Download JEE Main 2023 Question Paper (06 Apr - Shift 2) 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. Let $P$ be a square matrix such that $P^2$ = $I – P$. For $\alpha, \beta, \gamma, \delta \in IN$, if $P^{\alpha}$ + $P^{\beta}$ = $\gamma I – 29P$ and $P^{\alpha} – P^{\beta}$ = $\delta I – 13P$, then $\alpha+\beta+\gamma–\delta$ is equal to
   
   1. 40
   2. 24
   3. 18
   4. 22
2. Let the sets $A$ and $B$ denote the domain and range respectively of the function $f(x)$=$$\frac{1}{\sqrt{\lceil x \rceil-x}}$$, where $\lceil x \rceil$ denotes the smallest integer greater than or equal to $x$. Then among the statements (S1) : $A \cap B$ = $(1, \infty) – N$ and
   (S2) : $A \cup B$ = $(1, \infty)$
   
    
   1. both (S1) and (S2) are true
   2. only (S2) is true
   3. only (S1) is true
   4. neither (S1) nor (S2) is true
3.  If the coefficient of $x^7$ in $\left(ax^2+\frac{1}{2bx}\right)^{11}$ and $x^{-7}$ in $\left(ax-\frac{1}{3bx^2}\right)^{11}$ are equal. Then
   
    
   1. $729ab = 32$
   2. $64ab = 243$
   3. $32ab = 729$
   4. $243ab = 64$
4.  The area bounded by the curves $y$=$|x-1|$+$|x-2|$ and $y = 3$ is equal to
   
    
   1. 3
   2. 5
   3. 4
   4. 6
5.  All the letters of the word PUBLIC are written in all possible orders and these words are written as in a dictionary with serial number. The serial number of the word PUBLIC is
   
    
   1. 578
   2. 576
   3. 582
   4. 580
6. Let $f(x)$ be a function satisfying $f(x)$+$f(\pi–x)$ =$\pi^2$, $\forall x \in IR$. Then $\int \limits_{0}^{\pi}f(x)\sin x dx$ is equal to
   
    
   1. $\frac{\pi^2}{2}$
   2. $\pi^2$
   3. $\frac{\pi^2}{4}$
   4. $2 \pi^2$
7.  Among the statements:
   (S1) : $2023^{2022}$–$1999^{2022}$ is divisible by 8
   (S2) : $13(13)^n$ – $11n$ – 13 is divisible by 144 for infinitely many $n \in IN$
   
    
   1. only (S2) is correct
   2. only (S1) is correct
   3. both (S1) and (S2) are incorrect
   4. both (S1) and (S2) are correct
8. $\lim \limits_{n \to \infty} \left\{\left(2^{\frac{1}{2}}-2^{\frac{1}{3}}\right)\right.$$ \left(2^{\frac{1}{2}}-2^{\frac{1}{5}}\right)$......$\left. \left(2^{\frac{1}{2}}-2^{\frac{1}{2n+1}}\right)\right\}$ is equal to
   
    
   1. 0
   2. $\frac{1}{\sqrt{2}}$
   3. $\sqrt{2}$
   4. 1
9.  In a group of 100 persons 75 speak English and 40 speak Hindi. Each person speaks at least one of the two languages. If number of persons, who speak only English is $\alpha$ and the number of personas who speak only Hindi is $\beta$, then the eccentricity of the ellipse $25(\beta^2x^2+\alpha^2y^2)$=$\alpha^2\beta^2$ is
   
    
   1. $\frac{\sqrt{117}}{12}$
   2. $\frac{\sqrt{129}}{12}$
   3. $\frac{\sqrt{119}}{12}$
   4. $\frac{3\sqrt{15}}{12}$
10.  The Sum of all values of $\alpha$ for which the points whose position vectors are $\hat{i}-2\hat{j}+3\hat{k}$, $2 \hat{i}-3\hat{j}+4\hat{k}$, $(\alpha+1)\hat{i}+2\hat{k}$ and $9\hat{i}+(\alpha-8)\hat{j}+6\hat{k}$ are coplanar, is equal to ..........
    
     
    1. 6
    2. -2
    3. 4
    4. 2
11.  If $gcd (m, n)$ = 1 and
    $1^2$–$2^2$+$3^2$–$4^2$ + ….. + $(2021)^2$ – $(2022)^2$ + $(2023)^2$ = $1012 m^2 n$. The $m^2 – n^2$ is equal to
    
     
    1. 200
    2. 180
    3. 240
    4. 220
12. Let the vectors $\vec{a}$, $\vec{b}$, $\vec{c}$ represent three conterminous edges of a parallelepiped of volume $V$. Then the volume of the parallelepiped, whose coterminous edges are represented by $\vec{a}$, $\vec{b}$+$\vec{c}$ and $\vec{a}$+$2\vec{b}$+$3\vec{c}$ is equal to:
    
     
    1. $2V$
    2. $6V$
    3. $3V$
    4. $V$
13.  If the solution curve $f(x,y)$= 0 of the differential equation $(1+\log_ex)\frac{dx}{dy}-x\log_ex$=$e^y$, $x>0$ passes through the points (1, 0) and $(\alpha , 2)$, then $\alpha^{\alpha}$ is equal to
    
    1. ${{e^{2e}}^{\sqrt{2}}}$
    2. $e^{2e^2}$
    3. $e^{e^2}$
    4. $e^{\sqrt{2}e^2}$
14.  For the system of equations $x$ + $y$ + $z$ = 6 $x$ + 2$y$ + $\alpha z$ = 10 $x$ + 3$y$ + 5$z$ = $\beta$, which one the following is Not true?
    
