Download JEE Main 2024 Question Paper (04 Apr - Shift 2) Mathematics

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05 Apr 2024:Shift-1

05 Apr 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 the function $f(x)$=$\left\{\begin{array}{cc} \frac{72^x-9^x-8^x+1}{\sqrt{2}-\sqrt{1+\cos x}} & , x \neq 0 \\ a \log_e 2\log_e3 &, x = 0, \end{array}\right.$ is continuous at $x$ = 0, then the value of $a^2$ is equal to
   
      
   1. 968
   2. 1152
   3. 746
   4. 1250
2. If $\lambda$ > 0, let $\theta$ be the angle between the vectors $\vec{a}$=$\hat{i}$+$\lambda \hat{j}$$-3\hat{k}$ and $\vec{b}$=$3 \hat{i}$$-\hat{j}$+$2 \hat{k}$. If the vectors $\vec{a}+\vec{b}$ and $\vec{a}-\vec{b}$ are mutually perpendicular, thenthe value of $(14 cos \theta)^2$is equal to
   
      
   1. 25
   2. 20
   3. 50
   4. 40
3. Let $C$ be a circle with radius 10 units and centreat the origin. Let the line $x + y$ = 2 intersects thecircle $C$ at the points $P$ and $Q$. Let $MN$ be a chord of $C$ of length 2 unit and slope –1. Then, a distance (in units) between the chord $PQ$ and the chord $MN$ is
   
      
   1. $2-\sqrt{3}$
   2. $3-\sqrt{2}$
   3. $\sqrt{2}-1$
   4. $\sqrt{2}+1$
4. Let a relation $R$ on $N×N$be defined as :
   $(x_1,y_1) R(x_2,y_2)$ if and only if $x_1$ < $x_2$ or $y_1$ < $y_2$
   Consider the two statements :
   (I) R is reflexive but not symmetric.
   (II) R is transitive
   Then which one of the following is true ?
   
      
   1. Only (II) is correct.
   2. Only (I) is correct.
   3. Both (I) and (II) are correct.
   4. Neither (I) nor (II) is correct.
5. Let three real numbers $a$,$b$,$c$ be in arithmetic progression and $a + 1$, $b$, $c + 3$ be in geometric progression. If $a > 10$ and the arithmetic mean of $a$, $b$ and $c$ is 8, then the cube of the geometric mean of $a$, $b$ and $c$ is
   
      
   1. 120
   2. 312
   3. 316
   4. 128
6. Let $A$=$\begin{equation*}\begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix} \end{equation*}$ and $B$=1+$adj(A)$+$(adj A)^2$+...+$(adj A)^{10}$. Then, the sum of all the elements of the matrix $B$ is :
   
      
   1. -110
   2. 22
   3. -88
   4. -124
7. The value of $\frac{1×2^2+2×3^2+...+100×(101)^2}{1^2×2+2^2×3+...+100^2×101}$ is
   
      
   1. $\frac{306}{305}$
   2. $\frac{305}{301}$
   3. $\frac{32}{31}$
   4. $\frac{31}{30}$
8. Let $f(x)$=$\int \limits_{0}^{x} (t + \sin(1-e^t))dt$, $x \in R$. Then $\lim \limits_{x \to 0} \frac{f(x)}{x^3}$ is equal to
   
      
   1. $\frac{1}{6}$
   2. $-\frac{1}{6}$
   3. $-\frac{2}{3}$
   4. $\frac{2}{3}$
   
9.  The area (in sq. units) of the region described by {$(x,y) : y^2\leq 2x$, and $y \geq 4x –1$} is
   
      
   1. $\frac{11}{32}$
   2. $\frac{8}{9}$
   3. $\frac{11}{12}$
   4. $\frac{9}{32}$
10. The area (in sq. units) of the region $S$={$z\in C$; $|z-1| \leq 2$; $(z +\bar{z})+i(z-\bar{z}) \leq 2$; $Im (z) \geq 0$} is
    
    
       
    1. $\frac{7\pi}{3}$
    2. $\frac{3\pi}{2}$
    3. $\frac{17\pi}{8}$
    4. $\frac{7\pi}{4}$
11. If the value of the integral $\int \limits_{-1}^{1}\frac{\cos \alpha x}{1+3^x}dx$ is $\frac{2}{\pi}$. Then, a value of $\alpha$ is
    
       
    1. $\frac{\pi}{6}$
    2. $\frac{\pi}{2}$
    3. $\frac{\pi}{3}$
    4. $\frac{\pi}{4}$
12. Let $f(x)$=$3\sqrt{x-2}$+$\sqrt{4-x}$ be a real valued function. If $\alpha$ and $\beta$ are respectively the minimum and the maximum values of $f$, then $\alpha^2$+$2\beta^2$ is equal to
    
       
    1. 44
    2. 42
    3. 24
    4. 38
13. If the coefficients of $x^4$, $x^5$and $x^6$ in the expansion of $(1 + x)^n$ are in the arithmetic progression, then the maximum value of $n$ is :
    
       
    1. 14
    2. 21
    3. 28
    4. 7
14. Consider a hyperbola $H$ having centre at the origin and foci and the $x-$axis. Let $C_1$ be the circle touching the hyperbola $H$ and having the centre at the origin. Let $C_2$ be the circle touching the hyperbola $H$ at its vertex and having the centre at one of its foci. If areas (in sq. units) of $C_1$ and $C_2$are 36$\pi$ and 4$\pi$, respectively, then the length (in units) of latus rectum of $H$ is
    
       
    1. $\frac{28}{3}$
    2. $\frac{14}{3}$
    3. $\frac{10}{3}$
    4. $\frac{11}{3}$
15. If the mean of the following probability distribution of a random variable $X$;
    

    X	0	2	4	6	8
    P(X)	$a$	$2a$	$a+b$	$2b$	$3b$

    
    
    is$\frac{46}{9}$, then the variance of the distribution is
    
       
    1. $\frac{581}{81}$
    2. $\frac{566}{81}$
    3. $\frac{173}{27}$
    4. $\frac{151}{27}$
16. Let $PQ$ be a chord of the parabola $y^2= 12x$ and the midpoint of $PQ$ be at $(4,1)$. Then, which of the following point lies on the line passing through the points $P$ and $Q$ ?
    
       
    1. (3, -3)
    2. $\left(\frac{3}{2}, -16\right)$
    3. $(2, -9)$
    4. $\left(\frac{1}{2}, -20\right)$
17. Given the inverse trigonometric function assumes principal values only. Let $x$, $y$ be any two real numbers in [–1,1] such that $cos^{–1}x – sin^{–1}y$ = $\alpha$,$-\frac{\pi}{2} \leq \alpha \leq \pi$. Then, the minimum value of $x^2+ y^2$ + $2xy sin \alpha$ is
    
       
    1. -1
    2. 0
    3. $\frac{-1}{2}$
    4. $\frac{1}{2}$
18. Let $y$ = $y(x)$ be the solution of the differential equation $(x^2 + 4)^2dy$ + $(2x^3y + 8xy – 2)dx$ = 0. If $y(0)$ = 0, then $y(2)$ is equal to
    
       
    1. $\frac{\pi}{8}$
    2. $\frac{\pi}{16}$
    3. $2 \pi$
    4. $\frac{\pi}{32}$
19.  Let $\vec{a}$ = $\hat{i}$+$\hat{j}$+$\hat{k}$, $\vec{b}$=$2\hat{i}$+$4\hat{j}$$-5\hat{k}$ and $\vec{c}$=$x \hat{i}$+$2\hat{j}$+$3\hat{k}$, $x \in R$. If $\vec{d}$ is the unit vector in the direction of $\vec{b}+\vec{c}$ such that $\vec{a}•\vec{d}$=1, then $(\vec{a}×\vec{b})•\vec{c}$ is equal to
    
       
    1. 9
    2. 6
    3. 3
    4. 11
20. Let $P$ the point of intersection of the lines $\frac{x-2}{1}$=$\frac{y-4}{5}$=$\frac{z-2}{1}$ and $\frac{x-3}{2}$=$\frac{y-2}{3}$=$\frac{z-3}{2}$. Then, the shortest distance of $P$ from the line $4x$ = $2y$ = $z$ is
    
       
    1. $\frac{5\sqrt{14}}{7}$
    2. $\frac{\sqrt{14}}{7}$
    3. $\frac{3\sqrt{14}}{7}$
    4. $\frac{6\sqrt{14}}{7}$

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. Let $S$ = {$sin^2 2\theta$ : ($sin^4\theta$ + $cos^4\theta$)$x^2$ + $(sin2 \theta)x$ + $(sin^6\theta + cos^6\theta)$ = 0 has real roots}. If $\alpha$ and $\beta$ be the smallest and largest elements of the set $S$, respectively, then $3((\alpha – 2)^2$ + $(\beta – 1)^2)$ equals…..
2. If $\int cosec ^5 x dx$=$\alpha \cot x cosec x$$\left(cosec^2 x+\frac{3}{2}\right)$+$\beta \log_e\left|\tan \frac{x}{2}\right|$+C where $\alpha, \beta \in R$ and C is constant of integration, then the value of $8(\alpha + \beta)$ equals …..
3. Let $f : R \to R$ be a thrice differentiable function such that $f(0)$ = 0, $f(1)$ = 1, $f(2)$ = –1, $f(3)$ = 2 and $f(4)$ = –2. Then, the minimum number of zeros of $(3f^{,} f^{,,} + ff^{,,,}) (x)$  is …..
4. Consider the function $f: R \to R$ defined by $f(x)$=$\frac{2x}{\sqrt{1+9x^2}}$. If the composition of $\mathrm{f}, \underbrace{(\mathrm{fofofo} \mathrm{f} \text { of })}_{10 \text { times }}(\mathrm{x})$=$\frac{2^{10}x}{\sqrt{1+9\alpha x^2}}$, then value of $\sqrt{3 \alpha +1}$ is equal to.........
5. Let $A$ be a 2 × 2 symmetric matrix such that $\begin{equation*}A \begin{bmatrix} 1 \\ 1 \end{bmatrix}=\begin{bmatrix} 3 \\ 7 \end{bmatrix}\end{equation*}$ and the determinant of $A$ be 1. If $A^{–1}$ = $\alpha A$ + $\beta I$, where $I$ is an identity matrix of order 2 × 2, then $\alpha$ + $\beta$ equals …..
6. There are 4 men and 5 women in Group A, and 5 men and 4 women in Group B. If 4 persons are selected from each group, then the number of ways of selecting 4 men and 4 women is …..
7. In a tournament, a team plays 10 matches with probabilities of winning and losing each match as $\frac{1}{3}$ and $\frac{2}{3}$ respectively. Let $x$ be the number of matches that the team wins, and y be the number of matches that team loses. If the probability $P(|x – y| < 2)$ is $p$, then $3^9p$ equals……
8. Consider a triangle $ABC$ having the vertices $A(1,2)$, $B(\alpha, \beta)$ and $C(\gamma, \delta)$ and angles $\angle{ABC}$=$\frac{\pi}{6}$and $\angle{BAC}$=$\frac{2\pi}{3}$. If the points $B$ and $C$ lie on the line $y$ = $x + 4$, then $\alpha^2$ + $\gamma^2$ is equal to …..
9. Consider a line $L$ passing through the points $P(1,2,1)$ and $Q(2,1,–1)$. If the mirror image of the point $A(2,2,2)$ in the line $L$ is $(\alpha, \beta, \gamma)$, then $\alpha$ + $\beta$ + $6\gamma$ is equal to …..
10. Let $y = y(x)$ be the solution of the differentialequation $(x + y + 2)^2 dx$ = $dy$, $y(0)$ = –2. Let the maximum and minimum values of the function $y = y(x)$ in $\left[0, \frac{\pi}{3}\right]$ be $\alpha$ and $\beta$ respectively. If $(3\alpha+\pi)^2$+$\beta^2$=$\gamma$+$\delta \sqrt{3}$, $\gamma$, $\delta in Z$, then $\gamma+\delta$ equals.......

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