Download JEE Main 2023 Question Paper (30 Jan - Shift 2) Mathematics

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31 Jan 2023:Shift-1

31 Jan 2023:Shift-1

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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 $f$, $g$ and $h$ be the real valued functions defined on $R$ as
   $f(x)$=$\left\{\begin{array}{cc}\frac{x}{|x|}, & x \neq 0 \\ 1, & x =0\end{array}\right.$, $g(x)$=$\left\{\begin{array}{cc}\frac{\sin (x+1)}{(x+1)}, & x \neq -1 \\ 1, & x =-1\end{array}\right.$
   and $h(x)$=$2[x]-f(x)$, where $[x]$ is the greatest integer $\leq x$. Then the value of $\lim \limits_{x \to 1}g(h(x-1))$ is
   
    
   1. 0
   2. sin(1)
   3. -1
   4. 1
2.  Let $\vec{a}$ and $\vec{b}$ be two vectors. Let $|\vec{a}|$=1, $|\vec{b}|$=4 and $\vec{a}•\vec{b}$=2. If $\vec{c}$=$(2\vec{a}×\vec{b})-3\vec{b}$, then the value of $\vec{b}•\vec{c}$ is
   
    
   1. $-48$
   2. $-24$
   3. $-60$
   4. $-84$
3.  The solution of the differential equation $\frac{dy}{dx}$=$-\left(\frac{x^2+3y^2}{3x^2+y^2}\right)$, $y(1)$=0 is
   
    
   1. $\log_e|x+y|-$$\frac{2xy}{(x+y)^2}$=0
   2. $\log_e|x+y|+$$\frac{2xy}{(x+y)^2}$=0
   3. $\log_e|x+y|+$$\frac{xy}{(x+y)^2}$=0
   4. $\log_e|x+y|-$$\frac{xy}{(x+y)^2}$=0
4. Let $\lambda \in R$, $\vec{a}$=$\lambda \hat{i}$+$2 \hat{j}-3\hat{k}$, $\vec{b}$=$\hat{i}$$-\lambda \hat{j}+2\hat{k}$. If $((\vec{a}+\vec{b})×(\vec{a}×\vec{b}))×(\vec{a}-\vec{b})$=$8\hat{i}-40\hat{j}-24\hat{k}$, then $|\lambda (\vec{a}+\vec{b})×(\vec{a}-\vec{b})|^2$ is equal to
   
    
   1. 136
   2. 132
   3. 140
   4. 144
5.  If a plane passes through the points $(-1, k, 0)$, $(2, k, -1)$, $(1, 1, 2)$ and is parallel to the line $\frac{x-1}{1}$=$\frac{2y+1}{2}$=$\frac{z+1}{-1}$, then the value of $\frac{k^2+1}{(k-1)(k-2)}$ is
   
    
   1. $\frac{6}{13}$
   2. $\frac{13}{6}$
   3. $\frac{5}{17}$
   4. $\frac{17}{5}$
6. $\lim \limits_{n \to \infty} \frac{3}{n} \left\{4+\left(2+\frac{1}{n}\right)^2 \right.$+$\left. \left(2+\frac{2}{n}\right)^2+...+\left(3-\frac{1}{n}\right)^2 \right \}$ is equal to
   
    
   1. $\frac{19}{3}$
   2. 12
   3. 0
   4. 19
7. Let $a_1$=1, $a_2$, $a_3$, $a_4$, ..... be consecutive natural numbers. Then $\tan^{-1} \left(\frac{1}{1+a_1a_2}\right)$+$\tan^{-1} \left(\frac{1}{1+a_2a_3}\right)$+......+$\tan^{-1} \left(\frac{1}{1+a_{2021}a_{2022}}\right)$ is equal to
   
