Download JEE Main 2024 Question Paper (09 Apr - Shift 1) 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. Let the line $L$ intersect the lines $x – 2$ = $–y$ = $z – 1$, $2 (x + 1)$ = $2(y – 1)$ = $z + 1$ and be parallel to the line $\frac{x-2}{3}$=$\frac{y-1}{1}$=$\frac{z-2}{2}$. Then which of the following points lies on $L$ ?
   
      
   1. $\left(-\frac{1}{3}, 1, 1\right)$
   2. $\left(-\frac{1}{3}, 1, -1\right)$
   3. $\left(-\frac{1}{3}, -1, -1\right)$
   4. $\left(-\frac{1}{3}, -1, 1\right)$
2. The parabola $y^2$ = $4x$ divides the area of the circle$x^2$+ $y^2$ = 5 in two parts. The area of the smallerpart is equal to :
   
      
   1. $\frac{2}{3}+5\sin^{-1}\left(\frac{2}{\sqrt{5}}\right)$
   2. $\frac{1}{3}+5\sin^{-1}\left(\frac{2}{\sqrt{5}}\right)$
   3. $\frac{1}{3}+\sqrt{5}\sin^{-1}\left(\frac{2}{\sqrt{5}}\right)$
   4. $\frac{2}{3}+\sqrt{5}\sin^{-1}\left(\frac{2}{\sqrt{5}}\right)$
3. The solution curve, of the differential equation $2y\frac{dy}{dx}$+3=$5\frac{dy}{dx}$, passing through the point (0, 1) is a conic, whose vertex lies on the line :
   
      
   1. $2x+3y$=9
   2. $2x+3y$=-9
   3. $2x+3y$=-6
   4. $2x+3y$=6
4. A ray of light coming from the point $P (1, 2)$ gets reflected from the point $Q$ on the $x-$axis and then passes through the point R (4, 3). If the point $S (h, k)$ is such that $PQRS$ is a parallelogram, then $hk^2$ is equal to :
   
      
   1. 80
   2. 90
   3. 60
   4. 70
5. Let $\lambda, \mu \in R$. If the system of equations
   $3x$ + $5y$ + $\lambda z$ = 3
   $7x$ + $11y$ –$9z$ = 2
   $97x$ + $155y$ – $189z$ = $\mu$
   has infinitely many solutions, then $\mu +2\lambda$ is equal to :
   
      
   1. 25
   2. 24
   3. 27
   4. 22
6. The coefficient of $x^{70}$ in $x^2(1 + x)^{98}$ + $x^3(1 + x)^{97}$ + $x^4(1 + x)^{96} $+ ........ +$x^{54}(1 + x)^{46}$ is ${}^{99}C_p$ – ${}^{46}C_q$. Then a possible value to p + q is :
   
      
   1. 55
   2. 61
   3. 68
   4. 83
7. Let $\int \frac{2-\tan x}{3+\tan x}dx$=$\frac{1}{2}(\alpha x+\log_e|\beta \sin x+\gamma \cos x|)$+C, where C is the constant of integration. Then $\alpha +\frac{\gamma}{\beta}$ is equal to :
   
      
   1. 3
   2. 1
   3. 4
   4. 7
8. A variable line $L$ passes through the point (3, 5) and intersects the positive coordinate axes at the points $A$ and $B$. The minimum area of the triangle $OAB$, where $O$ is the origin, is :
   
      
   1. 30
   2. 25
   3. 40
   4. 35
9. Let $|\cos \theta \cos(60-\theta) \cos(60+\theta)|\leq \frac{1}{8}$, $\theta \in[0, 2\pi]$. Then, the sum of all $\theta \in [0,2\pi]$, where $\cos 3\theta$ attains its maximum value, is :
   
      
   1. $9\pi$
   2. $18\pi$
   3. $6\pi$
   4. $15\pi$
10. Let $\vec{OA}$=2$\vec{a}$, $\vec{OB}$=6$\vec{a}$+$5\vec{b}$ and $\vec{OC}$=$3\vec{b}$, where $O$ is the origin. If the area of the parallelogram with adjacent sides $\vec{OA}$ and $\vec{OC}$ is 15 sq. units, then the area (in sq. units) of the quadrilateral $OABC$ is equal to :
    
       
    1. 38
    2. 40
    3. 32
    4. 35
11. If the domain of the function
    $f(x)$=$\sin^{-1}\left(\frac{x-1}{2x+3}\right)$ is $R-(\alpha, \beta)$, then $12\alpha \beta$ is equal to
    
       
    1. 36
    2. 24
    3. 40
    4. 32
12. If the sum of series
    $\frac{1}{1•(1+d)}$+$\frac{1}{(1+d)(1+2d)}$+....+$\frac{1}{(1+9d)(1+10d)}$
    is equal to 5, then $50d$ is equal to :
    
       
    1. 20
    2. 5
    3. 15
    4. 10
13. Let $f(x)$=$ax^3$+$bx^2$+$ex$+41 be such that $f(1)$ = 40, $f'(1)$ = 2 and $f''(1)$ = 4. Then $a^2$ + $b^2$ + $c^2$ is equal to :
    
       
    1. 62
    2. 73
    3. 54
    4. 51
14. Let a circle passing through (2, 0) have its centre at the point $(h, k)$. Let $(x_c , y_c)$ be the point of intersection of the lines $3x + 5y$ = 1 and $(2 + c) x$ + $5c^2y$ = 1. If $h$=$\lim \limits_{c \to 1}x_c$ and $k$=$\lim \limits_{c \to 1}y_c$, then the equation of the circle is :
    
       
    1. $25x^2$+ $25y^2$– $20x$ + $2y$ – 60 = 0
    2. $5x^2$+ $5y^2$– $4x$ – $2y$ – 12 = 0
    3. $25x^2$+ $25y^2$– $2x$ + $2y$ – 60 = 0
    4. $5x^2$+ $5y^2$– $4x$ + $2y$ – 12 = 0
15. The shortest distance between the line $\frac{x-3}{4}$=$\frac{y+7}{-11}$=$\frac{z-1}{5}$ and $\frac{x-5}{3}$=$\frac{y-9}{-6}$=$\frac{z+2}{1}$ is:
    
       
    1. $\frac{187}{\sqrt{563}}$
    2. $\frac{178}{\sqrt{563}}$
    3. $\frac{185}{\sqrt{563}}$
    4. $\frac{179}{\sqrt{563}}$
16. The frequency distribution of the age of students in a class of 40 students is given below.
    
    

    Age	15	16	17	18	19	20
    No. of students	5	8	5	12	x	y

    
    
    
    If the mean deviation about the median is 1.25, then $4x + 5y$ is equal to :
    
       
    1. 43
    2. 44
    3. 47
    4. 46
17. The solution of the differential equation $(x^2+ y^2)dx$ – $5xy dy$ = 0, $y(1)$ = 0, is :
    
       
    1. $|x^2-4y^2|^5$=$x^2$
    2. $|x^2-2y^2|^6$=$x$
    3. $|x^2-4y^2|^6$=$x$
    4. $|x^2-2y^2|^5$=$x^2$
18. Let three vectors $\vec{a}$=$\alpha \hat{i}$+$4\hat{j}$+$2\hat{k}$, from a triangle such that $\vec{c}$=$\vec{a} – \vec{b}$and the area of the triangle is $5\sqrt{6}$. If $\alpha$ is a positive real number, then $|\vec{c}|^2$ is :
    
       
    1. 16
    2. 14
    3. 12
    4. 10
19.  Let $\alpha$, $\beta$ be the roots of the equation $x^2$+$2\sqrt{2}x-1=0$. The quadratic equation, whose roots are $\alpha^4+\beta^4$ and $\frac{1}{10}(\alpha^6+\beta^6)$, is
    
       
    1. $x^2 – 190x$ + 9466 = 0
    2. $x^2– 195x$ + 9466 = 0
    3. $x^2– 195x$ + 9506 = 0
    4. $x^2– 180x$ + 9506 = 0
20. Let $f(x)$=$x^2$+9, $g(x)$=$\frac{x}{x-9}$ and $a$=$fog(10)$, $b$=$gof(3)$. If $e$ and $l$ denote the eccentricity and the length of the latus rectum of the ellipse $\frac{x^2}{a}$+$\frac{y^2}{b}$=1, then $8e^2$+$I^2$ is equal to
    
       
    1. 16
    2. 8
    3. 6
    4. 12

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 $a$, $b$ and $c$ denote the outcome of three independent rolls of a fair tetrahedral die, whose four faces are marked 1, 2, 3, 4. If the probability that $ax^2$ + $bx$ + $c$ = 0 has all real roots is $\frac{m}{n}$, $gcd(m, n)$ = 1, then $m + n$ is equal to ________.
2. The sum of the square of the modulus of the elements in the set {$z=a + ib : a,b \in Z$, $z \in C$, $|z - 1|\leq 1$, $|z - 5| \leq |z - 5i|$} is ________.
3. Let the set of all positive values of $\lambda$, for which the point of local minimum of the function $(1 + x (\lambda^2–x^2))$ satisfies $\frac{x^2+x+2}{x^2+5x+6}$ < 0, be $(\alpha, \beta)$. Then $\alpha^2+\beta^2$ is equal to..........
4. Let $\lim \limits_{n \to \infty}\left(\frac{n}{\sqrt{n^4+1}}-\frac{2n}{(n^2+1)\sqrt{n^4+1}}\right.$$\left.\frac{n}{\sqrt{n^4+16}}-\frac{8n}{(n^2+4)\sqrt{n^4+16}}\right.$+.....+$\left. \frac{n}{\sqrt{n^4+n^4}}-\frac{2n. n^2}{(n^2+n^2)\sqrt{n^4+n^4}}\right)$ be $\frac{\pi}{k}$, using only the principal values of the inverse trigonometric functions. Then $k^2$ is equal to _____.
5. The remainder when $428^{2024}$ is divided by 21 is ________.
6. Let $f:(0, \pi) \in R$ be a function given by
   $\left\{\begin{array}{ccc}\left(\frac{8}{7}\right)^{\frac{\tan 8x}{\tan 7x}}&, 0 < x < \frac{\pi}{2}\\ a-8 &, x=\frac{\pi}{2}\\ (1+|\cot x|)^{\frac{b}{a}|\tan x|} &, \frac{\pi}{2} < x < \pi\end{array}\right.$ Where $a, b \in Z$. If $f$ is continuous at $x$=$\frac{\pi}{2}$, then $a^2$+ $b^2$is equal to ________.
7. Let $A$ be a non-singular matrix of order 3. If $det(3adj(2adj((detA)A)))$ = $3^{–13}•2^{–10}$ and $det (3adj(2A))$ = $2^m•3^n$, then $|3m + 2n|$ is equal to ________.
8. Let the centre of a circle, passing through the point (0, 0), (1, 0) and touching the circle $x^2 + y^2$ = 9, be $(h, k)$. Then for all possible values of the coordinates of the centre $(h, k), 4(h^2 + k^2)$ is equal to ________.
9. If a function $f$ satisfies $f(m + n)$ = $f(m)$ + $f(n)$ for all $m, n \in N$ and $f(1)$ = 1, then the largest natural number $\lambda$ such that $\sum \limits_{k=1}^{2022}f(\lambda+k)\leq (2022)^2$ is equal to.........
10. Let $A$ = {2, 3, 6, 7} and $B$ = {4, 5, 6, 8}. Let $R$ be a relation defined on $A × B$ by $(a_1, b_1) R (a_2, b_2)$ is and only if $a_1$ + $a_2$ = $b_1$ + $b_2$. Then the number of elements in $R$ is ________.

09 Apr 2023:Shift-2

09 Apr 2023:Ans-Keys

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