Download JEE Main 2023 Question Paper (10 Apr - Shift 2) Mathematics

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11 Apr 2023:Shift-1

11 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. If the points $P$ and $Q$ are respectively the circumcenter and the orthocentre of a $∆ABC$, then $\vec{PA}$+$\vec{PB}$+$\vec{PC}$ is equal to
   
   1. $\vec{QP}$
   2. 2$\vec{PQ}$
   3. $\vec{PQ}$
   4. 2$\vec{QP}$
2.  If $S_n$ = 4 + 11 + 21 + 34 + 50 + ……… to $n$ terms, then $\frac{1}{60}(S_{29} – S_{9})$ is equal to
   
    
   1. 220
   2. 226
   3. 227
   4. 223
3.  If $A$=$\frac{1}{5!6!7!}\begin{equation*}\begin{bmatrix}5! & 6! & 7! \\ 6! & 7! & 8! \\ 7! & 8! & 9! \end{bmatrix}\end{equation*}$, then $|adj(adj(2A))|$ is equal to
   
    
   1. $2^8$
   2. $2^{16}$
   3. $2^{12}$
   4. $2^{20}$
4.  The statement ~ [p ∨ (~ (p∧q))] is equivalent to
   
    
   1. ~(p∧q)
   2. ~(p∨q)
   3. (~(p∧q))∧q
   4. (p∧q)∧(~p)
5.  Eight persons are to be transported from city A to city B in three cars of different makes. If each car can accommodate at most three persons, then the number of ways, in which they can be transported, is
   
    
   1. 1120
   2. 1680
   3. 3360
   4. 560
6.  Let a circle of radius 4 be concentric to the ellipse $15x^2$+ $19y^2$ = 285. Then the common tangents are inclined to the minor axis of the ellipse at the angle
   
    
   1. $\frac{\pi}{3}$
   2. $\frac{\pi}{12}$
   3. $\frac{\pi}{6}$
   4. $\frac{\pi}{4}$
7.  Let the number $(22)^{2022}$ + $(2022)^{22}$ leave the remainder $\alpha$ when divided by 3 and $\beta$ when divided by 7 . Then $(\alpha^2 + \beta^2)$ is equal to :
   
    
   1. 5
   2. 20
   3. 10
   4. 13
8.  Let $S$=$\left\{z=x+iy:\frac{2z-3i}{4z+2i} \text{is real number}\right\}$. Then which of the following is NOT correct ?
   
    
   1. $(x, y)$=$\left(0, -\frac{1}{2}\right)$
   2. $y$+$x^2$+$y^2$$\neq -\frac{1}{4}$
   3. $x=0$
   4. $y \in \left(-\infty, -\frac{1}{2}\right)\cup\left(-\frac{1}{2}, \infty\right)$
9.  For $\alpha$, $\beta$, $\gamma$, $\delta \in N$, if $\int \left(\left(\frac{x}{e}^{2x}\right)+\left(\frac{e}{x}^{2x}\right)\right)\log_exdx$=$\frac{1}{\alpha}\left(\frac{x}{e}\right)^{\beta x}$$-\frac{1}{\gamma}\left(\frac{e}{x}\right)^{\delta x}$+$C$, where $e$=$\sum \limits_{n=0}^{\infty}\frac{1}{n!}$ and $C$ is constant of integration, then $\alpha$ + 2$\beta$ + 3$\gamma$ – 4$\delta$ is equal to
   
    
   1. 4
   2. -4
   3. -8
   4. 1
10.  If the coefficients of $x$ and $x^2$in $(1 + x)^p(1–x)^q$ are 4 and –5 respectively, then $2p + 3q$ is equal to
    
     
    1. 69
    2. 60
    3. 66
    4. 63
11.  Let $A$ = {2, 3, 4} and $B$ = {8, 9, 12}. Then the number of elements in the relation $R$ = {($(a_1, b_1)$, $(a_2 , b_2)$) $\in (A × B, A × B)$ : $a_1$ divides $b_2$ and $a_2$ divides $b_1$} is :
    
     
    1. 18
    2. 36
    3. 12
    4. 24
12.  Let $\mu$ be the mean and $\sigma$ be the standard deviation of the distribution
    

    $x_i$	0	1	2	3	4	5
    $f_i$	$k+2$	$2k$	$k^2-1$	$k^2-1$	$k^2+1$	$k-3$

    
    where $\sum f_i$=62. If $[x]$ denotes the greatest integer $\leq x$, then $[µ^2 + \sigma^2]$ is equal to
    
    
     
    1. 8
    2. 7
    3. 6
    4. 9
13.  Let $\vec{a}$=$2\hat{i}$+$7\hat{j}$$-\hat{k}$, $\vec{b}$=$3\hat{i}$+$5\hat{k}$ and $\vec{c}$=$\hat{i}-\hat{j}+2\hat{k}$. Let $\vec{d}$ be a vector which is perpendicular to both $\vec{a}$ and $\vec{b}$ and $\vec{c}•\vec{d}$=12. Then $(-\hat{i}+\hat{j}-\hat{k})•(\vec{c}×\vec{d})$ is equal to
    
     
    1. 48
    2. 42
    3. 44
    4. 24
14. Let a die be rolled $n$ times. Let the probability of getting odd numbers seven times be equal to the probability of getting odd numbers nine times. If the probability of getting even numbers twice is $\frac{k}{2^{15}}$, then $k$ is equal to :
    
     
    1. 90
    2. B
    3. 60
    4. 15
15.  Let $A$ be the point (1, 2) and $B$ be any point on the curve $x^2$ + $y^2$ = 16. If the centre of the locus of the point $P$, which divides the line segment $AB$ in the ratio 3 : 2 is the point $C (\alpha, \beta)$, then the length of the line segment $AC$ is
    
     
    1. $\frac{4\sqrt{5}}{5}$
    2. $\frac{2\sqrt{5}}{5}$
    3. $\frac{3\sqrt{5}}{5}$
    4. $\frac{6\sqrt{5}}{5}$
16.  Let $S$=$\left\{x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right):9^{1-\tan^2x}+9^{\tan^2x}=10\right\}$ and $\beta$=$\sum \limits_{x \in S}\tan^2\frac{x}{3}$, then $\frac{1}{6}(\beta-14)^2$ is equal to
    
     
    1. 8
    2. 16
    3. 32
    4. 64
17. Let $g(x)$=$f(x)$+$f(1-x)$ and $f''(x)>0$, $x \in (0, 1)$, If $g $is decreasing in the interval $(0, \alpha)$ and increasing in the interval $(\alpha, 1)$ then $\tan^{-1}(2\alpha)$+$\tan^{-1}\left(\frac{1}{\alpha}\right)$+$\tan^{-1}\left(\frac{\alpha+1}{\alpha}\right)$ is equal to
    
     
    1. $\frac{3\pi}{4}$
    2. $\frac{3\pi}{2}$
    3. $\pi$
    4. $\frac{5\pi}{4}$
18.  Let the line $\frac{x}{1}$=$\frac{6-y}{2}$=$\frac{z+8}{5}$ intersect the lines $\frac{x-5}{4}$=$\frac{y-7}{3}$=$\frac{z+2}{1}$ and $\frac{x+3}{6}$=$\frac{3-y}{3}$=$\frac{z-6}{1}$ at the points $A$ and $B$ respectively. Then the distance of the mid-point of the line segment $AB$ from the plane $2x $$– 2y$ + $z$ = 14 is :
    
     
    1. 3
    2. $\frac{10}{3}$
    3. 4
    4. $\frac{11}{3}$
19. Let $f$ be continuous function satisfying $\int \limits_{0}^{t^2}(f(x)+x^2)dx$=$\frac{4}{3}t^3$, $\forall t>0$. Then $f\left(\frac{\pi^2}{4}\right)$ is equal to
    
     
    1. $\pi\left(1-\frac{\pi^3}{16}\right)$
    2. $\pi^2\left(1-\frac{\pi^2}{16}\right)$
    3. $-\pi^2\left(1+\frac{\pi^2}{16}\right)$
    4. $-\pi\left(1+\frac{\pi^3}{16}\right)$
20.  Let the image of the point $P(1, 2, 6)$ in the plane passing through the points $A(1, 2, 0)$, $B(1, 4, 1)$ and $C (0, 5,1)$ be $Q (\alpha, \beta, \gamma)$. Then $(\alpha^2 + \beta^2 + \gamma^2)$ is equal to
    
     
    1. 65
    2. 70
    3. 62
    4. 76

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. In the figure, $\theta_1$+$\theta_2$=$\frac{\pi}{2}$ and $\sqrt{3}(BE)$=$4(AB)$. If the area of $\Delta CAB$ is $2\sqrt{3}-3 unit^2$, when $\frac{\theta_2}{\theta_1}$ is the largest, then the perimeter (in unit) of $\Delta CED$ is equal to ________.
   
2.  Let the foot of perpendicular from the point $A (4,3,1)$ on the plane $P$ : $x – y$ + $2z$ + 3 = 0 be $N$. If $B(5, \alpha, \beta), \alpha, \beta \in Z$ is a point on plane $P$ such that the area of the triangle $ABN$ is $3\sqrt{2}$, then $\alpha^2$+$\beta^2$ +$\alpha \beta$ is equal to ______.
3.  Let $S$ be the set of values of $\lambda$, for which the system of equations $6\lambdax$$ – 3y$ + $3z$ = $4\lambda^2$, $2x$ + $6\lambda y$ + $4z$ = 1, $3x$ + $2y$ + $3\lambda z$ = $\lambda$ has no solution. Then $12\sum \limits_{\lambda \in S}|\lambda|$ is equal to........
4.  Let the tangent at any point $P$ on a curve passing through the points (1,1) and $\left(\frac{1}{10}, 100\right)$, intersect positive $x-$axis and $y-$axis at the point $A$ and $B$ respectively. If $PA$ : $PB$ = $1 : k$ and $y = y(x)$ is the solution of the differential equation $e^{\frac{dy}{dx}}$=$kx$+$\frac{k}{2}$, $y(0)=k$, then $4y(1)-5\log_e3$ is equal to ........
5.  Let the equation of two adjacent sides of a parallelogram $ABCD$ be $2x – 3y$ = –23 and $5x$ + $4y$ = 23. If the equation of its one diagonal $AC$ is $3x$ + $7y$ = 23 and the distance of $A$ from the other diagonal is $d$, then $50 d^2$ is equal to _______.
6.  The sum of all the four-digit numbers that can be formed using all the digits 2, 1, 2, 3 is equal to _____.
7.  If the domain of the function $f(x)$ =$\sec^{-1}\left(\frac{2x}{5x+3}\right)$ is $[\alpha, \beta) \cup (\gamma, \delta]$, then $|3\alpha+10(\beta+\gamma)+21\delta|$ is equal to ...........
8.  Let the quadratic curve passing through the point (–1, 0) and touching the line $y = x$ at (1,1) be $y = f(x)$. Then the $x-$intercept of the normal to the curve at the point $(\alpha, \alpha + 1)$ in the first quadrant is ______.
9.  Suppose $a_1$, $a_2$, 2, $a_3$, $a_4$, be in an arithmetic co-geometric progression. If the common ratio of the corresponding geometric progression is 2 and the sum of all 5 terms of the arithmetic co-geometric progression is $\frac{49}{2}$, then $a_4$ is equal to ............
10.  If the area of the region {$x, y$ : $|x^2 – 2| \leq y \leq x$} is $A$, then 6$A$ + $16\sqrt{2}$ is equal to ______.

10 Apr 2023:Shift-1

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