Download JEE Main 2024 Question Paper (06 Apr - Shift 1) 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. If $f(x)$=$\left \{ \begin{array} {cc} x^3 \sin\left(\frac{1}{x}\right) &, x \neq 0 \\ 0 &, x=0 \end{array}\right.$, then
   
      
   1. $f"(0)$=1
   2. $f"\left(\frac{2}{\pi}\right)$=$\frac{24-\pi^2}{2\pi}$
   3. $f"\left(\frac{2}{\pi}\right)$=$\frac{12-\pi^2}{2\pi}$
   4. $f"(0)$=0
2. If $A(3, 1, -1)$, $B\left(\frac{5}{3}, \frac{7}{3}, \frac{1}{3}\right)$, $C(3, 1, -1)$ and $D\left(\frac{10}{3}, \frac{2}{3}, \frac{-1}{3}\right)$ are the vertices of a quadrilateral $ABCD$, then its area is
   
      
   1. $\frac{4\sqrt{2}}{3}$
   2. $\frac{5\sqrt{2}}{3}$
   3. $2\sqrt{2}$
   4. $\frac{2\sqrt{2}}{3}$
3. $\int \limits_{0}^{\pi/4}\frac{\cos^2x \sin^2x}{(\cos^3 x +\sin^3 x)^2}dx$ is equal to
   
      
   1. 1/12
   2. 1/9
   3. 1/6
   4. 1/3
4. The mean and standard deviation of 20 observations are found to be 10 and 2, respectively. On respectively, it was found that an observation by mistake was taken 8 instead of 12. The correctstandard deviation is
   
      
   1. $\sqrt{3.86}$
   2. 1.8
   3. $\sqrt{3.96}$
   4. 1.94
5. The function $f(x)$=$\frac{x^2+2x-15}{x^2-4x+9}$, $x \in R$ is
   
      
   1. both one-one and onto
   2. onto but not one-one.
   3. neither one-one nor onto.
   4. one-one but not onto.
6. Let $A$ = {$n \in [100, 700] \cap N$ : $n$ is neither a multiple of 3 nor a multiple of 4}. Then the number of elements in $A$ is
   
      
   1. 300
   2. 280
   3. 310
   4. 290
7. Let $C$ be the circle of minimum area touching the parabola $y = 6 – x^2$ and the lines $y = 3 x$. Then, which one of the following points lies on the circle $C$ ?
   
      
   1. (2, 4)
   2. (1, 2)
   3. (2, 2)
   4. (1, 1)
8. For $\alpha, \beta \in R$ and a natural number $n$, let $\begin{equation*}\begin{vmatrix} r & 1 & \frac{n^2}{2}+\alpha \\ 2r & 2 & n^2-\beta \\ 3r-2 & 3 & \frac{n(3n-1)}{2}\end{vmatrix}\end{equation*}$. Then $2A_{10}-A_8$ is
   
      
   1. $4\alpha+2\beta$
   2. $2\alpha+4\beta$
   3. $2n$
   4. 0
9.  The shortest distance between the lines $\frac{x-3}{2}$=$\frac{y+15}{-7}$=$\frac{z-9}{5}$ and $\frac{x+1}{2}$=$\frac{y-1}{1}$=$\frac{z-9}{-3}$ is
   
      
   1. $6\sqrt{3}$
   2. $4\sqrt{3}$
   3. $5\sqrt{3}$
   4. $8\sqrt{3}$
10. A company has two plants A and B to manufacture motorcycles. 60% motorcycles are manufactured at plant A and the remaining are manufactured at plant B. 80% of the motorcycles manufactured at plant A are rated of the standard quality, while 90% of the motorcycles manufactured at plant B are rated of the standard quality. A motorcycle picked up randomly from the total production is found to be of the standard quality. If p is the probability that it was manufactured at plant B, then 126p is
    
       
    1. 54
    2. 64
    3. 66
    4. 56
11. Let, $\alpha$, $\beta$ be the distinct roots of the equation $x^2-$$(t^2-5t+6)x$+1=0, $t \in R$and $a_n$=$\alpha^n +\beta^n$. Then the minimum value of $\frac{a_{2024}+a_{2025}}{a_{2024}}$ is
    
       
    1. 1/4
    2. -1/2
    3. -1/4
    4. 1/2
12. Let the relations $R_1$ and $R_2$ on the set
    $X$ = {1, 2, 3, ..., 20} be given by
    $R_1$ = {$(x, y)$ : $2x – 3y$ = 2} and
    $R_2$ = {$(x, y)$ : $–5x + 4y$ = 0}. If $M$ and $N$ be the minimum number of elements required to be added in $R_1$ and $R_2$, respectively, in order to make the relations symmetric, then $M + N$ equals
    
       
    1. 8
    2. 16
    3. 12
    4. 10
13. Let a variable line of slope m > 0 passing through the point (4, –9) intersect the coordinate axes at the points A and B. the minimum value of the sum of the distances of A and B from the origin is
    
       
    1. 25
    2. 30
    3. 15
    4. 10
14. The interval in which the function $f(x)$ = $x^x$, $x$ > 0, is strictly increasing is
    
       
    1. $(0, 1/e]$
    2. $[1/e^2, 1)$
    3. $(0, \infty)$
    4. $[1/e, \infty)$
15. A circle in inscribed in an equilateral triangle of side of length 12. If the area and perimeter of any square inscribed in this circle are $m$ and $n$, respectively, then $m + n^2$is equal to
    
       
    1. 396
    2. 408
    3. 312
    4. 414
16. The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is
    
       
    1. 24
    2. 56
    3. 16
    4. 48
17. Let $y = y(x)$ be the solution of the differential equation $(1+x^2)\frac{dy}{dx}$+$y$=$e^{\tan^{-1}x}$, $y(1)$=0. Then $y(0)$ is
    
       
    1. $\frac{1}{4}(e^{\pi/2}-1)$
    2. $\frac{1}{2}(1-e^{\pi/2})$
    3. $\frac{1}{4}(1-e^{\pi/2})$
    4. $\frac{1}{2}(e^{\pi/2}-1)$
18. Let $y = y(x)$ be the solution of the differential equation $(2x\log_ex)\frac{dy}{dx}$+$2y$=$\frac{3}{x}\log_ex$, $x >0$ and $y(e^{-1})$=0. Then, $y(e)$ is equal to
    
       
    1. $-\frac{3}{2e}$
    2. $-\frac{2}{3e}$
    3. $-\frac{3}{e}$
    4. $-\frac{2}{e}$
19.  Let the area of the region enclosed by the curves $y = 3x$, $2y$ = $27 – 3x$ and $y$= $3x – x \sqrt{x}$be $A$. Then $10 A$ is equal to
    
       
    1. 184
    2. 154
    3. 172
    4. 162
20. Let $f : (–\infty , \infty) – {0} \to R$ be a differentiable function such that $f'(1)$=$\lim \limits_{a \to \infty}a^2f\left(\frac{1}{a}\right)$. Then $\lim \limits_{a \to \infty} \frac{a(a+1)}{2}\tan^{-1}\left(\frac{1}{a}\right)$+$a^2-2\log_ea$ is equal to
    
       
    1. $\frac{3}{2}+\frac{\pi}{4}$
    2. $\frac{3}{8}+\frac{\pi}{4}$
    3. $\frac{5}{2}+\frac{\pi}{8}$
    4. $\frac{3}{4}+\frac{\pi}{8}$

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 $\alpha \beta \gamma$ = 45 ; $\alpha, \beta, \gamma \in R$. If $x(\alpha, 1, 2)$ + $y(1, \beta, 2)$ + $z(2, 3, \gamma)$ = (0, 0, 0) for some $x, y, z \in R$, $xyz \neq0$, then $6\alpha + 4\beta + \gamma$ is equal to_______
2. Let a conic $C$ pass through the point (4, –2) and $P(x, y)$, $x \geq 3$, be any point on $C$. Let the slope of the line touching the conic $C$ only at a single point $P$ be half the slope of the line joining the points $P$ and (3, –5). If the focal distance of the point (7, 1) on $C$ is $d$, then 12$d$ equals_______.
3. Let $r_k$=$\frac{\int \limits_{0}^{1}(1-x^7)^kdx}{\int \limits_{0}^{1}(1-x^7)^{k+1}}dx$, $k \in N$. Then the value of $\sum \limits_{k=1}^{10}\frac{1}{7(r_k-1)}$ is equal to.......
4. Let $x_1$, $x_2$, $x_3$, $x_4$ be the solution of the equation $4x^4$+ $8x^3$– $17x^2$– $12x $ + 9 = 0 and $(4+x_1^2)$$(4+x_2^2)$$(4+x_3^2)$$(4+x_4^2)$=$\frac{125}{16}m$. Then the value of $m$ is........
5. Let $L_1$, $L_2$ be the lines passing through the point $P(0, 1)$ and touching the parabola $9x^2$ + $12x$ + $18y$ – 14 = 0. Let $Q$ and $R$ be the points on the lines $L_1$ and $L_2$ such that the $\Delta PQR$ is an isosceles triangle with base $QR$. If the slopes of the lines $QR$ are $m_1$ and $m_2$. then 16$(m_1^2+m_2^2)$ is equal to _______.
6. If the second, third and fourth terms in the expansion of $(x + y)^n$ are 135, 30 and $\frac{10}{3}$,respectively, then $6(n^3+x^2+y)$ is equal to _____.
7. Let the first term of a series be $T_1$ = 6 and its $r^{th}$term $T_r$ = 3 $T_{r–1}$ + $6r$, $r$ = 2, 3, ....., $n$. If the sum of the first $n$ terms of this series is $\frac{1}{5}(n^2-12n+39)$ $(4.6^n-5.3^n+1)$. Then $n$ is equal to ______.
8. For $n \in N$, if $\cot^{–1}3$ + $\cot^{–1}4$ + $\cot^{–1}5$ + $\cot^{-1} n$=$\frac{\pi}{4}$, then $n$ is equal to ________.
9. Let $P$ be the point (10, –2, –1) and $Q$ be the foot of the perpendicular drawn from the point $R(1, 7, 6)$ on the line passing through the points (2, –5, 11) and (–6, 7, –5). Then the length of the line segment $PQ$ is equal to ________.
10. Let $\vec{a}$=$2\hat{i}$$-3\hat{j}$+$4\hat{k}$, $\vec{b}$=$3\hat{i}$+$4\hat{j}$$-5\hat{k}$, and a vector $\vec{c}$ be such that $\vec{a}×(\vec{b}+\vec{c})$+$\vec{b}×\vec{c}$=$\hat{i}$+$8\hat{j}$+$13\hat{k}$. If $\vec{a}•\vec{c}$=13, then (24-\vec{b}•\vec{c})$ is equal to.........

06 Apr 2024:Shift-2

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