Download JEE Main 2024 Question Paper (30 Jan - Shift 1) Mathematics

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31 Jan 2023:Shift-1

31 Jan 2023: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. A line passing through the point A(9, 0) makes an angle of 30º with the positive direction of x-axis. If this line is rotated about A through an angle of 15º in the clockwise direction, then its equation in the new position is
   
      
   1. $\frac{y}{\sqrt{3}-2}+x$=9
   2. $\frac{x}{\sqrt{3}-2}+y$=9
   3. $\frac{x}{\sqrt{3}+2}+y$=9
   4. $\frac{y}{\sqrt{3}+2}+x$=9
2. Let $S_a$ denote the sum of first $n$ terms an arithmetic progression. If $S_{20}$ = 790 and $S_{10}$ = 145, then $S_{15}$ –$S_5$ is :
   
      
   1. 395
   2. 390
   3. 405
   4. 410
3. If $z$ = $x + iy$, $xy \neq 0$, satisfies the equation $z^2+i\bar{z}$, then $|z^2|$ is equal to :
   
      
   1. 9
   2. 1
   3. 4
   4. $\frac{1}{4}$
4. Let $\vec{a}$=$a_1\hat{i}$+$a_2\hat{j}$+$a_3\hat{k}$ and $\vec{b}$=$b_1\hat{i}$+$b_2\hat{j}$+$b_3\hat{k}$ be two vectors such that $|\vec{a}|$=1; $\vec{a}•\vec{b}$=2 and $|\vec{b}|$=4. If $\vec{c}$=$2(\vec{a}×\vec{b})-3\vec{b}$, then the angle between $\vec{b}$ and $\vec{c}$ is equal to:
   
      
   1. $\cos^{-1}\left(\frac{2}{\sqrt{3}}\right)$
   2. $\cos^{-1}\left(-\frac{1}{\sqrt{3}}\right)$
   3. $\cos^{-1}\left(-\frac{\sqrt{3}}{2}\right)$
   4. $\cos^{-1}\left(\frac{2}{3}\right)$
5. The maximum area of a triangle whose one vertex is at (0, 0) and the other two vertices lie on the curve $y = -2x^2$ + 54 at points $(x, y)$ and $(-x, y)$where $y > 0$ is :
   
      
   1. 88
   2. 122
   3. 92
   4. 108
6. The value of $\lim \limits_{n \to \infty}\sum \limits_{k=1}^{n}\frac{n^3}{(n^2+k^2)(n^2+3k^2)}$ is:
   
      
   1. $\frac{(2+\sqrt{3})\pi}{24}$
   2. $\frac{13\pi}{8(4\sqrt{3}+3)}$
   3. $\frac{13(2\sqrt{3}-3)\pi}{24}$
   4. $\frac{\pi}{8(2\sqrt{3}+3)}$
7. Let $g : R \to R$ be a non constant twice differentiable such that $g'\left(\frac{1}{2}\right)$=$g'\left(\frac{3}{2}\right)$.  If a real valued function $f$ is defined as $f(x)$=$\frac{1}{2}[g(x)+g(2-x)]$, then
   
      
   1. $f”(x)$ = 0 for atleast two $x$ in (0, 2)
   2. $f”(x)$ = 0 for exactly one $x$ in (0, 1)
   3. $f”(x)$ = 0 for no $x$ in (0, 1)
   4. $f'\left(\frac{3}{2}\right)$+$f'\left(\frac{1}{2}\right)$=1
8. The area (in square units) of the region bounded by the parabola $y^2$ = $4(x – 2)$ and the line $y = 2x - 8$
   
      
   1. 8
   2. 9
   3. 6
   4. 7
9. Let $y = y (x)$ be the solution of the differential equation $\sec x dy$ + {$2(1 – x) \tan x$ + $x(2 – x)$} $dx$ = 0 such that $y(0)$ = 2.Then $y(2)$ is equal to :
   
      
   1. 2
   2. $2{1 – \sin (2)}$
   3. $2{\sin (2) + 1}$
   4. 1
10. Let $(\alpha, \beta, \gamma)$ be the foot of perpendicular from the point (1, 2, 3) on the line $\frac{x+3}{5}$=$\frac{y-1}{2}$=$\frac{z+4}{3}$. Then $19(\alpha+\beta+\gamma)$ is equal to :
    
       
    1. 102
    2. 101
    3. 99
    4. 100
11. Two integers $x$ and $y$ are chosen with replacement from the set {0, 1, 2, 3, ….., 10}. Then the probability that $| x - y| > 5$is :
    
       
    1. $\frac{30}{121}$
    2. $\frac{62}{121}$
    3. $\frac{60}{121}$
    4. $\frac{31}{121}$
12. If the domain of the function $f(x)$=$\cos^{-1}\left(\frac{2-|x|}{4}\right)$+$(\log_e(3-x))^{-1}$ is $[-\alpha, \beta)-{\gamma}$, then $\alpha$+$\beta$+$\gamma$ is equal to:
    
       
    1. 12
    2. 9
    3. 11
    4. 8
13. Consider the system of linear equation $x + y + z$ = $4\mu$, $x + 2y + 2 \lambda z$= $10\mu$, $x + 3y + 4\lambda^2z$ = $\mu^2$ +15, where $\lambda, \mu \in R$ . Which one of the following statements is NOT correct ?
    
       
    1. The system has unique solution if $\lambda \neq \frac{1}{2}$ and $\mu \neq 1, 15$
    2. The system is inconsistent if $\lambda=\frac{1}{2}$ and $\mu \neq 1$
    3. The system has infinite number of solutions if $\lambda=\frac{1}{2}$ and $\mu=15$
    4. The system is consistent if $\lambda \neq \frac{1}{2}$
14. If the circles $(x+1)^2$+$(y+2)^2$=$r^2$ and $x^2+y^2$$-4x-4y$+4=0 intersect at exactly two distinct points, then
    
       
    1. 5 < $r$ < 9
    2. 0 < $r$ < 7
    3. 3 < $r$ < 7
    4. $\frac{1}{2} < $r$ < 7
15. If the length of the minor axis of ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is :
    
       
    1. $\frac{\sqrt{5}}{3}$
    2. $\frac{\sqrt{3}}{2}$
    3. $\frac{1}{\sqrt{3}}$
    4. $\frac{2}{\sqrt{5}}$
16. Let $M$ denote the median of the following frequency distribution.
    

    Class	0-4	4-8	8-12	12-16	16-20
    Frequency	3	9	10	8	6

    
    
    Then 20 $M$ is equal to :
    
       
    1. 416
    2. 104
    3. 52
    4. 208
17. If $f(x)$=$\begin{equation*}\begin{vmatrix} 2\cos^4x & 2\sin^4x & 3+\sin^2 2x \\ 3+2\cos^4x & 2\sin^4x & \sin^2 2x \\ 2\cos^4x & 3+2\sin^4x & \sin^2 2x \end{vmatrix}\end{equation*}$ then $\frac{1}{5}f'(0)$ is equal to ................
    
       
    1. 0
    2. 1
    3. 2
    4. 6
18. Let $A (2, 3, 5)$ and $C(-3, 4, -2)$ be opposite vertices of a parallelogram $ABCD$ if the diagonal $\vec{BD}$ =$\hat{i}$+$2\hat{j}$+$3\hat{k}$ then the area of the parallelogram is equal to
    
       
    1. $\frac{1}{2}\sqrt{410}$
    2. $\frac{1}{2}\sqrt{474}$
    3. $\frac{1}{2}\sqrt{586}$
    4. $\frac{1}{2}\sqrt{306}$
19.  If $2\sin^3x$ + $\sin 2x cos x$ + $4sinx$ – 4 = 0 has exactly 3 solutions in the interval $\left[0, \frac{n\pi}{2}\right]$, $n \in N$, then the roots of the equation $x^2$+$nx$+$(n-3)$ belong to :
    
