CUET Mathematics 2023 22 May Shift 3 PYQs

If $P = \begin{bmatrix} 1 & x & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4 \end{bmatrix}$ is the adjoint of 3x3 matrix A and $|A|$ is 4, then $x$ is equal to :

If $f(x) = 2x$ and $g(x) = \frac{x^2}{2} + 1$, then which of the following can be a discontinuous function ?

If $f(x) = \begin{cases} ax^2 + b, & x < -1 \\ bx^2 + ax + 4, & x \geq -1 \end{cases}$ is everywhere differentiable, then :

Interval in which the function $f(x) = 2x^3 - 3x^2 - 12x + 10$ is decreasing is :

All points lying inside the triangle formed by the points (5, 0), (-1, 2) and (1, 3) satisfy : (A) $3x + 2y - 18 > 0$ (B) $3x + 2y > 0$ (C) $2x + y + 13 < 0$ (D) $2x - 3y - 12 < 0$ (E) $2x - 3y + 12 > 0$ Choose the **correct** answer from the options given below :

The feasible region for an LPP is shown below. Let $Z = 3x - 4y$ be the objective function. Maximum of Z occurs at : <img src="https://balti.afterboards.in/P4A0ckrzsn6mVTf" width="300px"/>

The probability that a student is not a swimmer is $\frac{1}{5}$. Then the probability that out of five students, four are swimmers is :

For the following probability distribution : | X | 1 | 2 | 3 | 4 | |---|---|---|---|---| | P(X) | 1/10 | 1/5 | 3/10 | 2/5 | $E(X^2)$ is equal to :

If $A = \begin{bmatrix} 1 & 2 \\ 4 & 2 \end{bmatrix}$, then the value of K for which $|2A| = K|A|$ is :

The differential equation of the family of curves $y = a \sin(bx + c)$, a and c are parameters, is :

Which of the following statements is incorrect regarding matrices ? For any matrices A and B of suitable orders,

The angle of intersection between the curves $y = 4 - x^2$ and $y = x^2$ is :

$\int_1^2 \frac{x \, dx}{(x+1)(x+2)} =$

The area enclosed by the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$ is given by :

$\int e^x \left(\frac{1-x}{1+x^2}\right)^2 dx =$

The position vector of a point R which divides the line joining two points P and Q whose position vectors are $\hat{i} + 2\hat{j} - \hat{k}$ and $-\hat{i} + \hat{j} + \hat{k}$ respectively in the ratio 2 : 1 externally is :

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

Match **List - I** with **List - II**. | | List - I | | List - II | |---|---|---|---| | (A) | $f(x) = \frac{1}{x}, f : \mathbf{R} - \{0\} \to \mathbf{R} - \{0\}$ | (I) | neither injective nor surjective | | (B) | $f(x) = x^2, f : \mathbf{N} \to \mathbf{N}$ | (II) | surjective but not injective | | (C) | $f(x) = x^2, f : \mathbf{R} \to \mathbf{R}$ | (III) | injective but not surjective | | (D) | $f : \{1, 2, 3\} \to \{1, 2\}$ defined as $f : \{(1, 1), (2, 2), (3, 1)\}$ | (IV) | injective and surjective | Choose the **correct** answer from the options given below :

The black and red die are rolled. The conditional probability of obtaining a sum greater than 9 given that the black die resulted in a 5 is :

Match **List - I** with **List - II**. | | List - I | | List - II | |---|---|---|---| | (A) | Area of triangle $\Delta$ with adjacent sides $\vec{a}$ and $\vec{b}$ | (I) | $\vec{a} \times \vec{b}$ | | (B) | Area of parallelogram with adjacent sides $\vec{a}$ and $\vec{b}$ | (II) | $\frac{1}{2}\lvert \vec{a} \times \vec{b} \rvert$ | | (C) | $(\vec{a} - \vec{b}) \times (\vec{a} + \vec{b})$ | (III) | $\lvert \vec{a} \times \vec{b} \rvert$ | | (D) | $\lvert \vec{a} \rvert \lvert \vec{b} \rvert \sin\theta \hat{n}$, where symbols have their usual meaning | (IV) | $2(\vec{a} \times \vec{b})$ | Choose the **correct** answer from the options given below :

The differential equation $y = xp + \sqrt{x^2 p^3 + 4}$ where $p = \frac{dy}{dx}$ is : (A) of order 1 (B) of degree 1 (C) of order 2 (D) of degree 3 Choose the **correct** answer from the options given below :

Distance between the point (3, 4, 5) and the point where the line $\frac{x-3}{1} = \frac{y-4}{2} = \frac{z-5}{2}$ meets the plane $x + y + z = 17$ is :

Urn I contains 6 red balls and 4 black balls and Urn II contains 4 red balls and 6 black balls. One ball is drawn at random from Urn I and placed in Urn II. If one ball is drawn at random from Urn II, then the probability that it is a red ball is :

Calculate the shaded area as given below : <img src="https://balti.afterboards.in/mD6m255LRScuQ6k" width="300px"/>

Integerating factor of $(x \log_e x) \frac{dy}{dx} + y = 2 \log_e x$ is :

The value of C, in Rolle's theorem for the function $f(x) = e^x \sin x$, when $x \in [0, \pi]$ is :

Match **List - I** with **List - II**. | | List - I | | List - II | |---|---|---|---| | (A) | $x = 2at^2, y = at^4$ | (I) | Inverse trignometric function | | (B) | $f(x) = (2x + 3)^3$ | (II) | Implicit function | | (C) | $xy + y^2 = \tan(x + y)$ | (III) | Parametric function | | (D) | $y = \tan^{-1}\left(\frac{3x - x^3}{1 - 3x^2}\right), -\frac{1}{\sqrt{3}} < x < \frac{1}{\sqrt{3}}$ | (IV) | Composite function | Choose the **correct** answer from the options given below :

$\int e^x (\tan x + \log_e \sec x) \, dx =$

Let $y = \log_e \left(\frac{a + b \sin x}{a - b \sin x}\right)$, then value of $\frac{dy}{dx}$ is :

Match **List - I** with **List - II**. | | List - I | | List - II | |---|---|---|---| | (A) | $y = \log(\sin x)$ | (I) | $\frac{d^2y}{dx^2} = -\frac{1}{x^2}$ | | (B) | $y = e^{(1 + \log x)}$ | (II) | $\frac{d^2y}{dx^2} = 2$ | | (C) | $y = \log\lvert x \rvert$ | (III) | $\frac{d^2y}{dx^2} = 0$ | | (D) | $y = x^2 + 4x - 1$ | (IV) | $\frac{d^2y}{dx^2} = -\csc^2 x$ | Choose the **correct** answer from the options given below :

The region represented by the system of inequalities $x, y \geq 0$ ; $2x + 3y \geq 4$ ; $x \geq 1$ is :

The equation of tangent to the curve $x = a \cos^3 t, y = a \sin^3 t$ at t is :

Let A = PQ. The elementary operation on A, that produces the same effect as it does on applying on P and keeping Q unchanged is : (A) $R_i \leftrightarrow R_j$ (B) $R_i \to R_i + KR_j$ (C) $C_i \to KC_i$ (D) $C_i \to C_i + KC_j$ Choose the **correct** answer from the options given below :

The set of values of K for which the system of equations $\begin{bmatrix} 2 & 3 & 1 \\ 4 & 5 & 0 \\ 1 & K & 3 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 5 \\ 6 \\ 7 \end{bmatrix}$ gives a unique solution is :

If the rate of change of area of a circle is equal to the rate of change of its diameter, then its radius is equal to :

### Match List–I with List–II <img src="https://balti.afterboards.in/mEkezKpygVWnmgS" width="400px"/> Choose the correct answer from the options given below

Let R be a relation on the set of natural numbers N defined by nRm if n divides m. Then R is : (A) Reflexive Relation (B) Symmetric Relation (C) Transitive Relation (D) Identity Relation Choose the **correct** answer from the options given below :

The area enclosed between the curve $x^2 + y^2 = 16$ and the coordinate axes in the first quadrant is :

Owner of a whole sale computers shop plans to sell 2 types of computers. A desktop and portable model. If $x$ is the number of desktops and $y$ is the number of portable model and the shop's capacity cannot exceed 250 units. Which of the following is correct ?

The value of the determinant $\Delta = \begin{vmatrix} 1! & 2! & 3! \\ 2! & 3! & 4! \\ 3! & 4! & 5! \end{vmatrix}$ is :

Cartesian equation of plane passing through the points (2, -4, 5) and perpendicular to the line with direction ratios (3, -1, 2) is :

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

The appropriate change in the volume V of a cube of side $x$ metres caused by increasing the side by 2% is :

If $A = \begin{pmatrix} \cos 2\theta & \sin 2\theta \\ -\sin 2\theta & \cos 2\theta \end{pmatrix}$, then $A^2 =$

Which of the following statements are **correct** ? (A) $|A'| = |A|$, where A is the transpose of matrix A (B) If $A = [a_{ij}]_{3 \times 3}$, then $|4A| = 64|A|$ (C) $|A| = |\text{adj } A|^{n-1}$, where n is the order of the matrix (D) If A is an invertible matrix of order 2, then $\det(A^{-1})$ is equal to $\frac{1}{\det(A)}$ Choose the **correct** answer from the options given below :

If $f(x) = \sqrt{x}$, $g(x) = 2x - 3$, then domain of $fog(x)$ is :

The given function $f(x) = [x]$ is discontinuous at :

The maximum value of $2x^3 - 24x + 107$ in the interval $[1, 3]$ is :

The equation of curve whose slope is given by $\frac{dy}{dx} = x$ and which passes through $\left(1, \frac{5}{2}\right)$ is :

If the shortest distance between the lines $l_1$ and $l_2$ given by $\vec{r} = a\hat{i} + 2\hat{j} - \hat{k} + \lambda(2\hat{i} - \hat{j} + \hat{k})$ and $\vec{r} = \hat{i} - \hat{j} + \hat{k} + \mu(2\hat{i} - \hat{j} + \hat{k})$ is $\sqrt{\frac{35}{6}}$ units, the values of 'a' can be :

If $x^{2/3} + y^{2/3} = a^{2/3}$, then $\frac{dy}{dx}$ is equal to :

If the objective function for an L.P.P. is $z = 3x + 4y$ and the corner points for unbounded feasible region are (9, 0), (4, 3), (2, 5) and (0, 8), then the minimum value of $z$ occurs at :

The digit in the unit's place of $13^{37}$ is :

If objective function $Z = 20x + 30y$ of an LPP is subject to the constraints $3x + 4y \geq 12$, $4x + y \geq 4$, $x \geq 0, y \geq 0$, then Z has : (A) Min at (0, 4) (B) Max at (0, 4) (C) Min at (4, 0) (D) Max at (4, 0) (E) Min at $\left(\frac{4}{13}, \frac{36}{13}\right)$ Choose the **correct** answer from the options given below :

Match **List - I** with **List - II**. | | List - I | | List - II | |---|---|---|---| | (A) | If $y = x^2 - 8$ and $\frac{dy}{dx} = 0$, then $x = ?$ | (I) | 1 | | (B) | If $p(x) = 3x + 1$, then $R(x)$ at $x = 2$ | (II) | 0 | | (C) | If $y = x^3$, then $\frac{dy}{dx}$ at $x = -1$ | (III) | 14 | | (D) | If $C(x) = 100 + 5x$, $R(x) = 102 + 3x$, then break-even point | (IV) | 3 | Choose the **correct** answer from the options given below :

A retailer has 250 kg of rice a part of which he sells at 10% profit. The remaining quantity of rice is of low quality and he sold it at 5% loss. Overall he made a profit of 7%. The quantity of rice sold at 5% loss is :

Which of the following statements are correct ? (A) Index number are free from units (B) Index number represents specialised averages in percentage (C) $P_{01} = \frac{\Sigma P_0}{\Sigma P_1} \times 100$ (D) Index numbers are helpful in formulating and adopting appropriate economic policies Choose the **correct** answer from the options given below :

Let $f : R \to R$ be a function defined as $f(x) = 2x^3 - 21x^2 + 36x - 20$, then : (A) maximum value of $f(x)$ is $-3$ (B) minimum value of $f(x)$ is $-128$ (C) maximum value exists at $x = 6$ (D) minimum value exists at $x = 1$ Choose the **correct** answer from the options given below :

Ram and Ankur make a partnership. Ram invests Rs. 35,000 for 6 months and Ankur invests some money for 7 months. Ankur claims $\frac{4}{7}$ of total profit. Then the money invested by Ankur is :

Pipes A and B can fill a tank in 20 hours and 30 hours respectively and pipe C can empty the full tank in 40 hours. If all the pipes are opened together, how much time will be needed to make the tank full ?

