Download JEE Main 2024 Question Paper (31 Jan - Shift 2) Mathematics

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01 Feb 2024:Shift-1

01 Feb 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. The number of ways in which 21 identical apples can be distributed among three children such that each child gets at least 2 apples, is
   
      
   1. 406
   2. 130
   3. 142
   4. 136
2. Let $A (a, b)$, $B(3, 4)$ and $C(–6, –8)$ respectively denote the centroid, circumcentre and orthocentre of a triangle. Then, the distance of the point $P(2a + 3, 7b + 5)$ from the line $2x + 3y – 4$ = 0 measured parallel to the line $x – 2y – 1$ = 0 is
   
      
   1. $\frac{15\sqrt{5}}{7}$
   2. $\frac{17\sqrt{5}}{6}$
   3. $\frac{17\sqrt{5}}{7}$
   4. $\frac{\sqrt{5}}{17}$
3. Let $z_1$ and $z_2$ be two complex number such that $z_1$+ $z_2$ = 5 and $z_1^3+z_2^3$=$20+15i$. Then $|z_1^4+z_2^4|$ equals
   
      
   1. $30\sqrt{3}$
   2. $75$
   3. $15\sqrt{15}$
   4. $25\sqrt{3}$
4. Let a variable line passing through the centre of the circle $x^2 + y^2– 16x – 4y$ = 0, meet the positive co-ordinate axes at the point $A$ and $B$. Then the minimum value of $OA$ + $OB$, where $O$ is the origin, is equal to
   
      
   1. 12
   2. 18
   3. 20
   4. 24
5. Let $f, g:(0, \infty) \to R$ be two functions defined by $f(x)$=$\int \limits_{-x}^{x}(|t|-t^2)e^{-t^2}dt$ and $g(x)$=$\int \limits_{0}^{x^2} t^{1/2}e^{-t}dt$. Then the value of $(f(\sqrt{\log_e9})+g(\sqrt{\log_e9})$ is equal to
   
      
   1. 6
   2. 9
   3. 8
   4. 10
6. Let $(\alpha, \beta, \gamma)$ be mirror image of the point (2, 3, 5) in the line $\frac{x-1}{2}$=$\frac{y-2}{3}$=$\frac{z-3}{4}$. Then $2\alpha$+$3\beta$+$4\gamma$ is equal to
   
      
   1. 32
   2. 33
   3. 31
   4. 34
7. Let $P$ be a parabola with vertex (2, 3) and directrix $2x + y$ = 6. Let an ellipse $E:\frac{x^2}{a^2}+\frac{y^2}{b^2}$=1, $a > b$ of eccentricity $\frac{1}{\sqrt{2}}$ pass through the focus of the parabola $P$. Then the square of the length of the latus rectum of $E$, is
   
      
   1. $\frac{385}{8}$
   2. $\frac{347}{8}$
   3. $\frac{512}{25}$
   4. $\frac{656}{25}$
8. The temperature $T(t)$ of a body at time $t$ = 0 is 160°$F$ and it decreases continuously as per the differential equation $\frac{dT}{dt}$=$-K(T-80)$, where $K$ is positive constant. If $T(15)$ = 120°$F$, then $T(45)$ is equal to
   
      
   1. $85°F$
   2. $95°F$
   3. $90°F$
   4. $80°F$
9. Let $2^{nd}$, $8^{th}$ and $44^{th}$, terms of a non-constant $A.P.$ be respectively the $1^{st}$, $2^{nd}$ and $3^{rd}$ terms of $G.P.$ If the first term of $A.P.$ is 1 then the sum of first 20 terms is equal to
   
      
   1. 980
   2. 960
   3. 990
   4. 970
10. Let $f : R \to (0, \infty)$be strictly increasing function such that $\lim \limits_{x \to \infty}\frac{f(7x)}{f(x)}$=1. Then, the value of $\lim \limits_{x \to \infty}\left[\frac{f(5x)}{f(x)}-1\right]$ is equal to
    
       
    1. 4
    2. 0
    3. 7/5
    4. 1
11. The area of the region enclosed by the parabola $y = 4x – x^2$and $3y = (x – 4)^2$ is equal to
    
       
    1. $\frac{32}{9}$
    2. 4
    3. 6
    4. $\frac{14}{3}$
12. Let the mean and the variance of 6 observation $a$, $b$, 68, 44, 48, 60 be 55 and 194, respectively if $a > b$, then $a + 3b$ is
    
       
    1. 200
    2. 190
    3. 180
    4. 210
13. If the function $f :(-\infty , 1] \to (a,b]$ defined by $f(x)$=$e^{x^3-3x+1}$ is one-one and onto, then the distance of the point $P(2b + 4, a + 2)$ from the line $x + e^{–3}y$ = 4 is :
    
       
    1. $2\sqrt{1+e^6}$
    2. $4\sqrt{1+e^6}$
    3. $3\sqrt{1+e^6}$
    4. $\sqrt{1+e^6}$
14. Consider the function $f :(0, \infty) \to R$ defined by $e^{-|\log_ex|}$. If $m$ and $n$ be respectively the number of points at which f is not continuous and $f$is not differentiable, then $m + n$ is
    
       
    1. 0
    2. 3
    3. 1
    4. 2
15. The number of solutions, of the equation $e^{\sin x}-2e^{-\sin x}$=2 is
    
       
    1. 2
    2. more than 2
    3. 1
    4. 0
16. If $a$=$\sin^{-1}(\sin(5))$1and $b= cos^{-1}\cos(5))$, then $a^2$+$b^2$ is equal to
    
