Download JEE Main 2023 Question Paper (01 Feb - Shift 1) Mathematics

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24 Jan 2023:Shift-1

24 Jan 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.  The area enclosed by the closed curve $C$ given by the differential equation $\frac{dy}{dx}$+$\frac{x+a}{y-2}$=0, $y(1)=0$ is $4\pi$. Let $P$ and $Q$ be the points of intersection of the curve $C$ and the $y-$axis. If normals at $P$ and and $Q$ on the curve $C$ intersect $x-$axis at points $R$ and $S$ respectively, then the length of the line segment RS is
   
   1. $2\sqrt{3}$
   2. 2
   3. $\frac{4\sqrt{3}}{3}$
   4. $\frac{2\sqrt{3}}{3}$
2.  The mean and variance of 5 observations are 5 and 8 respectively. If 3 observations are 1, 3, 5, then the sum of cubes of the remaining two observations is
   
    
   1. 1456
   2. 1792
   3. 1072
   4. 1216
3. Let $S$ be the set of all solutions of the equation $\cos^{-1}2x$$-2\cos^{-1}(\sqrt{1-x^2})$=$\pi$. $x \in \left[-\frac{1}{2}, \frac{1}{2}\right]$. Then $\sum \limits_{x \in S}2\sin^{-1}(x^2-1)$ is equal to
   
    
   1. $\pi-\sin^{-1}\left(\frac{\sqrt{3}}{4}\right)$
   2. $\frac{-2\pi}{3}$
   3. 0
   4. $\pi-\sin^{-1}\left(\frac{\sqrt{3}}{4}\right)$
4.  Let $R$ be a relation on $R$ , given by $R$={$(a, b):3a-3b+\sqrt{7}$ is an irrational number}. Then $R$ is
   
    
   1. reflexive but neither symmetric nor transitive
   2. reflexive and transitive but not symmetric
   3. reflexive and symmetric but not transitive
   4. an equivalence relation
5. The negation of the expression $q∨((~q)∧p)$ is equivalent to
   
   1. $(~p)∧(~q)$
   2. $(~p)∨q$
   3. $(~p)∨(~q)$
   4. $p∧(~q)$
6.  In a binomial distribution $B(n, p)$, the sum and the product of the mean and the variance are 5 and 6 respectively, then $6(n+p+q)$ is equal to
   
    
   1. 52
   2. 53
   3. 50
   4. 51
7.  For a triangle $ABC$, the value of $cos2A$+$cos2B$+$cos2C$ is least. If its inradius is 3 and incentre is $M$, then which of the following is NOT correct?
   
    
   1. $sin2A$+$sin2B$+$sin2C$=$sinA$+$sinB$+$sinC$
   2. perimeter of $∆ABC$ is $18\sqrt{3}$
   3. $\vec{MA}.\vec{MB}$=$-18$
   4. area of $∆ABC$ is $\frac{27\sqrt{3}}{2}$
8.  If $y$=$y(x)$ is the solution curve of the differential equation $\frac{dy}{dx}+y\tan x=x \sec x$, $0 \leq x \leq \frac{\pi}{3}$, $y(0)=1$, then $y\left(\frac{\pi}{6}\right)$ is equal to
   
    
   1. $\frac{\pi}{12}$+$\frac{\sqrt{3}}{2}\log_e\left(\frac{2}{e\sqrt{3}}\right)$
   2. $\frac{\pi}{12}$+$\frac{\sqrt{3}}{2}\log_e\left(\frac{2\sqrt{3}}{e}\right)$
   3. $\frac{\pi}{12}$$-\frac{\sqrt{3}}{2}\log_e\left(\frac{2\sqrt{3}}{e}\right)$
   4. $\frac{\pi}{12}$$-\frac{\sqrt{3}}{2}\log_e\left(\frac{2}{e\sqrt{3}}\right)$
9.  If the center and radius of the circle $\left|\frac{z-2}{z-3}\right|$=2 are respectively $(\alpha, \beta)$ and $\gamma$, then $3(\alpha, \beta, \gamma)$ is equal to
   