If A is square matrix of order 3 with $|A| = 3$, then $|4 \text{ adj} A|$ is equal to :

The purchase price P of a Rs. 50,000, 6% bond, dividends payable semi-annually, redeemable at par in 10 years, if the yield rate is to be 5% compounded semi-annually. Then P is equal : [Given $(1.025)^{-20} = 0.61027094$]

For the data : | Variable | Price - Base year | Price - Current year | Weights | |---|---|---|---| | X | 50 | 60 | 5 | | Y | 20 | 25 | 7 | | Z | 30 | 40 | 4 | The weighted aggregative index number is :

In an LPP if the objective function $z = ax + by$ has same maximum value on two corner points of the feasible region, then the number of points at which maximum value of $z$ occurs is :

If the probability distribution of X is : | X | 2 | 3 | 4 | 5 | 6 | |---|---|---|---|---|---| | P(X) | 1/15 | 2/15 | 3/15 | 4/15 | 5/15 | Then variance is equal to :

Five litres of water is added to a certain quantity of pure milk costing Rs. 60 per litre. If by selling the mixture at the same price as before, a profit of 20% is made, then the amount of pure milk in the mixture is :

If the matrix $\begin{bmatrix} 0 & 2 & 5x \\ -2 & 0 & 6 \\ 10 & -6 & y \end{bmatrix}$ is skew-symmetric matrix, then the value of $(y - 4x)$ is :

The wholesale price index (or price relative) of rice in 2018 compared to 2014 is 150. If the cost of rice was Rs. 16 per kg in 2014, then per kg cost of rice in 2018 is :

The assumption opposite of what is made in the null hypothesis is known as the _________ hypothesis.

If $y = 3e^{2x} + 2e^{3x}$, then which one of the following is true ?

Match **List - I** with **List - II**. | | List - I (Matrix) | | List - II (Type) | |---|---|---|---| | (A) | $\begin{bmatrix} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{bmatrix}$ | (I) | Lower Triangular Matrix | | (B) | $\begin{bmatrix} 1 & 5 & 0 & -1 \end{bmatrix}$ | (II) | Row Matrix | | (C) | $\begin{bmatrix} 3 & 0 & 0 \\ 1 & -1 & 0 \\ 2 & 5 & 4 \end{bmatrix}$ | (III) | Diagonal Matrix | | (D) | $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 4 \end{bmatrix}$ | (IV) | Scalar Matrix | Choose the **correct** answer from the options given below :

Match **List - I** with **List - II**. | | List - I | | List - II | |---|---|---|---| | (A) | Left tailed test | (I) | $H_0 : \mu = 50$ | | (B) | Two tailed test | (II) | $H_1 : \mu > 50$ | | (C) | Null hypothesis | (III) | $H_1 : \mu < 50$ | | (D) | Right tailed test | (IV) | $H_1 : \mu \neq 50$ | Choose the **correct** answer from the options given below :

Match **List - I** with **List - II**. | | List - I | | List - II | |---|---|---|---| | (A) | For break-even point | (I) | $< 0$ | | (B) | For maxima $\frac{d^2y}{dx^2}$ | (II) | $\frac{dy}{dx} = 0$ | | (C) | For points of maxima/minima | (III) | $R(x) - C(x)$ | | (D) | $P(x) = $ Profit function | (IV) | $R(x) = C(x)$ | Choose the **correct** answer from the options given below :

The effective rate equivalent to a nominal rate of 8% per annum compounded semi annually is :

If $x\%$ of $y + y\%$ of $30 = 61\%$ of $xy$, then $x =$

If $K \equiv 5 \pmod{11}$, then all the possible non-negative values of K are :

The investment ratio of P, Q and R if their profit ratio is 3 : 5 : 7 respectively and their investment time period ratio is 4 : 5 : 6 respectively is :

A point $P(3a, -2b)$ lies in region $4x + 7y \leq (-3)$ which of the following options is true ?

Ravi takes a loan of Rs. 40,000 at an interest of 12% per annum for a period of 4 year, then EMI by using flat method is :

For testing the difference between the means of two samples, the following data is available : | | Size | Mean | Variance | |---|---|---|---| | Sample 1 | 5 | 40 | 101 | | Sample 2 | 7 | 30 | 60 | The value of the t-statistics is :

The quantity of water that must be added to 36 litres of pure milk at $1\frac{1}{2}$ litres for Rs. 75 so as to have mixture worth Rs. 40 a litre is :

If $x = 2t^2 + 3, y = 3t^2 + 6t + 5$, then the value of $\frac{d^2y}{dx^2}$ is :

If a random variable X follows binomial distribution with mean 5 and variance $\frac{5}{2}$, then $P(X \leq 9)$ is :

A simple random sample consists of five observations 2, 4, 7, 12, 15. The point estimate of the population mean is :

A vehicle costing Rs. 10,50,000 has a final scrap value of Rs. 5,25,000. If annual depreciation charge is Rs. 75,000, then useful life of the vehicle is :