    
   1. $\cot^{-1}(2022)-\frac{\pi}{4}$
   2. $\tan^{-1}(2022)-\frac{\pi}{4}$
   3. $\frac{\pi}{4}-\tan^{-1}(2022)$
   4. $\frac{\pi}{4}-\cot^{-1}(2022)$
8.  A vector $\vec{v}$ in the first octant is inclined to the $x-$ axis at 60°, to the $y-$ axis at 45° and to the $z-$ axis at an acute angle. If a plane passing through the points $(\sqrt{2}, -1, 1)$ and $(a, b, c)$ is normal to $\vec{v}$, then
   
    
   1. $a+b+\sqrt{2}c$=1
   2. $\sqrt{2}a+b+c$=1
   3. $a+\sqrt{2}b+c$=1
   4. $\sqrt{2}a-b+c$=1
9. If the functions $f(x)$=$\frac{x^3}{3}$+$2bx$+$\frac{ax^2}{2}$ and $g(x)$=$\frac{x^3}{3}$+$ax$+$bx^2$, $a \neq 2b$ have a common extreme point, then $a$+$2b$+$7$ is equal to:
   
    
   1. 3
   2. 6
   3. $\frac{3}{2}$
   4. 4
10.  Let $q$ be the maximum integral value of $p$ in [0, 10] for which the roots of the equation $x^2-px$+$\frac{5}{4}p$=0 are rational. Then the area of the region $\left\{(x,y):0 \leq y \leq (x-q)^2, 0 \leq x \leq q \right\}$ is
    
     
    1. 164
    2. $\frac{125}{3}$
    3. $243$
    4. $25$
11.  Let $A$ be a point on the $x-$axis. Common tangents are drawn from $A$ to the curves $x^2$+$y^2$=8 and $y^2$=$16x$. If one of these tangents touches the two curves at $Q$ and $R$, then $(QR)^2$ is equal to
    
     
    1. 76
    2. 81
    3. 72
    4. 64
12.  The number of ways of selecting two numbers $a$ and $b$, $a \in {2, 4, 6, ..., 100}$ and $b \in {1, 3, 5, ... , 99}$ such that 2 is the remainder when $a+b$ is divided by 23 is
    
     
    1. 268
    2. 108
    3. 54
    4. 186
13.  For $\alpha, \beta \in R$, suppose the system of linear equations
    $x-$$y-$$z$=5
    $2x$+$2y$+$\alpha z$=8
    $3x-$$y$+$4z$=$\beta$
    has infinitely many solutions. Then $\alpha$ and $\beta$ are the roots of
    
     
    1. $x^2$+$18x$+56=0
    2. $x^2$$-10x$+16=0
    3. $x^2$$-18x$+56=0
    4. $x^2$+$14x$+24=0
14.  Let $x$=$(8\sqrt{3}+13)^{13}$ and $y$=$(7\sqrt{2}+9)^{9}$. If $[t]$ denotes the greatest integer $\leq t$, then
    
     
    1. $[x]$ is even but $[y]$ is odd
    2. $[x]$ is even but $[y]$ is odd
    3. $[x]$ and $[y]$ are both odd
    4. $[x]$+$[y]$ is even
15.  Let $S$ be the set of all values of $a_1$for which the mean deviation about the mean of 100 consecutive positive integers $a_1$, $a_2$, $a_3$,....,$a_{100}$ is 25. Then $S$ is
    
     
    1. {99}
    2. {9}
    3. N
    4. $\phi$
16.  The parabolas: $ax^2$+$2bx$+$cy$=0 and $dx^2$+$2ex$+$fy$=0 intersect on the line $y$=1. If $a$, $b$, $c$, $d$, $e$, $f$ are positive real numbers and $a$, $b$, $c$ are in G.P., then
    
     
    1. $\frac{d}{a}$, $\frac{e}{b}, $\frac{f}{c}$ are in G.P.
    2. $d$, $e$, $f$ are in G.P.
    3. $d$, $e$, $f$ are in A.P.
    4. $\frac{d}{a}$, $\frac{e}{b}, $\frac{f}{c}$ are in A.P.
17.  If $P$ is a 3 x 3 real matrix such that $P^T$=$aP$+$(a-1)l$, where $a$>1, then
    
