Download JEE Main 2024 Question Paper (05 Apr - Shift 1) Mathematics

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06 Apr 2024:Shift-1

06 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. Let $d$ be the distance of the point of intersection of the lines $\frac{x+6}{3}$=$\frac{y}{2}$=$\frac{z+1}{1}$ and $\frac{x-7}{4}$=$\frac{y-9}{3}$=$\frac{z-4}{2}$ from the point (7, 8, 9). Then$d^2+ 6$ is equal to :
   
      
   1. 72
   2. 69
   3. 75
   4. 78
2. Let a rectangle $ABCD$ of sides 2 and 4 be inscribed in another rectangle PQRS such that the vertices ofthe rectangle ABCD lie on the sides of therectangle $PQRS$. Let $a$ and $b$ be the sides of therectangle $PQRS$ when its area is maximum. Then$(a + b)^2$is equal to :
   
      
   1. 72
   2. 60
   3. 80
   4. 64
3. Let two straight lines drawn from the origin $O$ intersect the line $3x + 4y$ = 12 at the points $P$ and $Q$ such that $\Delta OPQ$ is an isosceles triangle and $\angle{POQ}$ = 90°. If $l$ = $OP^2$ + $PQ^2$ + $QO^2$, then the greatest integer less than or equal to $l$ is :
   
      
   1. 44
   2. 48
   3. 46
   4. 42
4. If $y = y(x)$ is the solution of the differential equation $\frac{dy}{dx}+2y$=$\sin (2x)$, $y(0)$=$\frac{3}{4}$, then $y\left(\frac{\pi}{8}\right)$ is equal to:
   
      
   1. $e^{-\pi/8}$
   2. $e^{-\pi/4}$
   3. $e^{\pi/4}$
   4. $e^{\pi/8}$
5. For the function $f(x)$ = $\sin x$ + $3x$$-\frac{2}{\pi}(x^2+x)$, where $x \in \left[0, \frac{\pi}{2}\right]$, consider the following two statements :
   (I) f is increasing in $\left(0, \frac{\pi}{2}\right)$.
   (II) $f^{,}$ is decreasing in $\left(0, \frac{\pi}{2}\right)$.
   Between the above two statements,
   
      
   1. only (I) is true.
   2. only (II) is true.
   3. neither (I) nor (II) is true.
   4. both (I) and (II) are true.
6. If the system of equations
   $11x$ + $y$ + $\lambda z$ = –5
   $2x$ + $3y$ + $5z$ = 3
   $8x –$$ 19y –$$ 39z$ = µ
   has infinitely many solutions, then $\lambda^4– µ$ is equal to :
   
      
   1. 49
   2. 45
   3. 47
   4. 51
7. Let $A$ = {1, 3, 7, 9, 11} and $B$ = {2, 4, 5, 7, 8, 10, 12}. Then the total number of one-one maps $f : A \to B$, such that $f (1)$ + $f(3)$ = 14, is :
   
      
   1. 180
   2. 120
   3. 480
   4. 240
8. If the function $f(x)$=$\frac{sin3x+ \alpha sin x-\beta cos3x}{x^3}$, $x \in R$, is continuous at $x$ = 0, then $f(0)$ is equal to :
   
      
   1. 2
   2. -2
   3. 4
   4. -4
   
9.  The integral $\int \limits_{0}^{\frac{\pi}{4}}\frac{136 \sin x}{3 \sin x+5 \cos x}dx$ is equal to :
   
      
   1. $3\pi$$ – 50 log_e 2$ + $20 log_e 5$
   2. $3\pi$$ – 25 log_e 2$ + $10 log_e 5$
   3. $3\pi$$ – 10 log_e (2\sqrt{2})$ + $10 log_e 5$
   4. $3\pi$$ – 30 log_e 2$ + $20 log_e 5$
10. The coefficients $a$, $b$, $c$ in the quadratic equation $ax^2$+ $bx$ + $c$ = 0 are chosen from the set {1, 2, 3, 4, 5, 6, 7, 8}. The probability of this equation having repeated roots is :
    
    
       
    1. $\frac{3}{256}$
    2. $\frac{1}{128}$
    3. $\frac{1}{64}$
    4. $\frac{3}{128}$
11. Let $A$ and $B$ be two square matrices of order 3 such that $|A|$ = 3 and $|B|$ = 2. Then $|A^T A(adj(2A))^{–1}(adj(4B))(adj(AB))^{–1}AA^T| $ is equal to :
    
       
    1. 64
    2. 81
    3. 32
    4. 108
12. Let a circle C of radius 1 and closer to the origin be such that the lines passing through the point (3, 2) and parallel to the coordinate axes touch it. Then the shortest distance of the circle C from the point (5, 5) is :
    
       
    1. $2\sqrt{2}$
    2. 5
    3. $4\sqrt{2}$
    4. 4
13. Let the line $2x$ + $3y – k$ = 0, $k$ > 0, intersect the $x-$axis and $y-$axis at the points $A$ and $B$, respectively. If the equation of the circle having the line segment $AB$ as a diameter is $x^2$ + $y^2$$– 3x$$ – 2y$ = 0 and the length of the latus rectum of the ellipse $x^2$ + $9y^2$ = $k^2$ is $\frac{m}{n}$, where $m$ and $n$ are coprime, then $2m$ + $n$ is equal to
    
       
    1. 10
    2. 11
    3. 13
    4. 12
14. Consider the following two statements :
    Statement I : For any two non-zero complex numbers $z_1$, $z_2$
    $\left(|z_1|+|z_2|\right)$$\left|\frac{z_1}{|z_1|}+\frac{z_2}{|z_2|}\right|\leq$$2(|z_1|+|z_2|)$ and
    Statement II : If $x$, $y$, $z$ are three distinct complex numbers and $a$, $b$, $c$ are three positive real numbers such that $\frac{a}{|y-z|}$=$\frac{b}{|z-x|}$=$\frac{c}{|x-y|}$, then $\frac{a^2}{y-z}$+$\frac{b^2}{y-z}$+$\frac{c^2}{y-z}$=1.
    Between the above two statements,
    
       
    1. both Statement I and Statement II are incorrect.
    2. Statement I is incorrect but Statement II is correct.
    3. Statement I is correct but Statement II is incorrect.
    4. both Statement I and Statement II are correct
15. Suppose $\theta \in \left[0, \frac{\pi}{4}\right]$ is a solution of $4 cos\theta – 3 sin\theta$= 1. Then $cos\theta$ is equal to :
    
       
    1. $\frac{4}{3\sqrt{6}-2}$
    2. $\frac{6-\sqrt{6}}{3\sqrt{6}-2}$
    3. $\frac{6+\sqrt{6}}{3\sqrt{6}+2}$
    4. $\frac{4}{3\sqrt{6}+2}$
16. If $\frac{1}{\sqrt{1}+\sqrt{2}}$+$\frac{1}{\sqrt{2}+\sqrt{3}}$+...+$\frac{1}{\sqrt{99}+\sqrt{100}}$=$m$ and $\frac{1}{1•2}$+$\frac{1}{2•3}$+...+$\frac{1}{99•100}$=$n$, then the point $(m, n)$ lies on the line
    
       
    1. $11(x-1)$$-100(y-2)$=0
    2. $11(x-2)$$-100(y-1)$=0
    3. $11x$$-100y$=0
    4. $11(x-1)$$-100(y-2)$=0
17. Let $f(x)$= $x^5$ + $2x^3$ + $3x$ + 1, $x \in R$, and $g(x)$ be a function such that $g(f(x))=x$ for all $x \in R$. Then $\frac{g(7)}{g^{,}(7)}$ is equal to........
    
       
    1. 7
    2. 42
    3. 1
    4. 14
18. If $A(l, –1, 2)$, $B(5, 7, –6)$, $C(3, 4, –10)$ and $D(–l, –4, –2)$ are the vertices of a quadrilateral $ABCD$, then its area is :
    
       
    1. $12\sqrt{29}$
    2. $24\sqrt{29}$
    3. $24\sqrt{7}$
    4. $48\sqrt{7}$
19.  The value of $\int \limits_{-\pi}^{\pi}\frac{2y(1+\sin y)}{1+\cos^2 y}dy$ is :
    
       
    1. $\pi^2$
    2. $\frac{\pi^2}{2}$
    3. $\frac{\pi}{2}$
    4. $2\pi^2$
20. If the line $\frac{2-x}{3}$=$\frac{3y-2}{4\lambda+1}$=$4-z$ makes a right angle with the line $\frac{x+3}{3\mu}$=$\frac{1-2y}{6}$=$\frac{5-z}{7}$, then $4\lambda + 9µ$ is equal to :
    
       
    1. 13
    2. 4
    3. 5
    4. 6

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. From a lot of 10 items, which include 3 defective items, a sample of 5 items is drawn at random. Let the random variable X denote the number of defective items in the sample. If the variance of X is $\sigma^2$, then $96\sigma^2$ is equal to__________.
2. If the constant term in the expansion of $(1+2x-3x^3)$$\left(\frac{3}{2}x^2-\frac{1}{3x}\right)^9$ is $p$, then $108p$ is equal to
3. The area of the region enclosed by the parabolas $y = x^2– 5x$ and $y = 7x – x^2$is __________.
4. The number of ways of getting a sum 16 on throwing a dice four times is __________.
5. If $S$ = {$a \in R : |2a – 1| = 3[a]$+2{$a$}}, where $[t]$ denotes the greatest integer less than or equal to $t$ and {$t$} represents the fractional part of $t$, then $\sum \limits_{a \in S}a$ is equal to............
6. Let $f$ be a differentiable function in the interval $(0, \infty)$ such that $f(l)$ = 1 and $\lim \limits_{t \to x}\frac{t^2f(x)-x^2f(t)}{t-x}$=1 for each $x$ > 0. Then $2 f(2) + 3 f(3)$ is equal to __________.
7. Let $a_1$, $a_2$, $a_3$, ... be in an arithmetic progression of positive terms.
   Let $A_k$=$a_1^2-a_2^2+a_3^2-a_4^2+...+a_{2k-1}^2-a_{2k}^2$.
   If $A_3$ = –153, $A_5$ = –435 and $a_1^2$ + $a_2^2$ + $a_3^2$= 66, then $a_{17} – A_7$ is equal to __________.
8. Let $\vec{a}$=$\hat{i}$$-3\hat{j}$+$7\hat{k}$, $\vec{b}$=$2\hat{i}$$-\hat{j}$+$\hat{k}$ and $\vec{c}$ be a vector such that $(\vec{a}+2\vec{b})×\vec{c}$=$3(\vec{c}×\vec{a})$. If $\vec{a}•\vec{c}$=130, then $\vec{b}•\vec{c}$ is equal to.........
9. The number of distinct real roots of the equation $|x| |x + 2| – 5|x + l| – 1$ = 0 is __________.
10. Suppose $AB$ is a focal chord of the parabola $y^2 = 12x$ of length $l$ and slope $m < \sqrt{3}$. If the distance of the chord $AB$ from the origin is $d$, then $ld^2$is equal to __________.

05 Apr 2024:Shift-2

05 Apr 2024:Ans-Keys

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