CUET Mathematics 2025 PYQs21 May Shift 2. Free, no login required.

Let A be a 3 × 7 matrix, then each column of A contains:

If $x = t^{1/2}$, $y = t^{3/2}$, then $\frac{dy}{dx}$ =

If $A$ is a $3 \times 3$ matrix such that $|adj A| = 9$ and $|kA^{-1}| = 9$, then the value of $k$ are:

Value of $\int \left(\frac{1}{logx} - \frac{1}{(logx)^2}\right)dx$ is

The value of $\int_1^3 \frac{x^2}{x^3+1}dx$

If $X$ is a random variable which can assume values $0, 1, 2, 3$ or $4$ such that $P(X = 1) = P(X = 2)$ and $3P(X = 3) = 4P(X = 4) = P(X = 0) = \frac{1}{8}$, then $P(X > 0)$ is:

The nearest integral value of the shaded area shown below is: <img src="https://balti.afterboards.in/RpM45g4gxR34DDz" width="400px"/>

Match List-I with List-II | List-I | List-II | |---|---| | (A) Degree of the differential equation $\frac{d^2y}{dx^2} = e^{dy/dx}$ is | (I) 2 | | (B) Order of the differential equation $(\frac{dy}{dx})^2 + \frac{d^3y}{dx^3} = 0$ is | (II) not defined | | (C) Degree of the differential equation $\frac{d^2y}{dx^2} + (\frac{dy}{dx})^2 - 5x^2 = 0$ | (III) 3 | | (D) If p is the order and q is the degree of the differential equation $\frac{dy}{dx} + 3y = e^x$, then p + q is | (IV) 1 | Choose the correct answer from the options given below:

The solution of the differential equation $(x + 1)\frac{dy}{dx} + 1 - 2e^{-y} = 0$, $y(0) = 0$ is

Which one of the following represents the correct feasible region determined by the following constraints $x - y \geq 5$, $5x - 5y \leq 16$

The maximum value of $z = 5x + 7y$ subjected to constraints $x + y \leq 5$, $x \geq 0$, $y \geq 0$ is:

The matrix $X$ in the equation $AX = B$, such that $A = \begin{bmatrix} 1 & 3 \\ 0 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & -1 \\ 0 & 2 \end{bmatrix}$ is given by

The function $f(x) = 6 - 6x - 2x^2$

Let A be any skew- symmetric matrix (where $A^T$ is Transpose of matrix A). Then which of the following statements are correct? (A) $A^2$ is a symmetric matrix (B) $A^2$ is a skew- symmetric matrix (C) $A^T A = -A^2$ (D) $A^T A - AA^T = O$ Choose the correct answer from the options given below:

The function $f(x) = x + \frac{1}{x}$ has

The vector in the direction of the vector $2\hat{i} - \hat{j} - 2\hat{k}$ that has magnitude 9 units is:

$\sin^{-1}(\cos\frac{3\pi}{5})$ equals

If the maximum value of the function $f(x) = \frac{2\log_e x}{x}$, $x > 0$ occurs at $x = e$, then $e^3 f''(e)$ is equal to

A couple has 3 children each child is equally likely to be a boy or a girl. The probability that the eldest child is a girl given that they have atleast one boy is:

If $\begin{bmatrix} x-y & 0 \\ x+y & 1 \end{bmatrix}$ is an identity matrix and $\begin{bmatrix} x & y \\ z & x \end{bmatrix}$ is a singular matrix then:

Which of the region shown in the given figures represents the feasible region bounded by the following constraints? $4x + y \geq 80$, $2x + y \geq 60$, $x + y \leq 80$, $x \geq 0$, $y \geq 0$ <img src="https://balti.afterboards.in/amdVa7RhS90gsFU" width="300px"/>

If $f(x) = \begin{cases} mx + 1,\ x \geq \pi/2 \\sin x + n, x \leq \pi/2, & \end{cases}$ is continuous at $x = \pi/2$, where $m \in \mathbb{Z}$ (set of integers), then $\sin 2n =$

Match List-I with List-II | List-I | List-II | |---|---| | (A) Integrating factor of $xdy - (y + x^2)dx = 0$ | (I) $x^2$ | | (B) Integrating factor of $xdy + (2y + x^2)dx = 0$ | (II) $x^3$ | | (C) Integrating factor of $(3y - x^2)dx + xdy = 0$ | (III) $x$ | | (D) Integrating factor of $(y + 3x^2)dx + xdy = 0$ | (IV) $\frac{1}{x}$ | Choose the correct answer from the options given below:

