Download JEE Main 2024 Question Paper (29 Jan - Shift 1) Mathematics

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30 Jan 2024:Shift-1

30 Jan 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. If in a G.P. of 64 terms, the sum of all the terms is 7 times the sum of the odd terms of the G.P, then the common ratio of the G.P. is equal to
   
      
   1. 7
   2. 4
   3. 5
   4. 6
2. In an A.P., the sixth terms $a_6$ = 2. If the $a_1a_4a_5$ is the greatest, then the common difference of the A.P., is equal to
   
      
   1. $\frac{3}{2}$
   2. $\frac{8}{5}$
   3. $\frac{2}{3}$
   4. $\frac{5}{8}$
3. If $f(x)$=$\left\{\begin{array}{cc}2+2x &, -1 \leq x < 0 \\ 1-\frac{x}{3} &, 0 \leq x \leq 3\end{array} \right.$; $g(x)$=$\left\{\begin{array}{cc}-x &, -3 \leq x \leq 0 \\ x &, 0 < x \leq 1\end{array} \right.$, then range of $(fog(x))$ is
   
      
   1. (0, 1]
   2. [0, 3)
   3. [0, 1]
   4. [0, 1)
4. A fair die is thrown until 2 appears. Then the probability, that 2 appears in even number of throws , is
   
      
   1. $\frac{5}{6}$
   2. $\frac{1}{6}$
   3. $\frac{5}{11}$
   4. $\frac{6}{11}$
5. If $z$=$\frac{1}{2}-2i$, is such that $|z+1|$=$\alpha z$+$\beta(1+i)$, $i$=$\sqrt{-1}$ and $\alpha, \beta \in R$, then $\alpha +\beta$ is equal to
   
      
   1. -4
   2. 3
   3. 2
   4. -1
6. $\lim \limits_{x \to \frac{\pi}{2}}\left(\frac{1}{\left(x-\frac{\pi}{2}\right)^2}\int \limits_{x^3}^{\left(\frac{\pi}{2}\right)^3}\cos \left(\frac{1}{t^3}\right)dt\right)$ is equal to
   
      
   1. $\frac{3\pi}{8}$
   2. $\frac{3\pi^2}{4}$
   3. $\frac{3\pi^2}{8}$
   4. $\frac{3\pi}{4}$
7. In a $\Delta ABC$, suppose $y = x$ is the equation of the bisector of the angle $B$ and the equation of the side $AC$ is $2x –y$ =2. If $2AB$ = $BC$ and the point $A$ and $B$ are respectively (4, 6) and $(\alpha, \beta)$, then $\alpha+2\beta$ is equal to
   
      
   1. 42
   2. 39
   3. 48
   4. 45
8. Let $\vec{a}$, $\vec{b}$ and $\vec{c}$be three non-zero vectors such that $\vec{b}$ and $\vec{c}$ are non-collinear. If$\vec{a}+5\vec{b}$ is collinear with $\vec{c}•\vec{b}+6\vec{c}$ is collinear with $\vec{a}$and $\vec{a}$+$\alpha \vec{b}$+$\beta \vec{c}$=$\vec{0}$, then $\alpha +\beta$ is equal to
   
      
   1. 35
   2. 30
   3. -30
   4. -25
9. Let $\left(5, \frac{a}{4}\right)$, be the circumcenter of a triangle with vertices $A(a, -2)$, $B(a, 6)$ and $C\left(\frac{a}{4}, -2\right)$. Let $\alpha$ denote the circumradius, $\beta$ denote the area and $\gamma$ denote the perimeter of the triangle. Then $\alpha$+$\beta$+$\gamma$is
   
      
   1. 60
   2. 53
   3. 62
   4. 30
10. For $x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, if $y(x)$=$\int \frac{cosec x+\sin x}{cosec x \sec x+\tan x \sin^2x}dx$ and $\lim \limits_{x \to \left(\frac{\pi}{2}\right)^{-}}y(x)=0$ then $y\left(\frac{\pi}{4}\right)$ is equal to
    
       
    1. $\tan^{-1}\left(\frac{1}{\sqrt{2}}\right)$
    2. $\frac{1}{2}\tan^{-1}\left(\frac{1}{\sqrt{2}}\right)$
    3. $-\frac{1}{\sqrt{2}}\tan^{-1}\left(\frac{1}{\sqrt{2}}\right)$
    4. $\frac{1}{\sqrt{2}}\tan^{-1}\left(-\frac{1}{2}\right)$
11. If $\alpha$, $-\frac{\pi}{2} < \alpha < \frac{\pi}{2}$ is the solution of $4 \cos \theta 5 \sin \theta$=1, then the value of $tan \alpha$is
    
       
    1. $\frac{10-\sqrt{10}}{6}$
    2. $\frac{10-\sqrt{10}}{12}$
    3. $\frac{\sqrt{10}-10}{12}$
    4. $\frac{\sqrt{10}-10}{6}$
12. A function $y = f(x)$ satisfies $f(x) \sin 2x$+$\sin x$$-(1+\cos^2 x)f'(x)$=0 with condition $f(0)$ = 0 . Then $f\left(\frac{\pi}{2}\right)$ is equal to
    
       
    1. 1
    2. 0
    3. -1
    4. 2
13. Let $O$ be the origin and the position vector of $A$ and $B$ be $2\hat{i}$+$2\hat{j}$+$\hat{k}$ and $2\hat{i}$+$4\hat{j}$+$4\hat{k}$ respectively. If the internal bisector of $\angle{AOB}$ meets the line $AB$at $C$, then the length of $OC$ is
    
       
    1. $\frac{2}{3}\sqrt{31}$
    2. $\frac{2}{3}\sqrt{34}$
    3. $\frac{3}{4}\sqrt{34}$
    4. $\frac{3}{2}\sqrt{31}$
14. Consider the function $f:\left[\frac{1}{2}, 1\right] \to R$ defined by $f(x)$=$4\sqrt{2}x^3-3\sqrt{2}x-1$ Consider the statements
    (I) The curve $y = f(x)$ intersects the $x-$axis exactly at one point
    (II) The curve $y = f(x)$ intersects the $x-$axis at $x$=$\frac{\pi}{12}$
    Then
    
       
    1. Only (II) is correct
    2. Both (I) and (II) are incorrect
    3. Only (I) is correct
    4. Both (I) and (II) are correct
15. Let $A$=$\begin{equation*}\begin{bmatrix} 1 & 0 & 0 \\ 0 & \alpha & \beta \\ 0 & \beta & \alpha \end{bmatrix} \end{equation*}$ and and $|2A|^3$=$2^{21}$ where $\alpha, \beta \in Z$, Then a value of $\alpha$ is
    
       
    1. 3
    2. 5
    3. 17
    4. 9
16. Let $PQR$ be a triangle with $R(-1,4, 2)$. Suppose $M(2, 1, 2)$ is the mid point of $PQ$. The distance of the centroid of $\Delta PQR$ from the point of intersection of the line $\frac{x-2}{0}$=$\frac{y}{2}$=$\frac{z+3}{-1}$ and $\frac{x-1}{1}$=$\frac{y+3}{-3}$=$\frac{z+1}{1}$ is
    
       
    1. 69
    2. 9
    3. $\sqrt{69}$
    4. $\sqrt{99}$
17. Let $R$ be a relation on $Z × Z$ defined by $(a, b)R(c, d)$ if and only if $ad – bc$ is divisible by 5. Then $R$ is
    
       
    1. Reflexive and symmetric but not transitive
    2. Reflexive but neither symmetric not transitive
    3. Reflexive, symmetric and transitive
    4. Reflexive and transitive but not symmetric
18. If the value of the integral
    $\int \limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{x^2\cos x}{1+\pi^x}+\frac{1+\sin^2x}{1+e^{\sin x^{2024}}}\right)dx$=$\frac{\pi}{4}(\pi+a)-2$, then the value of $a$ is
    
