Download JEE Main 2024 Question Paper (05 Apr - Shift 2) Mathematics

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06 Apr 2024:Shift-1

06 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 $ƒ: [–1, 2] \to R$ be given by $ƒ(x)$ = $2x^2$+ $x$ + $[x^2]$ – $[x]$, where $[t]$ denotes the greatest integer less than or equal to t. The number of points, where $ƒ$ is not continuous, is :
   
      
   1. 6
   2. 3
   3. 4
   4. 5
2. The differential equation of the family of circles passing the origin and having center at the line $y = x$ is :
   
      
   1. $(x^2 – y^2+ 2xy)dx$ = $(x^2– y^2+ 2xy)dy$
   2. $(x^2+ y^2+ 2xy)dx$ = $(x^2 + y^2– 2xy)dy$
   3. $(x^2– y^2+ 2xy)dx$ = $(x^2– y^2 – 2xy)dy$
   4. $(x^2+ y^2– 2xy)dx$ = $(x^2 + y^2+ 2xy)dy$
3. Let $S_1$ = {$z \in C$ : $|z| \leq 5$},
   $S_2$=$\left\{z \in C: Im \left(\frac{z+1-\sqrt{3}i}{1-\sqrt{3}i}\right)\geq 0 \right\}$ and $S_3$ = {$z \in C : Re (z) \geq 0$}. Then the area of region $S_1 \cap S_2 \cap S_3$ is
   
      
   1. $\frac{125\pi}{6}$
   2. $\frac{125\pi}{24}$
   3. $\frac{125\pi}{4}$
   4. $\frac{125\pi}{12}$
4. The area enclosed between the curves $y = x|x|$ and $y = x – |x|$ is :
   
      
   1. $\frac{8}{3}$
   2. $\frac{2}{3}$
   3. 1
   4. $\frac{4}{3}$
5. 60 words can be made using all the letters of the word $BHBJO$, with or without meaning. If these words are written as in a dictionary, then the $50^{th}$ word is :
   
      
   1. OBBHJ
   2. HBBJO
   3. OBBJH
   4. JBBOH
6. Let $\vec{a}$=$2\hat{i}$+$5\hat{j}$$-\hat{k}$, $\vec{b}$=$2\hat{i}$$-2\hat{j}$+$2\hat{k}$ and $\vec{c}$ be three vectors such that $(\vec{c}+\hat{i})$×$(\vec{a}+\vec{b}+\hat{i})$=$\vec{a}×(\vec{c}+\hat{i})$. $\vec{a}•\vec{c}$=-29, then $\vec{c}•(-2\hat{i}+\hat{j}+\hat{k})$ is equal to
   
      
   1. 10
   2. 5
   3. 15
   4. 12
7. Consider three vectors $\vec{a}$, $\vec{b}$, $\vec{c}$. Let $|\vec{a}|$=2, $|\vec{b}|$=3 and $\vec{a}$=$\vec{b}$×$\vec{c}$. If $\alpha \in \left[0, \frac{\pi}{3}\right]$ is the angle between the vectors $\vec{b}$ and $\vec{c}$, then the minimum value of $27|\vec{c}-\vec{a}|^2$ is equal to:
   
      
   1. 110
   2. 105
   3. 124
   4. 121
8. Let $A(–1, 1)$ and $B(2, 3)$ be two points and $P$ be a variable point above the line $AB$ such that the area of $\Delta PAB$ is 10. If the locus of $P$ is $ax + by$ = 15, then $5a + 2b$ is :
   
      
   1. $-\frac{12}{5}$
   2. $-\frac{6}{5}$
   3. 4
   4. 6
9.  Let $(\alpha, \beta, \gamma)$ be the image of the point (8, 5, 7) in the line $\frac{x-1}{2}$=$\frac{y+1}{3}$=$\frac{z-2}{5}$. Then $\alpha$+$\beta$+$\gamma$ is equal to
   
      
   1. $16$
   2. $18$
   3. $14$
   4. $20$
10. If the constant term in the expansion of $\left(\frac{\sqrt{5}{3}}{x}+\frac{2x}{\sqrt{3}{5}}\right)^{12}$, $x \neq 0$, is $\alpha × 2^8 × \sqrt{5}{3}$, then $25\alpha$ is equal to:
    
    
       
    1. 639
    2. 724
    3. 693
    4. 742
11. Let $ƒ, g : R \to R$ be defined as : $ƒ(x) = |x – 1|$ and $g(x)$=$\left\{\begin{array} {cc} e^x &, x \geq 0 \\ x+1 &, x \leq 0 \end{array}\right.$.Then the function $ƒ(g(x))$ is
    
       
    1. neither one-one nor onto.
    2. one-one but not onto.
    3. both one-one and onto.
    4. onto but not one-one.
12. Let the circle $C_1$ : $x^2+ y^2$$– 2(x + y)$ + 1 = 0 and $C_2$be a circle having centre at (–1, 0) and radius 2. If the line of the common chord of $C_1$ and $C_2$intersects the $y-$axis at the point $P$, then the square of the distance of $P$ from the centre of $C_1$ is :
    
       
    1. 2
    2. 1
    3. 6
    4. 4
13. Let the set $S$ = {2, 4, 8, 16, ....., 512} be partitioned into 3 sets $A$, $B$, $C$ with equal number of elements such that $A \cup B \cup C$ = $S$ and $A \cap B$ = $B \cap C$ = $A \cap C$ = $\phi$. The maximum number of such possible partitions of $S$ is equal to :
    
       
    1. 1680
    2. 1520
    3. 1710
    4. 1640
14. The values of $m$, $n$, for which the system of equations
    $x$ + $y$ + $z$ = 4, $2x$ + $5y$ + $5z$ = 17, $x$ + $2y$ + $mz$ = $n$
    has infinitely many solutions, satisfy the equation :.
    
       
    1. $m^2 + n^2– m – n$ = 46
    2. $m^2+ n^2 + m + n$ = 64
    3. $m^2+ n^2 + mn$ = 68
    4. $m^2 + n^2– mn$ = 39
15. The coefficients $a$, $b$, $c$ in the quadratic equation $ax^2$ + $bx$ + $c$ = 0 are from the set {1, 2, 3, 4, 5, 6}. If the probability of this equation having one real root bigger than the other is $p$, then $216p$ equals :
    
       
    1. 57
    2. 38
    3. 19
    4. 76
16. Let $ABCD$ and $AEFG$ be squares of side 4 and 2 units, respectively. The point $E$ is on the line segment $AB$ and the point $F$ is on the diagonal $AC$. Then the radius $r$ of the circle passing through the point $F$ and touching the line segments $BC$ and $CD$ satisfies :
    
       
    1. $r$ = 1
    2. $r^2– 8r$ + 8 = 0
    3. $2r^2– 4r$ + 1 = 0
    4. $2r^2– 8r $+ 7 = 0
17. Let $\beta(m, n)$=$\int \limits_{0}^{1}x^{m-1}(1-x)^{n-1}dx$, $m$, $n$ > 0. If $\int \limits_{0}^{1}(1-x^{10})^{20}dx$=a×$\beta(b, c)$, then $100(a + b + c)$ equals ______.
    
       
    1. 1021
    2. 1120
    3. 2012
    4. 2120
18. Let $\alpha \beta \neq 0$ and $A$=$\begin{equation*}\begin{bmatrix}\beta & \alpha & 3\\ \alpha & \alpha & \beta \\ -\beta & \alpha & 2\alpha \end{bmatrix} \end{equation*}$. If $B$=$\begin{equation*} \begin{bmatrix} 3\alpha & -9 & 3\alpha \\ -\alpha & 7 & -2\alpha \\ -2\alpha & 5 & -2\beta \end{bmatrix}\end{equation*}$ is the matrix of cofactors of the elements of $A$, then $det(AB)$ is equal to :
    
       
    1. 343
    2. 125
    3. 64
    4. 216
19.  If $y(\theta)$=$\frac{2\cos \theta+\cos 2\theta}{\cos 3\theta+4\cos 2\theta +5\cos \theta +2}$, then at $\theta$=$\frac{\pi}{2}$, $y" + y' + y$ is equal to:
    
       
    1. $\frac{3}{2}$
    2. 1
    3. $\frac{1}{2}$
    4. 2
20. For $x \geq 0$, the least value of $K$, for which $4^{1+x}$ + $4^{1–x}$, is $\frac{K}{2}$, $16^x + 16^{–x}$ are three consecutive terms of an $A.P.$ is equal to :
    
       
    1. 10
    2. 4
    3. 8
    4. 16

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. Let the mean and the standard deviation of the probability distribution
   
   

   X	$\alpha$	1	0	-3
   P(X)	$\frac{1}{3}$	$K$	$\frac{1}{6}$	$\frac{1}{4}$

   
   
   
   be $\mu$ and $\sigma$, respectively. If $\sigma – \mu$ = 2, then $\sigma + \mu$ is equal to _____.
2. Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}$+$\frac{2x}{(1+x^2)^2}y$=$xe^{\frac{1}{(1+x^2)}}$; $y(0)$=0. Then the area enclosed by the curve $ƒ(x)$=$y(x)e^{\frac{1}{1+x^2)}}$ and the line $y – x$ = 4 is ____.
3. The number of solutions of $sin^2x $+$ (2 + 2x – x^2)$$\sin x – 3(x – 1)^2$ = 0, where $–\pi \leq x \leq x$, is
4. Let the point $(–1, \alpha, \beta)$ lie on the line of the shortest distance between the lines $\frac{x+2}{-3}$=$\frac{y-2}{4}$=$\frac{z-5}{2}$ and $\frac{x+2}{-1}$=$\frac{y+6}{2}$=$\frac{z-1}{0}$. Then $(\alpha – \beta)^2$is equal to _____.
5. If $1+\frac{\sqrt{3}-\sqrt{2}}{2\sqrt{3}}$+$\frac{5-2\sqrt{6}}{18}$+$\frac{9\sqrt{3}-11\sqrt{2}}{36\sqrt{3}}$+$\frac{49-20\sqrt{6}}{180}$+.... upto $\infty$=$2\left(\sqrt{\frac{b}{a}}+1\right)\log_e\left(\frac{b}{a}\right)$, where $a$ and $b$ are integers with $gcd(a, b)$ = 1, then $11a + 18b$ is equal to ______.
6. Let $a$ > 0 be a root of the equation $2x^2 + x – 2$ = 0. If $\lim \limits_{x \to \frac{1}{a}}\frac{16(1-\cos(2+x-2x^2))}{(1-ax^2)}$=$\alpha+\beta\sqrt{17}$, where $\alpha, \beta \in Z$ then $\alpha+\beta$ is equal to......
7. If $f(t)$=$\int \limits_{0}^{\pi}\frac{2x dx}{1-\cos^2t \sin^2x dx}$, $0 < t < \pi$, then the value of $\int \limits_{0}^{\frac{\pi}{2}}\frac{\pi^2dt}{f(t)}$ equals.......
8. Let the maximum and minimum values of $(\sqrt{8x-x^2-12}-4)^2$+$(x-7)^2$, $x \in R$ be $M$ and $m$ respectively. Then $M^2 – m^2$is equal to _____.
9. Let a line perpendicular to the line $2x – y$ = 10 touch the parabola $y^2 = 4(x – 9)$ at the point $P$. The distance of the point $P$ from the centre of the circle $x^2+ y^2$$ – 14x – 8y$ + 56 = 0 is ______.
10. The number of real solutions of the equation $x |x + 5| + 2|x + 7|$ – 2 = 0 is _____.

05 Apr 2024:Shift-1

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