Download JEE Main 2023 Question Paper (24 Jan - Shift 1) Mathematics

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25 Jan 2023:Shift-1

25 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. $\tan^{-1}\left(\frac{1+\sqrt{3}}{3+\sqrt{3}}\right)$+$\sec^{-1}\left(\sqrt{\frac{8+4\sqrt{3}}{6+3\sqrt{3}}}\right)$ is equal to :
   1. $\frac{\pi}{6}$
   2. $\frac{\pi}{3}$
   3. $\frac{\pi}{4}$
   4. $\frac{\pi}{2}$
2.  $\lim \limits_{t \to 0}\left(1^{\frac{1}{\sin^2t}}+2^{\frac{1}{\sin^2t}}+....+n^{\frac{1}{\sin^2t}}\right)^{\sin^2t}$ is equal to
   1. $n^2+n$
   2. $n^2$
   3. $\frac{n(n+1)}{2}$
   4. $n$
3.  Let a tangent to the curve $y^2$=$24x$ meet the curve $xy=2$ at the points $A$ and $B$. Then the mid points of such line segments $AB$ lie on a parabola with the
   1. directrix $4x=-3$
   2. length of latus rectum 2
   3. directrix $4x=3$
   4. length of latus rectum $\frac{3}{2}$
4. The value of $\sum \limits_{r=0}^{22} {}^{22}C_{r} {}^{23}C_{r}$ is
   1. ${}^{44}C_{23}$
   2. ${}^{45}C_{23}$
   3. ${}^{44}C_{22}$
   4. ${}^{45}C_{24}$
5.  If $A$ and $B$ are two non-zero $n×n$ matrices such that $A^2$+$B$=$A^2B$, then
   1. $A^2B=BA^2$
   2. $A^2B$=$I$
   3. $A^2$=$I$ or $B$=$I$
   4. $AB=I$
6. Let $N$ denote the number that turns up when a fair die is rolled. If the probability that the system of equations $x$+$y$+$z$=1
   $2x$+$Ny$+$2z$=2
   $3x$+$3y$+$Nz$=3
   has unique solution is $\frac{k}{6}$, then the sum of value of $k$ and all possible values of $N$ is
   1. 18
   2. 21
   3. 19
   4. 20
7.  Let $PQR$ be a triangle. The points $A$, $B$ and $C$ are on the sides $QR$, $RP$ and $PQ$ respectively such that $\frac{QA}{AR}$=$\frac{RB}{BP}$=$\frac{PC}{CQ}$=$\frac{1}{2}$. Then $\frac{Area(\Delta PQR)}{Area(\Delta ABC)}$ is equal to
   1. 4
   2. 3
   3. $\frac{5}{2}$
   4. 2
8.  Let $\vec{u}$=$\hat{i}$$-\hat{j}$$-2\hat{k}$, $\vec{v}$=$2\hat{i}$+$\hat{j}$$-\hat{k}$, $\vec{v}•\vec{w}$=2 and $\vec{v}×\vec{w}$=$\vec{u}$+$\lambda \vec{v}$. Then $\vec{u}•\vec{w}$ is equal to
   1. 2
   2. $\frac{3}{2}$
   3. $-\frac{2}{3}$
   4. $1$
9. Let $\alpha$ be the roots of the equation $(a-c)x^2$+$(b-a)x$+$(c-b)$=0 where $a$, $b$, $c$ are distinct real numbers such that matrix $\begin{equation*} \begin{bmatrix} \alpha^2 & \alpha & 1\\ 1 & 1 & 1\\ a & b & c \end{bmatrix} \end{equation*}$ is singular. Then, the value of
   $\frac{(a-c)^2}{(b-a)(c-b)}$+$\frac{(b-a)^2}{(a-c)(c-b)}$+$\frac{(c-b)^2}{(a-c)(b-a)}$ is
   1. 12
   2. 3
   3. 9
   4. 6
10.  Let $p, q \in R$ and $(1-\sqrt{3}i)^{200}$=$2^{199}(p+iq)$, $i$=$\sqrt{-1}$. Then $p$+$q$+$q^2$ and $p-q$+$q^2$ are roots of the equation.
    1. $x^2-4x+1$=0
    2. $x^2+4x-1$=0
    3. $x^2+4x+1$=0
    4. $x^2-4x-1$=0
11. The distance of the point $(7, -3, -4)$ from the plane passing through the points $(2, -3,1)$, $(-1,1, -2)$ and $(3, -4, 2)$ is :
    1. $4\sqrt{2}$
    2. 5
    3. $5\sqrt{2}$
    4. 4
12.  Let $y=y(x)$ be the solution of the differential equation $x^3dy$+$(xy-1)dx$=0, $x>0$, $y\left(\frac{1}{2}\right)$=$3-e$. Then $y(1)$ is equal to
    1. 1
    2. $2-e$
    3. $e$
    4. 3
13. Let $\Omega$ be the sample space and $A⊆\Omega$ be an event. Given below are two statements : (S1) : If $P(A)$=0, then $A=\phi$
    (S2) : If $P(A)$=1, then $A=\Omega$
    Then
    1. both (S1) and (S2) are true
    2. only (S2) is true
    3. both (S1) and (S2) are false
    4. only (S1) is true
14.  The relation $R$={$(a,b) : gcd(a,b)$=1, $2a\neq b$, $a,b \in Z$} is :
    1. symmetric but not transitive
    2. neither symmetric nor transitive
    3. reflexive but not symmetric
    4. transitive but not reflexive
15. For three positive integers $p$, $q$, $r$, $x^{pq^2}$=$y^{qr}$=$z^{p^2r}$ and $r$=$pq+1$ such that 3, $3\log_yx$, $3\log_zy$, $7\log_xz$ are in A.P. with common difference $\frac{1}{2}$. The $r-p-q$ is equal to
    1. 12
    2. 6
    3. 2
    4. -6
16.  The distance of the point $(-1, 9, -16)$ from the plane $2x+3y-z$=5 measured parallel to the line $\frac{x+4}{3}$=$\frac{2-y}{4}$=$\frac{z-3}{12}$
    1. $13\sqrt{2}$
    2. $20\sqrt{2}$
    3. $31$
    4. $26$
17.  The area enclosed by the curves $y^2+4x$=4 and $y-2x$=2 is :
    1. $\frac{25}{3}$
    2. $\frac{23}{3}$
    3. 9
    4. $\frac{22}{3}$
18.  The equation $x^2-4x$+$[x]$+3=$x[x]$, where $[x]$ denotes the greatest integer function, has :
    1. a unique solution in $(-\infty, \infty)$
    2. no solution
    3. a unique solution in $(-\infty, 1)$
    4. exactly two solutions in $(-\infty, \infty)$
19.  The compound statement (~(P∧Q))∨((~P)∧Q)=>((~P)∧(~Q)) is equivalent to
    1. (~P)∨Q
    2. ((~P)∨Q)∧((~Q)∨P)
    3. ((~P)∨Q)∧(~Q)
    4. (~Q)∨P
20. Let $f(x)$=$\begin{cases}x^2\sin\left(\frac{1}{x}\right), x \neq 0\\0, x=0 \end{cases}$ then at $x=0$
    1. $f'$ is continuous but not differentiable
    2. $f$ and $f'$ both are continuous
    3. $f$ is continuous but not differentiable
    4. $f$ is continuous but $f'$ is not continuous

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 value of 12$\int \limits_{0}^{3}|x^2-3x+2|dx$ is............
2. Suppose $\sum \limits_{r=0}^{2023}r^2 {}^{2023}C_r$=2023×$\alpha$×$2^{2022}$. Then the value of $\alpha$ is.........
3. The value of $\frac{8}{\pi} \int \limits_{0}^{\frac{\pi}{2}}\frac{(\cos x)^{2023}}{(\sin x)^{2023}+(\cos x)^{2023}}dx$ is
4.  Let a tangent to the curve $9x^2$+$16y^2$=144 intersect the coordinate axes at the points $A$ and $B$. Then, the minimum length of the line segment $AB$ is………
5.  Let $\lambda \in R$ and let the equation $E$ be $|x^2|$$-2|x|$+$|\lambda-3|$=0. Then the largest element in the set $S$={$ x+\lambda : x$ is an integer solution of $E$ is………..
6. The $4^{th}$ term of $GP$ is 500 and its common ratio is $\frac{1}{m}$, $m \in N$. Let $S_n$ denote the sum of the first $n$ terms of this $GP$. If $S_6$>$S_5$+1 and $S_7$ < $S_6$+$\frac{1}{2}$, then the number of possible values of $m$ is……….
7.  The shortest distance between the lines $\frac{x-2}{3}$=$\frac{y+1}{2}$=$\frac{z-6}{2}$ and $\frac{x-6}{3}$=$\frac{1-y}{2}$=$\frac{z+8}{0}$ is equal to.........
8.  A boy needs to select five courses from 12 available courses, out of which 5 courses are language course. If he can choose at most two language courses, then the number of ways he can choose five courses is ……….
9.  The number of 9 digit numbers, that can be formed using all the digits of the number 123412341 so that the even digits occupy only even places, is……
10.  Let $C$ be the largest circle centered at (2,0) and inscribed in the ellipse $\frac{x^2}{36}$+$\frac{y^2}{16}$=1. If $(1, \alpha)$ lies on $C$, then $10\alpha^2$ is equal to..........

24 Jan 2023:Shift-2

24 Jan 2023:Ans-Key

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