    
   1. 11
   2. 9
   3. 10
   4. 12
10. If the orthocentre of the triangle, whose vertices are (1, 2), (2, 3) and (3,1) is $(\alpha, \beta)$, then the quadratic equation whose roots are $\alpha+4\beta$ and $4\alpha+\beta$, is
    
     
    1. $x^2-$$19x$+90=0
    2. $x^2-$$19x$+80=0
    3. $x^2-$$20x$+99=0
    4. $x^2-$$22x$+120=0
11. Let $f(x)$=$2x$+$\tan^{-1}x$ and $g(x)$=$\log_e(\sqrt{1+x^2}+x)$, $x \in [0, 3]$. Then
    
    1. there exists $\hat{x} \in [0, 3]$ such that $f'(\hat{x})$ < $g'(\hat{x})$
    2. min $f'(x)$=1+max $g'(x)$
    3. there exist $0 < x_1 < x_2 < x_3$ such that $f(x) < g(x)$, $\forall x \in (x_1,x_2)$
    4. max $f(x)$>max $g(x)$
12.  Let $S$=$\left\{x:x \in R and (\sqrt{3}+\sqrt{2})^{x^2-4}+(\sqrt{3}-\sqrt{2})^{x^2-4}=10\right\}$. Then $n(S)$ is equal to
    
     
    1. 4
    2. 2
    3. 6
    4. 0
13. Let $S$ denote the set of all real values of $\lambda$ such that the system of equations
    $\lambda x$+$y$+$z$=1
    $x$+$\lambda y$+$z$=1
    $x$+$y$+$\lambda z$=1
    is in consistent, then $\sum \limits_{\lambda \in S}\left(|\lambda|^2+|\lambda|\right)$ is equal to
    
     
    1. 6
    2. 4
    3. 2
    4. 12
14.  The sum to 10 terms of the series
    $\frac{1}{1+1^2+1^4}$+$\frac{1}{2+2^2+2^4}$+$\frac{1}{3+3^2+3^4}$+........ is
    
     
    1. $\frac{56}{111}$
    2. $\frac{55}{111}$
    3. $\frac{59}{111}$
    4. $\frac{58}{111}$
15.  The combined equation of the two lines $ax$+$by$+$c$=0 and $a'x$+$b'y$+$c'$=0 can be written as $(ax+by+c)$$(a'x+b'y+c')$=0. The equation of the angle bisectors of the lines represented by the equation $2x^2$+$xy-$$3y^2$=0 is
    
     
    1. $x^2$$-y^2$$-10xy$=0
    2. $3x^2$+$5xy$+$2y^2$=0
    3. $3x^2$+$xy$$-2y^2$=0
    4. $x^2-$$y^2$+$10xy$=0
16. Let $\begin{\equation*} f(x)=\begin{vmatrix} 1+\sin^2 x & \cos^2 x & \sin 2x \\ \sin^2 x & 1+\cos^2 x & \sin 2x \\ \sin^2 x & \cos^2 x \\ 1+\sin 2x \end{vmatrix}\end{equation*}$, $x \in $\left[\frac{\pi}{6}, \frac{\pi}{3}$. If $\alpha$ and $\beta$ respectively are the maximum and the minimum values of $f$, then
    
    1. $\alpha^2$$-\beta^2$=$4\sqrt{3}$
    2. $\alpha^2$+$\beta^2$=$\frac{9}{2}$
    3. $\beta^2$+$2\sqrt{\alpha}$=$\frac{19}{4}$
    4. $\beta^2$$-2\sqrt{\alpha}$=$\frac{19}{4}$
17. The value of $\frac{1}{1!50!}$+$\frac{1}{3!48!}$+$\frac{1}{5!46!}$+......+$\frac{1}{49!2!}$+$\frac{1}{51!1!}$ is:
    
     
    1. $\frac{2^{50}}{51!}$
    2. $\frac{2^{51}}{50!}$
    3. $\frac{2^{51}}{51!}$
    4. $\frac{2^{50}}{50!}$
18.  Let the image of the point $P(2, -1, 3)$ in the plane $x$+$2y-$$z$=0 be $Q$. Then the distance of the plane $3x$+$2y$+$z$+29=0 from the point $Q$ is
    
     
    1. $\frac{22\sqrt{2}}{7}$
    2. $3\sqrt{14}$
    3. $\frac{24\sqrt{2}}{7}$
    4. $2\sqrt{14}$
19.  The shortest distance between the lines $\frac{x-5}{1}$=$\frac{y-2}{2}$=$\frac{z-4}{-3}$ and $\frac{x+3}{1}$=$\frac{y+5}{4}$=$\frac{z-1}{-5}$ is
    
     
    1. $5\sqrt{3}$
    2. $6\sqrt{3}$
    3. $7\sqrt{3}$
    4. $4\sqrt{3}$
20.  $\lim \limits_{n \to \infty} \left[\frac{1}{1+n}+\frac{1}{2+n}+\frac{1}{3+n}+.....+\frac{1}{2n}\right]$ is equal to
    
     
    1. 0
    2. $\log_e\left(\frac{2}{3}\right)$
    3. $\log_e2$
    4. $\log_e\left(\frac{3}{2}\right)$

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. The number of 3-digit numbers, that are divisible by either 2 or 3 but not divisible by 7, is…………
2. Let $\vec{v}$=$\alpha \hat{i}$+$2\hat{j}$$-3\hat{k}$, $\vec{w}$=$2\alpha \hat{i}$+$\hat{j}$$-\hat{k}$ and $\vec{u}$ be a vector such that $|\vec{u}|$=$\alpha >0$. If the minimum value of the scalar triple product $\begin{equation*} \begin{bmatrix} \vec{u} & \vec{v} & \vec{w} \end{bmatrix} \end{equation*}$ is $-\alpha \sqrt{3401}$, and $|\vec{u}•\vec{i}|^2$=$\frac{m}{n}$ where $m$ and $n$ are coprime natural numbers, then $m+n$ is equal to ..............
3.  The remainder, when $19^{200}$+$23^{200}$ is divisible by $49$, is........
4.  $A(2, 6, 2)$, $B(-4, 0, \lambda)$, $C(2, 3, -1)$ and $D(4, 5, 0)$, $|\lambda| \leq 5$ are the vertices of a quadrilateral $ABCD$. If its area is 18 square units, then $5-6\lambda$ is equal to ..............
5.  Let $f:R \to R$ be a differentiable function such that $f'(x)$+$f(x)$=$\int \limits_{0}^{2}f(t)dt$. If $f(0)$=$e^{-2}$, then $2f(0)-f(2)$ is equal to............
6. If $\int \limits_{0}^{1}(x^{21}+x^{14}+x^7)(2x^{14}+3x^7+6)^{1/7}dx$=$\frac{1}{l}(11)^{m/n}$ where $l, m, n \in N, m$ and $n$ are coprime then $l+m+n$ is equal to........
7.  Let $A$ be the area bounded by the curve $y$=$x|x-3|$, the $x-$axis and the ordinates $x=-1$ and $x=2$. Then $12A$ is equal to..........
8.  Let $a_1$=8, $a_2$, $a_3$,......., $a_n$ be an A.P. If the sum of its first four terms is 50 and the sum of its last four terms is 170, then the product of its middle two terms is ............
9.  The number of words, with or without meaning, that can be formed using all the letters of the word ASSASSINATION so that the vowels occur together, is…….
10.  If $f(x)$=$x^2$+$g'(1)x$+$g''(2)$ and $g(x)$=$f(1)x^2$+$xf'(x)$+$f''(x)$, then the value of $f(4)-g(4)$ is equal to .............

01 Feb 2023:Shift-2

01 Feb 2023:Ans-Key

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