CUET Mathematics 2025 26 May Shift 2 PYQs

If the objective function z = 4x + 3y has maximum value on a line joining points (3, a) and (b, 2) where a > 0, b > 0 such that a - b = 2, then the maximum value of z is:

In the following differential equation $\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 = 2x^2 \log\left(\frac{d^2y}{dx^2}\right)$ order and degree is:

If P and Q are non-singular square matrices of the same order, then $(PQ^{-1})^{-1}$ equals

$\frac{d}{dx}\left(e^{2\log_e x^3}\right)$ equals

If the random variable X has the following probability distribution: | X | 0 | 1 | 2 | otherwise | |---|---|---|---|---| | P(X) | k | 3k | 5k | 0 | Match List-I with List-II | List-I | List-II | |---|---| | (A) k | (I) $\frac{13}{9}$ | | (B) E (X) | (II) $\frac{4}{9}$ | | (C) P (X ≤ 1) | (III) $\frac{8}{9}$ | | (D) P (1 ≤ X ≤ 2) | (IV) $\frac{1}{9}$ | Choose the correct answer from the options given below: 1. (A) - (II), (B) - (I), (C) - (IV), (D) - (III) 2. (A) - (IV), (B) - (I), (C) - (II), (D) - (III) 3. (A) - (IV), (B) - (II), (C) - (I), (D) - (III) 4. (A) - (III), (B) - (II), (C) - (I), (D) - (IV)

With respect to the following shaded feasible region (ABCDEFA), the maximum value of the objective function z = 3x + 4y – 2 is at point(s): <img src="https://balti.afterboards.in/gUAK5hc16W6wryv" width="300px"/>

The area of the region bounded by the curve $y = x + 1$, $x = axis$ and the lines $x = 2$ and $x = 3$ is

$\int_{1}^{2} \frac{1}{x(x+1)} dx, x > 0$ equals

If $A = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}, B = \begin{bmatrix} 0 & 0 \\ 3 & 0 \end{bmatrix}$ then

The function $f(x) = x^3 + 3x^2 + 4x + 4$, $x \in \mathbb{R}$ (set of real numbers) :

If $A = \begin{bmatrix} x+z & 2 & -3 \\ x & 0 & 4 \\ 3 & x-y & 0 \end{bmatrix}$ is a skew-symmetric matrix, then which of the following are true? (A) $y > z > x$ (B) $x > y$ (C) $x + y + z > 0$ (D) $z > x$ Choose the correct answer from the options given below:

The solution of the differential equation $xdy - ydx = 0$ represents

The maximum value of the function $f(x) = x^2(60 - x)$ in [20, 80] is:

$\int \frac{(x-1)e^x}{x^2} dx, x > 0$ equals (where C is an arbitrary constant)

If $\begin{bmatrix} 1 & 0 & 0 \\ 0 & y+1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 2x & \\ -2 & \\ z-3 & \end{bmatrix} = \begin{bmatrix} 6 \\ 4 \\ 1 \end{bmatrix}$ then $x + y + z$ is

The particular solution of the differential equation $\frac{dy}{dx} = e^{x^2/2} + xy$, when $x = 0$, $y = 1$, is

If the points (-1, -1, 2), (2, m, 5) and (3, 11, 6) are collinear, then m equals

Let $A = [a_{ij}]_{3 \times 3}$ be a matrix, defined by $a_{ij} = \begin{cases} 2i+3j & , i < j \\6 &, i=j\\ 3i-2j & , i > j \end{cases}$. The number of elements in A which are greater than 6, is

Derivative of $x^x$ with respect to $x\log x$ is

If $x = a\sec^3 \theta$, $y = a \tan^3 \theta$, then $\frac{dy}{dx}$ at $ \theta = \frac{\pi}{3}$ is

If $y = -4$ is a root of $\begin{vmatrix} y & 2 & 3 \\ 1 & y & 1 \\ 3 & 2 & y \end{vmatrix} = 0$, then the product of the other two roots is

