Download JEE Main 2023 Question Paper (06 Apr - Shift 1) Mathematics

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08 Apr 2023:Shift-1

08 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. 1Let $A$={$x \in R:[x+3]+[x+4] \leq 3$}, $B$=$\left\{x \in R:3^x\left(\sum \limits_{r=1}^{\infty}\frac{3}{10^r}\right)^{x-3} < 3^{-3x}\right\}$, where $[t]$ denotes greatest integer functions. Then
   
   1. $A=B$
   2. $A⊂B$, $A \neq B$
   3. $B⊂C$, $A \neq B$
   4. $A \cap B =\phi$
2. $I(x)$=$\int \frac{x^2(x \sec^2x+\tan x)}{(x \tan x+1)^2}dx$. If $I(0)=0$, then $I\left(\frac{\pi}{4}\right)$ is equal to
   
    
   1. $\log_e\frac{(\pi+4)^2}{32}-\frac{\pi^2}{4(\pi+4)}$
   2. $\log_e\frac{(\pi+4)^2}{16}+\frac{\pi^2}{4(\pi+4)}$
   3. $\log_e\frac{(\pi+4)^2}{32}+\frac{\pi^2}{4(\pi+4)}$
   4. $\log_e\frac{(\pi+4)^2}{16}-\frac{\pi^2}{4(\pi+4)}$
3.  The mean and variance of a set of 15 numbers are 12 and 14 respectively. The mean and variance of another set of 15 numbers are 14 and $\sigma^2$ respectively. If the variance of all the 30 numbers in the two sets is 13, then $\sigma^2$ is equal to
   
    
   1. 11
   2. 12
   3. 10
   4. 9
4.  Let $a_1$, $a_2$, $a_3$,……… an, be $n$ positive consecutive terms of an arithmetic progression If $d$ > 0 is its common difference, then $\lim \limits_{n \to \infty} \sqrt{\frac{d}{n}}\left(\frac{1}{\sqrt{a_1}+\sqrt{a_2}}+.......+\frac{1}{\sqrt{a_{n-1}}+\sqrt{a_n}}\right)$ is
   
    
   1. $\frac{1}{\sqrt{d}}$
   2. $\sqrt{d}$
   3. 0
   4. 1
5.  From the top $A$ of a vertical wall $AB$ of height 30$m$, the angles of depression of the top $P$ and bottom $Q$ of a vertical tower $PQ$ are 15° and 60°respectively, $B$ and $Q$ are on the same horizontal level. If $C$ is a point on $AB$ such that $CB$ = $PQ$, then the are (in $m^2$) of the quadrilateral $BCPQ$ is equal to
   
    
   1. $300(\sqrt{3}+1)$
   2. $200(3-\sqrt{3})$
   3. $600(\sqrt{3}-1)$
   4. $300(\sqrt{3}-1)$
6.  Statement (P=>Q)∧(R=>Q) is logically equivalent to :
   
    
   1. (P=>R)∧(Q=>R)
   2. (P=>R)∨(Q=>R)
   3. (P∧R)=>Q
   4. (P∨R)=>Q
7.  Sum of the first 20 terms of the series : 5 + 11 + 19 + 29 + 41+...........is
   
    
   1. 3520
   2. 3420
   3. 3450
   4. 3250
8.  The sum of all the roots of the equation $|x^2–8x+15|$$-2x$+7=0 is :
   
    
   1. $11+\sqrt{3}$
   2. $9+\sqrt{3}$
   3. $9-\sqrt{3}$
   4. $11-\sqrt{3}$
9.  Let $A$ = $[a_{ij}]_{2×2}$, where $a_{ij} \neq 0$ for all $i, j$ and $A^2 = I$. Let $a$ be the sum of all diagonal elements of $A$ and $b = |A|$. Then $3a^2+ 4b^2$ is equal to
   
    
   1. 4
   2. 14
   3. 7
   4. 3
10.  One vertex of a rectangular parallelepiped is at the origin $O$ and the lengths of its edges along $x$, $y$ and $z$ axes are 3, 4 and 5 units respectively. . Let $P$ be the vertex (3, 4, 5) . Then the shortest distance between the diagonal $OP$ and an edge parallel to $z$ axis, not passing through $O$ or $P$ is :
    
     
    1. $12\sqrt{5}$
    2. $\frac{12}{5\sqrt{5}}$
    3. $\frac{12}{5}$
    4. $\frac{12}{\sqrt{5}}$
11.  If the system of equations
    $x$ + $y$ + $az$ = $b$
    2$x$ + 5$y$ + 2$z$ = 6
    $x$ + 2$y$ + 3$z$ = 3
    has infinitely many solutions, then $2a + 3b$ is equal to
    
     
    1. 25
    2. 28
    3. 23
    4. 20
12.  If the equation of the plane passing through the line of intersection of the planes $2x – y$ + $z$ = 3, $4x – 3y$ + $5z$ + 9 = 0 and parallel to the line $\frac{x+1}{-2}$=$\frac{y+3}{4}$=$\frac{z-2}{5}$ is $ax$ + $by$ + $cz$ + 6 = 0, then $a + b + c$ is equal to
    
     
    1. 14
    2. 15
    3. 13
    4. 12
13. The straight lines $l_1$ and $l_2$ pass through the origin and trisect the line segment of the line $L$ : $9x + 5y$ = 45 between the axes. If $m_1$ and $m_2$ are the slopes of the lines $I_1$ and $I_2$ then the point of intersection of the line $y$ = $(m_1 + m_2)x$ with $L$ lies on
    
     
    1. $y-2x$=15
    2. $6x-y$=15
    3. $6x+y$=10
    4. $y-x$=5
14. If $^{2n}C_3$ : $^nC_3$ = 10 : 1, then the ratio $(n^2 + 3n)$ : $(n^2 – 3n + 4)$ is
    
     
    1. 2:1
    2. 35:16
    3. 27:11
    4. 65:37
15.  Let $\vec{a}$=$2\hat{i}+3\hat{j}+4\hat{k}$, $\vec{b}$=$\hat{i}-2\hat{j}-2\hat{k}$ and $\vec{c}$=$-\hat{i}$+$4\hat{j}$+$3\hat{k}$ if $\vec{d}$ is a vector perpendicular to both $\vec{b}$ and $\vec{c}$ and $\vec{a}$•$\vec{d}$=18, then $|\vec{a}×\vec{d}|^2$ is equal to
    
     
    1. 720
    2. 680
    3. 760
    4. 640
16.  A pair of dice is thrown 5 times. For each throw, a total of 5 is considered a success. If the probability of at least 4 successes is $\frac{k}{3^{11}}$, then $k$ is equal to.........
    
     
    1. 123
    2. 164
    3. 82
    4. 75
17. Let the position vectors of the points $A$, $B$ , $C$ and $D$ be $5\hat{i}$+$5\hat{j}$+$2\lambda \hat{k}$, $\hat{i}$+$2\hat{j}$+$3 \hat{k}$, $-2\hat{i}$+$\lambda \hat{j}$+$4 \hat{k}$ and $-\hat{i}+5 \hat{j}$+$6 \hat{k}$. Let the set $S$={$\lambda \in R$ : the points $A$, $B$, $C$ and $D$ are coplanar } . Then $\sum \limits_{\lambda \in x}(\lambda+2)^2$ is equal to
    
     
    1. 41
    2. $\frac{37}{2}$
    3. 13
    4. 25
18.  Let $5f(x)$+$4f\left(\frac{1}{x}\right)$=$\frac{1}{x}$+3, $x >0$. Then $18 \int \limits_{1}^{2}f(x)dx$ is equal to
    
     
    1. $5\log_e2-3$
    2. $5\log_e2+3$
    3. $10\log_e2+6$
    4. $10\log_e2-6$
19.  If $2x^y$ + $3y^x$= 20, then $\frac{dy}{dx}$ at (2, 2) is equal to :
    
     
    1. $-\left(\frac{3+\log_e16}{4+\log_e8}\right)$
    2. $-\left(\frac{2+\log_e8}{3+\log_e4}\right)$
    3. $-\left(\frac{3+\log_e4}{2+\log_e8}\right)$
    4. $-\left(\frac{3+\log_e8}{2+\log_e4}\right)$
20.  If the ratio of the fifth term from the beginning to the fifth term form the end in the expansion of $$\left(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}}\right)^n$$ is $\sqrt{6}$:1, then the third term from the beginning is :
    
     
    1. $30 \sqrt{2}$
    2. $60 \sqrt{3}$
    3. $60 \sqrt{2}$
    4. $30 \sqrt{3}$

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 $a \in Z$ and $[t]$ be the greatest integer $\leq t$. Then the number of points, where the function $f(x)$=$[a+13sinx]$, $x \in (0, \pi)$ is not differentiable, is _________.
2.  A circle passing through the point $P(\alpha,\beta)$ in the first quadrant touches the two coordinate axes at the points $A$ and $B$. The point $P$ is above the line $AB$. The point $Q$ on the line segment $AB$ is the foot of perpendicular from $P$ on $AB$. If $PQ$ is equal to 11 units, then the value of $\alpha \beta$ is ___________.
3.  Let $A$ = {1,2,3, 4,…….10} and $B$ = {0, 1,2,3,4} . The number of elements in the relation $R$ = {$(a,b) \in A×A$ : $2(a–b)^2$+$3(a–b) \in B$} is ________.
4.  Let the image of the point $P(1,2,3)$ in the plane $2x – y + z$ = 9 be $Q$. If the coordinates of the point $R$ are (6, 10, 7) , then the square of the area of the triangle $PQR$ is _______.
5. Let the point $(p, p + 1)$ lie inside the region $\left\{E={x,y}:3-x \leq y \leq \sqrt{9-x^2}\right.$, $\left. 0 \leq x \leq 3 \right\}$. If the set of all values of $p$ is the interval $(a, b)$, then $b^2+b–a^2$is equal to _______.
6. If the area of the region $S$=$\left\{(x, y):2y-y^2 \leq x^2 \leq 2y, x \geq y \right\}$ is equal to $\frac{n+2}{n+1}-\frac{\pi}{n-1}$ then the natural number $n$ is equal to _________.
7.  The coefficient of $x^{18}$ in the expansion of $\left(x^4-\frac{1}{x^3}\right)^{15}$ is...........
8.  Let $y = y(x)$ be a solution of the differential equation $(xcosx)dy$ + $(xysinx+ycosx–1)dx$ = 0, $0
9.  Let the tangent to curve $x^2$+$2x – 4y$+$9x$= 0 at the point $P(1,3)$ on it meet the $y-$axis at $A$. Let the line passing through $P$ and parallel to the line $x – 3y$ = 6 meet the parabola $y^2$ = $4x$ at $B$. If $B$ lies on the line $2x – 3y$ = 8, then $(AB)^2$ is equal to _______________.
10.  The number of ways of giving 20 distinct oranges to 3 children such that each child gets at least one orange is _________

06 Apr 2023:Shift-2

06 Apr 2023:Ans-Key

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