CUET Mathematics 2025 16 May Shift 1 PYQs

The corner points of the bounded feasible region determined by a set of constraints (linear inequalities) are A(0, 5), B(3, 5), C(5, 0) and D(4, 1) and the objective function is z = $px$ + 2$qy$ where p,q > 0. The condition on $p$ and $q$ such that the maximum z occurs at B and D, is:

The random variable $x$ has a probability distribution $p(x)$ of the form $P(x=r)=\begin{cases} rk, & \text{if } r \le 2, \\ (r-1)k, & \text{if } 2<r\le 4, \\ 0, & \text{otherwise,} \end{cases}$ where $r \in \mathbb{N}\cup\{0\}$ and $k\in\mathbb{R}$, where $\mathbb{N}$ is the set of natural numbers. Then: (A) $k=\frac{1}{9}$ (B) $P(2\le x\le 3)=\frac{1}{2}$ (C) $P(x=4)=\frac{1}{3}$ (D) $P(x>1)=\frac{7}{8}$ Choose the correct answer from the options given below.

If $A^T = \begin{bmatrix} -2 & 3 \\ 1 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} -1 & 0 \\ 1 & 2 \end{bmatrix}$, then the matrix $(A + 2B)^T$ is

Consider the following L.P.P. Minimize z = 400x + 300y subject to 100x + 200y ≥ 12000, 300x + 400y ≥ 20000, 200x + 100y ≥ 15000 and x, y ≥ 0. Then

If A is a square matrix of order 3 and |A| = -5, then |3A| is equal to

The value of $\int_0^1 [\log x - \log(1-x)] dx$ is

Match List-I with List-II Consider the function f(x) = 2x³ - 21x² + 36x + 80, x∈[0, 6]. Then | List-I | List-II | |---|---| | (A) one of its critical points is at x = | (I) -28 | | (B) Its absolute maximum value is | (II) -42 | | (C) Its absolute minimum value is | (III) 97 | | (D) Its second derivative at x = 0 is | (IV) 6 | Choose the correct answer from the options given below:

The degree of the differential equation $\left(2 + \left(\frac{dy}{dx}\right)^2\right)^{\frac{3}{2}} = a^2 \frac{d^2y}{dx^2}$ is:

If the interval in which f(x) = $\frac{x}{4}$ + $\frac{4}{x}$, x ≠ 0 is strictly increasing is (-∞, a) ∪ (b, ∞), then

If a matrix has 12 elements, then the possible orders it can have, are (A) 1 × 12 (B) 4 × 3 (C) 6 × 3 (D) 6 × 2 Choose the correct answer from the options given below:

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

If $y = (log x)^{(log x)}$, $x > 1$ then $\frac{dy}{dx}$ is equal to

The area of the region bounded by y² = 9x, x = 2, x = 4 and the x-axis in the first quadrant, is

If A is an invertible symmetric matrix, then A⁻¹ is

$\int \frac{(x-3)e^x}{(x-1)^3} dx$ is equal to

Match List-I with List-II | List-I | List-II | |---|---| | (A) Angle between î and -ĵ is | (I) $\frac{\pi}{6}$ | | (B) Angle between 2î + k̂ and 10î + 5k̂ is | (II) $\frac{\pi}{4}$ | | (C) Angle between î and î + ĵ is | (III) 2π | | (D) Angle between √3ĵ - k̂ and ĵ is | (IV) $\frac{\pi}{2}$ | Choose the correct answer from the options given below:

The area enclosed between the graph of y = x³ and the lines x = 0, y = 1, y = 8 is

Match List-I with List-II Where ℝ is set of real numbers | List-I | List-II | |---|---| | (A) f: ℝ → ℝ s.t f(x) = x⁴ is | (I) one-one, Into | | (B) f: ℝ → [0, ∞) s.t f(x) = x⁴ is | (II) many-one, into | | (C) f: [0, ∞) → ℝ s.t f(x) = x⁴ is | (III) one-one, onto | | (D) f: [0, ∞) → [0, ∞) s.t f(x) = x⁴ is | (IV) many-one, onto | Choose the correct answer from the options given below:

For x ∈ ℝ - {0}, the function f(x) = $\frac{3}{x}$ + 7 is decreasing when

If a line makes angles α, β and γ with positive x-axis, y-axis and z-axis respectively, then the value of $sin^2\frac{α}{2}cos^2\frac{α}{2} + sin^2\frac{β}{2}cos^2\frac{β}{2} + sin^2\frac{γ}{2}cos^2\frac{γ}{2}$ is

If $g(x) = \begin{cases} \frac{αx}{|x|}, & \text{if } x < 0 \\ 5, & \text{if } x ≥ 0 \end{cases}$ is continuous at x = 0, then the value of α is

If $\vec{a}$ and $\vec{b}$ are two unit vectors and $\vec{a} + \vec{b}$ is also unit vector, the magnitude of $\vec{a} - \vec{b}$ is

If the points (a₁, b₁), (a₂, b₂) and (a₁ + a₂, b₁ + b₂) are collinear, then

If A and B are square matrices of order 3 such that |A| = -1 and |B| = 5, then the value of |3AB| is

In class XII, suppose 5% of boys and 0.25% of girls are physically fit for a game. A fit student is selected at random from this class having same number of boys and girls. If the probability that the selected student is a girl is $\frac{m}{n}$, gcd(m, n) = 1, then m + n is equal to

Match List-I with List-II | List-I | List-II | | --- | --- | | Functions | Integrals | | --- | --- | | (A) $\int \frac{dx}{x^2 - 4}, x \neq \pm 2$ | (I) $\log \vert x + \sqrt{4 + x^2}\vert + C$, where $C$ is an arbitrary constant | | (B) $\int \frac{1}{\sqrt{16 - x^2}} dx; \vert x\vert < 4$ | (II) $\sin^{-1} \left( \frac{x}{4} \right) + C$, where $C$ is an arbitrary constant | | (C) $\int \frac{1}{16 + x^2} dx$ | (III) $\frac{1}{4} \log \left\vert \frac{x-2}{x+2} \right\vert + C$, where $C$ is an arbitrary constant | | (D) $\int \frac{1}{\sqrt{4 + x^2}} dx$ | (IV) $\frac{1}{4} \tan^{-1} \left( \frac{x}{4} \right) + C$, where $C$ is an arbitrary constant | Choose the correct answer from the options given below:

The maximum value of f(x) = $\left(\frac{1}{x}\right)^x$ is

If $y = e^{acos^{-1}x}, -1 < x < 1$, then $(1-x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx}$ is equal to

If the interval in which the function f(x) = $\frac{x}{x^2+1}$ is strictly increasing is (-a, a), then a is equal to

If R and S are two equivalence relations on a set A, then

A random variable X has the following probability distribution | X | 0 | 1 | 2 | 3 | 4 | 5 | |---|---|---|---|---|---|---| | P(X) | 0.1 | k | 0.2 | 2k | 0.3 | k | Then P (X < 3) is

Given differential equation, (1 + y²)dx = (tan⁻¹y - x)dy, then which of the following is/are true? (A) Integrating factor = tan⁻¹x (B) Integrating factor = tan⁻¹y (C) Integrating factor = $e^{tan⁻¹y}$ (D) Degree = 1 Choose the correct answer from the options given below:

Match List-I with List-II Let $A$ be any invertible square matrix. Then | List-I | List-II | | --- | --- | | (A) $A - A^T$ | (I) $\vert A\vert A^{-1}$ | | (B) $A A^T$ | (II) Skew-symmetric | | (C) $\det (A^{-1})$ | (III) Symmetric | | (D) $\text{adj} A$ | (IV) $[\det(A)]^{-1}$ | Choose the correct answer from the options given below:

Let A = [aᵢⱼ]ₙₓₙ and B = [bᵢⱼ]ₙₓₙ. Then which of the following is/are true? (A) AB = BA (B) (AB)⁻¹ = B⁻¹ A⁻¹ (C) $(AB)^T = B^T A^T$ (D) AB = 0 ⇒ A = 0 or B = 0 Choose the correct answer from the options given below:

Consider the line $\frac{x-2}{2} = \frac{2y-5}{-3}, z = -1$. Then which of the following is/are true? (A) It has Direction ratios (2, -3, -1) (B) It has Direction cosines $\left(\frac{4}{5}, \frac{-3}{5}, \frac{-1}{5}\right)$ (C) It has Direction ratios $\left(2, \frac{-3}{2}, 0\right)$ (D) It has Direction cosines $\left(\frac{4}{5}, \frac{-3}{5}, 0\right)$ Choose the correct answer from the options given below:

$\int \frac{\cos 2x - \cos 2α}{\cos x - \cos α} dx$ is equal to

A bag contains 4 red and 6 green balls. A ball is drawn at random. Its colour is noted and is returned to the bag. One additional ball of the colour drawn is put in the bag. Again a ball is then drawn from the bag. The probability of this ball to be of green colour is

If $\det \begin{pmatrix} 2x & 5 \\ 8 & x \end{pmatrix} = \det \begin{pmatrix} 6 & -2 \\ 1 & 1 \end{pmatrix}$, then the value of $x$ is

$\int_{3\pi/8}^{\pi/8} \frac{ \tan^{2025} x}{ \tan^{2025} x + \cot^{2025} x} dx$ is equal to

Match List-I with List-II | List-I | List-II | | --- | --- | | (A) $f(x) = x \sin x$ | (I) is not continuous at $x = -3$ | | (B) $f(x) = \frac{\vert x\vert }{x}, x \neq 0$ and $f(x) = 1 \text{ at } x = 0$ | (II) is continuous everywhere | | (C) $f(x) = x - [x]$, $[x]$ denotes greatest integer function | (III) is not differentiable at $x = 1$ | | (D) $f(x) = e^{\vert x - 1\vert }$ | (IV) is not continuous at $x = 0$ | Choose the correct answer from the options given below:

The angle between the pair of lines given by $\vec{r} = \hat{i} + 2\hat{j} - 3\hat{k} + \lambda (\hat{i} - 2\hat{j} + 2\hat{k})$ and $\vec{r} = 5\hat{i} + \hat{j} + \hat{k} + \mu (3\hat{i} - 2\hat{j} + 6\hat{k})$ is

The general solution of the differential equation $\frac{dy}{dx} + y \tan x = \sec x$

If $\vec{a}$ is a non-zero vector, then always

Consider the following L.P.P Max. z = 5x + 2y; subject to -2x - 3y ≤ -6, x - 2y ≤ 2, 3x + 2y ≤ 12, -3x + 2y ≤ 3 and x, y ≥ 0 then

If $3\begin{bmatrix} x & y \\ z & w \end{bmatrix} = \begin{bmatrix} x & 6 \\ -1 & 2w \end{bmatrix} + \begin{bmatrix} 4 & x + y \\ z + w & 3 \end{bmatrix}$, then the values of $x, y, z$ and $w$ are

Match List-I with List-II | List-I | List-II | |---|---| | Inverse Trigonometric function | Principal values of arguments | | (A) $sin^{-1}\left(\frac{-1}{2}\right)$ | (I) $\frac{-\pi}{3}$ | | (B) $cos^{-1}\left(\frac{-1}{2}\right)$ | (II) $\frac{3\pi}{4}$ | | (C) $tan^{-1}(-\sqrt{3})$ | (III) $\frac{-\pi}{6}$ | | (D) $sec^{-1}(-\sqrt{2})$ | (IV) $\frac{2\pi}{3}$ | Choose the correct answer from the options given below:

Consider the following L.L.P. Minimize z = 30x - 30y + 1800; subject to x + y ≤ 30, x ≤ 15, y ≤ 20, x + y ≥ 15 and x, y ≥ 0. Then it attains its optimal value at the point

Given that $\vec{a} = -3\hat{i} - 6\hat{j} + 4\hat{k}$, $\vec{b} = 9\hat{i} - λ\hat{j} - 12\hat{k}$. If $\vec{a} \times \vec{b} = \vec{0}$, then the value of λ is

The area (sq.units) bounded by the curve y = sinx, π ≤ x ≤ 2π and the x-axis is

A box contains 2 black and 4 red balls and another box contains 4 black and 3 red balls. If a ball drawn at random from one of the two boxes, then the probability of getting a black ball is

Which of the following are normal equations to fit a straight line trend y = a + bx by the method of least squares?