If $|\vec{a}| = a$, then the value of $|\vec{a} \times \hat{i}|^2 + |\vec{a} \times \hat{j}|^2 + |\vec{a} \times \hat{k}|^2$ is

If the events A and B are independent, then which of the following statements are true? (A) P(A'B) = [1-P(A)] P(B) (B) A and B are mutually exclusive (C) P(A) = P(B) (D) P(A'B') = [1-P(A)] [1-P(B)] Choose the correct answer from the options given below:

Which of the following statements are true? (A) The vector equation of the line through the point (5, 2, -4) and parallel to the vector $3\hat{i} + 2\hat{j} - 8\hat{k}$ is $\vec{r} = (5\hat{i} + 2\hat{j} - 4\hat{k}) + \lambda(3\hat{i} + 2\hat{j} - 8\hat{k})$ (B) Vector form of the equation of line $\frac{x-5}{3} = \frac{y+4}{7} = \frac{z-6}{2}$ is $\vec{r} = (5\hat{i} - 4\hat{j} + 6\hat{k}) + \lambda(3\hat{i} + 7\hat{j} + 2\hat{k})$ (C) The direction cosines of z-axis are (1, 1,0). (D) If a line has direction ratios 2, -1, -2, then its direction cosines are -2/3, -1/3, -2/3. Choose the correct answer from the options given below:

The edge of a cube is increasing at a rate of 7 cm/s. The rate of change of area of the cube when its side is 3 cm is:

$\int e^x \left(\frac{1-x}{1+x^2}\right)^2 dx$ is equal to

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined as $f(x) = [x]$, where [x] denotes the greatest integer less than or equal to x. Then which of the following statements are correct? (A) f is one-one but not onto (B) f is not onto (C) f is not one-one (D) f is one-one and onto Choose the correct answer from the options given below:

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

The function $f(x) = \begin{cases} \frac{(\sin 2x)}{x} + \cos x & , if \ x \neq 0 \\ K & , if \ x = 0 \end{cases}$ is continuous at $x = 0$, then the value of K is:

The corner points of a bounded feasible region determined by the following system of linear inequalities $x + 3y \leq 60, x + y \geq 10$, $x \leq y$, $x \geq 0$, $y \geq 0$ are (0,10), (5,5), (15, 15) and (0, 20). Let $z = 2px + qy$, $p, q > 0$. If maximum of z occurs at both (15, 15) and (0, 20), then the relation between p and q is

If $e^x + e^y = e^{x+y}$, then $\frac{dy}{dx}$ equals

Area (in sq. units) of the region bounded by the curves $y = -1$, $y = 2$, $x = y^3$ and $x = 0$ is

The value of p so that the lines $\frac{x-1}{-3} = \frac{2y-2}{2p} = \frac{z-3}{2}$ and $\frac{x-1}{-3p} = \frac{y-1}{4} = \frac{6-z}{5}$ are at right angles is

If it is given that at $x = 1$, the function $f(x) = x^4 - 62x^2 + 2ax + b$ attains its maximum value on the interval [0, 2], then the value of a is:

Arrange the principal values of the following functions in ascending order (A) $\cosec^{-1}(2)$ (B) $\tan^{-1}(-\sqrt{3})$ (C) $\tan^{-1}(1)$ (D) $\tan^{-1}\left(\cos\frac{3\pi}{7}\right)$ Choose the correct answer from the options given below:

If A and B are two distinct events such that P(A|B) = P(B|A), then which of the following is /are possible? (A) A= B (B) P (A) = P(B) (C) A ⊂ B but A ≠ B (D) A∩ B = ɸ Choose the correct answer from the options given below:

If $|\vec{a}| = 10$, $|\vec{b}| = 2$ and $\vec{a} \cdot \vec{b} = 12$, then value of $|\vec{a} \times \vec{b}|$ is :

If $f(x)$ and $g(x)$ are continuous functions in [0, a] such that $f(x) = f(a - x)$ and $g(x) + g(a - x) = a$ then $\int_{0}^{a} f(x)g(x)dx =$

A and B throw a die alternatively till one of them gets 3 or 6 and wins the game. If B starts the game, then the probability of winning the game by A is

Let A = {1, 2, 3}. The number of equivalence relations containing (1, 3) is

If $A = \begin{bmatrix} 2 & -1 & 0 \\ 1 & 1 & 2 \\ -1 & 0 & 1 \end{bmatrix}$, then which of the following statement(s) is/are correct? (A) A is singular matrix (B) |3A| = 135 (C) |adj A| = 125 (D) $|A^{-1}| = \frac{1}{5}$ Choose the correct answer from the options given below:

The area (in sq. units) of the region bounded by the curve $x^2 = 250y$, $y = 0$ and $x = 50$ is

The value of $\int_{0}^{\pi/2} \frac{\tan^7 x}{\cot^7 x + \tan^7 x} dx$ is

Match List-I with List-II | List-I | List-II | |---|---| | Differential Equations | Order and degree | | (A) $ydx + x\log(y/x)dy - 2xdy = 0$ | (I) Order : 2, degree:1 | | (B) $\left(\frac{d^3y}{dx^3}\right)^2 + 3\frac{d^2y}{dx^2} + 2\left(\frac{dy}{dx}\right)^4 = y^2$ | (II) Order :1, degree:1 | | (C) $\frac{dy}{dx} + \log\left(\frac{dy}{dx}\right) + x = y$ | (III) Order : 3, degree:2 | | (D) $\left(\frac{ds}{dt}\right)^4 + 2s\frac{d^2s}{dt^2} = 0$ | (IV) Order : 1, degree: Not defined | Choose the correct answer from the options given below:

The corner points of the bounded feasible region of the LPP: Maximize $z = x + y$ subject to constraints $2x + 5y \leq 100$, $8x + 5y \leq 200$, $x \geq 0$, $y \geq 0$ are

Let A, B, C be three events. If the probability of occurring exactly one out of A and B is $\frac{3}{5}$, exactly one of B and C is $\frac{1}{5}$, exactly one of C and A is $\frac{3}{5}$ and that of occurring of three events is $\frac{4}{25}$, then the probability of occurring at least one of them is

The value of $\begin{vmatrix} 2^x & 1 & 6^x \\ 4^x & 1 & 3^x \\ 2^x & 1 & 6^x \end{vmatrix}$, where $x \neq 0$ is:

For the matrix $A = \begin{bmatrix} 2 & -1 & -1 \\ 0 & 2 & 3 \\ 1 & -2 & 1 \end{bmatrix}$, which of the following statements are correct? (A) The order of the matrix is 3 × 3 (B) |A| = 21 (C) $|adj\ A| = 225$ (D) A is skew symmetric matrix Choose the correct answer from the options given below:

A line passes through the point with position vector $2\hat{i} - \hat{j} + 4\hat{k}$ and is in the direction of the vector $\hat{i} + \hat{j} - 2\hat{k}$. The equation of the line in Cartesian form is:

For two matrices $A = \begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 & 1 \end{bmatrix}$ and $B^T = \begin{bmatrix} -1 & 2 & 1 \\ 1 & 2 & 3 \end{bmatrix}$, A - B equals

Anisha invested Rs.20000 in a mutual fund in the year 2016, which increased to Rs.36000 in the year 2024. The percentage compounded annual growth rate(CAGR) of her investment is: (Given: $(1.8)^{1/8} = 1.076$)

In a Binomial distribution, the probability of getting a success is $\frac{3}{4}$ and the variance is $\frac{3}{8}$ then the probability of no success is:

Match List-I with List-II | List-I | List-II | |---|---| | Time Series Component | Example | | (A) Secular Variation | (I) Pandemic | | (B) Seasonal Variation | (II) Recession in business | | (C) Cyclic Variation | (III) Monthly sale of woolen cloths | | (D) Irregular variation | (IV) Data regarding National income | Choose the correct answer from the options given below:

The volume of spherical balloon is increasing at the rate of $4 \text{ cm}^3/ \text{sec}$. The rate of increase of its surface area, when the radius is 3cm will be :-

The least non-negative remainder when $2^{75}$ is divided by 5 will be:-

Inlet Pipe A can fill a tank in 30 minutes, and outlet pipes B and C can empty the tank in 2 hours each. If all 3 pipes operate together, the tank will be filled in:

Match List-I with List-II | List-I | List-II | |---|---| | Terms | definition | | (A) POPULATION | (I) Measurable characteristics of the population such as mean, variance, standard deviation etc. of population | | (B) SAMPLE | (II) Measurable characteristics of the sample such as mean, variance, standard deviation etc. of a sample | | (C) PARAMETER | (III) Finite set of statistical individuals drawn from a population for investigation. | | (D) STATISTIC | (IV) Collection of objects having the same characteristics | Choose the correct answer from the options given below:

At what rate will the present value of a perpetuity of Rs.1000 payable at the end of each quarter be Rs.50000?

If A and B are two matrices of order 2 × 2 such that A is a symmetric matrix and B is a skew-symmetric matrix, then:

The interval(s), where the function $f(x) = \begin{cases} \frac{1-e^x}{e^{2x}-1} & : x \neq 0 \\ \frac{-1}{2} & : x = 0 \end{cases}$ is increasing, is/ are:

The solution of $\frac{7x+12}{x-9} < 4$; $ \neq 9$ is:

Consider the following data | Year (x) | 2010 | 2011 | 2012 | 2013 | 2014 | |---|---|---|---|---|---| | Profit (Rs. in thousands) (y) | 10 | 12 | 14 | 16 | 13 | The equation of straight line trend by method of least square for the above data is given by

If the corner points of bounded feasible region for an LPP are (0,2) (3,0) (6,0) (6,8) and (0, 5) then the minimum value of the objective function f=4x+6y occur at

Curd is at 80° F, five minutes later it came down at 60°F. After another 5 minutes, its temperature became 50° F. Given that the rate of change of temperature is proportional to (T - S), where S is temperature of the surroundings and T is temperature of the curd at any time t. Then the temperature of the surroundings is :

The value of $\left|\begin{array}{cc}\log_5 10 & 2 \\[4pt] 2 & \log_{10} 5\end{array}\right|$ is

With reference to sampling, which of the following are correct? (A) Simple random sampling is probability sampling (B) Snow-ball sampling is non-probability sampling (C) Stratified sampling is probability sampling (D) Cluster sampling is non-probability sampling Choose the correct answer from the options given below:

Match List-I with List-II | List-I | List-II | |---|---| | Matrix/equations | Values | | (A) $\begin{bmatrix} 2x+1 & 3y \\ 0 & y^2-5y \end{bmatrix} = \begin{bmatrix} x+3 & y^2+2 \\ 0 & -6 \end{bmatrix}$ | (I) $x = 2, y = -1$ | | (B) $\begin{bmatrix} 1 & 2 & -1 \\ x & 0 & 3 \\ y & 3 & 4 \end{bmatrix}$ is symmetric | (II) $x = 2, y = 2$ | | (C) $[x \ \ 1]\begin{bmatrix} 1 & 0 \\ -2 & -3 \end{bmatrix}\begin{bmatrix} 5 & 2 \\ 0 & y \end{bmatrix} = O$ | (III) $x = -2, y = 2$ | | (D) $\begin{bmatrix} x & 0 \\ 1 & 1 \end{bmatrix}\begin{bmatrix} x & 0 \\ 1 & 1 \end{bmatrix} = \begin{bmatrix} 4 & 0 \\ -1 & y/2 \end{bmatrix}$ | (IV) $x = 2, y = 0$ | Choose the correct answer from the options given below:

The integral $\int \frac{2dx}{e^{2x}-1}$ is equal to:

