CUET Mathematics 2024 16 May Shift 1 PYQs

The corner points of the feasible region determined by $x+y \leq 8, 2 x+y \geq 8, x \geq 0, y \geq 0$ are $A(0,8), B(4,0)$ and $C(8,0)$. If the objective function $Z=a x+$ by has its maximum value on the line segment $A B$, then the relation between $a$ and $b$ is :

If $t=e^{2 x}$ and $y=\log _{e} t^{2}$, then $\frac{d^{2} y}{d x^{2}}$ is :

An objective function $Z=a x+b y$ is maximum at points $(8,2)$ and $(4,6)$. If $a \geq 0$ and $b \geq 0$ and $a b=25$, then the maximum value of the function is equal to :

The area of the region bounded by the lines $x+2 y=12, x=2, x=6$ and $x$-axis is :

A die is rolled thrice. What is the probability of getting a number greater than $4$ in the first and the second throw of dice and a number less than $4$ in the third throw ?

$\int \dfrac{\pi}{x^{n+1}-x} d x=$

The value of $\int_{0}^{1} \frac{a-b x^{2}}{\left(a+b x^{2}\right)^{2}} d x$ is :

The second order derivative of which of the following functions is $5^{\mathrm{x}}$ ?

The degree of the differential equation $\left(1-\left(\frac{d y}{d x}\right)^{2}\right)^{\frac{3}{2}}=k \frac{d^{2} y}{d x^{2}}$ is :

If $A$ and $B$ are symmetric matrices of the same order, then $A B-B A$ is a :

If $A$ is a square matrix of order $4$ and $|A|=4$, then $|2 A|$ will be :

If $[\mathrm{A}]_{3 \times 2}[\mathrm{~B}]_{\mathrm{x} \times \mathrm{y}}=[\mathrm{C}]_{3 \times 1}$, then :

If a function $f(x)=x^{2}+b x+1$ is increasing in the interval $[1,2]$, then the least value of $b$ is :

Two dice are thrown simultaneously. If X denotes the number of fours, then the expectation of X will be :

For the function $f(x) = 2x^3 - 9x^2 + 12x - 5$, $x \in [0, 3]$, match List-I with List-II : | List-I | List-II | | --- | --- | | A. Absolute maximum value | (I) $ 3 $ | | B. Absolute minimum value | (II) $ 0 $ | | C. Point of maxima | (III) $ -5 $ | | D. Point of minima | (IV) $ 4 $ |

The rate of change (in $\mathrm{cm}^{2} / \mathrm{s}$ ) of the total surface area of a hemisphere with respect to radius $r$ at $r=\sqrt[3]{1.331} \mathrm{~cm}$ is :

The area of the region bounded by the lines $\frac{x}{7 \sqrt{3} a}+\frac{y}{b}=4, x=0$ and $y=0$ is :

If $A$ is a square matrix and $I$ is an identity matrix such that $A^{2}=A$, then $A(I-2 A)^{3}+2 A^{3}$ is equal to :

The value of the integral $\int_{\log _{e} 2}^{\log _{e} 3} \frac{e^{2 x}-1}{e^{2 x}+1} d x$ is :

If $\vec{a}, \vec{b}$ and $\vec{c}$ are three vectors such that $\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}$, where $\vec{a}$ and $\vec{b}$ are unit vectors and $|\vec{c}|=2$, then the angle between the vectors $\vec{b}$ and $\vec{c}$ is :

Let $[x]$ denote the greatest integer function. Then match List-I with List-II:</p> | List-I | List-II | | --- | --- | | (A) $ \vert x - 1\vert + \vert x - 2\vert $ | (I) is differentiable everywhere except at $ x = 0 $ | | (B) $ x - \vert x\vert $ | (II) is continuous everywhere | | (C) $ x - [x] $ | (III) is not differentiable at $ x = 1 $ | | (D) $ x \, \vert x\vert $ | (IV) is differentiable at $ x = 1 $ |

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

If the function $f: \mathbb{N} \rightarrow \mathbb{N}$ is defined as $f(n)=\left\{\begin{array}{ll}n-1, & \text { if } n \text { is even } \\ n+1, & \text { if } n \text { is odd }\end{array}\right.$, then (A) f is injective (B) f is into (C) f is surjective (D) f is invertible