The following data shows the percentage of rural and urban Indian households who have a high speed internet connection. | Year (x) | Rural Household (y%) | Urban Household (y%) | |---|---|---| | 2020 | 2 | 6 | | 2021 | 3 | 7 | | 2022 | 3 | 9 | | 2023 | 4 | 15 | | 2024 | 5 | 20 | Based on the above data, which of the following statements are correct? (A) A straight line trend by the method of the least squares for rural Indians is $y^t$ = 3.4 + 0.7 (x - 2022) (B) A straight line trend by the method of the least squares for urban Indians is $y^t$ = 11.4 + 4.3 (x - 2022) (C) The forecast for the year 2025 for urban group using the trend equation is 22.2 %. (D) The forecast for the year 2025 for rural group using the trend equation is 5.5 %. Choose the correct answer from the options given below:

The point on the curve y = (x - 2)² at which the tangent is parallel to the chord joining the points (2, 0) and (4, 4) is:

The point estimate of the population standard deviation for the random sample 5, 8, 10, 7, 10, 14 is:

If the random variable X is normally distributed with mean 16 and standard deviation 4, then the standard normal variable Z corresponding to X = 17 is

Which of the following are correct? (A) For a square matrix A, if A³ = I, then A⁻¹ = A². (B) The determinant of only a square matrix can be defined. (C) If A is a square matrix of order 3, then the number of minors of the matrix A is 3. (D) If A and B are two non-singular matrices of the same order, then (AB)⁻¹ = B⁻¹A⁻¹. Choose the correct answer from the options given below:

A container contains 100 liters of milk. From this container, 10 liters of milk is taken out and replaced by water. This process is repeated two more times.The amount of milk left in the container now is:

A person invested ₹ 2,50,000 in a fund. At the end of the year, the value of the fund is ₹ 3,00,000. If the market price is the same at the end of the year, then the nominal rate of return is

Which of the following are correct? (A) If a ≡ b(mod n), then -a ≡ -b (mod n) (B) If a + b = c, then a(mod n) + b(mod n) ≡ (a + b + c)(mod n). (C) If a ≡ b (mod n), then ka ≡ kb(mod n), ∀ k ∈ I. (D) If a ≡ b (mod n), then (a + k) ≡ (b + k)(mod n), ∀ k ∈ I. Choose the correct answer from the options given below:

The fund created to accumulate money over the years to discharge a future obligation is known as:

A boat covers 32 km upstream and 36 km downstream in 7 hours. Also, it covers 40 km upstream and 48 km downstream in 9 hours. The speed of the boat in still water is:

If a moneylender charges 'interest' at the rate of 10 rupees per 100 rupees per half year, payable in advance, then the effective rate of interest per annum is

The corner points of the bounded feasible region for a linear programming problem (LPP) are (0, 3/2), (1, 2) and (4, 0). If the objective function is Z = ax + by, where 'a' and 'b' are positive, then the condition on 'a' and 'b' so that the maximum of Z occurs at (1, 2) and (4, 0) is:

If $y = \left(\log\left(x + \sqrt{x^2+a^2}\right)\right)^2$ and $x \neq \frac{1-a^2}{2}$, then $(x^2+a^2)\frac{d^2y}{dx^2} + x\frac{dy}{dx}$ is equal to:

A box contains 10 balls, each marked with one of the digits 0 to 9. If four balls are drawn successively with replacement from the bag, then the probability that none is marked with the digit 0 is:

Irregular trends in a time series are caused by (A) Epidemics (B) Strikes (C) Festival season (D) Floods Choose the correct answer from the options given below:

Mr. X purchased a house from a company for ₹ 7,00,000 and made a down payment of ₹ 1,50,000. He repays the balance in 25 years by equal monthly installments at 9 % per annum compounded monthly. The equated monthly installment (EMI) is: [Given that : (1.0075)⁻³⁰⁰ = 0.106]

If, in a pair of consecutive positive integers, both numbers are greater than 5 and their sum is less than 23, then the number of such pairs are:

In an influenza epidemic, the number of cases diagnosed in a week were as follows: | Day | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday | Sunday | |---|---|---|---|---|---|---|---| | Number of Cases | 10 | 12 | 8 | 16 | 9 | 14 | 4 | The 3-day moving averages are:

Shyam invested ₹ 2,00,000 in 2019 for 5 years. If the compound annual growth rate (CAGR) for his investment is 10 %, then the end balance of his investment is:

Consider the linear programming problem (LPP): Maximize Z = 6x + 3y subject to the conditions, 4x + y ≥ 80, x + 5y ≥ 115, 3x + 2y ≤ 150, x, y ≥ 0. In reference to the above LPP, which of the following are correct? (A) The feasible region is bounded. (B) The corner points of the feasible region are (15, 20), (40, 15) and (0, 75). (C) The maximum value of the objective function is 285. (D) The LPP does not have optimal solution. Choose the correct answer from the options given below:

If the random variable X follows the Poisson distribution such that P[X = k] = P[X = k+1], then the mean value of X is:

In a 500 m race, the ratio of speeds of two participants, A and B, is 4:5 respectively. If A has a start of 180 m, then the distance by which A wins is:

If $A = \begin{bmatrix} 0 & a & 2 \\ -2 & 0 & b \\ -2 & 2 & c \end{bmatrix}$ is a skew symmetric matrix, then the value of $(a + b + c)^3$ is

Pipe A can fill a tank three times faster than pipe B. If both pipes A and B are running together, they can fill the tank in 12 minutes, then the time taken by pipe B alone to fill the tank is:

The value of $\begin{vmatrix} 1 & bc & bc(b+c) \\ 1 & ca & ca(c+a) \\ 1 & ab & ab(a+b) \end{vmatrix}$ is

Which of the following are correct? (A) A statistical hypothesis is a statement about the population parameter which may be true or false. (B) In a statistical hypothesis testing procedure, a Type I error is made when the null hypothesis is accepted when it is false. (C) In a statistical hypothesis testing procedure, a Type II error is made when the null hypothesis is rejected when it is true. (D) The number of degrees of freedom of a statistic is the number of independent variates used to compute that statistic. Choose the correct answer from the options given below:

In a game, a man wins ₹ 8 for getting a number greater than 3 and loses ₹ 3 otherwise, when a fair die is thrown. The man decided to throw a die 4 times but to quit as and when he gets a number greater than 3. If X denotes the amount which the man wins or loses, then which of the following are correct? (A) All the possible values of X are 8, 5, 2 and -1. (B) The probability distribution of X is: | X | 8 | 5 | 2 | -1 | -12 | |---|---|---|---|---|---| | P(X) | 1/2 | 1/4 | 1/8 | 1/16 | 1/16 | (C) The mean value of X is 75/16. (D) The variance of X is 6615/256. Choose the correct answer from the options given below:

Match List-I with List-II | List-I | List-II | |---|---| | (Matrix A) | (Determinant of adj A) | | (A) $\begin{bmatrix} 2 & 1 \\ -1 & 2 \end{bmatrix}$ | (I) 3 | | (B) $\begin{bmatrix} 3 & 4 \\ 3 & 6 \end{bmatrix}$ | (II) 6 | | (C) $\begin{bmatrix} 3 & 7 \\ -2 & -4 \end{bmatrix}$ | (III) 5 | | (D) $\begin{bmatrix} 4 & 3 \\ 3 & 3 \end{bmatrix}$ | (IV) 2 | Choose the correct answer from the options given below:

If the integral $I = \int \frac{x^2}{\sqrt{1+x}} dx = \frac{1}{\alpha} (1+x)^\alpha - \frac{8\alpha}{15} (1+x)^{\alpha-1} + 2(1+x)^{\alpha-2} + C$, $C$ is constant of integration, then the value of $\alpha$ is:

If A is a square matrix of order 3 and $|A| = 4$, then the value of $|2A^T|$ is

A machine has been producing steel tubes of mean inner diameter of 2 cm. A sample of 10 tubes gives an inner diameter of 2.01 cm and a standard deviation of 0.063 cm. Test the hypothesis that the machine is working in the proper order at 5 % level of significance and answer, which of the following statements are correct? [Given that : t₉(0.05) = 2.262] (A) The value of the test statistic is t = 0.476. (B) The null hypothesis is rejected at 5 % level of significance. (C) The null hypothesis is accepted at 5 % level of significance. (D) The degree of freedom is 10. Choose the correct answer from the options given below:

The particular solution of the differential equation x(1 + y²)dx - y(1 + x²)dy = 0, y(0) = 1, is:

Match List-I with List-II | List-I | List-II | |---|---| | (Parametric equations) | $\left(\frac{dy}{dx}\right)$ | | (A) $x = \frac{2}{t}, y = 2t$ | (I) $4t^2$ | | (B) $x = t^3, y = 3t + 2$ | (II) $2(t+1)$ | | (C) $x = \log t, y = 2t^2$ | (III) $-t^2$ | | (D) $x = e^t, y = 2te^t$ | (IV) $t^{-2}$ | Choose the correct answer from the options given below:

If $C$ is the original value of the asset, $S$ is the scrap value, n is the useful life of the asset and $D$ is the annual depreciation, then