Download JEE Main 2024 Question Paper (08 Apr - Shift 2) Mathematics

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09 Apr 2024:Shift-1

09 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 image of the point (-4, 5) in the line $x + 2y$ = 2 lies on the circle $(x + 4)^2$ + $(y-3)^2$ = $r^2$,then $r$ is equal to :
   
      
   1. 1
   2. 2
   3. 75
   4. 3
2. Let $\vec{a}$=$\hat{i}$+$2\hat{j}$+$3\hat{k}$, $\vec{b}$=$2\hat{i}$+$3\hat{j}$$-5\hat{k}$ and $\vec{c}$=$3\hat{i}$$-\hat{j}$+$\lambda \hat{k}$ be three vectors. Let $\vec{r}$ be a unit vector along $\vec{b}$+$\vec{c}$. If $\vec{r}$•$\vec{a}$=3, then $3\lambda$ is equal to:
   
      
   1. 27
   2. 25
   3. 23
   4. 21
3. Let $\alpha \neq a$, $\beta \neq b$, $\gamma \neq c$ and $\begin{equation*} \begin{vmatrix} \alpha & b & c\\ a & \beta & c\\ a & b & \gamma \end{vmatrix}=0 \end{equation*}$, then $\frac{a}{\alpha-a}$=$\frac{b}{\beta-b}$=$\frac{c}{\gamma-c}$ is equal to:
   
      
   1. 2
   2. 3
   3. 0
   4. 1
4. In an increasing geometric progression ol positive terms, the sum of the second and sixth terms is $\frac{70}{3}$and the product of the third and fifth terms is 49. Then the sum of the $4^{th}$, $6^{th}$ and $8^{th}$ terms is :
   
      
   1. 96
   2. 78
   3. 91
   4. 84
5. The number of ways five alphabets can be chosen from the alphabets of the word MATHEMATICS, where the chosen alphabets are not necessarily distinct, is equal to :
   
      
   1. 175
   2. 181
   3. 177
   4. 179
6. The sum of all possible values of $\theta \in [– \pi, 2\pi]$, for which $\frac{1+i\cos\theta}{1-2i\cos\theta}$ is purely imaginary, is equal to:
   
      
   1. $2\pi$
   2. $3\pi$
   3. $5\pi$
   4. $4\pi$
7. If the system of equations $x + 4y – z$ = $\lambda$, $7x + 9y + \mu z$ = –3, $5x + y + 2z$ = –1 has infinitely many solutions, then $(2\mu + 3\lambda)$ is equal to :
   
      
   1. 2
   2. -3
   3. 3
   4. -2
8. If the shortest distance between the lines
   $\frac{x-\lambda}{2}$=$\frac{y-4}{3}$=$\frac{z-3}{4}$ and $\frac{x-2}{4}$=$\frac{y-4}{6}$=$\frac{z-7}{8}$ is $\frac{13}{\sqrt{29}}$, then a value of $\lambda$ is
   
      
   1. $-\frac{13}{25}$
   2. $\frac{13}{25}$
   3. 1
   4. -1
9. If the value of $\frac{3\cos 36°+5\sin 18°}{5\cos 36°-3\sin 18°}$ is $\frac{a\sqrt{5}-b}{c}$, where $a, b, c$ are natural numbers and $gcd(a, c)$ = 1, then $a + b + c$ is equal to :
   
      
   1. 50
   2. 40
   3. 52
   4. 54
10. Let $y = y(x)$ be the solution curve of the differential equation $\sec y \frac{dy}{dx}$+ $2x\sin y$ = $x^3\cos y$, $y(1)$ = 0. Then $y(\sqrt{3})$ is equal to :
    
       
    1. $\frac{\pi}{3}$
    2. $\frac{\pi}{6}$
    3. $\frac{\pi}{4}$
    4. $\frac{\pi}{12}$
11. The area of the region in the first quadrant inside the circle $x^2$ + $y^2$ = 8 and outside the parabola $y^2$ = $2x$ is equal to :
    
       
    1. $\frac{\pi}{2}-\frac{1}{3}$
    2. $\pi-\frac{2}{3}$
    3. $\frac{\pi}{2}-\frac{2}{3}$
    4. $\pi-\frac{1}{3}$
12. If the line segment joining the points $(5, 2)$ and $(2, a)$ subtends an angle $\frac{\pi}{4}$ at the origin, then the absolute value of the product of all possible values of $a$ is :
    
       
    1. 6
    2. 8
    3. 2
    4. 4
13. Let $\vec{a}$=$4\hat{i}$$-\hat{j}$+$\hat{k}$, $\vec{b}$=$11\hat{i}$$-\hat{j}$+$\hat{k}$ and $\vec{c}$ be a vector such that $(\vec{a}+\vec{b})×\vec{c}$=$\vec{c}×(-2\vec{a}+3\vec{b})$. If $(2\vec{a}+3\vec{b})•\vec{c}$=1670, then $|\vec{c}|^2$ is equal to:
    
       
    1. 1627
    2. 1618
    3. 1600
    4. 1609
14. If the function $f(x)$ = $2x^3– 9ax^2$+ $12a^2x$ + 1, $a$ > 0 has a local maximum at $x = \alpha$ and a local minimum $x = \alpha^2$, then $\alpha$ and $\alpha^2$are the roots of the equation :
    
       
    1. $x^2– 6x$ + 8 = 0
    2. $8x^2+ 6x$ – 8 = 0
    3. $8x^2– 6x$ + 1 = 0
    4. $x^2 + 6x$ + 8 = 0
15. There are three bags $X$, $Y$ and $Z$. Bag $X$ contains 5 one-rupee coins and 4 five-rupee coins; Bag $Y$ contains 4 one-rupee coins and 5 five-rupee coins and Bag $Z$ contains 3 one-rupee coins and 6 five-rupee coins. A bag is selected at random and a coin drawn from it at random is found to be a one-rupee coin. Then the probability, that it came from bag $Y$, is :
    
       
    1. $\frac{1}{3}$
    2. $\frac{1}{2}$
    3. $\frac{1}{4}$
    4. $\frac{5}{12}$
16. Let $\int \limits_{\alpha}^{\log_e4}\frac{dx}{\sqrt{e^x-1}}$=$\frac{\pi}{6}$. Then $e^{\alpha}$ and $e^{-\alpha}$ are the roots of the equation :
    
       
    1. $2x^2 – 5x$ + 2 = 0
    2. $x^2 – 2x$ - 8 = 0
    3. $2x^2 – 5x$ – 2 = 0
    4. $x^2 + 2x$ – 8 = 0
17. Let $f(x)$=$\left\{\begin{array}{cc}-a & \text { if }-a \leq x \leq 0 \\ x+a & \text { if } 0 < x \leq a\end{array}\right.$ where $a$ > 0 and $g(x)$ = $(f |x|) – | f (x)|)/2$. Then the function $g : [ –a, a] \to [ –a, a]$ is
    
       
    1. neither one-one nor onto.
    2. both one-one and onto.
    3. one-one.
    4. onto
18. Let $A$= {2, 3, 6, 8, 9, 11} and $B$ = {1, 4, 5, 10, 15} Let $R$ be a relation on $A × B$ define by $(a, b)R(c, d)$ if and only if $3ad – 7bc$ is an even integer. Then the relation $R$ is
    
       
    1. reflexive but not symmetric.
    2. transitive but not symmetric.
    3. reflexive and symmetric but not transitive.
    4. an equivalence relation
19.  For $a, b$ > 0, let
    $f(x)$=$\left\{ \begin{array} {cc} \frac{\tan((a+1)x)+b \tan x}{x}&, x < 0\\ 3 &, x=0 \\ \frac{\sqrt{ax+b^2x^2}-\sqrt{ax}}{b\sqrt{a}x\sqrt{x}}&, x > 0 \end{array}\right.$ be a continous function at $x$ = 0. Then $\frac{b}{a}$ is equal to
    
       
    1. 5
    2. 4
    3. 8
    4. 6
20. If the term independent of x in the expansion of $\left(\sqrt{a}x^2+\frac{1}{2x^3}\right)^{10}$ is 105, then $a^2$ is equal to
    
       
    1. 4
    2. 9
    3. 6
    4. 2

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 $A$ be the region enclosed by the parabola $y^2$ = $2x$ and the line $x$ = 24. Then the maximum area of the rectangle inscribed in the region $A$ is _________ .
2. If $\alpha$=$\lim \limits_{x \to 0^+}\left(\frac{e^{\sqrt{\tan x}}-e^{\sqrt{x}}}{\sqrt{\tan x}-\sqrt{x}}\right)$ and $\beta$=$\lim \limits_{x \to 0}(1+\sin x)^{\frac{1}{2}\cot x}$ are the roots of quadratic equation $ax^2+bx-\sqrt{e}$=0, then 12 $log_e(a + b)$ is equal to _________ .
3. Let $S$ be the focus of the hyperbola $\frac{x^2}{3}-\frac{y^2}{5}=1$, on the positive $x-$axis. Let $C$ be the circle with its centre at $A(\sqrt{6}, \sqrt{5})$ and passing through the point $S$. If $O$ is the origin and $SAB$ is a diameter of $C$ then the square of the area of the triangle $OSB$ is equal to
4. Let $P(\alpha, \beta, \gamma)$ be the image of the point $Q(l, 6, 4)$ in the line $\frac{x}{1}$=$\frac{y-1}{2}$=$\frac{z-2}{3}$. Then $2\alpha$ + $\beta$ + $\gamma$ is equal to _______ .
5. An arithmetic progression is written in the following way
   
   The sum of all the terms of the $10^th$ row is ______ .
6. The number of distinct real roots of the equation $|x + 1|$ $|x + 3|$ $– 4 |x + 2|$ + 5 = 0, is ________ .
7. Let a ray of light passing through the point (3, 10) reflects on the line $2x + y$ = 6 and the reflected ray passes through the point (7, 2). If the equation of the incident ray is $ax + by$ + l = 0, then $a^2 + b^2+ 3ab$ is equal to_.
8. Let $a, b, c \in N$ and $a < b < c$. Let the mean, the mean deviation about the mean and the variance of the 5 observations 9, 25, $a$, $b$, $c$ be 18, 4 and $\frac{136}{5}$, respectively. Then $2a + b – c$ is equal to _______ .
9. Let $\alpha|x|$=$|y|^{xy-\beta}$, $\alpha, \beta \in N$ be the solution of the differential equation $xdy – ydx$ + $xy(xdy+ydx)$ = 0, $y(1)$ = 2. Then $\alpha + \beta$ is equal to _
10. If $\int \frac{1}{\sqrt{5}{(x-1)^4(x+3)^6}}dx$=$A\left(\frac{\alpha x-1}{\beta x+3}\right)^{B}+C$, where $C$ is the constant of integration, then the value of $\alpha$ + $\beta$ + $20AB$ is _______ .

08 Apr 2024:Shift-1

08 Apr 2024:Ans-Keys

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