$\int_{0}^{\frac{\pi}{2}} \frac{1-\cot x}{\operatorname{cosec} x+\cos x} d x=$

If the random variable $ X $ has the following distribution: | $X$ | 0 | 1 | 2 | otherwise | | --- | --- | --- | --- | --- | | $P(X)$ | $ k $ | $ 2k $ | $ 3k $ | $ 0 $ | Match List-I with List-II: | List-I | List-II | | --- | --- | | (A) $ k $ | (I) $ \frac{5}{6} $ | | (B) $ P(X < 2) $ | (II) $ \frac{4}{3} $ | | (C) $ E(X) $ | (III) $ \frac{1}{2} $ | | (D) $ P(1 \leq X \leq 2) $ | (IV) $ \frac{1}{6} $ |

For a square matrix $A_{n \times n}$ (A) $|\operatorname{adj} \mathrm{A}|=|\mathrm{A}|^{\mathrm{n}-1}$ (B) $|\mathrm{A}|=|\operatorname{adj} \mathrm{A}|^{\mathrm{n}-1}$ (C) $\mathrm{A}(\operatorname{adj} \mathrm{A})=|\mathrm{A}|$ (D) $\left|\mathrm{A}^{-1}\right|=\frac{1}{|\mathrm{~A}|}$

The matrix $\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$ is a : (A) scalar matrix (B) diagonal matrix (C) skew-symmetric matix (D) symmetric matrix Choose the correct answer from the options given below :

<img src="https://balti.afterboards.in/QKMiqcaA8YsIhak" width="500px"/> The feasible region represented by the constraints $4 x+y \geq 80, x+5 y \geq 115,3 x+2 y \leq 150, x, y \geq 0$ of an LPP is

The area of the region enclosed between the curves $4 x^{2}=y$ and $y=4$ is :

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

If $f(x)$, defined by $f(x)=\left\{\begin{array}{lll}k x+1 & \text { if } & x \leq \pi \\ \cos x & \text { if } & x>\pi\end{array}\right.$ is continuous at $x=\pi$, then the value of $k$ is :

If $P=\left[\begin{array}{r}-1 \\ 2 \\ 1\end{array}\right]$ and $Q=\left[\begin{array}{lll}2 & -4 & 1\end{array}\right]$ are two matrices, then $(P Q)^{\prime}$ will be :

$\Delta=\left|\begin{array}{ccc}1 & \cos x & 1 \\ -\cos x & 1 & \cos x \\ -1 & -\cos x & 1\end{array}\right|$ (A) $\Delta=2\left(1-\cos ^{2} x\right)$ (B) $\Delta=2\left(2-\sin ^{2} x\right)$ (C) Minimum value of $\Delta$ is $2$ (D) Maximum value of $\Delta$ is $4$ Choose the correct answer from the options given below :

$f(x)=\sin x+\frac{1}{2} \cos 2 x \text { in }\left[0, \frac{\pi}{2}\right]$ (A) $f^{\prime}(x)=\cos x-\sin 2 x$ (B) The critical points of the function are $x=\frac{\pi}{6}$ and $x=\frac{\pi}{2}$ (C) The minimum value of the function is $2$ (D) The maximum value of the function is $\frac{3}{4}$ Choose the correct answer from the options given below :

The direction cosines of the line which is perpendicular to the lines with direction ratios $1,-2,-2$ and $0,2,1$ are :

Let $X$ denote the number of hours you play during a randomly selected day. The probability that $X$ can take values $x$ has the following form, where $c$ is some constant. $\mathrm{P}(\mathrm{X}=\mathrm{x})=\left\{\begin{array}{lll} 0.1, & \text { if } \mathrm{x}=0 \\ \mathrm{cx}, & \text { if } \mathrm{x}=1 \text { or } \mathrm{x}=2 \\ \mathrm{c}(5-\mathrm{x}), & \text { if } \mathrm{x}=3 \text { or } \mathrm{x}=4 \\ 0, & \text { otherwise } \end{array}\right.$ Match List-I with List-II : | List-I | List-II | | --- | --- | | (A) $ c $ | (I) 0.75 | | (B) $ P(X \leq 2) $ | (II) 0.3 | | (C) $ P(X = 2) $ | (III) 0.55 | | (D) $ P(X \geq 2) $ | (IV) 0.15 |

If $\sin y=x \sin (a+y)$, then $\frac{d y}{d x}$ is :

The unit vector perpendicular to each of the vectors $\vec{a}+\vec{b}$ and $\vec{a}-\vec{b}$, where $\vec{a}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{b}=\hat{i}+2 \hat{j}+3 \hat{k}$, is :

The distance between the lines $\vec{r}=\hat{i}-2 \hat{j}+3 \hat{k}+\lambda(2 \hat{i}+3 \hat{j}+6 \hat{k})$ and $\vec{r}=3 \hat{i}-2 \hat{j}+1 \hat{k}+\mu(4 \hat{i}+6 \hat{j}+12 \hat{k})$ is :

If $f(x)=2\left(\tan ^{-1}\left(e^{x}\right)-\frac{\pi}{4}\right)$, then $f(x)$ is :

For the differential equation $\left(x \log _{e} x\right) d y=\left(\log _{e} x-y\right) d x$ (A) Degree of the given differential equation is $1$. (B) It is a homogeneous differential equation. (C) Solution is $2y \log _{\mathrm{e}} \mathrm{x}+A=\left(\log _{\mathrm{e}} \mathrm{x}\right)^{2}$, where $A$ is an arbitrary constant (D) Solution is $2 y \log _{e} x+A=\log _{e}\left(\log _{e} x\right)$, where $A$ is an arbitrary constant Choose the correct answer from the options given below :

There are two bags. Bag-1 contains $4$ white and $6$ black balls and Bag-2 contains $5$ white and $5$ black balls. A die is rolled, if it shows a number divisible by 3, a ball is drawn from Bag-1, else a ball is drawn from Bag-2. If the ball drawn is not black in colour, the probability that it was not drawn from Bag-2 is :

Which of the following cannot be the direction ratios of the straight line $\frac{x-3}{2}=\frac{2-y}{3}=\frac{z+4}{-1}$ ?

Which one of the following represents the correct feasible region determined by the following constraints of an LPP? $x+y \geq 10,2 x+2 y \leq 25, x \geq 0, y \geq 0$

Let R be the relation over the set A of all straight lines in a plane such that $l_{1} \mathrm{R} l_{2} \Leftrightarrow l_{1}$ is parallel to $l_{2}$. Then R is :

The probability of not getting $53$ Tuesdays in a leap year is :

The angle between two lines whose direction ratios are propotional to $1,1,-2$ and $(\sqrt{3}-1),(-\sqrt{3}-1),-4$ is :

If $(\vec{a}-\vec{b}) \cdot(\vec{a}+\vec{b})=27$ and $|\vec{a}|=2|\vec{b}|$, then $|\vec{b}|$ is :

If $\tan ^{-1}\left(\frac{2}{3^{-x}+1}\right)=\cot ^{-1}\left(\frac{3}{3^{x}+1}\right)$, then which one of the following is true ?

If $A, B$ and $C$ are three singular matrices given by $A=\left[\begin{array}{ll}1 & 4 \\ 3 & 2 a\end{array}\right], B=\left[\begin{array}{ll}3 b & 5 \\ a & 2\end{array}\right]$ and $C=\left[\begin{array}{cc}a+b+c & c+1 \\ a+c & c\end{array}\right]$, then the value of $a b c$ is :

For which one of the following purposes is CAGR (Compounded Annual Growth Rate) <b>not</b> used ?

A flower vase costs ₹ $36,000$. With an annual depreciation of ₹ $2,000$, its cost will be ₹ $6,000$ in $_____$ years.

Arun's speed of swimming in still water is $5 \mathrm{~km} / \mathrm{hr}$. He swims between two points in a river and returns back to the same starting point. He took $20$ minutes more to cover the distance upstream than downstream. If the speed of the stream is $2 \mathrm{~km} / \mathrm{hr}$, then the distance between the two points is :

If $\mathrm{e}^{\mathrm{y}}=\mathrm{x}^{\mathrm{x}}$, then which of the following is true ?

The probability of a shooter hitting a target is $\frac34$. How many minimum number of times must he fire so that the probability of hitting the target at least once is more than $90 \%$ ?

Ms. Sheela creates a fund of ₹ $1,00,000$ for providing scholarships to needy children. The scholarship is provided in the beginning of the year. This fund earns an interest of $r \%$ per annum. If the scholarship amount is taken as ₹ $8,000$, then $r=$

A person wants to invest an amount of ₹ $75,000$. He has two options A and B yielding $8 \%$ and $9 \%$ return respectively on the invested amount. He plans to invest at least ₹ $15,000$ in Plan A and at least ₹ $25,000$ in Plan B. Also he wants that his investment in Plan A is less than or equal to his investment in Plan B. Which of the following options describes the given LPP to maximize the return (where $x$ and $y$ are investments in Plan A and Plan B respectively) ?

In a $700 \mathrm{~m}$ race, Amit reaches the finish point in $20$ seconds and Rahul reaches in $25$ seconds. Amit beats Rahul by a distance of :

For the given five values $12,15,18,24,36$; the three-year moving averages are :

The corner points of the feasible region for an L.P.P. are $(0,10),(5,5),(5,15)$ and $(0,30)$. If the objective function is $Z=\alpha x+\beta y, \alpha, \beta>0$, the condition on $\alpha$ and $\beta$ so that maximum of $Z$ occurs at corner points $(5,5)$ and $(0,20)$ is :

The solution set of the inequality $|3 x| \geq|6-3 x|$ is :

If the matrix $\left[\begin{array}{rrr}0 & -1 & 3 x \\ 1 & y & -5 \\ -6 & 5 & 0\end{array}\right]$ is skew-symmetric, then the value of $5 x-y$ is:

A company is selling a certain commodity $x$. The demand function for the commodity is linear. The company can sell $2000$ units when the price is ₹$8$ per unit and it can sell $3000$ units when the price is ₹$4$ per unit. The Marginal revenue at $x=5$ is:

If the lengths of the three sides of a trapezium other than the base are $10 \mathrm{~cm}$ each, then the maximum area of the trapezium is:

Three defective bulbs are mixed with $8$ good ones. If three bulbs are drawn one by one with replacement, the probabilities of getting exactly $1$ defective, more than $2$ defective, no defective and more than $1$ defective respectively are:

If $\mathrm{A}=\left[\begin{array}{ll}2 & 4 \\ 4 & 3\end{array}\right], \mathrm{X}=\left[\begin{array}{l}\mathrm{n} \\ 1\end{array}\right], \mathrm{B}=\left[\begin{array}{c}8 \\ 11\end{array}\right]$ and $A X=B$, then the value of $n$ will be :

The equation of the tangent to the curve $\mathrm{x}^{\frac{5}{2}}+\mathrm{y}^{\frac{5}{2}}=33$ at the point $(1,4)$ is :

| $ X $ | -2 | -1 | 0 | 1 | 2 | | --- | --- | --- | --- | --- | --- | | $ P(X) $ | $ 0.2 $ | $ 0.1 $ | $ 0.3 $ | $ 0.2 $ | $0.2$ | The variance of $X$ will be :

A Multinational company creates a sinking fund by setting a sum of ₹ $12,000$ annually for $10$ years to pay off a bond issue of ₹ $72,000$. If the fund accumulates at $5 \%$ per annum compound interest, then the surplus after paying for bond is : (Use $\left.(1.05)^{10} \approx 1.6\right)$

The least non-negative remainder when $3^{51}$ is divided by $7$ is :

If $\left[\begin{array}{cc}5 x+8 & 7 \\ y+3 & 10 x+12\end{array}\right]=\left[\begin{array}{cc}2 & 3 y+1 \\ 5 & 0\end{array}\right]$, then the value of $5 x+3 y$ is equal to :

There are $6$ cards numbered $1$ to $6$, one number on one card. Two cards are drawn at random without replacement. Let X denote the sum of the numbers on the two cards drawn. Then $\mathrm{P}(\mathrm{X}>3)$ is :

Which of the following are components of a time series ? (A) Irregular component (B) Cyclical component (C) Chronological Component (D) Trend Component Choose the <b>correct</b> answer from the options given below :

The following data is from a simple random sample : $15,23, x, 37,19,32$ If the point estimate of the population mean is $23$, then the value of $x$ is :

For an investment, if the nominal rate of interest is $10 \%$ compounded half yearly, then the effective rate of interest is :

A mixture contains apple juice and water in the ratio $10: \mathrm{x}$. When $36$ litres of the mixture and $9$ litres of water are mixed, the ratio of apple juice and water becomes $5: 4$. The value of $x$ is :

For $I=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$, if $X$ and $Y$ are square matrices of order $2$ such that $X Y=X$ and $Y X=Y$, then $\left(\mathrm{Y}^{2}+2 \mathrm{Y}\right)$ equals to :

A coin is tossed K times. If the probability of getting $3$ heads is equal to the probability of getting $7$ heads, then the probability of getting $8$ tails is :

If $95 \%$ confidence interval for the population mean was reported to be $160$ to $170$ and $\sigma=25$, then size of the sample used in this study is: (Given $\mathrm{Z}_{0.025}=1.96$ )

Two pipes A and B together can fill a tank in $40$ minutes. Pipe A is twice as fast as pipe B. Pipe A alone can fill the tank in :