Download JEE Main 2024 Question Paper (04 Apr - Shift 1) Mathematics

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05 Apr 2024:Shift-1

05 Apr 2024:Shift-2

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Important Instructions

• Section-A : Attempt all questions.
• Section-B : Do any 5 questions out of 10 Questions.
• Section-A (01 – 20) contains 20 multiple choice questions which have only one correct answer. Each question carries +4 marks for correct answer and –1 mark for wrong answer.
• Section-B (1 – 10) contains 10 Numerical based questions. The answer to each question is rounded off to the nearest integer value. Each question carries +4 marks for correct answer and –1 mark for wrong answer

SECTION - A
(One Option Correct Type)

This section contains 20 multiple choice questions. Each question has four choices (A), (B), (C) and (D), out of which ONLY ONE option is correct

     
1. Let $f : R \to R$ be a function given by
   $f(x)$=$\left\{\begin{array}{cc} \frac{1-\cos 2x}{x^2} & , x < 0 \\ \alpha &, x = 0, \\ \frac{\beta \sqrt{1-\cos x}}{x} & , x > 0 \end{array}\right.$$\text{where } \alpha, \beta \in R$. If $ƒ$ is continuous at $x$ = 0, then $\alpha^2$+$\beta^2$ is equal to :
   
      
   1. 48
   2. 12
   3. 3
   4. 6
2. Three urns $A$, $B$ and $C$ contain 7 red, 5 black;5 red, 7 black and 6 red, 6 black balls, respectively.One of the urn is selected at random and a ball isdrawn from it. If the ball drawn is black, then theprobability that it is drawn from urn $A$ is :
   
      
   1. $\frac{4}{17}$
   2. $\frac{5}{18}$
   3. $\frac{7}{18}$
   4. $\frac{5}{16}$
3. The vertices of a triangle are $A(–1, 3)$, $B(–2, 2)$ and$C(3, –1)$. A new triangle is formed by shifting the sidesof the triangle by one unit inwards. Then the equationof the side of the new triangle nearest to origin is :
   
      
   1. $x-y$$-(2+\sqrt{2})$=0
   2. $-x$+$y$$-(2+\sqrt{2})$=0
   3. $x$+$y$$-(2+\sqrt{2})$=0
   4. $x$+$y$+$(2+\sqrt{2})$=0
4. If the solution $y$ = $y(x)$ of the differential equation $(x^4 + 2x^3 + 3x^2 + 2x + 2)dy$$ – (2x^2 + 2x + 3)dx$ = 0 satisfies $y(–1)$ = $-\frac{\pi}{4}$, then y(0) is equal to :
   
      
   1. $-\frac{\pi}{12}$
   2. 0
   3. $\frac{\pi}{4}$
   4. $\frac{\pi}{2}$
5. Let the sum of the maximum and the minimum values of the function $f(x)$ = $\frac{2x^2-3x+8}{2x^2+3x+8}$ be $\frac{m}{n}$, where $gcd(m, n)$ = 1. Then $m + n$ is equal to :
   
      
   1. 182
   2. 217
   3. 195
   4. 201
6. One of the points of intersection of the curves $y$ = 1 + $3x – 2x^2$ and $y$ = $\frac{1}{x}$is $\left(\frac{1}{2}, 2 \right)$. Let the area of the region enclosed by these curves be $\frac{1}{24}(l \sqrt{5}+m)$$-n \log_e(1+\sqrt{5})$, where $l$, $m$, $n \in N$. Then $l + m + n$ is equal to
   
      
   1. 32
   2. 30
   3. 29
   4. 31
7. If the system of equations
   $x$+$(\sqrt{2}\sin \alpha)y$+$(\sqrt{2}\cos \alpha)z$=0
   $x$ + $(cos \alpha)y$ + $(sin \alpha)z$ = 0
   $x$ + $(sin \alpha)y$ $– (cos \alpha)z$ = 0
   has a non-trivial solution, then $\alpha \in \left(0, \frac{\pi}{2}\right)$ is equal to :
   
      
   1. $\frac{3\pi}{4}$
   2. $\frac{7\pi}{24}$
   3. $\frac{5\pi}{24}$
   4. $\frac{11\pi}{24}$
8. There are 5 points $P_1$, $P_2$, $P_3$, $P_4$, $P_5$ on the side $AB$, excluding $A$ and $B$, of a triangle $ABC$. Similarly there are 6 points $P_6$,$P_7$, …, $P_{11}$ on the side $BC$ and 7 points $P_{12}$, $P_{13}$, …, $P_{18}$ on the side $CA$ of the triangle. The number of triangles, that can be formed using the points $P_1$, $P_2$, …, $P_{18}$ as vertices, is :
   
      
   1. 776
   2. 751
   3. 796
   4. 771
   
9.  $\left\{\begin{array}{cc} -2, & -2 \leq x \leq 0 \\ x-2, & 0 < x \leq 2\end{array}\right.$ and $h(x)$=$f(|x|)$+$|f(x)|$. Then $\int \limits_{-2}^{2}h(x)dx$ is equal to :
   
      
   1. 2
   2. 4
   3. 1
   4. 6
10. The sum of all rational terms in the expansion of $\left(2^{\frac{1}{5}}+5^{\frac{1}{3}}\right)^{15}$ is equal to
    
    
       
    1. 3133
    2. 633
    3. 931
    4. 6131
11. Let a unit vector which makes an angle of 60° with $2\hat{i}+2\hat{j}$$-\hat{k}$ and an angle of 45° with $\hat{i}-\hat{k}$ be $\vec{C}$. Then $\vec{C}$+$\left(-\frac{1}{2}\hat{i} +\frac{1}{3\sqrt{2}}\hat{j}-\frac{\sqrt{2}}{3}\hat{k}\right)$ is
    
       
    1. $-\frac{\sqrt{2}}{3}\hat{i}$+$\frac{\sqrt{2}}{3}\hat{j}$+$\left(\frac{1}{2}+\frac{2\sqrt{2}}{3}\right)\hat{k}$
    2. $\frac{\sqrt{2}}{3}\hat{i}$+$\frac{1}{3\sqrt{2}}\hat{j}$$-\frac{1}{2}\hat{k}$
    3. $\left(\frac{1}{\sqrt{3}}+\frac{1}{2}\right)\hat{i}$+$\left(\frac{1}{\sqrt{3}}-\frac{1}{3\sqrt{2}}\right)\hat{j}$+$\left(\frac{1}{\sqrt{3}}+\frac{\sqrt{2}}{3}\right)\hat{k}$
    4. $\frac{\sqrt{2}}{3}\hat{i}$$-\frac{1}{2}\hat{k}$
12. Let the first three terms 2, $p$ and $q$, with $q \neq 2$, of a $G.P.$ be respectively the $7^{th}$, $8^{th}$ and $13^{th}$ terms of an $A.P.$ If the $5^{th}$ term of the $G.P.$ is the $n^{th}$ term of the $A.P.$, then $n$ is equal to
    
       
    1. 151
    2. 169
    3. 177
    4. 163
13. Let $a, b \in R$. Let the mean and the variance of 6 observations –3, 4, 7, –6, a, b be 2 and 23, respectively. The mean deviation about the mean of these 6 observations is :
    
       
    1. $\frac{13}{3}$
    2. $\frac{16}{3}$
    3. $\frac{11}{3}$
    4. $\frac{14}{3}$
14. If 2 and 6 are the roots of the equation $ax^2$ + $bx$ + 1 = 0, then the quadratic equation, whose roots are $\frac{1}{2a+b}$+$\frac{1}{6a+b}$, is
    
       
    1. $2x^2$+ $11x$ + 12 = 0
    2. $4x^2$+ $14x$ + 12 = 0
    3. $x^2$+ $10x$ + 16 = 0
    4. $x^2$+ $8x$ + 12 = 0
15. Let $\alpha$ and $\beta$ be the sum and the product of all the non-zero solutions of the equation $(\bar{z})^2$+$|z|$=0, $z \in C$. Then $4(\alpha^2+\beta^2)$ is equal to :
    
       
    1. 6
    2. 8
    3. 4
    4. 2
16. Let the point, on the line passing through the points $P(1, –2, 3)$ and $Q(5, –4, 7)$, farther from the origin and at a distance of 9 units from the point $P$, be $(\alpha, \beta, \gamma)$. Then $\alpha^2+\beta^2+\gamma^2$ is equal to :
    
       
    1. 155
    2. 150
    3. 160
    4. 165
17. A square is inscribed in the circle $x^2$ + $y^2$$– 10x$$ – 6y$ + 30 = 0. One side of this square is parallel to $y$ = $x$ + 3. If $(x_i , y_i)$ are the vertices of the square, then $\sum (x_i^2+y_i^2)$ is equal to
    
       
    1. 148
    2. 156
    3. 160
    4. 152
18. If the domain of the function
    $\sin^{-1}\left(\frac{3x-22}{2x-19}\right)$+$\log_e\left(\frac{3x^2-8x+5}{x^2-3x-10}\right)$ is $(\alpha, \beta]$, then $3\alpha+10\beta$ is equal to
    
       
    1. 97
    2. 100
    3. 95
    4. 98
19.  Let $ƒ(x)$ = $x^5 + 2e^{x/4}$ for all $x \in R$. Consider a function $g(x)$ such that $(gof) (x) = x$ for all $x \in R$. Then the value of $8g'(2)$ is :
    
       
    1. 16
    2. 4
    3. 8
    4. 2
20. Let $\alpha \in (0, \infty)$ and $A$=$\begin{equation*} \begin{bmatrix} 1 & 2 & \alpha \\ 1 & 0 & 1 \\ 0 & 1 & 2\end{bmatrix}\end{equation*}$. If $det(adj(2A – A^T).adj(A – 2A^T))$ = 28, then $(det(A))^2$ is equal to :
    
       
    1. 1
    2. 49
    3. 16
    4. 36

SECTION - B
(Numerical Answer Type)

This section contains 10 Numerical based questions. The answer to each question is rounded off to the nearest integer value.

1. If $\lim \limits_{x \to 1} \frac{(5x+1)^{1/3}-(x+5)^{1/3}}{(2x+3)^{1/2}-(x+4)^{1/2}}$=$\frac{m\sqrt{5}}{n(2n)^{2/3}}$, where $gcd(m, n)$ = 1, then $8m$ + $12n$ is equal to _____
2. In a survey of 220 students of a higher secondaryschool, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and at most 95 studied Physics; at least 75 and at most 90 studied Chemistry; 30 studied both Physics and Chemistry; 50 studied both Chemistry and Mathematics; 40 studied both Mathematics and Physics and 10 studied none of these subjects. Let m and n respectively be the least and the most number of students who studied all the three subjects. Then m + n is equal to _____
3. Let the solution $y = y(x)$ of the differential equation $\frac{dy}{dx}-y$=1+$4\sin x$ satisfy $y(\pi)$=1. Then $y\left(\frac{\pi}{2}\right)$+10 is equal to
4. If the shortest distance between the lines $\frac{x+2}{2}$=$\frac{y+3}{3}$=$\frac{z-5}{4}$ and $\frac{x-3}{1}$=$\frac{y-2}{-3}$=$\frac{z+4}{2}$ is $\frac{38}{3\sqrt{5}}k$ and $\int \limits_0^k[x^2]dx$=$\alpha-\sqrt{\alpha}$, where $[x]$ denotes the greatest integer function, then $6\alpha^3$is equal to _____
5. Let $A$ be a square matrix of order 2 such that $|A|$ = 2 and the sum of its diagonal elements is –3. If the points $(x, y)$ satisfying $A^2$ + $xA$ + $yI$ = 0 lie on a hyperbola, whose transverse axis is parallel to the $x-$axis, eccentricity is $e$ and the length of the latus rectum is $l$, then $e^4+ l^4$is equal to ______
6. Let $a$=1+$\frac{{}^2C_2}{3!}$+$\frac{{}^3C_2}{4!}$+$\frac{{}^4C_2}{5!}$+...,
   $b$=1+$\frac{{}^1C_0+{}^1C_1}{1!}$+$\frac{{}^2C_0+{}^2C_1+{}^2C_2}{2!}$+$\frac{{}^3C_0+{}^3C_1+{}^3C_2+{}^3C_3}{3!}$+...
   Then $\frac{2b}{a^2}$ is equal to ............
7. Let $A$ be a 3 × 3 matrix of non-negative real elements such that $\begin{equation*} A \begin{bmatrix} 1\\ 1\\ 1 \end{bmatrix}=3\begin{bmatrix} 1\\ 1\\ 1 \end{bmatrix} \end{equation*}$. Then the maximum value of $det(A)$ is _____
8. Let the length of the focal chord $PQ$ of the parabola $y^2$ = $12x$ be 15 units. If the distance of $PQ$ from the origin is $p$, then $10p^2$is equal to ____
9. Let $ABC$ be a triangle of area $15\sqrt{2}$ and the vectors $AB$=$\hat{i}$+$2\hat{j}$$- 7\hat{k}$, $BC$= $a\hat{i}$+$b\hat{j}$+$c\hat{k}$ and $AC$= $6\hat{i}$+$d\hat{j}$$-2\hat{k}$, $d$ > 0. Then the square of the length of the largest side of the triangle $ABC$ is
10. If $\int \limits_{0}^{\frac{\pi}{4}}\frac{\sin^2 x}{1+\sin x \cos x}dx$=$\frac{1}{a}\log_e\left(\frac{a}{3}\right)$+$\frac{\pi}{b\sqrt{3}}$, where $a$, $b \in N$, then $a + b$ is equal to ____

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