CUET Mathematics 2025 15 May Shift 2 PYQs

If A is a matrix of order $m × n$ and $B$ is a matrix such that $AB^T$ and $B^TA$ are both well-defined matrices, then order of matrix B is

$\int \frac{f'(x)}{f(x) \log_e[f(x)]} dx$ is equal to

Match List-I with List-II | List-I | List-II | | :--- | :--- | | **Differential equation** | **Order and degree** | | (A) $(y'')^3 + (y')^4 - 6 = (y''')^2$ | (I) Order = 1, Degree = 2 | | (B) $\sqrt{(y')^2 + 5} = y''$ | (II) Order = 2, Degree = 3 | | (C) $(y')^2 = (2 + y'')^{3/2}$ | (III) Order = 2, Degree = 2 | | (D) $y = xy' + \sqrt{a^2(y')^2 + b^2}$ | (IV) Order = 3, Degree = 2 | Choose the correct answer from the options given below:

If A is square matrix of order 3 × 3 and |adj A| = 64, then the value of |5A| is

$\int \frac{(x^4 - x)^{1/4}}{x^5} dx$ is equal to

If $\begin{bmatrix} x - 2 & 3 & -2 \\ y & 0 & -4 \\ 2 & z & 0 \end{bmatrix}$ is a skew symmetric matrix, then the value of $x + y + z$ is

If a random variable X has the following probability distribution: | X | 0 | 1 | 2 | 3 | |---|---|---|---|---| | P(X) | K | K/2 | K/4 | K/8 | then, Match List-I with List-II | List-I | List-II | |---|---| | (A) The value of K is | (I) 2/15 | | (B) P(0 < X < 2) is | (II) 1/15 | | (C) P(1 < X < 3) is | (III) 8/15 | | (D) P(X > 2) is | (IV) 4/15 | Choose the correct answer from the options given below:

Consider the differential equation $xdy = (x + y) dx$. Which of the following are true? (A) It is a homogenous differential equation (B) It is a differential equation of order 2 (C) The general solution of the differential equation contains 2 arbitrary constants (D) Integrating factor of differential equation is $\frac{1}{x}$ (E) Degree of the differential equation is not defined Choose the correct answer from the options given below:

The area of the region enclosed between the parabola $y = \frac{3x^2}{4}$ and the line $3x - 2y = 12$ is,

The real valued function $f(x) = 12x^\frac{4}{3} - 6x^\frac{1}{3}, x \in [-8, 8]$ has absolute maximum value equal to

For $x > y > 0$, if $x^5 y^6 = (x + y)^{11}$, then $\frac{d^2y}{dx^2}$ is

The feasible region represented by the constraints $x + y \leq 50, 3x + y \leq 90, x \geq 0, y \geq 0$ of an LPP is <img src="https://balti.afterboards.in/PT2fmLOw2WWECQ6" width="300px"/>

If $A = \begin{bmatrix} 2 & -3 \\ -4 & 7 \end{bmatrix}$ and $2A^{-1} = KI - A$, where K is a real number and I is the identity matrix of order 2, then the value of K is:

For the given linear programming problem $z = ax + by; a, b > 0$ subject to the constraints $2x + y \leq 10, x + 3y \leq 15, x, y \geq 0$. If the corner points are (0,0), (5,0), (3,4) and (0,5) and z is maximum at both (3,4) and (0,5), then the relationship between a and b is

If the interval in which the function $f(x) = 4x^3 - 6x^2 - 72x + 30$ is strictly decreasing, is (a,b) then a+b is equal to

Consider the equation of the line $\vec{r} = -\hat{i} + 2\hat{k} + \mu(4\hat{i} - \hat{j} + 2\hat{k})$. Match List-I with List-II | List-I | List-II | |---|---| | (A) It passes through the point | (I) 4, -1, 2 | | (B) Its direction ratios are | (II) $\frac{4}{\sqrt{21}}, \frac{-1}{\sqrt{21}}, \frac{2}{\sqrt{21}}$ | | (C) Its Cartesian form is | (III) (-1, 0, 2) | | (D) Its direction cosines are | (IV) $\frac{x+1}{4} = \frac{y}{-1} = \frac{z-2}{2}$ | Choose the correct answer from the options given below:

The co-ordinates of the point at which the line $\frac{x-3}{3} = \frac{y+1}{2} = \frac{z-4}{-2}$ crosses x-y plane, are

If A and B are two square symmetric matrices of same order, then AB-BA is

If A speaks truth in 75% cases and B speaks truth in 80% cases, then the probability that they contradict each other in a statement, is:

If the lines $\frac{x-5}{7} = \frac{y+2}{-5} = \frac{z}{\lambda}$ and $\frac{x}{1} = \frac{y}{2\lambda} = \frac{z}{3}$ are perpendicular to each other, then $\lambda$ is equal to

Which of the following statements are correct? (A) If $\vec{a}$ and $\vec{b}$ represent the adjacent sides of a triangle, then its area is $\frac{1}{2}|\vec{a} \times \vec{b}|$ (B) If $\vec{a}$ and $\vec{b}$ represent the adjacent sides of a parallelogram, then its area is $|\vec{a} \times \vec{b}|$ (C) $|\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \cos\theta$ (D) If $\vec{a}$ and $\vec{b}$ represent the 'diagonals' of a parallelogram, then its area is $\frac{1}{2}|\vec{a} \times \vec{b}|$ Choose the correct answer from the options given below:

If $x = a\cos\alpha + b\sin\alpha$ and $y = a\sin\alpha - b\cos\alpha$, then $\left(x\frac{dy}{dx} - y^2\frac{d^2y}{dx^2}\right)$ is equal to:

The domain of the function $\cos^{-1}(2x - 3)$ is

The sides of an equilateral triangle are increasing at the rate of 2 cm/sec. The rate at which the area increases when the side is 10 cm, is

If the vectors $\vec{a} = 3\hat{i} - p\hat{j} + 5\hat{k}$ and $\vec{b} = -6\hat{i} + 14\hat{j} + q\hat{k}$ are collinear, then the value of p and q are:

If $\hat{a}, \hat{b}$ and $\hat{c}$ are three unit vectors and $\hat{a} + \hat{b} + \hat{c} = \vec{0}$, then the angle between $\hat{a}$ and $(-\hat{b})$ is

The area bounded by the curve $y = \log x, y = 0$ and $x = e$, is

An objective function $z = ax + by$ is maximum at points (15,15) and (0, 20). If $a, b \geq 0$ and $ab = 27$, then the maximum value of the objective function is

The value of $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (\sin|x| + \cos|x|) dx$ is

The area of the region bounded by $y = -1$, $y = 2$, $x = y^3$ and $x = 0$ is $\frac{m}{n}$ sq. units, where $\gcd(m, n) = 1$, then $m - n$ is equal to:

If $A = \{1, 2, 3, 4, ..., n\}$ and $B = \{x, y\}$, then the number of surjections from A to B is

Match List-I with List-II The function $f(x) = (x - 1)(x + 1)^2$ has | List-I | List-II | |---|---| | (A) A local maxima at $x = $ ____ | (I) $\frac{1}{3}$ | | (B) A local minima at $x = $ ____ | (II) 0 | | (C) The local minimum value of $f(x) = $ ____ | (III) -1 | | (D) The local maximum value of $f(x) = $ ____ | (IV) $-\frac{32}{27}$ | Choose the correct answer from the options given below:

If A and B are two events such that $P(A) = \frac{1}{2}$, $P(B) = \frac{1}{3}$, $P(B|A) = \frac{1}{4}$, then $P(A|B)$ is:

Two events A and B will be independent, then

The following system of equations $2x - y + 3z = 5, 3x + 2y - z = 7, 4x + 5y - \lambda z = \mu$ is consistent. Then

If $A = \begin{bmatrix} 3 & -2 & 3 \\ 2 & 1 & -1 \\ 4 & -3 & 2 \end{bmatrix}$, then the matrix (adj A)A is equal to

The value of k for which the function, defined by, $f(x) = \begin{cases} \frac{3x + 4 \tan x}{x} & : x \neq 0 \\ k & : x = 0 \end{cases}$ is continuous at $x = 0$, is

Match List-I with List-II [.] denotes the greatest integer function. | List-I | List-II | |---|---| | (A) $\int_0^3 [x]dx$ | (I) $\frac{1}{2}$ | | (B) $\int_0^1 [2x]dx$ | (II) 1 | | (C) $\int_0^1 [3x]dx$ | (III) $\frac{3}{2}$ | | (D) $\int_0^1 [4x]dx$ | (IV) 3 | Choose the correct answer from the options given below:

If $A = \begin{bmatrix} 5 & 3 \\ 2 & 4 \end{bmatrix}$, then the matrix $A^2 - 6A + 14$ I is (where I is an identity matrix of order 2)

A player participates in 3 matches against three teams T₁, T₂ and T₃.The probability of winning a match against teams T₁, T₂ and T₃ are 0.2, 0.3 and 0.9 respectively. If 'wins' can be regarded as independent events, then the probability that he (A) wins all the 3 matches is 0.054 (B) wins no match is 0.054 (C) wins exactly two matches is 0.348 (D) wins exactly one match is 0.542 Choose the correct answer from the options given below:

A relation R in the set A = {1, 2,3, 4} is given by R = {(1,1), (2,2), (1,2), (2,3), (3,4), (4,4), (1,3), (2,4), (1,4)} is

If $A = \begin{bmatrix} 2 & 0 & 3 \\ -1 & 1 & 3 \\ 0 & -4 & 0 \end{bmatrix}$, then the value of det (2A) is

Consider the following L.P.P minimize $z = x - 7y + 190$ subject to $x + y \le 8, x + y \ge 4, x \le 5, y \le 5$ and $x, y \ge 0$. Then which of the following is/are true? (A) It's feasible region is unbounded (B) It's feasible region is bounded (C) It's feasible region has 5 corner points (D) It's feasible region has 6 corner points Choose the **correct** answer from the options given below:

The general solution of the differential equation $\frac{xdy}{dx} + 4y = x^3, (x \neq 0)$ is:

If $\vec{a}, \vec{b}$ and $\vec{c}$ are three unit vectors such that $\vec{a} + 2\vec{b} - 3\vec{c} = \vec{0}$, then the value of $2\vec{a}.\vec{b} - 6\vec{b}.\vec{c} - 3\vec{c}.\vec{a}$ is

For $x > 1$, $\int \frac{e^{7\log x} - e^{5\log x}}{e^{5\log x} - e^{4\log x}} dx$ equals.

If $\begin{vmatrix} -a^2 & ab & ac \\ ba & -b^2 & bc \\ ac & bc & -c^2 \end{vmatrix} = k \cdot a^l b^m c^n$, then Match List-I with List-II | List-I | List-II | |---|---| | (A) $l = m = n =$ | (I) 10 | | (B) $k + l + m + n =$ | (II) 6 | | (C) $k^2 + l^2 + (m - n)^2 =$ | (III) 2 | | (D) $l^2 + m^2 + (n - k) =$ | (IV) 20 | Choose the correct answer from the options given below:

Which of the following functions $f(x)$ are differentiable at $x = 0$? (A) $|x|$ (B) $|x - 1|$ (C) $|\sin x|$ (D) $|\cos x|$ (E) $x^2$ Choose the correct answer from the options given below:

The interval on which the function $f(x) = x^4 - \frac{x^3}{3}$ is strictly decreasing, is:

Consider the curve which is represented by the differential equation $\frac{dy}{dx} = 1 + x + y + xy$. If it passes through the point $(0,0)$, then which of the following is/are true? (A) it is a straight line. (B) it is a parabola. (C) it also passes through the point $(-1, \frac{1}{\sqrt{e}} - 1)$ (D) Its equation is $xy(x + 1)\left(y - \frac{1}{\sqrt{e}} + 1\right) = 0$ Choose the **correct** answer from the options given below:

Let two independent random samples of sizes $n_1$ and $n_2$ respectively have been drawn from the same normal population. Let $\overline{X_1}$ and $\overline{X_2}$ be the means and let $s_1$ and $s_2$ be their standard deviations. In order to test whether the the two sample means $\overline{X_1}$ and $\overline{X_2}$ differ significantly or not, the $t$-test statistic is given by

If $A = [a_{ij}]$ be square matrix of order 3, such that $a_{ij} = i + j$, $\forall i, j$ then which of the following are correct? (A) A is a skew-symmetric matrix. (B) A is a non-singular matrix. (C) The inverse of A does not exist. (D) A is a symmetric matrix. Choose the correct answer from the options given below:

| Year (x) | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | |---|---|---|---|---|---|---|---| | Sales (in lakh Rs.) (y) | 65 | 68 | 70 | 72 | 75 | 67 | 73 | If $y = 70 + 0.964(x - 2014)$ be a straight line trend fitted by the method of the least squares to the above data, then the trend value for the year 2015 is:

If X is a random variable and a, b are real numbers, then which of the following statements are correct? (A) $E[aX+b] = a E(X) + b$ (B) $Var (aX + b) = a^2 Var (X) + b$ (C) $Var (aX + b) = a Var (X)$ (D) $Var (X) = E(X^2) - [E(X)]^2$ Choose the correct answer from the options given below:

A die is tossed 6 times and getting "1 or 5" is considered a success. The probability of getting at least one success in six tosses is:

The value of the definite integral $I = \int_{-1}^{1} \frac{1}{1 + \sqrt{e^x}} dx$ is:

If the probability that an individual suffers a bad reaction from an injection of a given serum is 0.001. The probability that out of 2000 individuals, more than two individuals suffer from bad reaction is: [Given that $e^{-2} \approx 0.13534$]

If $e^y(x + 1) = 1$, then

Which of the following are similarities between the sinking fund and the savings account? (A) The sinking fund and the savings account are both financial tools. (B) Both can be used in any emergency. (C) They both involve setting aside an amount of money for the future. (D) Both are long-term accounts which can be closed any time. Choose the correct answer from the options given below:

A T.V. panel costing Rs 18000 has a useful life of 12 years. If the annual depreciation is Rs 1000, then its scrap value by linear method is:

In reference to the Inferential Statistics, which of the following is NOT correct?

If $\begin{bmatrix} 1 & 0 \\ b & 5 \end{bmatrix} + 2\begin{bmatrix} a & 0 \\ 1 & -2 \end{bmatrix} = I$, where $I$ is a unit matrix of order 2, then the value of $(a - b)$ is:

The equation of the tangent to the curve $y = \frac{(x - 3)}{(x - 1)(x - 2)}$ at the point, where it cuts x-axis is:

If a random variable $X$ has the following probability distribution: | X | 0 | 1 | 2 | 3 | 4 | |---|---|---|---|---|---| | P(X) | k | 2k | 3k | k² | 6k² | , then Match List-I with List-II | List-I | List-II | |---|---| | (A) k | (I) 3/7 | | (B) $P(X < 2)$ | (II) 6/49 | | (C) $P(X > 3)$ | (III) 1/7 | | (D) $P(2 \leq X \leq 3)$ | (IV) 22/49 | Choose the correct answer from the options given below:

Which of the following is NOT correct?

A person has invested Rs.20,000 in 2020 for 5 years. If CAGR for his investment is 11.84%. The end balance of his investment is (Given $(1.1184)^5 \approx 1.7498$)

The average salary per head of the entire staff of a small factory including the managers and technicians is Rs.5750. The average salary per head of the manager is Rs. 20000 and that of technicians is Rs. 5000. If there are four managers, then the number of technicians in the factory is:

The solution set of $6 \leq -3(2x - 4) < 12$, $x \in R$ is:

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 pipe B is turned off. If the tank is filled in 15 minutes, then after how many minutes pipe B is turned off?