       
    1. $(0, \infty)$
    2. $(-\infty, 0)$
    3. $\left(-\frac{\sqrt{17}}{2}, \frac{\sqrt{17}}{2}\right)$
    4. Z$
20. Let $f:\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \to R$ be a differentiable function such that $f(0)$=$\frac{1}{2}$. If the $\lim \limits_{x \to 0} \frac{x \int \limits_{0}^{x}f(t)dt}{e^{x^2}-1}$=$\alpha$, then $8\alpha^2$ is equal to:
    
       
    1. 16
    2. 2
    3. 1
    4. 4

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. A group of 40 students appeared in an examination of 3 subjects – Mathematics, Physics & Chemistry. It was found that all students passed in at least one of the subjects, 20 students passed in Mathematics, 25 students passed in Physics, 16 students passed in Chemistry, at most 11 students passed in both Mathematics and Physics, at most 15 students passed in both Physics and Chemistry, at most 15 students passed in both Mathematics and Chemistry. The maximum number of students passed in all the three subjects is ________.
2. If $d_1$ is the shortest distance between the lines $x + 1$ = $2y$ = $-12z$, $x = y$ + 2 = $6z$ – 6 and $d_2$ is the shortest distance between the lines $\frac{x-1}{2}$=$\frac{y+8}{-7}$=$\frac{z-4}{5}$, $\frac{x-1}{2}$=$\frac{y-2}{1}$=$\frac{z-6}{-3}$, then the value of $\frac{32\sqrt{3}d_1}{d_2}$ is:
3. Let the latus rectum of the hyperbola $\frac{x^2}{9}$$-\frac{y^2}{b^2}$=1 subtend an angle of $\frac{\pi}{3}$ at the centre of the hyperbola. If $b^2$is equal to $\frac{l}{m}(l+\sqrt{n})$, where $l$and $m$ are co-prime numbers, then $l^2$ + $m^2$ + $n^2$ is equal to __________
4. Let $A$ = {1, 2, 3,….7} and let $P(1)$ denote the power set of $A$. If the number of functions $f : A \to P (A)$ such that $a \in f(a)$, $\forall a \in A$ is $m^n$, $m$ and $n \in N$ and $m$ is least, then $m + n$ is equal to _______.
5. The value 9$\int \limits_{0}^{9}\left[\sqrt{\frac{10x}{x+1}}\right]dx$, where $[t]$ denotes the greatest integer less than or equal to $t$, is _____.
6. Number of integral terms in the expansion of $\left\{7^{\left(\frac{1}{2}\right)}+11^{\left(\frac{1}{6}\right)}\right\}^{824}$ is equal to..........
7. Let $y = y(x)$ be the solution of the differential equation $(1 – x^2) dy$=$\left[xy+(x^3+2)\sqrt{3(1-x^2)}\right]dx$, $-1 < x < 1$, $y(0)$=0. If $y\left(\frac{1}{2}\right)$=$\frac{m}{n}$, $m$ and $n$ are co-prime numbers, then m + n is equal to ________.
8. Let $\alpha, \beta \in N$ be roots of equation $x^2 – 70x$ + $\lambda$= 0, where $\frac{\lambda}{2}$, $\frac{\lambda}{3} \notin N$. If $\lambda$ assumes the minimum possible value, then $\frac{(\sqrt{\alpha-1}+\sqrt{\beta-1})(\lambda+35)}{|\alpha-\beta|}$ is equal to:
9. If the function $f(x)$=$\left\{ \begin{array}{cc}\frac{1}{|x|} &, |x|\geq 2\\ax^2+2b &, |x| < 2 \end{array} \right.$ is differentiable on $R$, then $48 (a + b)$ is equal to _____.
10. Let $\alpha$=$1^2$+$4^2$+$8^2$+$13^2$+$19^2$+$26^2$+...... upto 10 terms and $\beta$=$\sum \limits_{n=1}^{10}n^4$. If $4\alpha-\beta$=$55k$+40, then $k$ is equal to...............

30 Jan 2023:Shift-2

30 Jan 2023:Ans-Keys

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