CUET Mathematics 2025 PYQs3 June Shift 1. Free, no login required.

If the feasible region of an LPP is bounded and the corresponding objective function is $z = 5x - 9y$, then the objective function attains:

A random variable X has the following probability distribution: | X | 2 | 3 | 4 | 5 | |---|---|---|---|---| | P(X) | 5/k | 7/k | 9/k | 11/k | Then the value of $\frac{k}{4}$ is:

The function $f(x) = 4x^3 - 7x^2$ has point(s) of local minima at

If $A = [a_{ij}]$ is skew symmetric matrix of order 'n', then

$\int \sqrt{1 + \frac{x^2}{9}} dx$ is equal to (Where C is an arbitrary constant)

If $y = \sqrt{ax + b}$ then $y\frac{d^2y}{dx^2} + (\frac{dy}{dx})^2 =$

$\int_{1}^{2} \frac{\sqrt{x}}{\sqrt{3 - x} + \sqrt{x}} dx$ is equal to

If the points (a, b), (c, d) and (a + c, b + d) are collinear, then

For the function $f(x) = x^x, x > 0$, which of the following are TRUE? (A) $f'(x) = x^x(1 + \log x)$ (B) $x = e$ is the critical point (C) $f$ is increasing in $(\frac{1}{e}, \infty)$ (D) $f$ is increasing in $(0, \infty)$ Choose the *correct* answer from the options given below:

Let A be a matrix such that $A = \begin{bmatrix} 1 & 2 \\ -2 & 3 \end{bmatrix}$. Then which of the following are TRUE? (A) A is non-singular matrix (B) $A^T = A$ (C) A is not invertible matrix (D) A is not skew-symmetric matrix Choose the *correct* answer from the options given below:

The area of the region bounded by the line $y = 2x$ and the x-axis between $x = -2$ and $x = 2$ is

The corner points of a bounded feasible region are (0, 5), (6, 1), (17, 2) and (4, 29). If the maximum value of objective function $z = px + qy$ where $p$ and $q > 0$ occurs at two points (17, 2) and (4, 29), then the relation between $p$ and $q$ is:

If the system of equations $2x + 5y = 7, 6x + \lambda y = 28$ is inconsistent, then

The particular solution of the differential equation $xdy = (2x^2 + 1)dx, x \neq 0$, given that $y = 1$ when $x = 1$ is:

If m and n are respectively the order and degree of the differential equation $(\frac{d^2y}{dx^2})^{2} + (\frac{dy}{dx})^3 + y= 4x$, then the value of $m + n$ is:

The shortest distance between the lines $\vec{r} = \hat{i} + \hat{j} + \lambda(2\hat{i} - \hat{j} + \hat{k})$ and $\vec{r} = 2\hat{i} + \hat{j} - \hat{k} + \mu(4\hat{i} - 2\hat{j} + 2\hat{k})$ is

The function $f(x) = 4 - 3x + 3x^2 - x^3$ is (Here $\mathbb{R}$ is set of real numbers)

Let $A = [a_{ij}]$ be a square matrix of order 3 with $|A| = 2$ and let $C = [c_{ij}]$ where $c_{ij} =$ cofactor of $a_{ij}$ in A. Then $|C|$ is equal to:

Let $\vec{a} = \hat{i} + 4\hat{j} + 2\hat{k}$, $\vec{b} = 3\hat{i} - 2\hat{j} + 7\hat{k}$ and $\vec{c} = 2\hat{i} + \hat{j} + 4\hat{k}$. A vector $\vec{d}$ which is perpendicular to both $\vec{a}$ and $\vec{b}$, and $\vec{c} \cdot \vec{d} = 14$, is:

The range of function $f(x) = 4x^2 + 12x + 7, x \in \mathbb{R}$ is

For a square matrix $A$ of order 3, if $|A| = 2$, then $|adj\ 2A| =$

The sum of the x-coordinates of the corner points of the feasible region for the LPP: Minimize $z = 3x + 2y$ subject to constraints $x + y \leq 14$, $x \geq 4$, $x \leq 8, y \geq 0$ is

If $y = \log_e(\sec e^{x^2})$ then $\frac{dy}{dx} =$

Let $\vec{a} = 3\hat{i} + \hat{j} - 4\hat{k}$ and $\vec{b} = 6\hat{i} + 5\hat{j} - 2\hat{k}$ be two vectors. Then a vector perpendicular to $\vec{a}$ and $\vec{b}$ with magnitude 3 units is

The Cartesian equation of the line passing through the point (1, 2, -1) and parallel to the line $5x - 25 = 14 - 7y = 35z$ is

Area of the region bounded by the curve $y = \sin x$ and x-axis between $x = \frac{\pi}{2}$ and $x = \frac{3\pi}{2}$ is