Which of the following are the properties of Normal Distribution function f(x) and Normal probability curve: (A) The probability of success remains the same in each trial and the number of trials is small in number. (B) The curve is bell-shaped and is symmetrical about the mean. (C) If set of n trials are repeated N times, then frequency f(r) of r successes is given by f(r) = N.p(r) = N$e^{-m\frac{m^r}{r!}}$, r=0,1,2,... (D) As x increases numerically, f(x) decreases rapidly and the maximum value of f(x) occurs at x=μ(mean) Choose the correct answer from the options given below:

The random variable X has the following probability distribution | X | 0 | 1 | 2 | 3 | |---|---|---|---|---| | P(X) | a | a | b | b | such that E(x²) = 2E(x), then the value of b is:

The annual depreciation of an asset is independent of:-

The amount should be deposited at the end of every 6 months to accumulate Rs.50,000 in 8 years if money is worth 6% p.a. compounded semiannually, is: [Given $(1.03)^{16} = 1.6047$]

A square board of side 36cm is made into a box without top by cutting a square from each corner and folding up the flaps to form a box then maximum volume of the box is

A boat covers a distance 24 km upstream and returns to the same point in a total of 4 hours. If the speed of boat in downstream is twice its speed in upstream, then the speed of boat in upstream is:

The point estimate of the population standard deviation as per the below mentioned data from a simple random sample 6,10,15,12,9,8 will be :-

The demand function P for maximising a profit monopolist is given by P=274-x², while the marginal cost is 4+3x for x units of commodity. The consumer surplus is

The probability that in a year of the 22nd century choosen at random, there will be 53 Sundays is:

In a game, A can give 36 points to B, A can give 42 point to C, B can give 10 points to C. How many points make the game ?

If a matrix $A = \begin{bmatrix} 5 & -8 \\ -3 & 5 \end{bmatrix}$ then which of the following is / are TRUE? (A) $|A| = 1$ (B) $A$ is a singular matrix. (C) $-2A = \begin{bmatrix} 10 & -16 \\ -6 & 10 \end{bmatrix}$ (D) $AI = \begin{bmatrix} 5 & 0 \\ 0 & 5 \end{bmatrix}$ $I$ is an identity matrix of order 2. Choose the correct answer from the options given below:

If $\begin{bmatrix} a-b & 0 & 0 \\ 0 & b-c & 0 \\ 0 & 0 & c-2 \end{bmatrix}$ is a scalar matrix such that $a + b + c = 0$, then, which of the following are TRUE? (A) $a = 0$ (B) $b = 0$ (C) $a = 1$ (D) $c = 1$ Choose the correct answer from the options given below:

For the objective function Z=-4x + 6y subject to the constraints 3x + 2y ≥ 5, 7x + 2y ≤ 9, x ≥ 0, y ≥ 0, the maximum value of Z occurs at $(a, b)$ and the minimum value of Z occurs at $(p, q)$ then the value of $\frac{a}{p} + \frac{b}{q}$ is:

Consider the following test: H₀: μ ≤ 12 H₁: μ > 12 A sample of 36 provided a sample mean $\bar{x} = 16$ and a sample standard deviation S=4.2. Then the value of the t- test statistic is:

From a container full of orange juice, 7.5 liters was drawn out and replaced by soda water. This process is repeated 5 more time. The ratio of quantity of orange juice and soda water left in the container is 4:5. How much liter of orange juice did the container originally had? [(Use:0.44) 1/6 = 0.802]

Vatsala buys a car for Rs.7,00,000 and pays upfront Rs.2,50,000 through her credit card. The balance is to be paid in 5 years by equal monthly installments at an interest of 7% per annum as reducing balance. The EMI to be paid by Vatsala will be :- [given (1.0058)⁻⁶⁰=0.7068]

A man wishes to ensure that he gets Rs. 75,000/- at the end of each year indefinitely. The amount that he invest now to produce the desired cash flow, if money is worth 2.5% compounded annually is: