Download JEE Main 2024 Question Paper (06 Apr - Shift 2) Mathematics

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08 Apr 2024:Shift-1

08 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 $ABC$ be an equilateral triangle. A new triangleis formed by joining the middle points of all sidesof the triangle $ABC$ and the same process isrepeated infinitely many times. If $P$ is the sum ofperimeters and $Q$ is be the sum of areas of all thetriangles formed in this process, then:
   
      
   1. $P^2$=$36\sqrt{3}$
   2. $P^2$=$6\sqrt{3}$
   3. $P$=$36\sqrt{3}Q^2$
   4. $P^2$=$72\sqrt{3}$
2. Let $A$ = {1, 2, 3, 4, 5}. Let $R$ be a relation on $A$defined by $xRy$ if and only if $4x \leq 5y$. Let $m$ be thenumber of elements in $R$ and $n$ be the minimumnumber of elements from $A × A$ that are requiredto be added to $R$ to make it a symmetric relation. Then $m + n$ is equal to:
   
      
   1. 24
   2. 23
   3. 25
   4. 26
3. If three letters can be posted to any one of the 5different addresses, then the probability that thethree letters are posted to exactly two addresses is:
   
      
   1. $\frac{12}{25}$
   2. $\frac{18}{25}$
   3. $\frac{4}{25}$
   4. $\frac{6}{25}$
4. Suppose the solution of the differential equation $\frac{dy}{dx}$=$\frac{(2+\alpha)x-\beta y +2}{\beta x -2\alpha y-(\beta\gamma-4\alpha)}$  represents a circle passing through origin. Then the radius of this circle is :
   
      
   1. $\sqrt{17}$
   2. $\frac{1}{2}$
   3. $\frac{\sqrt{17}}{2}$
   4. 2
5. If the locus of the point, whose distances from the point (2, 1) and (1, 3) are in the ratio 5 : 4, is $ax^2 + $by^2$ + $cxy$ + $dx$ + $ey$ + 170 = 0, then the value of $a^2$+ $2b$ + $3c$ + $4d$ + $e$ is equal to:
   
      
   1. 5
   2. -27
   3. 37
   4. 437
6. $\lim \limits_{n \to \infty} \frac{1^2-1)(n-1)+(2^2-2)(n-2)+....+((n-1)^2-(n-1))}{(1^3+2^3+....+n^3)-(1^2+2^2+...+n^2)}$ is equal to:
   
      
   1. $\frac{2}{3}$
   2. $\frac{1}{3}$
   3. $\frac{3}{4}$
   4. $\frac{1}{2}$
7. Let $0 \leq r \leq n$. If ${}^{n+1}C_{r+1}:{}^nC_r:{}^{n-1}C_{r-1}$=55:35:21, then $2n + 5r$ is equal to:
   
      
   1. 60
   2. 62
   3. 50
   4. 55
8. A software company sets up $m$ number of computer systems to finish an assignment in 17 days. If 4 computer systems crashed on the start of the second day, 4 more computer systems crashed on the start of the third day and so on, then it took 8 more days to finish the assignment. The value of $m$ is equal to :
   
      
   1. 125
   2. 150
   3. 180
   4. 160
9.  $z_1$, $z_2$ are two distinct complex number such that $\left|\frac{z_1-2z_2}{\frac{1}{2}-z_1\bar{z_2}}\right|$=2, then
   
      
   1. either $z_1$ lies on a circle of radius 1 or $z_2$ lies on a circle of radius $\frac{1}{2}
   2. either $z_1$ lies on a circle of radius $\frac{1}{2}$ or $z_2$ lies on a circle of radius 1.
   3. $z_1$ lies on a circle of radius $\frac{1}{2}$ and $z_2$ lies on a circle of radius 1.
   4. both $z_1$ and $z_2$ lie on the same circle.
10. If the function $f(x)$=$\left(\frac{1}{x}\right)^{2x}$; $x > 0$ attains the maximum value at $x$=$\frac{1}{e}$ then :
    
       
    1. $e^{\pi} < \pi^e$
    2. $e^{2\pi} < (2\pi)^e$
    3. $e^{\pi} > \pi^e$
    4. $(2e)^{\pi} > \pi^{(2e)}$
11. Let, $\vec{a}$=$6\hat{i}$+$\hat{j}$$-\hat{k}$ and $\vec{b}$=$\hat{i}$+$\hat{j}$. If $\vec{c}$ is a vector such that $|\vec{c}|\geq 6$, $\vec{a}•\vec{c}$=6$|\vec{c}|$, $|\vec{c}-\vec{a}|$=$2\sqrt{2}$ and the angle between $\vec{a}×\vec{b}$ and $\vec{c}$ is 60°, then $|(\vec{a}×\vec{b})×\vec{c}|$ is equal to:
    
       
    1. $\frac{9}{2}(6-\sqrt{6})$
    2. $\frac{3}{2}\sqrt{3}$
    3. $\frac{3}{2}\sqrt{6}$
    4. $\frac{9}{2}(6+\sqrt{6})$
12. If all the words with or without meaning made using all the letters of the word $"NAGPUR"$ are arranged as in a dictionary, then the word at $315^{th}$ position in this arrangement is :
    
       
    1. NRAGUP
    2. NRAGPU
    3. NRAPGU
    4. NRAPUG
13. Suppose for a differentiable function $h$, $h(0)$ = 0, $h(1)$ = 1 and $h'(0)$ = $h'(1)$ = 2. If $g(x)$ = $h(e^x) e^{h(x)}$, then $g'(0)$ is equal to:
    
       
    1. 5
    2. 3
    3. 8
    4. 4
14. Let $P (\alpha, \beta, \gamma)$ be the image of the point $Q(3, –3, 1)$ in the line $\frac{x-0}{1}$=$\frac{y-3}{1}$=$\frac{z-1}{-1}$ and $R$ be the point (2, 5, –1). If the area of the triangle $PQR$ is $\lambda$ and $\lambda^2$= $14K$, then $K$ is equal to:
    
       
    1. 36
    2. 72
    3. 18
    4. 81
15. If $P(6, 1)$ be the orthocentre of the triangle whosevertices are $A(5, –2)$, $B(8, 3)$ and $C(h, k)$, then the point $C$ lies on the circle.
    
       
    1. $x^2$ + $y^2$– 65 = 0
    2. $x^2$ + $y^2$– 74 = 0
    3. $x^2$ + $y^2$– 61 = 0
    4. $x^2$ + $y^2$– 52 = 0
16. Let $f(x)$=$\frac{1}{7-\sin 5x}$ be a function defined on $R$. Then the range of the function $f(x)$ is equal to:
    
       
    1. $\left[\frac{1}{8}, \frac{1}{5}\right]$
    2. $\left[\frac{1}{7}, \frac{1}{6}\right]$
    3. $\left[\frac{1}{7}, \frac{1}{5}\right]$
    4. $\left[\frac{1}{8}, \frac{1}{6}\right]$
17. Let $\vec{a}$=$2\hat{i}$+$\hat{j}$$-\hat{k}$, $\vec{b}$=$((\vec{a}×(\hat{i}+\hat{j}))×\hat{i})×\hat{i}$. Then the square of the projection of $\vec{a}$ on $\vec{b}$ is:
    
       
    1. $\frac{1}{5}$
    2. $2$
    3. $\frac{1}{3}$
    4. $\frac{2}{3}$
18. If the area of the region
    $\left\{(x, y):\frac{a}{x^2}\leq y\leq\frac{1}{x}, 1\leq x \leq 2, 0 < a < 1 \right\}$ is $(\log_e2)-\frac{1}{7}$ then the value of $7a – 3$ is equal to:
    
       
    1. 2
    2. 0
    3. -1
    4. 1
19.  If $\int \frac{1}{a^2\sin^2x+b^2\cos^2x}dx$=$\frac{1}{12}\tan^{-1}(3\tan x)$+constant, then the maximum value of $a\sin x + b\cos x$, is :
    
       
    1. $\sqrt{40}$
    2. $\sqrt{39}$
    3. $\sqrt{42}$
    4. $\sqrt{41}$
20.   If $A$ is a square matrix of order 3 such that $det(A)$ = 3 and $det(adj(–4 adj(–3 adj(3 adj((2A)^{–1})))))$ = $2^m3^n$, then $m +| 2n$ is equal to:
    
       
    1. 3
    2. 2
    3. 4
    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. Let $[t]$ denote the greatest integer less than or equal to $t$. Let $f: [0, \infty) \in R$ be a function defined by $f(x)$=$\left[\frac{x}{2}+3\right]$$-[\sqrt{x}]$. Let S be the set of all points in the interval [0, 8] at which $f$ is not continuous. Then $\sum \limits_{a \in S}a$ is equal to .............
2. The length of the latus rectum and directrices of a hyperbola with eccentricity $e$ are 9 and $x$=±$\frac{4}{\sqrt{3}}$, respectively. Let the line $y – \sqrt{3}x$ + $\sqrt{3}$ = 0 touch this hyperbola at $(x_0, y_0)$. If $m$ is the product of the focal distances of the point $(x_0, y_0)$, then $4e^2 + m$ is equal to _______.
3. If $S(x)$ = $(1 + x)$ + $2(1 + x)^2$ + $3(1 + x)^3$+ ..... + $60(1 + x)^{60}$, $x \in 0$, and $(60)^2 S(60)$ = $a(b)^b$ + $b$, where $a, b \in N$, then $(a + b)$ equal to ______
4. Let $[t]$ denote the largest integer less than or equal to $t$. If
   $\int \limits_0^3\left([x^2]+\left[\frac{x^2}{2}\right]\right)dx$=$a+b\sqrt{2}$$-\sqrt{3}-\sqrt{5}$+$c\sqrt{6}-\sqrt{7}$, where $a, b, c \in z$, then $a + b + c$ is equal to ______
5. From a lot of 12 items containing 3 defectives, a sample of 5 items is drawn at random. Let the random variable $X$ denote the number of defective items in the sample. Let items in the sample be drawn one by one without replacement. If variance of $X$ is $\frac{m}{n}$, where $gcd(m, n)$ = 1, then $n – m$ is equal to _______.
6. In a triangle $ABC$, $BC$ = 7, $AC$ = 8, $AB$ = $\alpha \in N$ and $\cos A$ = $\frac{2}{3}$. If $49\cos(3C)$ + 42 = $\frac{m}{n}$, where $gcd(m, n)$ = 1, then $m + n$ is equal to _______
7. If the shortest distance between the lines $\frac{x-\lambda}{3}$=$\frac{y-2}{-1}$=$\frac{z-1}{1}$ and $\frac{x+2}{-3}$=$\frac{y+5}{2}$=$\frac{z-4}{4}$ is $\frac{44}{\sqrt{30}}$, then the largest possible value of $|\lambda|$ is equal to _______.
8. Let $\alpha, \beta$ be roots of $x^2$+$\sqrt{2}x$$-8$=0. If $U_n$=$\alpha^n+\beta^n$, then $\frac{U_{10}+\sqrt{12}U_9}{2U_8}$ is equal to _______.
9. If the system of equations
   $2x$ + $7y$ + $\lambda z$ = 3
   $3x$ + $2y$ + $5z$ = 4
   $x$ + $\mu y$ + $32z$ = –1
   has infinitely many solutions, then $(\lambda – \mu)$ is equal to ________ :
10. If the solution $y(x)$ of the given differential equation $(e^y + 1) \cos x dx$ + $e^y \sin x dy$ = 0 passes through the point $\left(\frac{\pi}{2}, 0\right)$, then the value of $e^{y\left(\frac{\pi}{6}\right)}$ is equal to ______.

06 Apr 2024:Shift-1

06 Apr 2024:Ans-Keys

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