 heads is $\dfrac{9}{64}$. Choose the **correct** answer from the options given below:

For what value of $\mu$, $\begin{bmatrix}5 & \mu-7\\ 12 & 9\end{bmatrix}$ is a singular matrix?

Mr. Bean purchased a laptop for ₹75,000. The Laptop is estimated to have a scrap value of ₹10,000 after a span of 8 years. The book value of the laptop at the end of 5 years is

The feasible region for an L.P.P is always a/an

Rama takes a personal loan for a car worth ₹5,00,000 at an interest rate of 7% per annum to be repaid by equal monthly instalments (EMI) in 5 years. Using flat rate method, EMI will be:-

Identify the formula used for List I from List II Match **List-I** with **List-II** | List-I | List-II | |---|---| | (A) Present value of perpetuity payable at the end of each payment period | (I) $R+\dfrac{R}{i}$ (₹R payable at the beginning of each payment period, $i$ = interest per rupee per payment period) | | (B) Present value of perpetuity payable at the beginning of each payment period | (II) $\left(\dfrac{V_n}{V_0}\right)^{1/n}-1$, $V_n$ = Final value of investment, $V_0$ = Beginning value of the investment, $n$ = number of years | | (C) Computation of CAGR | (III) $\dfrac{C-s}{n}$, $C$ = Original cost, $s$ = Scrap value, $n$ = number of years | | (D) Annual Depreciation | (IV) $\dfrac{R}{i}$ (₹R payable at the end of each period, $i$ = interest per rupee per payment period) | Choose the **correct** answer from the options given below:

The value of the integral $I=\displaystyle\int\dfrac{e^x(x-2)}{(x+1)^4}dx$ is, [$C$ is an arbitrary constant]

Ajay is a vendor who informs that there are 2% rotten apples in the fruit bag. Find the probability that a sample of 10 apples will include not more than one rotten apple.