     
    1. System has a unique solution for $\alpha$ = – 3, $\beta$ = 14
    2. System has infinitely many solutions for $\alpha$ = 3, $\beta$ = 14.
    3. System has a unique solution for $\alpha$ = 3, $\beta \neq$14.
    4. System has no solution for $\alpha$ = 3, $\beta$ = 24.
15.  Among the statements
    (S1) : (p=>q)∨((~p)∧q) is a tautology
    (S2) : (q=>p) => ((~p)∧q) is a contradiction
    
     
    1. only (S2) is True
    2. neither (S1) and (S2) is True
    3. Both (S1) and (S2) are True
    4. only (S1) is True
16.  Let the line $L$ pass through the point (0, 1, 2), intersect the line $\frac{x-1}{2}$=$\frac{y-2}{3}$=$\frac{z-3}{4}$ and be parallel to the plane 2$x$ + $y – 3z$ = 4. Then the distance of the point $P(1, –9 ,2)$ from the line $L$ is.
    
     
    1. $\sqrt{74}$
    2. $\sqrt{69}$
    3. $\sqrt{54}$
    4. 9
17.  Three dice are rolled. If the probability of getting different numbers on the three dice is $\frac{p}{q}$, where $p$ and $q$ are co-prime, then $q – p$ is equal to
    
     
    1. 4
    2. 2
    3. 3
    4. 1
18.  A plane $P$ contains the line of intersection of planes $\vec{r}•(\hat{i}+\hat{j}+\hat{k})$=6 and $\vec{r}•(2\hat{i}+3\hat{j}+4\hat{k})$=-5. If $P$ passes through the point (0,2,-2), then square of distance of point (12,12,18) from the plane $P$ is
    
     
    1. 310
    2. 1240
    3. 620
    4. 155
19.  Let $a \neq b$ be two non-zero real numbers. Then the number of elements in the set $X$= {$z \in C$ : $Re(az^2 + bz)$ = $a$ and $Re(bz^2 + az) = b$} is equal to
    
     
    1. 2
    2. 0
    3. 1
    4. 3
20.  If the tangents at the points $P$ and $Q$ on the circle $x^2$ + $y^2$ – $2x$ + $y$ = 5 meet at the point $R\left(\frac{9}{4}, 2\right)$, thenthe area of the triangle $PQR$ is:
    
    1. $\frac{5}{4}$
    2. $\frac{5}{8}$
    3. $\frac{13}{8}$
    4. $\frac{13}{4}$

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 $f(x)$=$\frac{x}{\left(1+x^2\right)^{\frac{1}{n}}}$, $x \in IR -{-1}$, $n \in N$, $n >2$. If $f^n(x)$= ($fofof$…. upto $n$ times) $(x)$, then $\lim \limits_{n \to \infty} \int \limits_{0}^{1}x^{n-2}\left(f^n(x)\right)dx$ is equal to..........
2.  If the mean and variance of the frequency distribution
   

   $x_i$	2	4	6	8	10	12	14	16
   $f_i$	4	4	$\alpha$	15	8	$\beta$	4	5

   
   are 9 and 15.08 respectively, then the value of $\alpha^2+\beta^2-\alpha \beta$ is.............
3.  If $(20)^{19}$+ $2 (21)(20)^{18}$ + $3(21)^2(20)^{17}$ + ….. + $20(21)^{19}$ = $k (20)^{19}$, the $k$ is equal to ______ .
4.  Let the eccentricity of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}$=1 is reciprocal to that of the hyperbola $2x^2 – 2y^2$ = 1. If the ellipse intersects the hyperbola at right angles, then square of length of the latus-rectum of the ellipse is…….
5.  For $\alpha, \beta, z \in C$ and $\lambda >1$, if $\sqrt{\lambda-1}$ is the radius of the circle $|z-\alpha|^2$+$|z-\beta|^2$=$2\lambda$, then $|\alpha-\beta|$ is equal to............
6.  If the lines $\frac{x-1}{2}$=$\frac{2-y}{-3}$=$\frac{z-3}{\alpha}$ and $\frac{x-4}{5}$=$\frac{y-1}{2}$=$\frac{z}{\beta}$ intersect, then the magnitude of the minimum value of $8\alpha \beta$ is _____
7.  The number of points, where the curve $y$ = $x^5$– $20x^3$ + $50x$ +2 crosses the $x–$axis, is ______ .
8.  The value of tan9º – tan 27º – tan63º + tan81º is____________
9.  Let a curve $y = f(x)$, $x \in (0, \infty)$ pass through the points $P\left(1, \frac{3}{2}\right)$ and $Q\left(\alpha, \frac{1}{2}\right)$. If the tangent at any point $R(b, f(b))$ to the given curve cuts the $y-$axis at the point $S(0, c)$ such that $bc$ = 3, then $(PQ)^2$is equal to ________ .
10.  The number of 4-letter words, with or without meaning, each consisting of 2 vowels and 2 consonants, which can be formed from the letters of the word UNIVERSE without repetition is ______.

06 Apr 2023:Shift-1

06 Apr 2023:Ans-Key

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