Download JEE Main 2024 Question Paper (27 Jan - Shift 2) Mathematics

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29 Jan 2024:Shift-1

29 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. Considering only the principal values of inverse trigonometric functions, the number of positive real values of $x$ satisfying $\tan^{-1}(x)$+$\tan^{-1}(2x)$ is :
   
      
   1. more than 2
   2. 1
   3. 2
   4. 0
2. Consider the function $f :(0,2) \to R$ defined by $f (x)$=$\frac{x}{2}$+$\frac{2}{x}$ and the function g(x) defined by
   $g(x)$=$\left\{\begin{array}{cc}min{f(t)} &, 0 < t \leq x \text{ and } 0 < x \leq 1\\ \frac{3}{2}+x &, 1 < x < 2 \end{array} \right.$. Then
   
      
   1. $g$ is continuous but not differentiable at $x$ = 1
   2. $g$ is not continuous for all $x \in (0,2)$
   3. $g$ is neither continuous nor differentiable at $x$ = 1
   4. $g$ is continuous and differentiable for all $x \in (0,2)$
3. Let the image of the point (1, 0, 7) in the line $\frac{x}{1}$=$\frac{y-1}{2}$=$\frac{z-2}{3}$be the point $(\alpha, \beta, \gamma)$. Then which one of the following points lies on the line passing through $(\alpha, \beta, \gamma)$ and making angles with $y-$axis and $z-$axis respectively and an acute angle with $x-$axis ?
   
      
   1. $(1, -2, 1+\sqrt{2})$
   2. $(1, 2, 1-\sqrt{2})$
   3. $(3, 4, 3-2\sqrt{2})$
   4. $(3, -4, 3+2\sqrt{2})$
4. Let $R$ be the interior region between the lines $3x-y$+1=0 and $x+2y-5$=0 containing the origin. The set of all values of $a$, for which the points $(a^2, a + 1)$ lie in $R$, is :
   
      
   1. $(-3, -1) \cup \left(-\frac{1}{3}, 1\right)$
   2. $(-3, 0) \cup \left(\frac{1}{3}, 1\right)$
   3. $(-3, 0) \cup \left(\frac{2}{3}, 1\right)$
   4. $(-3, -1) \cup \left(\frac{1}{3}, 1\right)$
5. The $20^{th}$ term from the end of the progression 20, $19\frac{1}{4}$, $18\frac{1}{2}$, $17\frac{3}{4}$, ... ,$-129\frac{1}{4}$ is:-
   
      
   1. -118
   2. -110
   3. -115
   4. -100
6. Let $f:R-\left\{\frac{-1}{2}\right\}\to R$ and $g:R-\left\{\frac{-5}{2}\right\}\to R$ be defined as $f(x)$=$\frac{2x+3}{2x+1}$ and $g(x)$=$\frac{|x|+1}{2x+5}$. Then the domain of the function $fog$ is :
   
      
   1. $R-\left\{-\frac{5}{2}\right\}$
   2. $R$
   3. $R-\left\{-\frac{7}{4}\right\}$
   4. $R-\left\{-\frac{5}{2}, -\frac{7}{4}\right\}$
7. For $0 < a < 1$, the value of the integral $\int \limits_{0}^{\pi}\frac{dx}{1-2a\cos x+a^2}$ is
   
      
   1. $\frac{\pi^2}{\pi+a^2}$
   2. $\frac{\pi^2}{\pi-a^2}$
   3. $\frac{\pi}{1-a^2}$
   4. $\frac{\pi}{1+a^2}$
8. Let $g(x)$=$3f \left(\frac{x}{3}\right)$+$f(3-x)$ and $f"(x) > 0$ for all $x \in (0,3)$. If $g$ is decreasing in $(0, \alpha)$ and increasing in $(\alpha, 3)$, then $8\alpha$ is
   
      
   1. 24
   2. 0
   3. 18
   4. 20
9. If $\lim \limits_{x \to 0}\frac{3+\alpha \sin x+\beta \cos x+\log_e(1-x)}{3\tan^2x}$=$\frac{1}{3}$, then $2\alpha-\beta$ is equal to:
   
      
   1. 2
   2. 7
   3. 5
   4. 1
10. If $\alpha$, $\beta$ are the roots of the equation, $x^2-x-1$=0 and $S_n$=$2023\alpha^n+2024\beta^n$, then
    
       
    1. $2S_{12}$=$S_{11}$+$S_{10}$
    2. $S_{12}$=$S_{11}$+$S_{10}$
    3. $2S_{11}$=$S_{12}$+$S_{10}$
    4. $S_{11}$=$S_{10}$+$S_{12}$
11. Let $A$ and $B$ be two finite sets with $m$ and $n$ elements respectively. The total number of subsets of the set $A$ is 56 more than the total number of subsets of $B$. Then the distance of the point $P(m, n)$ from the point $Q(–2, –3)$ is
    
       
    1. 10
    2. 6
    3. 4
    4. 8
12. The values of $\alpha$, for which
    $\begin{equation*}\begin{vmatrix} 1 & \frac{3}{2} & \alpha + \frac{3}{2} \\ 1 & \frac{1}{3} & \alpha+\frac{1}{3} \\ 2\alpha+3 & 3\alpha+1 & 0\end{vmatrix}\end{equation*}$=0, lie in the interval
    
       
    1. (-2, 1)
    2. (-3, 0)
    3. $\left(-\frac{3}{2}, \frac{3}{2}\right)$
    4. (0, 3)
13. An urn contains 6 white and 9 black balls. Two successive draws of 4 balls are made without replacement. The probability, that the first draw gives all white balls and the second draw gives all black balls, is :
    
       
    1. $\frac{5}{256}$
    2. $\frac{5}{715}$
    3. $\frac{3}{715}$
    4. $\frac{3}{256}$
14. The integral $\int \frac{(x^8-x^2)dx}{(x^{12}+3x^6+1)\tan^{-1}\left(x^3+\frac{1}{x^3}\right)}$ equal to:
    
       
    1. $\log_e\left(\left|\tan^{-1}\left(x^3+\frac{1}{x^3}\right)\right|\right)^{1/3}+C$
    2. $\log_e\left(\left|\tan^{-1}\left(x^3+\frac{1}{x^3}\right)\right|\right)^{1/2}+C$
    3. $\log_e\left(\left|\tan^{-1}\left(x^3+\frac{1}{x^3}\right)\right|\right)+C$
    4. $\log_e\left(\left|\tan^{-1}\left(x^3+\frac{1}{x^3}\right)\right|\right)^{3}+C$
15. If $2 \tan^2 \theta$$-5 \sec \theta$=1 has exactly 7 solutions in the interval $\left[0, \frac{n\pi}{2}\right]$,for the least value of $n \in N$,then $\sum \limits_{k=1}^{n}\frac{k}{2^{k}}$ is equal to :
    
       
    1. $\frac{1}{2^{15}}(2^{14}-14)$
    2. $\frac{1}{2^{14}}(2^{15}-14)$
    3. $1-\frac{15}{2^{13}}$
    4. $\frac{1}{2^{13}}(2^{14}-15)$
16. The position vectors of the vertices $A$, $B$ and $C$ of a triangle are $2\hat{i}$$-3\hat{j}$+$3\hat{k}$, $2\hat{i}$+$2\hat{j}$+$3\hat{k}$ and $-\hat{i}$+$\hat{j}$+$3\hat{k}$ respectively. Let $l$ denotes the length of the angle bisector $AD$ of $\angle{BAC}$ where $D$ is on the line segment $BC$, then $2l^2$ equals :
    
       
    1. 49
    2. 42
    3. 50
    4. 45
17. If $y = y(x)$ is the solution curve of the differential equation $(x^2-4)dy-$$(y^2-3y)dx$=0, and the slope of the curve is never zero, then the value of $y(10)$ equals :
    
       
    1. $\frac{3}{1+(8)^{1/4}}$
    2. $\frac{3}{1+2\sqrt{2}}$
    3. $\frac{3}{1-2\sqrt{2}}$
    4. $\frac{3}{1-(8)^{1/4}}$
18. Let $e_1$ be the eccentricity of the hyperbola $\frac{x^2}{16}-\frac{y^2}{9}$=1 and $e_2$ be the eccentricity of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}$=1, $a > b$, which passes through the foci of the hyperbola. If $e_1e_2$ = 1, then the length of the chord of the ellipse parallel to the x-axis and passing through (0, 2) is :
    
       
    1. $4\sqrt{5}$
    2. $\frac{8\sqrt{5}}{3}$
    3. $\frac{10\sqrt{5}}{3}$
    4. $3\sqrt{5}$
19.  Let $\alpha$=$\frac{(4!)!}{(4!)^{3!}}$ and $\beta$=$\frac{(5!)!}{(5!)^{4!}}$
    
       
    1. $\alpha \in N and $\beta \notin N$
    2. $\alpha \notin N and $\beta \in N$
    3. $\alpha \in N and $\beta \in N$
    4. $\alpha \notin N and $\beta \notin N$
20. Let the position vectors of the vertices $A$, $B$ and $C$ of a triangle be $2\hat{i}$+$2\hat{j}$+$\hat{k}$, $\hat{i}$+$2\hat{j}$+$2\hat{k}$ and $2\hat{i}$+$\hat{j}$+$2\hat{k}$ respectively. Let $l_1$, $l_2$ and $l_3$ be the lengths of perpendiculars drawn from the ortho center of the triangle on the sides $AB$, $BC$ and $CA$ respectively, then $l_1^2$+$l_2^2$+$l_3^2$ equals :
    
       
    1. $\frac{1}{5}$
    2. $\frac{1}{2}$
    3. $\frac{1}{4}$
    4. $\frac{1}{3}$

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. The mean and standard deviation of 15 observations were found to be 12 and 3 respectively. On rechecking it was found that an observation was read as 10 in place of 12. If $µ$ and $\sigma^2$ denote the mean and variance of the correct observations respectively, then $15(\mu+\mu^2+\sigma^2)$ is equal to ……………………..
2. If the area of the region
   {$(x,y):0 \leq y \leq min{2x,6x - x^2 }$} is $A$, then $12 A$ is equal to………………
3. Let $A$ be a 2 × 2 real matrix and $I$ be the identity matrix of order 2. If the roots of the equation $| A - xI|=0$ be –1 and 3, then the sum of the diagonal elements of the matrix $A^2$ is……………
4. If the sum of squares of all real values of $\alpha$, for which the lines $2x – y$ + 3 = 0, $6x + 3y$ + 1 = 0 and $\alpha x + 2y$ –2 = 0 do not form a triangle is $p$, then the greatest integer less than or equal to $p$ is ………
5. The coefficient of $x^{2012}$ in the expansion of $(1-x)^{2008}(1+x+x^2)^{2007}$ is equal to
6. If the solution curve, of the differential equation $\frac{dy}{dx}$=$\frac{x+y-2}{x-y}$ passing through the point (2, 1) is $\tan^{-1}\left(\frac{y-1}{x-1}\right)$$-\frac{1}{\beta}\log_e\left(\alpha+\left(\frac{y-1}{x-1}\right)^2\right)$=$\log_e|x-1|$, then $5\beta+\alpha$ is equal to:
7. Let $f(x)$=$\int \limits_{0}^{x}g(t)\log_e\left(\frac{1-t}{1+t}\right)dt$, where $g$ is a continuous odd function. If $\int \limits_{-\pi/2}^{\pi/2}\left(f(x)+\frac{x^2 \cos x}{1+e^x}\right)dx$=$\left(\frac{\pi}{\alpha}\right)^2$$-\alpha$, then $\alpha$ is equal to...............
8. Consider a circle $(x - \alpha)^2$+$(y-\beta)^2$=50, where $\alpha, \beta > 0$. If the circle touches the line $y + x$ = 0 at the point $P$, whose distance from the origin is $4 \sqrt{2}$, then $(\alpha+\beta)^2$ is equal to…………….
9. The lines $\frac{x-2}{2}$=$\frac{y}{-2}$=$\frac{z-7}{16}$ and $\frac{x+3}{4}$=$\frac{y+2}{3}$=$\frac{z+2}{1}$ intersect at the point $P$. If the distance of $P$ from the line $\frac{x+1}{2}$=$\frac{y-1}{3}$=$\frac{z-1}{1}$ is $l$, then $14l^2$ is equal to……………..
10. Let the complex numbers $\alpha$ and $\frac{1}{\bar{\alpha}}$ lie on the circles $|z-z_0|^2$=4 and $|z-z_0|^2$=16 respectively, where $z_0=1+i$. Then, the value of $100|\alpha|2$ is……………

27 Jan 2023:Shift-1

27 Jan 2023:Ans-Keys

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