       
    1. 3
    2. $-\frac{3}{2}$
    3. 2
    4. $\frac{3}{2}$
19.  Suppose
    $f(x)$=$\frac{(2^x+2^{-x})\tan x \sqrt{\tan^{-1}(x^2-x+1)}}{(7x^2+3x+1)^3}$, Then the value of $f '(0)$ is equal to
    
       
    1. $\pi$
    2. 0
    3. $\sqrt{\pi}$
    4. $\frac{\pi}{2}$
20. Let $A$ be a square matrix such that $AA^T$=$I$. Then $\frac{1}{2}A\left[(A+A^T)^2+(A-A^T)^2\right]$ is equal to
    
       
    1. $A^2+I$
    2. $A^3+I$
    3. $A^2+A^T$
    4. $A^3+A^T$

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. Equation of two diameters of a circle are $2x - 3y$ = 5 and $3x - 4y$ = 7. The line joining the points $\left(-\frac{22}{7}, -4\right)$ and $\left(-\frac{1}{7}, 3\right)$ intersects the circle at only one point $P(\alpha, \beta)$. Then $17\beta-\alpha$is equal to
2. All the letters of the word "GTWENTY" are written in all possible ways with or without meaning and these words are written as in a dictionary. The serial number of the word "GTWENTY" is
3. Let $\alpha, \beta$ be the roots of the equation $x^2 - x + 2 = 0$ with $Im(\alpha) > Im(\beta)$. Then $\alpha^6$+$\alpha^4$+$\beta^4$$-5\alpha^2$ is equal to
4. Let $f(x)$=$2^x-x^2$, $x \in R$. If $m$ and $n$ are respectively the number of points at which the curves $y = f(x)$ and $y = f '(x)$ intersects the $x-$axis, then the value of $m + n$ is
5. If the points of intersection of two distinct conics $x^2+y^2$=$4b$ and $\frac{x^2}{16}$+$\frac{y^2}{b^2}$=1 lie on the curve $y^2$=$3x^2$, then $3\sqrt{3}$ times the area of the rectangle formed by the intersection points is __
6. If the solution curve $y=y(x)$ of the differential equation $(1+y^2)(1+\log_ex)dx$+$x dy$=0, $x >0$ passes through the point (1, 1) and $y(e)$=$\frac{\alpha-\tan\left(\frac{3}{2}\right)}{\beta+\tan \left(\frac{3}{2}\right)}$, then $\alpha+2\beta$ is
7. If the mean and variance of the data 65, 68, 58, 44, 48, 45, 60, $\alpha$, $beta$, 60 where $\alpha > \beta$ are 56 and 66.2 respectively, then $\alpha^2+\beta^2$ is equal to
8. The area (in sq. units) of the part of circle $x^2+y^2$=169 which is below the line $5x - y$=13 is $\frac{\pi \alpha}{2\beta}-\frac{65}{2}$+$\frac{\alpha}{\beta}\sin^{-1}\left(\frac{12}{13}\right)$ where $\alpha$, $\beta$ are coprime numbers. Then $\alpha$+$\beta$ is equal to
9. If $\frac{{}^{11}C_1}{2}$+$\frac{{}^{11}C_2}{3}$+...+$\frac{{}^{11}C_9}{10}$=$\frac{n}{m}$ with $gcd(n, m)$ = 1, then $n +m$ is equal to
10. A line with direction ratios 2, 1, 2 meets the lines $x$ = $y +2$ = $z$ and $x + 2$ = $2y$ = $2z$ respectively at the point $P$ and $Q$. if the length of the perpendicular from the point (1, 2, 12) to the line $PQ$ is $l$, then $l^2$ is

29 Jan 2024:Shift-2

29 Jan 2024:Ans-Keys

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