If $y = \sin^{-1} \sqrt\frac{x}{x+1} + \sec^{-1}\sqrt{\frac{x+1}{x}}$, then $\frac{dy}{dx}$ is

The shortest distance between the following lines: $\vec{r} = (\hat{i} + \hat{j} - \hat{k}) + s(2\hat{i} + \hat{j} + \hat{k})$ $\vec{r} = (\hat{i} + \hat{j} + 2\hat{k}) + t(4\hat{i} + 2\hat{j} + 2\hat{k})$, where s and t are scalars, is:

The solution of the differential equation $\frac{dy}{dx} = \frac{ax + c}{by + d}$ represents a circle when

If the area of a triangle with vertices $(-3,0)$, $(3, 0)$ and $(0, k)$ is 9 sq. units, then k equals

If $x, y$ and $z$ are non-zero distinct numbers, then $\begin{vmatrix} x+y & y+z & z+x \\ z & x & y \\ 1 & 1 & 1 \end{vmatrix}$ is equal to

A letter is known to have come from either TATAPUR or from CHAKRATA. On the envelope, only two letters 'TA' are visible consecutively. The probability that the letter has come from CHAKRATA is:

The system of linear equations $kx + 5y = 5$, $2x + 3y = 5$ will be consistent if

Value of $\int_2^3 \frac{\sqrt{x}}{\sqrt{x} + \sqrt{5-x}}dx$ is

The corner points of the bounded feasible region determined by the system of linear constraints are $(0, 0)$, $(5, 0)$, $(6, 5)$, $(6, 8)$, $(4, 10)$, $(0, 8)$. Let $Z = 3x - 4y$ be the objective function. The minimum value of Z occurs at

$\int_0^{\pi/2} \sqrt{1 - \sin 2x}\,dx$ is equal to:

A relation $f: N \rightarrow N$ be defined by $f(x) = x^2$, $x \in N$ (Set of Natural numbers). Then $f(x)$ is

Let A and B be independent events such that P (A) = 0.3 and P (B) = 0.4, then Match List-I with List-II | List-I | List-II | | ----------------- | ---------- | | (A) $P(A \cap B)$ | (I) 0.3 | | (B) $P(A \cup B)$ | (II) 0.4 | | (C) $P(A \mid B)$ | (III) 0.12 | | (D) $P(B \mid A)$ | (IV) 0.58 | Choose the correct answer from the options given below:

If $\begin{vmatrix} 1 & \cos \theta & 0 \\ \sin \theta & 1 & \cos \theta \\ |\cos \theta & 1 & -\sin \theta| \end{vmatrix} = A\sin \theta + B\cos \theta + C\sin \theta\cos \theta$ then:

If $A = \begin{bmatrix} 4 & 3 \\ 2 & -1 \\ 1 & 0 \end{bmatrix}$ and $B^T = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 1 & -1 \end{bmatrix}$, then $A - B$ is equal to

The area of triangle with vertices P, Q, R is given by (where $\vec{AB}$ = position vector of point B – position vector of point A)

$\int \frac{\sin x - x\cos x}{x(x + \sin x)}dx =$ (where C is an arbitrary constant)

Let $R = \{(L_1, L_2): L_1 \perp L_2\ $ where $L_1, L_2 \in L$ (set of straight line in a plane)}, then

For the function $f(x) = 2x^3 - 3x^2 - 12x + 5$, the difference of maximum and minimum value of $f(x)$ is

The angle between the line $2x = 3y = z$ and $x$- axis is:

If lines $\frac{x+5}{5\lambda+2} = \frac{4-2y}{10} = \frac{1-3z}{-3}$ and $\frac{x-2}{1} = \frac{1+2y}{4\lambda} = \frac{2+z}{3}$ are perpendicular, than value of '$\lambda$' is

The probability that it will rain on any particular day is 50%. The probability that it rains only on the first 4 days of the week is:

If $\vec{a}$ and $\vec{b}$ are two non-zero vectors such that $|\vec{a} \cdot \vec{b}| = |\vec{a} \times \vec{b}|$, then the angle $\theta$ between $\vec{a}$ and $\vec{b}$ is

Area (in sq. units) of the region bounded by curves $y^2 = x$ and $x = 4$ is

If $\vec{a} = 2\hat{j} - \hat{k}$, $\vec{b} = 2\hat{i} - 3\hat{j} + \hat{k}$ and $\vec{c} = -\hat{i} + \hat{k}$ are three vectors, then the area (in sq. units) of the parallelogram whose diagonals are $(\vec{b} + \vec{c})$ and $(\vec{a} + \vec{c})$ is

Interval in which the function $f$ given by $f(x) = \tan x - 4x$, $x \in (0, \frac{\pi}{2})$ is strictly decreasing is

If $\int \frac{x^4}{x-2}dx = px + qx^2 + rx^3 + sx^4 + t\log |x - 2| + C$, where C is an arbitrary constant and p, q, r, s, t are real numbers, then the correct arrangement of p, q, r, s, t is:

Differentiation of $\log[\log(\log x^5)]$ with respect to $x$ is

Mr. X wishes to purchase a flat for Rs. 44,65,000 with a down payment of Rs. 10,00,000 and balance in equated monthly installments (EMI) for 25 years. If the bank charges 6 % per annum compounded monthly, the EMI is: [Given: $(1.005)^{300} =4.4650$]

A boat can row at the speed of 16 km/hr in still water. If the river is flowing at 8 km/hr, and it takes 8 hours for a round trip, then the distance between the two places is:

It is known that 3% of plastic bags manufactured in a factory are defective. Using the Poisson distribution on a sample of 100 bags, the probability of at most one defective bag is:

The behavior and pattern of the data in a time series is NOT based on which of the following component?

Which of the following is not correct about the Compound Annual Growth Rate (CAGR)?

The marginal cost (MC) and marginal revenue (MR) functions of a product are $MC = 20 + \frac{x}{20}$ and $MR = 30$ respectively. If the fixed cost is 200, then the maximum value of the profit is:

For the linear programming problem (LPP): Maximize $Z = x + 1.5y$, subject to constraints, $x + 2y \leq 40$, $2x + y \leq 40$, $x + y \leq 25$, $x \geq 0$, $y \geq 0$. Which of the following is NOT correct?

A die is rolled in such a way that an even number is twice likely to occur as an odd number. If the die is rolled twice, then the mean of the number of perfect squares in two tosses is:

If $x = t^3$, $y = t^2$ then $\frac{d^2y}{dx^2}$ is equal to:

The probability distribution function of a normal variate with mean $\mu$ and variance $\sigma^2$ is given by: $f(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}$, $-\infty < x < \infty$, $-\infty < \mu < \infty$, $\sigma > 0$ If $y = f(x)$ be the normal probability curve, then which of the following is correct? (A) The normal curve is symmetrical about the line $x = \mu$. (B) Mean, median and mode of the distribution coincide. (C) Y- axis is an asymptote to the normal curve. (D) If x increases numerically, $f(x)$ decreases rapidly. Choose the correct answer from the options given below:

A money lender charges Rs10 for Rs100 per month in advance then effective rate of interest per annum charged by money lender is: [given $\left(\frac{10}{9}\right)^{12} \approx 3.541$]

The number of all possible matrices of order 3 with each entry either 0 or 1 is:

Consider a random sample of 10 students having 116 cm as mean height and standard deviation as 9.798 cm. If the suggested mean height of the students population is 110 cm then the t-test statistic of the sample is: [given $\frac{\sqrt{10}}{9.798} = 0.3227$]

Three varieties A, B and C of rice are mixed together in the ratio 1:1:3 respectively. The price of rice A is Rs 127 per kg and that of rice B is Rs 135 per kg. If the price of the mixture is Rs. 152 per kg, then the price per kg of rice of type C is:

Match List-I with List-II | List-I (Matrix A) | List-II (Determinant of Adjoint of A) | |---|---| | (A) $\begin{bmatrix} 3 & 1 \\ 4 & 2 \end{bmatrix}$ | (I) 9 | | (B) $\begin{bmatrix} 5 & -1 \\ 4 & 2 \end{bmatrix}$ | (II) 8 | | (C) $\begin{bmatrix} 6 & -1 \\ 2 & 1 \end{bmatrix}$ | (III) 14 | | (D) $\begin{bmatrix} 4 & 1 \\ 3 & 3 \end{bmatrix}$ | (IV) 2 | Choose the correct answer from the options given below:

Let $y = 138.86 + 7.64(x - 2021)$ be a straight line of best fit by using least square method to the following data: | Year(x) | 2018 | 2019 | 2020 | 2021 | 2022 | 2023 | 2024 | |---|---|---|---|---|---|---|---| | Profit(y) (in Rs. '000) | 114 | 130 | 126 | 144 | 138 | 156 | 164 | Then the trend value for the year 2024 is:

Which of the following statements are correct? (A) A fund which is created to accumulate money over the years to discharge a future obligation is called a sinking fund. (B) The amount or future value of perpetuity is well-defined. (C) The sinking fund be used in any emergency. (D) An equated monthly installment is a fixed payment made by a borrower to a lender at a specific date every month to clear off the loan. Choose the correct answer from the options given below:

If $A$ and $B$ are square matrices of the same order, then which of the following statements are correct? (A) $|A^{-1}| = |A|^{-1}$ (B) $adj(A) = |A|A^{-1}$ (C) $(A + B)^{-1} = B^{-1} + A^{-1}$ (D) $(AB)^{-1} = B^{-1}A^{-1}$ Choose the correct answer from the options given below:

For the function, $f(x) = \frac{-3}{4}x^4 - 8x^3 - \frac{45}{2}x^2 - 350$, which of the following statements are correct? (A) $x = -3$ and $x = -5$ are the only critical points of the given function. (B) $x = -3$ is a point of local minimum. (C) The local minimum value at $x = -3$ is 23.1. (D) $x = -5$ is a point of local maximum. Choose the correct answer from the options given below:

Two pipes A and B can fill a tank in 20 minutes and 10 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 find after how many minutes pipe B is turned off?

For the given five values, 13, 17, 21, 22, 32; the 3-year moving averages are:

A motorcycle has a scrap value of Rs. 22,500 after 15 years of its purchase. If the annual depreciation charge is Rs. 8,500, then the original cost by linear method is:

If A and B are two non-singular matrices of order n, then which of the following statement/statements is/are not correct? (A) AB is non-singular. (B) AB is singular. (C) $(AB)^{-1} = A^{-1}B^{-1}$ (D) $(AB)^{-1}$ does not exist. Choose the correct answer from the options given below:

Match List-I with List-II | List-I (Curve) | List-II (Slope of tangent at $x = 4$) | |---|---| | (A) $y = \sqrt{x^3}$ | (I) -1 | | (B) $y = \sqrt{x}$ | (II) 1 | | (C) $y = x^3 - 47x$ | (III) 1/4 | | (D) $xy = 16$ | (IV) 3 | Choose the correct answer from the options given below:

About linear programming problem (LPP), which of the following statements are correct? (A) In a LPP, the linear inequalities or restrictions on the variables are called linear constraints. (B) If the feasible region for an LPP is unbounded, then the maximum or minimum value of the objective function $Z = ax + by$ never exists. (C) The feasible region for an LPP is always convex. (D) The common region determined by all the linear constraints of an LPP is called the feasible region. Choose the correct answer from the options given below:

If a, b, c are positive real numbers, then the least value of $(a+b+c)(ab+bc+ca)$ is:

If two dice are rolled 12 times and getting a total greater than 4 is considered as a success, then which of the following statements are correct? (A) The probability of getting a total greater than 4 in a single throw of the pair of dice is 5/6. (B) Mean = 10 (C) Variance = 3/5 (D) The probability of getting a total less than or equal to 4 in a single throw of the pair of dice is 1/6. Choose the correct answer from the options given below:

If the following data is obtained from a simple random sample: 6, 7, 9, 10, 11, 17 Then the point estimate of population standard deviation 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

Which of the following statements are correct about the "Central Limit Theorem"? (A) The sampling distribution of the sample mean approaches the normal distribution as the sample size gets larger. (B) A sample size of 30 or more is considered to be sufficient to hold the "Central Limit Theorem". (C) As the sample size becomes larger, the prediction of characteristics of the population becomes more accurate. (D) The sampling distribution of the sample mean approaches a bell shaped curve as the sample size gets larger. Choose the correct answer from the options given below:

The value of the definite integral $\int_0^1 e^x \frac{(1-x)^2}{(1+x^2)^2}dx$ is:

The remainder when $(672 + 541 + 383 + 295 + 101 + 86)$ is divided by 3, is:

At what rate of interest will the present value of a perpetuity of Rs. 1000 payable at the end of every six months be Rs. 40000?

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

In a time series, the variations which occur due to general tendency of the data to increase or decrease over a long term are known as: