Download JEE Main 2024 Question Paper (09 Apr - Shift 2) Mathematics

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04 Apr 2023:Shift-1

04 Apr 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. $\lim \limits_{x \to 0}\frac{e-(1+2x)^{\frac{1}{2x}}}{x}$ is equal to:
   
      
   1. $e$
   2. $\frac{-2}{e}$
   3. 0
   4. $e-e^2$
2. Consider the line L passing through the points (1, 2, 3) and (2, 3, 5). The distance of the point $\left(\frac{11}{3}, \frac{11}{3}, \frac{19}{3}\right)$ from the line $L$ along the line $\frac{3x-11}{2}$=$\frac{3y-11}{1}$=$\frac{3z-19}{2}$ is equal to.........
   
      
   1. 3
   2. 5
   3. 4
   4. 6
3. Let $\int \limits_{0}^{x}\sqrt{1-(y'(t))^2}dt$=$\int \limits_{0}^{x}y(t)dt$, $0 \leq x \leq 3$, $y \geq 0$, $y(0)$=0. Then at $x$ = 2, $y" + y$ + 1 is equal to :
   
      
   1. 1
   2. 2
   3. $\sqrt{2}$
   4. 1/2
4. Let $z$ be a complex number such that the real part of $\frac{z-2i}{z+2i}$ is zero. Then, the maximum value of $|z –(6+8i)|$ is equal to :
   
      
   1. 12
   2. $\infty$
   3. 10
   4. 8
5. The area (in square units) of the region enclosed by the ellipse $x^2$ + $3y^2$= 18 in the first quadrant below the line $y = x$ is :
   
      
   1. $\sqrt{3}\pi+\frac{3}{4}$
   2. $\sqrt{3}\pi$
   3. $\sqrt{3}\pi-\frac{3}{4}$
   4. $\sqrt{3}\pi+1$
6. Let the foci of a hyperbola $H$ coincide with the foci of the ellipse $E$=$\frac{(x-1)^2}{100}$+$\frac{(y-1)^2}{75}$=1 and the eccentricity of the hyperbola $H$ be the reciprocal of the eccentricity of the ellipse $E$. If the length of the transverse axis of $H$ is $\alpha$ and the length of its conjugate axis is $\beta$, then $3\alpha^2 + 2\beta^2$ is equal to :
   
      
   1. 242
   2. 225
   3. 237
   4. 205
7. Two vertices of a triangle $ABC$ are $A(3, –1)$ and $B (–2, 3)$, and its orthocentre is $P(1, 1)$. If the coordinates of the point $C$ are $(\alpha, \beta)$ and the centre of the circle circumscribing the triangle $PAB$ is $(h, k)$, then the value of $(\alpha + \beta)$ + $2 (h + k)$ equals :
   
      
   1. 51
   2. 81
   3. 5
   4. 15
8. If the variance of the frequency distribution is 160, then the value of $c \in N$ is
   

   x	c	2c	3c	4c	5c	6c
   f	2	1	1	1	1	1

   
   
   
   
      
   1. 5
   2. 8
   3. 7
   4. 6
9. Let the range of the function $f (x)$ =$\frac{1}{2+\sin 3x+ \cos 3x}$, $x \in IR$ be $[a, b]$. If $\alpha$ and $\beta$ are respectively the A.M. and the G.M. of $a$ and $b$, then $\frac{\alpha}{\beta}$ is equal to :
   
      
   1. $\sqrt{2}$
   2. $2$
   3. $\sqrt{\pi}$
   4. $\pi$
10. Between the following two statements :
    Statement-I : Let $\vec{a}$=$\hat{i}$+$2\hat{j}$$-3\hat{k}$ and $\vec{b}$=$2\hat{i}$+$\hat{j}$$-\hat{k}$. Then the vector $\vec{r}$ satisfying $\vec{a}$×$\vec{r}$=$\vec{a}$×$\vec{b}$and $\vec{a}•\vec{r}$=0 is of magnitude $\sqrt{10}$.
    Statement-II : In a triangle $ABC$, $\cos 2A$ + $\cos 2B$ + $\cos 2C \geq -\frac{3}{2}$.
    
       
    1. Both Statement-I and Statement-II are incorrect
    2. Statement-I is incorrect but Statement-II is correct
    3. Both Statement-I and Statement-II are correct
    4. Statement-I is correct but Statement-II is incorrect
11. $\lim \limits_{x \to \frac{\pi}{2}}\left(\frac{\int \limits_{x^3}^{(\pi/2)^3}(\sin (2t^{1/3})+\cos (t^{1/3}))dt}{\left(x-\frac{\pi}{2}\right)^2}\right)$ is equal to :
    
       
    1. $\frac{9\pi^2}{8}$
    2. $\frac{11\pi^2}{10}$
    3. $\frac{3\pi^2}{2}$
    4. $\frac{5\pi^2}{9}$
12. The sum of the coefficient of $x^{2/3}$ and $x^{–2/5}$ in the binomial expansion of $\left(x^{2/3}+\frac{1}{2}x^{-2/5}\right)^9$ is :
    
       
    1. 21/4
    2. 69/16
    3. 63/16
    4. 19/4
13. Let $B$=$\begin{equation*}\begin{bmatrix} 1 & 3\\ 1 & 5 \end{bmatrix} \end{equation*}$ and $A$ be a 2 × 2 matrix such that $AB^{–1}$ = $A^{–1}$. If $BCB^{–1}$ = $A$ and $C^4$ + $\alpha C^2$ + $\beta I$ = O, then $2\beta – \alpha$ is equal to :
    
       
    1. 16
    2. 2
    3. 8
    4. 10
14. If $log_e y$ = $3 \sin^{–1}x$, then $(1 – x)^2y" –xy'$ at $x$ = $\frac{1}{2}$ is equal to :
    
       
    1. $9e^{\pi/6}$
    2. $3e^{\pi/6}$
    3. $3e^{\pi/2}$
    4. $9e^{\pi/2}$
15. The integral $\int \limits_{1/4}^{3/4}\cos\left(2\cot^{-1}\sqrt{\frac{1-x}{1+x}}\right)dx$ is equal to :
    
       
    1. $-1/2$
    2. $1/4$
    3. $1/2$
    4. $-1/4$
16. Let $a$, $ar$, $ar^2$, ........be an infinite $G.P.$ If $\sum \limits_{n=0}^{\infty}ar^n$=57  and $\sum \limits_{n=0}^{\infty}a^3r^{3n}$=9747, then $a+18r$ is equal to :
    
       
    1. 27
    2. 46
    3. 38
    4. 31
17. If an unbiased dice is rolled thrice, then the probability of getting a greater number in the $i^{th}$ roll than the number obtained in the $(i–1)^{th}$ roll, $i$ = 2, 3, is equal to :
    
       
    1. 3/54
    2. 2/54
    3. 5/54
    4. 1/54
18. The value of the integral $\int \limits_{-1}^{2}\log_e(x+\sqrt{x^2+1})dx$ is
    
       
    1. $\sqrt{5}-\sqrt{2}$+$\log_e\left(\frac{9+4\sqrt{5}}{1+\sqrt{2}}\right)$
    2. $\sqrt{2}-\sqrt{5}$+$\log_e\left(\frac{9+4\sqrt{5}}{1+\sqrt{2}}\right)$
    3. $\sqrt{5}-\sqrt{2}$+$\log_e\left(\frac{7+4\sqrt{5}}{1+\sqrt{2}}\right)$
    4. $\sqrt{2}-\sqrt{5}$+$\log_e\left(\frac{7+4\sqrt{5}}{1+\sqrt{2}}\right)$
19.  Let $\alpha$, $\beta$:$\alpha > \beta$,  be the roots of the equation $x^2$$-\sqrt{2}x-\sqrt{3}$=0. Let $P_n$=$\alpha^n-\beta^n$, $n \in N$. Then $(11\sqrt{3}-10\sqrt{2})P_{10}$+$(11\sqrt{2}+10)P_{11}$$-11P_{12}$ is equal to:
    
       
    1. $10\sqrt{2}P_9$
    2. $10\sqrt{3}P_9$
    3. $11\sqrt{2}P_9$
    4. $11\sqrt{3}P_9$
20. Let $\vec{a}$=$2\hat{i}$+$\alpha \hat{j}$+$\hat{k}$, $\vec{b}$=$-\hat{i}+\hat{k}$, $\vec{c}$=$\beta \hat{j}-\hat{k}$, where $\alpha$ and $\beta$ are integers and $\alpha \beta$=-6. Let the values of the ordered pair $(\alpha, \beta)$ for which the area of the parallelogram of diagonals $\vec{a}$+$\vec{b}$ and $\vec{b}$+$\vec{c}$ is $\frac{\sqrt{21}}{2}$, be $(\alpha_1, \beta_1)$ and $(\alpha_2, \beta_2)$. Then $\alpha_1^2$+$\beta^2$$-\alpha_2\beta_2$ is equal to
    
       
    1. 17
    2. 24
    3. 21
    4. 19

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. Consider the circle $C$ : $x^2 + y^2$ = 4 and the parabola $P : y^2$ = $8x$. If the set of all values of $\alpha$, for which three chords of the circle $C$ on three distinct lines passing through the point $(\alpha, 0)$ are bisected by the parabola $P$ is the interval $(p, q)$, then $(2q – p)^2$ is equal to _____.
2. Let the set of all values of $p$, for which $f(x)$ = $(p^2–6p + 8)$ $(sin^22x – cos^22x)$ + $2(2 – p)x$ + 7 does not have any critical point, be the interval $(a, b)$. Then $16ab$ is equal to _____ .
3. For a differentiable function $f : IR \to IR$, suppose $f '(x)$ = $3f(x)$ + $\alpha$, where $\alpha \in IR$, $f(0)$ = 1 and $\lim \limits_{x \to nifty}f (x)$ = 7. Then $9f (–log_e3)$ is equal to______.
4. The number of integers, between 100 and 1000having the sum of their digits equals to 14, is ______.
5. Let $A$ = {$(x, y) : 2x + 3y$ = 23, $x, y \in N$} and $B$ = {$x : (x, y) \in A$}. Then the number of one-one functions from $A$ to $B$ is equal to _____.
6. Let $A$, $B$ and $C$ be three points on the parabola $y^2$= $6x$ and let the line segment $AB$ meet the line $L$ through $C$ parallel to the $x-$axis at the point $D$. Let $M$ and $N$ respectively be the feet of the perpendiculars from $A$ and $B$ on $L$. Then $\left(\frac{AM•BN}{CD}\right)^2$ is equal to .........
7. The square of the distance of the image of the point (6, 1, 5) in the line $\frac{x-1}{3}$=$\frac{y}{2}$=$\frac{z-2}{4}$, from the origin is ______.
8. $\left(\frac{1}{\alpha+1}+\frac{1}{\alpha+2}+...+\frac{1}{\alpha+1012}\right)$$-\left(\frac{1}{2•1}+\frac{1}{4•3}+\frac{1}{6•5}+...+\frac{1}{2024•2023}\right)$=$\frac{1}{2024}$, then $\alpha$ is equal to-
9. Let the inverse trigonometric functions take principal values. The number of real solutions of the equation $2 \sin^{–1}x$ + $3 \cos^{–1}x$ = $\frac{2}{5}$, is ______.
10. Consider the matrices: $A$=$\begin{equation*}\begin{bmatrix}2 & -5\\3 & m\end{bmatrix}\end{equation*}$, $B$=$\begin{equation*}\begin{bmatrix} 20 \\ m \end{bmatrix}\end{equation*}$ and $X$=$\begin{equation*}\begin{bmatrix} x \\ y \end{bmatrix} \end{equation*}$. Let the set of all $m$, for which the system of equations $AX = B$ has a negative solution (i.e., $x$ < 0 and $y$ < 0), be the interval $(a, b)$. Then $8\int \limits_{a}^{b}|A|dm$ is equal to........

09 Apr 2023:Shift-1

09 Apr 2023:Ans-Keys

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