Let $A = [a_{ij}]$ be a square matrix, where $a_{ij} = \begin{cases} 0, & \text{when } i = j \\ 1, & \text{otherwise} \end{cases}$. If |adj A| = |A|², then which of the following statements are correct? (A) A is a skew symmetric matrix. (B) A is a non-singular matrix. (C) A is a square matrix of order 4. (D) A is a symmetric matrix. Choose the correct answer from the options given below:

In a 10 km race, A, B and C each run at uniform speed, get the first, second and third position respectively. If A beats B by 1 km and B beats C by 1 km, then, by how much distance A beat C?

If A and B are symmetric matrices of the same order, then

Mr. X wishes to purchase a flat for Rs. 44,60,800 with a down payment of Rs 10,00,000 and balance in equal monthly installments (EMI) for 20 years. If bank charges 7.5% per annum compounded monthly, then the EMI is: [Given that $(1.00625)^{240} \approx 4.4608$]

If the corner points of the bounded feasible region for a linear programming problem (LPP) are (0, 2), (3, 0), (6, 0), (6, 8) and (0,5), then which of the following are correct for the objective function $Z = 4x + 6y$? (A) The minimum value of the objective function occurs at (0, 2) and (3, 0) only. (B) The minimum value of the objective function occurs at the mid-point of the line segment joining the points (0, 2) and (3, 0) only. (C) The minimum value of the objective function occurs at every point of the line segment joining the points (0, 2) and (3, 0). (D) The difference between the maximum value and minimum value of the objective function is 60. Choose the correct answer from the options given below:

A fisherman is rowing a boat. He takes 6 hours to row 48 km upstream whereas he takes 3 hours to go to the same distance downstream. Based on the above information, which of the following are correct? (A) His speed of rowing in still water is 10 km/hour. (B) The speed of the stream is 3 km/hour. (C) The average speed of the fisherman is 32/3 km/hour. (D) The speed of the stream is 4 km/hour. Choose the correct answer from the options given below:

A company 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 variance of 0.0004 cm². The value of test statistic is :

If the objective function for a linear programming problem (LPP) is $Z = 4x + 5y$ and the corner points of the bounded feasible region are (9, 0), (4, 3), (2, 5), and (0,8), then the minimum value of Z is:

The unit's digit of $12^{12}$ is equal to:

In which of the following interval, the function $f(x) = \frac{x}{\log x}$ is decreasing?

If $I = \int \frac{x}{x - \sqrt{x^2 - 4}} dx = \alpha x^3 + \beta(x^2 - 4)^{\frac{3}{2}} + \gamma$, where $\gamma$ is constant of integration, then

The general solution of the differential equation $(x^2 - yx^2)dy + (y^2 + x^2y^2)dx = 0$ is:

A measure of an average yearly growth of an investment over a certain time period when returns are reinvested is

Match List-I with List-II | List-I | List-II | |---|---| | (Matrix A) | (Determinant of adj A) | | (A) $\begin{bmatrix} 2 & 1 \\ 0 & -1 \end{bmatrix}$ | (I) 6 | | (B) $\begin{bmatrix} 0 & 1 \\ 4 & -1 \end{bmatrix}$ | (II) 5 | | (C) $\begin{bmatrix} 1 & 2 \\ -3 & -1 \end{bmatrix}$ | (III) -4 | | (D) $\begin{bmatrix} 4 & -2 \\ 3 & 0 \end{bmatrix}$ | (IV) -2 | Choose the correct answer from the options given below:

The pattern and behavior of the data in any time series are based on which of the following components: (A) Secular trend component (B) Seasonal component (C) Cyclical component (D) Irregular component Choose the correct answer from the options given below:

The number of letters posted in a certain city on each day for a week is given as follows: 40, 55, 28, 25, 31, 52, 43. Which of the following is not an entry in the three-day moving averages?