       
    1. $4\pi^2$+25
    2. $8\pi^2-40\pi$+50
    3. $4\pi^2-20\pi$+50
    4. 25
17. If for some $m$, $n$; ${}^6C_m$+$2({}^6C_{m+1})$+${}^6C_{m+2}$ > ${}^8C_3$ and ${}^{n-1}P_3:{}^nP_4$=1:8, then ${}^nP_{m+1}$+${}^{n+1}C_m$ is equal to
    
       
    1. 380
    2. 376
    3. 384
    4. 372
18. A coin is based so that a head is twice as likely to occur as a tail. If the coin is tossed 3 times, then the probability of getting two tails and one head is
    
       
    1. $\frac{2}{9}$
    2. $\frac{1}{9}$
    3. $\frac{2}{27}$
    4. $\frac{1}{27}$
19. Let $A$ be real matrix such that $A\begin{equation*}\begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}=2\begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}\end{equation*}$, $A\begin{equation*}\begin{pmatrix} -1 \\ 0 \\ 1 \end{pmatrix}=4\begin{pmatrix} -1 \\ 0 \\ 1 \end{pmatrix}\end{equation*}$, $A\begin{equation*}\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}=2\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}\end{equation*}$. Then, the system $\begin{equation*}(A-3I)\begin{pmatrix} x \\ y \\ z \end{pmatrix}=\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} \end{equation*}$ has
    
       
    1. unique solution
    2. exactly two solutions
    3. no solution
    4. infinitely many solutions
20. The shortest distance between lines $L_1$ and $L_2$, where $L_1$:$\frac{x-1}{2}$=$\frac{y+1}{-3}$=$\frac{z+4}{2}$ and $L_2$ is the line passing through the points $A(-4, 4, 3)$, $B(-1, 6, 3)$ and perpendicular to the line $\frac{x-3}{-2}$=$\frac{y}{3}$=$\frac{z-1}{1}$, is
    
       
    1. $\frac{121}{\sqrt{221}}$
    2. $\frac{24}{\sqrt{117}}$
    3. $\frac{141}{\sqrt{221}}$
    4. $\frac{42}{\sqrt{117}}$

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. $\left|\frac{120}{\pi^3}\int \limits_{0}^{\pi}\frac{x^2 \sin x \cos x}{\sin^4 x+\cos^4 x}dx\right|$ is equal to ................
2. Let $a$, $b$, $c$ be the length of three sides of a triangle satisfying the condition $(a^2 + b^2)x^2$– $2b(a + c)$. $x + (b^2 + c^2)$ = 0. If the set of all possible values of $x$ is the interval $(\alpha, \beta)$, then $12(\alpha^2+\beta^2)$ is equal to ______.
3. Let $A(–2, –1)$, $B(1, 0)$, $C(\alpha, \beta)$ and $D(\gamma, \delta)$, be the vertices of a parallelogram $ABCD$. If the point $C$ lies on $2x – y$ = 5 and the point $D$ lies on $3x – 2y$ = 6, then the value of  $|\alpha +\beta +\gamma+\delta|$ is equal to ________.
4. Let the coefficient of $x^r$ in the expansion of $(x+3)^{n-1}$+$(x+3)^{n-2}(x+2)$+$(x+3)^{n-3}(x+2)^2$+....+$(x+2)^{n-1}$ be $\alpha$. If $\sum \limits_{r=0}^n \alpha_r$=$\beta^n - \gamma^n$, $\beta$, $\gamma \in N$, then the value of $\beta^2+\gamma^2$ equals __________.
5. Let $A$ be a 3×3 matrix and $det (A)$ = 2. If $\operatorname{det}(\underbrace{\operatorname{adj}(\operatorname{adj}(\ldots \ldots \cdot(\operatorname{adj} A))))}_{2024-\text { times }}$ Then the remainder when n is divided by 9 is equal to ___________.
6. Let $\vec{a}$=$3\hat{i}$+$2\hat{j}$+$\hat{k}$, $\vec{b}$=$2\hat{i}$$-\hat{j}$+$3\hat{k}$ and $\vec{c}$ be a vector such that $(\vec{a}+\vec{b})×\vec{c}$=$2(\vec{a}×\vec{b})$+$24\hat{j}$$-6\hat{k}$ and $(\vec{a}-\vec{b}+\hat{i})•\vec{c}$=$-3$. Then $|\vec{c}|^2$ is equal to..............
7. If $\lim \limits_{x \to 0} \frac{ax^2e^x-b\log_e(1+x)+cxe^{-x}}{x^2 \sin x}$=1, then $16(a^2 + b^2 + c^2)$ is equal to ______.
8. A line passes through $A(4, –6, –2)$ and $B(16, –2,4)$. The point $P(a, b, c)$ where $a$, $b$, $c$ are non-negative integers, on the line $AB$ lies at a distance of 21 units, from the point $A$. The distance between the points $P(a, b, c)$ and $Q(4, –12, 3)$ is equal to ____.
9. Let $y = y(x)$ be the solution of the differential equation
   $\sec^2 x dx$+$(e^{2y} \tan^2 x +\tan x)dy$=0, $0 < x < \frac{\pi}{2}$, $y\left(\frac{\pi}{2}\right)$=0. If $y\left(\frac{\pi}{6}\right)$=$\alpha$. Then $e^{8\alpha}$ is equal to ..........
10. Let $A$ = {1, 2, 3, ………100}. Let $R$ be a relation on $A$ defined by $(x, y) \in R$ if and only if $2x = 3y$. Let $R_1$ be a symmetric relation on $A$ such that $R \subset R_1$and the number of elements in $R_1$ is $n$. Then, the minimum value of $n$ is ___________.

31 Jan 2024:Shift-1

31 Jan 2024:Ans-Keys

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