CUET Mathematics 2023 23 May Shift 3 PYQs

In the context of differential equation Match List I with List II | LIST I | LIST II | |---|---| | A. $\frac{dy}{dx} = \frac{1+y^2}{1+x^2}$ | I. Not a differential equation | | B. $x^2 \frac{dy}{dx} = x^2 - 2y^2 + xy$ | II. Linear first order | | C. $\sin x + y = \cos(x+y)$ | III. Variable separable | | D. $(x+y)\frac{dy}{dx} = 1$ | IV. Homogenous | Choose the correct answer from the options given below:

The critical points of $f(x) = x^3 + x^2 + x + 1$ are

The volume of a cube is increasing at the rate of 27 cm$^3$/s. How fast is the surface area increasing when the length of the cube is 12 cm.

If a fair coin is tossed 10 times, then the probability of obtaining at least one head is :

Two cards are drawn simultaneously from a well shuffled pack of 52 cards. Then variance of the number of kings is

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

The order of a null matrix is :

If $y = x^x$, $\frac{dy}{dx}$ will be:

If $5x + y \leq 100$, $x + y \leq 60$, $x \geq 0$, $y \geq 0$. Then one of the corner points of the feasible region is :

The value of $\int \left(\sqrt{x} + \frac{1}{\sqrt{x}}\right)^2 dx$ is:

Which of the following differential equation represents the family of circles touching the x-axis at the origin ?

Let $A = \begin{bmatrix} 3 & -1 \\ 2 & 4 \end{bmatrix}$, then adjoint (A) is:

In a LPP, let R be the feasible region. A. If R is unbounded then a max./min. value of objective function may not exist. B. If R is bounded then a max. and min. value of objective function will always exist. C. If a solution exists, it must occur at a corner point. D. If R is bounded then max. will exist but min. may or may not exist for an objective function. Choose the correct answer from the options given below:

Let A be a square matrix of order 3 then |3A| is equal to

The inverse of the matrix $A = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}$ is:

The angle between the lines $\vec{r} = 3\hat{i} + 2\hat{j} - 4\hat{k} + \lambda(\hat{i} + 2\hat{j} + 2\hat{k})$ and $\vec{r} = 5\hat{j} - 2\hat{k} + \mu(3\hat{i} + 2\hat{j} + 6\hat{k})$ is :

If $y = \sin^{-1}\left(\frac{1-x^2}{1+x^2}\right)$ then $\frac{dy}{dx} =$

The number of all onto functions from the set {1, 2, .......n} to itself is

The value of $\lambda$, so that the lines $\frac{1-x}{3} = \frac{7y-14}{2\lambda} = \frac{z-3}{2}$ and $\frac{7-7x}{3\lambda} = \frac{y-5}{1} = \frac{6-z}{5}$ are perpendicular is:

The intervals for which $f(x) = x^4 - 2x^2$ is increasing are :

The area enclosed between the curves $y = x^2$ and $x = y^2$ is

Match List I with List II | LIST I | LIST II | |---|---| | A. $lx + my + nz = d$ is | I. Equation of plane passing through a given point and normal to given vector | | B. $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$ is | II. Equation of plane in normal form | | C. $(\vec{r} - \vec{a}) \cdot \vec{n} = 0$ | III. Plane passing through the intersection of two planes | | D. $(a_1x + b_1y + c_1z + d_1) + \lambda(a_2x + b_2y + c_2z + d_2) = 0$ | IV. Intercept from of plane | Choose the correct answer from the options given below:

If $\vec{a} = 5\hat{i} - \hat{j} - 3\hat{k}$ & $\vec{b} = \hat{i} - 3\hat{j} + 5\hat{k}$ the angle between $\vec{a} + \vec{b}$ and $\vec{a} - \vec{b}$ is :

The slope of the normal to the curve $y = 2x^2 - 4$ at P (1, -2) is :

The principal value of $\cot^{-1}\left(-\frac{1}{\sqrt{3}}\right)$

The relation $R = \{(a, b) : a \leq b^2\}$ on the set of real numbers is:

If $\begin{vmatrix} 2x & 2 \\ 4 & x \end{vmatrix} = 10$, then $x$ is:

A doctor is to visit a patient. It is known that the probabilities that he will come by train, bus, scooter or by other means of transport are respectively $\frac{3}{10}, \frac{1}{5}, \frac{1}{10}$ and $\frac{2}{5}$. The probabilities that he will be late are $\frac{1}{4}, \frac{1}{3}$ and $\frac{1}{12}$, if he comes by train, bus and scooter respectively, but if he comes by other means of transport, then he will not be late. When he arrives, he arrives late. The probability that he comes by bus is:

The matrix $\begin{bmatrix} 2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ is a

The solution of $y' - y'' = 2x$ is: A. $y = x^2 + 2x + 2$ B. $y = x^2 + 2x + 1$ C. $y = x + 2$ D. $y = x^2 - 2x + 1$ Choose the correct answer from the options given below:

The minimum value of $f(x) = |2x - 1|$ is

Let A be the square matrix of order 3, then |kA|, where k is a scalar, is equal to:

Points of discontinuity of the greatest integer function $f(x) = [x]$, where $[x]$ denotes integer less than or equal to $x$, are

The simplest form of $\tan^{-1}\left\{\frac{x}{\sqrt{a^2 - x^2}}\right\}$ is, where $-a < x < a$.

The area of the region $\{(x, y) : y \geq x^2 \text{ and } y \leq |x|\}$ is

The general solution of $\frac{dy}{dx} = 1 + x^2 + y^2 + x^2y^2$ is: (given that $C$ is the constant of integration)

If a fair coin is tossed 10 times the probability of atleast 6 heads is:

Which one of the following options is incorrect? For a square matrix A in the matrix equation AX = B.

If $x, y$ & $z$ are non zero real numbers, the inverse of matrix $A = \begin{bmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{bmatrix}$ is :

The unit vector in the direction of $\vec{a} + \vec{b}$ if $\vec{a} = 2\hat{i} - \hat{j} + 2\hat{k}$ & $\vec{b} = -\hat{i} + \hat{j} + -\hat{k}$ is :

The maximum value of $\sin x + \cos x, x \in R$ is:

If $y = x^{(x \sin x)}$ then $\frac{dy}{dx} = ?$

The simplest form of $\tan^{-1} \frac{\sqrt{1+x^2} - 1}{x}$, $x \neq 0$ is:

The value of $\begin{vmatrix} 1 & a & b+c \\ 1 & b & c+a \\ 1 & c & a+b \end{vmatrix}$ is

$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{dx}{1 + \sqrt{\tan x}}$ is equal to :

The value of integral $\int \sqrt{4x^2 + 9}\, dx$ is

Let $f(x) = x^3$ be a function with domain {0, 1, 2, 3} then domain of $f^{-1}$ is :

The area of the parallelogram determined by the vectors $\hat{i} + 2\hat{j} + 3\hat{k}$ and $3\hat{i} - 2\hat{j} + \hat{k}$ is

A manufacturing company makes two models M$_1$ and M$_2$ of a product. Each piece of M$_1$ requires 9 labour hours for fabricating and one labour hour for finishing. Each piece of M$_2$ require 12 labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available are 180 and 30 respectively. The company makes a profit of Rs.800 on each piece of M$_1$ and Rs.1200 on each piece of M$_2$ The above Linear Programming Problem [LPP] is given by

A manufacturing company makes two models M$_1$ and M$_2$ of a product. Each piece of M$_1$ requires 9 labour hours for fabricating and one labour hour for finishing. Each piece of M$_2$ require 12 labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available are 180 and 30 respectively. The company makes a profit of Rs.800 on each piece of M$_1$ and Rs.1200 on each piece of M$_2$ The maximum profit will be at the point

The random variable X has a probability distribution P(X) of the following form where k is a scalar and $P(X = x) = \begin{cases} k, & \text{if } x = 0 \\ 2k, & \text{if } x = 1 \\ 3k, & \text{if } x = 2 \\ 0, & \text{otherwise} \end{cases}$ then value of P(X < 2) = _______.

If $A' = \begin{bmatrix} -2 & 3 \\ 1 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} -1 & 0 \\ 2 & 3 \end{bmatrix}$ then $(3A + 2B)'$ is :

For a manufacturer, total cost function is given by $C = \frac{x^2}{25} + 2x$. Which of the following are correct ? A. 2.6 is the marginal cost when 5 units are produced. B. $\frac{2x}{25} + 2$ is the marginal cost function. C. $\frac{x}{25} + 2$ is the average cost function. D. 2.4 is the marginal cost when 5 units are produced. E. $\frac{x}{25} + 1$ is the average cost function. Choose the correct answer from the options given below:

If the cash equivalent of a perpetuity of Rs.1200 payable at end of each half year is Rs.96,000, the annual rate of interest compounded half yearly is :

The following data is available for two independent sample: | | SAMPLE-1 | SAMPLE-2 | |---|---|---| | sample size | 8 | 10 | | sample mean | 191 | 199 | | sample standard deviation | 10 | 12 | Then the value of t-statistic is :

Two positive numbers $x$ and $y$ such that $x + y = 60$ and $xy^3$ is maximum are :

The total cost of producing $x$ generators is given by TC = $x^3 - 60x^2 + 1500x + 2000$. The Marginal Cost (MC), when $x = 10$ units is:

A book consisting of 2000 pages has 540 misprints distributed randomly throughout the book. The average number of misprints in one page of the book is:

Let us consider an annuity whose periodic payment is Rs. R payable at the end of each payment period for 'n' periods, interest paid r% per period or $i = \frac{r}{100}$, so the amount of obligation can be given as _______.

A dishonest delivery man purchases 20 liters of pure milk and dilutes it by adding 4 liters of water. If he sells the diluted milk at Rs.50/L, then what is the cost price of pure milk?

Mr. X takes a personal loan of Rs.10,00,000 at the rate of 12% per annum for 6 years. Calculate his EMI by using flat rate method

The variance of the Binomial Distribution $B\left(5, \frac{1}{4}\right)$ is:

The number of all possible matrices of order 3 $\times$ 3 with each entry 2 or 3 is.

Pipe A can independently empty a half-filled tank in 3 hours whereas pipe B can alone fill the same tank in 4 hours, when the tank is completely empty. In how much time will the empty tank be completely filled, if both the pipes are opened together?

The corner points of the feasible region determined by the system of linear constraints are (0, 10), (5, 5), (15, 15), (0, 20). Let $Z = ax + by$, where a, b > 0. Condition on a and b so that the maximizing value of Z occurs at both the points (15, 15) and (0, 20) is:

The feasible region for an LPP is shown in the figure given below: If objective is maximizing $Z = 22x + 18y$ find $(x, y)$ for the optimal $Z$.<img src="https://balti.afterboards.in/uzsNn04Qzqljaui" width="400px"/>

The price index of sugar in 2019 compared to 2016 is 120. If the cost of sugar was Rs.25 per kg in 2016 then the cost in 2019 is:

If the life expectancy of an asset is 10 years, scrap value Rs.5,000 and annual depreciation of Rs.3000 per annum, then original cost of the asset is:

For a 3 $\times$ 3 matrix A = [a$_{ij}$], whose element a$_{ij}$ is given by $a_{ij} = \sqrt{i^2 + 3j}$ then what is element a$_{31}$?

A runs 5 times faster than B. If A gives B a start of 100 m, how far must the goal be so that A and B reach there at the same time.

A shopkeeper wants to check the average number of cars sold per call. Past record of sales is shown below: | Sale of cars (Units) | 0 | 1 | 2 | 3 | |---|---|---|---|---| | Probabilities | $\frac{1}{6}$ | $\frac{1}{2}$ | $\frac{3}{10}$ | $\frac{1}{30}$ | The expected number of cars sold is :

In a game of carrom of 100 points, A can give B 20 points and C 40 points, then points B can give C is:

Given $\sum p_0 q_0 = 800$, $\sum p_0 q_1 = 1500$, $\sum p_1 q_1 = 1300$ and $\sum p_1 q_0 = 900$, where subscripts 0 and 1 are used for base year and current year respectively. The Paasche's index number is:

Given that p$_0$ and p$_1$ represent base and current prices of a commodity, q$_0$ and q$_1$ refer to the base and current quantities, then the formula used to calculate price relatives is :

In the formula for Z-test statistic $Z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}}$, mark the incorrect statement.

The set of values of $x$ satisfying $26 \equiv 5 \pmod{x}$ is _______.

Mr. X invested Rs.20,000 in year 2022 for 5 years. If Compound annual growth rate for that investment turned out to be 12%, the end value of the investment will be : {Use $(1.12)^5 = 1.76$}

A swimmer's speed in swimming pool is 6 km/hr. He swims between two points in a river and returns back to the starting point. He took 24 minutes more to cover the distance upstream than downstream. If the speed of the stream is 4 km/hr, then the distance between two points is:

The rise in prices before Christmas is an example of

Three partners Ramesh, Suresh and Mahesh shared the profit in a business in the ratio 5:4:7 respectively, they invested the money for 10 months, 8 months and 6 months respectively then ratio of their investment is :

The solution set of the inequation $\frac{7+5x}{4} \geq \frac{x}{3} - 10$, $x \in \mathbb{R}$ is :

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

If the mean of a binomial distribution is 12 and its standard deviation is 2, then the number of trials is :

A simple random sample consists of three observations 1, 3, 5. The point estimate of the population standard deviation is :

The effective rate that is equivalent to a nominal rate of 16% compounded semi-annually is :