     
    1. $|Adj P|$>1
    2. $P$ is a singular matrix
    3. $|Adj P|$=$\frac{1}{2}$
    4. $|Adj P|$=1
18.  The range of the function $f(x)$=$\sqrt{3-x}$+$\sqrt{2+x}$ is:
    
     
    1. $[\sqrt{5}, \sqrt{10}]$
    2. $[\sqrt{5}, \sqrt{13}]$
    3. $[2\sqrt{2}, \sqrt{11}]$
    4. $[\sqrt{2}, \sqrt{7}]$
19.  Consider the following statements :
    P: I have fever
    Q: I will not take medicine
    R: I will take rest
    The statement “If I have fever, then I will take medicine and I will take rest” is equivalent to :
    
     
    1. (P∨Q)∧((~ P)∨R)
    2. ((~ P)∨~ Q)∧((~ P)∨R)
    3. ((~ P)∨~ Q)∧((~ P)∨~ R)
    4. (P∨~ Q)∧(P∨~ R)
20.  Let $a$, $b$, $c$ > 1, $a^3$, $b^3$ and $c^3$ be in A.P., and $\log_ab$, $\log_ca$ and $\log_bc$ be in G.P. If the sum of first 20 terms of an A.P., whose first term is $\frac{a+4b+c}{3}$ and the common difference is $\frac{a-8b+c}{10}$ is -444, then $abc$ is equal to:
    
    1. 343
    2. 216
    3. $\frac{125}{8}$
    4. $\frac{343}{8}$

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.  If the value of the real number $a$ > 0 for which $x^2-5ax$+1=0 and $x^2-ax-5$ have a common real root is $\frac{3}{\sqrt{2\beta}}$ then the $\beta$ is equal to .......
2.  The number of seven digits odd numbers, that can be formed using all the seven digits 1, 2, 2, 2, 3, 3, 5 is…………
3.  Let A be the area of the region {$(x, y):y \geq x^2$, $y \geq (1-x)^2$, $y \leq 2x(1-x)$}. Then 540 A is equal to…….
4.  A bag contains six balls of different colours. Two balls are drawn in succession with replacement. The probability that both the balls are of the same colour is $p$. Next four balls are drawn in succession with replacement and the probability that exactly three balls are of the same colour is $q$. If $p : q$=$ m : n $, where $m$ and $n$ are coprime, then $m+n$ is equal to……..
5.  If $\int \sqrt{sec 2x-1}dx$=$\alpha \log_e \left|\cos 2x+\beta+\sqrt{\cos2x\left(1+\cos \frac{1}{\beta}x\right)}\right|$+constant, then $\beta-\alpha$ is equal to ..........
6.  Let a line $L$ passes through the point $P(2, 3, 1)$ and be parallel to the line $x+3y-2z-$2=0=$x-y$+$2z$. If the distance of $L$ from the point $(5, 3, 8)$ is $\alpha$, then $3 \alpha^2$ is equal to.........
7. Let A={1, 2, 3, 5, 8, 9}. Then the number of possible functions $f : A \to A$ such that $f(m•n)$=$f(m)$•$f(n)$ for every $m, n \in A$ with $m•n \in A$ is equal to………….
8.  $50^{th}$ root of a number $x$ is 12 and $50^{th}$ root of another number $y$ is 18. Then the remainder obtained on dividing $(x+y)$ by 25 is……..
9.  Let $P(a_1 ,b_1)$ and $Q(a_2 ,b_2)$ be two distinct points on a circle with center $C(\sqrt{2}, \sqrt{3})$. Let $O$ be the origin and $OC$ be perpendicular to both $CP$ and $CQ$. If the area of the triangle $OCP$ is $\frac{\sqrt{35}}{2}$, then $a_1^2$+$a_2^2$+$b_1^2$+$b_2^2$ is equal to...........
10.  The $8^{th}$ common terms of the series
    $S_1$:3+7+11+15+19+......., $S_2$:1+6+11+16+21+.......
    is..............

30 Jan 2023:Shift-1

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