Consider the differential equation $xdy = (y + 2x^3)dx$. Then which of the following are TRUE? (A) It is a homogeneous differential equation. (B) Product of the order and degree of the differential equation in one. (C) Integrating factor is x. (D) General solution of the differential equation is $y = x^3 + Cx$, where C is an arbitary constant. Choose the *correct* answer from the options given below:

If $A = \begin{bmatrix} 1 & 2 & 3 \\ -4 & -5 & -2 \end{bmatrix}$, $B = \begin{bmatrix} 2 & -3 \\ 4 & -5 \\ 2 & -1 \end{bmatrix}$ and $BA = [b_{ij}]$, then $(b_{23} - b_{31})$ is equal to

The sum of order and degree of the differential equation $(x^2\frac{d^2y}{dx^2})^{3/4} = 5(\frac{dy}{dx})^2 - 3$ is equal to

60% members of a committee favour a certain proposal and 40% members oppose the proposal. A member is selected and let the random variable X = 0 if he opposes and X = 1 if he is in favour. Then the variance of the random variable X is

The value of k for which the system of equations $x + y + z = 1$ $x - ky + z = 1$ $x - y + z = 1$ has more than one solutions is

$\int \frac{dx}{\sqrt{5 - 4x - x^2}}$ is equal to

If A and B are independent events, then which of the following is **not** true?

$\int \frac{\log_e x}{(1 + \log_e x)^2} dx$ is equal to

The projection of the vector $2\hat{i} - \hat{j} + 3\hat{k}$ on the vector $3\hat{i} + 2\hat{j} + 6\hat{k}$ is

A vector $\vec{a}$ of magnitude $3\sqrt{2}$ making an angle of $\frac{\pi}{3}$ with $\hat{i}$, $\frac{\pi}{4}$ with $\hat{j}$ and an actue angle $\theta$ with $\hat{k}$, is

The objective function of an LPP is $z = ax + \beta y, (a, \beta > 0)$ in that has to be maximized/minimized subject to constraints $x + y \leq 2$, $x \geq 0$, $y \geq 0$. Then max (z) $-$ min (z) is equal to

The value of $-\cosec^2(\cot^{-1}y) + \sec^2( \tan^{-1}x)$ is equal to

Match List-I with List-II | List-I | List-II | | --- | --- | | Integral | Value | | --- | --- | | (A) $\int_{-1}^{1} (\vert x\vert + 1) dx$ | (I) 0 | | (B) $\int_{-2}^{2} \vert x + 1\vert dx$ | (II) 2 | | (C) $\int_{-1}^{1} 3\vert x^2\vert dx$ | (III) 5 | | (D) $\int_{-1}^{1} x\vert x\vert dx$ | (IV) 3 | Choose the correct answer from the options given below:

If the lines $\frac{x - 5}{5\lambda + 2} = \frac{2 - y}{5} = \frac{1 - z}{-1}$ and $x = \frac{y + 1/2}{2\lambda} = \frac{z - 1}{3}$ are perpendicular, then the value of $\lambda$ is equal to

If $A = \begin{bmatrix} 1 & 2 \\ 4 & 5 \end{bmatrix}$, then Match **List-I** with **List-II** | List-I | List-II | |---|---| | (A) det (A) | (I) $-\frac{1}{3}$ | | (B) det $(A^{-1})$ | (II) $-12$ | | (C) det (2A) | (III) $-3$ | | (D) det $(3A^T)$ | (IV) $-27$ | Choose the **correct** answer from the options given below:

The area of the smaller region of the circle $x^2 + y^2 = 8$ cut off by the line $x = 2$ is

Let $f(x) = x^2 + \frac{250}{x}$ be any function defined on $\mathbb{R} - \{0\}$, where $\mathbb{R}$ is the set of real numbers. Then which of the following are TRUE? (A) $f'(x) = 2x + \frac{250}{x^2}$ (B) $x = 5$ in the only critical point of $f(x)$ (C) minimum value of $f(x)$ is 75 (D) maximum value of $f(x)$ is 50. Choose the **correct** answer from the options given below:

Let P and Q be any two invertible matrices of the same order. Then Match **List-I** with **List-II** | List-I | List-II | |---|---| | **Matrix** | **Equivalent matrix** | | (A) $(P Q)^{-1}$ | (I) $Q^{-1}P$ | | (B) $(P^{-1}Q)^{-1}$ | (II) $Q P^{-1}$ | | (C) $(P Q^{-1})^{-1}$ | (III) $Q^{-1}P^{-1}$ | | (D) $(P^{-1}Q^{-1})^{-1}$ | (IV) Q P | Choose the **correct** answer from the options given below:

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

Match **List-I** with **List-II** | List-I | List-II | | :--- | :--- | | **Function** | **Points of discontinuity** | | (A) $f(x) = \frac{x^2 + 1}{x}$ | (I) $x = 4$ | | (B) $f(x) = \frac{\vert x - 1 \vert}{x - 1}$ | (II) $x = 2$ | | (C) $f(x) = \begin{cases} x - 1, & x < 2 \\ x + 1, & x \ge 2 \end{cases}$ | (III) $x = 0$ | | (D) $f(x) = \frac{1 - x}{(x - 4)}$ | (IV) $x = 1$ | Choose the **correct** answer from the options given below:

If $\frac{d}{dx}[ax^3 + ax^2 + ax + 1] = 9x^2 + 6x + 3$, then $a$ is equal to

The probability that a leap year selected at random will have 53 Mondays is

For the relation $R = \{(a, b): a \leq b\}$ in $\mathbb{R}$, which of the following is correct?

Two persons A and B throw a die alternately till one of them gets a 'three' and wins the game. The probability of A's winning if A starts first is

From a container full of milk, 10 liters (l) was drawn and replaced by water. This process is repeated one more time. The ratio of quantity of milk and water left in the container is 4:5. Then the capacity of the container is:

'A' can run 1km in 5 minutes 20 seconds and 'B' can run the same distance in 6 minutes. How many meters start can 'A' give 'B' in a kilometer race so that they finish the race together?

If $e^y(x + 1) = 1$ and $\frac{d^2y}{dx^2} = k(\frac{dy}{dx})^2$, then k is equal to

Two pipes A and B can fill a tank in 20 minutes and 30 minutes respectively. Both pipes A and B are opened together for some time and then pipe B is turned off. If the tank is filled in 15 minutes, then time for which the pipe B works is:

A small start-up started making wafers and distributing them to the retailers. After a week, average sales per week were found to be 150 packets. So, to increase the sales, a strategy was used to change the packaging and add a chocolate worth Rs. 5 as a free gift with the pack. After this, a sample of 17 shops was taken, which showed that sales went up with mean 165 and a standard deviation of 25. Check whether the strategy was effective @5%, level of significance ? [Given $ t_{16} (0.05) =2.12$]

The average cost function for a commodity is given by $AC = 0.05x^2 - 5x + 1000 + \frac{3000}{x}$ in terms of output x. The fixed cost is

The value of $\begin{vmatrix} x & x+y & x+y+z \\ 2x & 3x+2y & 4x+3y+2z \\ 3x & 6x+3y & 10x+6y+3z \end{vmatrix}$ is

Match **List-I** with **List-II** | List-I | List-II | |---|---| | (A) Perpetuity | (I) A person deposits a fixed amount every year in his bank account to renovate his house after 10 yrs. | | (B) EMI | (II) A person depositis an amount regularly in his bank account and withdraws in case of need. | | (C) Sinking Fund | (III) A fixed amount is debited from the bank account of a person, every month, against a personal loan. | | (D) Saving Account | (IV) A person purchased a house and rents it out. | Choose the **correct** answer from the options given below:

The effective rate, which is equivalent to a nominal rate of 12% compounded semi-annually, is

If the difference between mean and variance of a Binomial distribution is 1 and the difference of their squares is 5, then the probability of success is

Ajesh purchased a printer ₹ 15,000. The printer is estimated to have a scrap value of ₹ 3,000 after a span of 6 years. Then the book value of the printer at the end of 3 years will be:-

Let X denote the number of hours a student studies on a selected day. The probability distribution of X is given by (where k is some unknown constant) $P(X = x_i) = \begin{cases} 0.5, & \text{if } x_i = 0, \\ kx_i, & \text{if } x_i = 1, \\ k(4 - x_i), & \text{if } x_i = 2 \text{ or } 3, \\ 0, & \text{otherwise}. \end{cases}$ Then the value of k is

A man rows 15 km upstream in 5 hours and 25 km downstream in 5 hours each time, then the speed of the stream is

A company purchased a machine for ₹ 15,00,000 and its effective life is estimated to be 10 years. A sinking fund is created for replacing the machine at the end of its effective life when its scrap value is ₹ 2,42,000. What amount company should provide, at the end of every year out of profits for the sinking fund if it accumulates an interest of 5% per annum? [Given(1.05)¹⁰=1.629]

The value of $\int \frac{(x^4 - x)^{1/4}}{x^5} dx$ is equal to (where C is an arbitrary constant)

The solution set of the linear inequation $|4x - 3| \leq \frac{3}{4}$ is:

A man plans to take a housing loan of Rs 99,53,000 from a bank costing 18% per annum compounded monthly. The loan is to be paid back in 30 years in equal monthly installments (EMI). The EMI by reducing balance method is: [Given $(1.015)^{-360} = 0.0047$]

Components of Time Series are (A) Secular Trend Component (B) Seasonal Component (C) Moving Average Component (D) Cyclical Component Choose the **correct** answer from the options given below:

The remainder, when $5^{60}$ is divided by 7, is

The integral value of k for which the system of linear equations $kx + y + 2z = 0$, $ky = x - 3z$ and $2x + y + kz = 0$ has a non-zero solution is

If $A = \begin{bmatrix} 5 & 6 \\ 3 & 2 \end{bmatrix}$ then which of the following is correct? (A) $|A|$ is positive (B) $|adj\ A| = -8$ (C) Cofactor of 3 is 6 (D) $|2A| = -32$ Choose the **correct** answer from the options given below:

Based on the data available for the production ($y_i$ in thousand tons) of a cloth factory for 7 years ($x_i$) using the method of least squares, the straight line trend is given by $y - a + bx$ with $\sum y_i = 608, \sum x_i = 0, \sum x_iy_i = 116, \sum x_i^2 = 28$. Then, the increase in production per year is:

As per the below-mentioned graph of shaded bounded feasible region of the LPP, the maximum value of the objective function $z = 2x + y$ is <img src="https://balti.afterboards.in/2vhvDBCiMgfU1Wc" width="400px"/>

Which of the following statements are correct? (A) Inverse of a matrix, if it exists, is unique (B) $(kA)' = -kA'$ (where k is any real number) (C) For an invertible matrix $A$, $(A^{-1})^{-1} = A$ (D) For an invertible matrix $A$, $(A')^{-1} = (A^{-1})'$ Choose the **correct** answer from the options given below:

If the matrix $\begin{bmatrix} -1 & x-y & 4 \\ 2 & 0 & 5 \\ x+y & z & 6 \end{bmatrix}$ is symmetric, then $x + 3y + 2z$ is equal to

Two percent of the bolts manufactured in a factory are found to be defective. Using the Poisson distribution, the probability that in a sample of 100 bolts chosen at random, exactly two will be defective, is: [Given $e^{-2}=0.135$]

Match **List-I** with **List-II** | List-I | List-II | |---|---| | **Differential Equation** | **Sum of order and degree** | | (A) $\frac{d^2y}{dx^2} + \frac{dy}{dx} + 3y = \sin x$ | (I) 2 | | (B) $\frac{dy}{dx} = \sin(x + y)$ | (II) 3 | | (C) $\sqrt{1 + (\frac{dy}{dx})^2} = \frac{d^2y}{dx^2}$ | (III) 4 | | (D) $x^2(\frac{d^2y}{dx^2})^3 + y(\frac{dy}{dx})^4 + y^5 = 0$ | (IV) 5 | Choose the **correct** answer from the options given below:

Let f be a function defined by $f(x) = 2x^3 - 3x^2 - 36x + 2$, then which of the following are correct? (A) The critical points of f(x) are -2 and 3. (B) The function f(x) increases in the interval $(3, \infty)$ (C) The function f(x) decreases in the interval (-2,3) (D) The function f(x) increases in the interval (-2,3) Choose the **correct** answer from the options given below:

Solution of the differential equation $y\log_e y dx - x dy = 0$ is (Where c is an arbitrary constant)

A startup company invested ₹ 5,00,000 in shares for 4 years. The value of the investment was ₹ 5,50,000 at the end of first year, ₹ 5,25,000 at the end of third year, and on maturity, the final value stood ₹ 6,25,000. The CAGR on the investment will be :- [Given : $(1.25)^{\frac{1}{4}} = 1.06$]

The marks obtained by five students in a test of Applied Mathematics carrying 100 marks are 49, 58, 67, 92, 99. Then the point estimate of the population mean is

Consider the following hypothesis $H_0: \mu = 315$ and $H_a: \mu \neq 315$ A sample of 60 provided a sample mean of 324.6. The standard deviation ($\sigma$) is 14 and level of significance $\alpha = 0.05$. Then the confidence interval is: [Given: $Z_{a/2}{\frac{14}{\sqrt60}} = 3.54$]

Which of the following statements are correct? (A) The mean and variance of the Poisson distribution are equal. (B) The mean and variance of a Binomial distribution are equal. (C) An unbiased die is thrown again and again until two sixes are obtained, then the probability of obtaining the second six in the 3rd throw is $\frac{5}{108}$. (D) If the variance of a Poisson distribution is 2, then P(X = 2) = $2e^{-2}$ Choose the **correct** answer from the options given below:

The first m-year moving average of the data 10, 20, 30, 40, 50 is 30. The value of m is

A Linear Programming Problem (LPP) consists of which of the following components? (A) Decision variables (B) The graphical compliment (C) The objective function (D) The linear constraints Choose the **correct** answer from the options given below: