Download JEE Main 2023 Question Paper (30 Jan - Shift 1) Mathematics

Please wait! Loading...

 Download as PDF

31 Jan 2023:Shift-1

31 Jan 2023:Shift-2

All the Papers

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. If an unbiased die, marked with - 2, - 1, 0, 1, 2, 3 on its faces, is thrown five times, then the probability that the product of the outcomes is positive, is :
   
   1. $\frac{881}{2592}$
   2. $\frac{27}{288}$
   3. $\frac{521}{2592}$
   4. $\frac{440}{2592}$
2.   The minimum number of elements that must be added to the relation $R$=${(a,b) , (b,c)}$ on the set ${a,b,c}$ so that it becomes symmetric and transitive is :
   
    
   1. 4
   2. 5
   3. 7
   4. 3
3.   Let a unit vector $\hat{OP}$ make angles $\alpha$, $\beta$, $\gamma$ with the positive directions of the co-ordinate axes $OX$, $OY$, $OZ$ respectively, where $\beta \in \left(0, \frac{\pi}{2}\right)$. If $\hat{OP}$ is perpendicular to the plane through points $(1, 2, 3)$, $(2, 3, 4)$ and $(1, 5, 7)$, then which of the following is true?
   
    
   1. $\alpha \in \left(\frac{\pi}{2}, \pi \right)$ and $\gamma \in \left(\frac{\pi}{2}, \pi \right)$
   2. $\alpha \in \left(0, \frac{\pi}{2} \right)$ and $\gamma \in \left(\frac{\pi}{2}, \pi \right)$
   3. $\alpha \in \left(\frac{\pi}{2}, \pi \right)$ and $\gamma \in \left(0, \frac{\pi}{2}\right)$
   4. $\alpha \in \left(0,\frac{\pi}{2}\right)$ and $\gamma \in \left(0, \frac{\pi}{2}\right)$
4. If $\tan 15°$+$\frac{1}{\tan 75°}$+$\frac{1}{\tan 105°}$+$\tan 195°$=2$a$, then the value of $\left(a+\frac{1}{a}\right)$ is:
   
    
   1. 2
   2. $4-2\sqrt{3}$
   3. $5-\frac{3}{2}\sqrt{3}$
   4. 4
5. Let the solution curve $y$=$y(x)$ of the differential equation $\frac{dy}{dx}-\frac{3x^5 \tan^{-1}(x^3)}{(1+x^6)^{3/2}}y$=$2x$ exp.$\left \{\frac{x^3-\tan^{-1}x^3}{\sqrt{(1+x^6)}}\right\}$ pass through the origin. Then $y(1)$ is equal to:
   
    
   1. exp$\left(\frac{1-\pi}{4\sqrt{2}}\right)$
   2. exp$\left(\frac{\pi-4}{4\sqrt{2}}\right)$
   3. exp$\left(\frac{4-\pi}{4\sqrt{2}}\right)$
   4. exp$\left(\frac{4+\pi}{4\sqrt{2}}\right)$
6.   Let the system of linear equations
   $x$+$y$+$kz$=2 $2x$+$3y$$-z$=1 $3x$+$4y$+$2z$=$k$
   have infinitely many solutions. Then the system
   $(k+1)x$+$(2k-1)y$=7
   $(2k+1)x$+$(k+5)y$=10
   has :
   
    
   1. unique solution satisfying $x-y 1$
   2. unique solution satisfying $x+y$=1
   3. no solution
   4. infinitely many solutions
7.  If $\vec{a}$, $\vec{b}$, $\vec{c}$ are three non-zero vectors and $\hat{n}$ is a unit vector perpendicular to $\vec{c}$ such that $\vec{a}$=$\alpha \vec{b}-\hat{n}$, $(\alpha \neq 0)$ and $\vec{b}$.$\vec{c}$=12, then $|\vec{c} × (\vec{a} × \vec{b})|$ is equal to :
   
    
   1. 9
   2. 12
   3. 6
   4. 15
8.  Let $y$=$x+2$, $4y$=$3x+6$ and $3y$=$4x+1$ be three tangent lines to the circle $(x-h)^2$+$(y-k)^2$=$r^2$. Then $h+k$ is equal to:
   
    
   1. 6
   2. $5\sqrt{2}$
   3. $5(1+\sqrt{2})$
   4. 5
9.  If the coefficient of $x^15$ in the expansion of $\left(ax^3+\frac{1}{bx^{1/3}}\right)^{15}$ is equal to the coefficient of $x^{-15}$ in the expansion of $\left(ax^{1/3}+\frac{1}{bx^{3}}\right)^{15}$, where $a$ and $b$ are positive real numbers, then for each such ordered pair $(a,b)$:
   
    
   1. $a=b$
   2. $a=3b$
   3. $ab$=3
   4. $ab$=1
10.  If $a_n$=$\frac{-2}{4n^2-16n+15}$, then $a_1$+$a_2$+....+$a_{25}$ is equal to :
    
    1. $\frac{49}{138}$
    2. $\frac{50}{141}$
    3. $\frac{52}{147}$
    4. $\frac{51}{144}$
11. Among the statements :
    (S1) ((p v q) => r) <=>(p=>r)
    (S2) ((p v q) => r) <=>((p=>r) v (q=>r))
    
     
    1. neither (S1) nor (S2) is a tautology
    2. only (S2) is a tautology
    3. both (S1) and (S2) are tautologies
    4. only (S1) is a tautology
12.  If $P(h,k)$ be a point on the parabola $x$=$4y^2$, which is nearest to the point $Q(0, 33)$, then the distance of $P$ from the directrix of the parabola $y^2$=$4(x+y)$ is equal to :
    
     
    1. 8
    2. 4
    3. 2
    4. 6
13.  If $[t]$ denotes the greatest integer $\leq t$, then the value of $\frac{3(1-e)}{e} \int \limits_1^2x^2e^{[x]+[x^2]}dx$ is :
    
     
    1. $e^8-e$
    2. $e^8-1$
    3. $e^7-1$
    4. $e^9-e$
14.  The number of points on the curve $y$=$54x^5-$$135x^4$$-70x^3$+$180x^2$+$210x$ at which the normal lines are parallel to $x$+$90y$+2=0 is :
    
     
    1. 3
    2. 2
    3. 0
    4. 4
15.  The line $l_1$ passes through the point (2,6,2) and is perpendicular to the plane $2x$+$y$$-2z$=10. Then the shortest distance between the line $l_1$ and the line $\frac{x+1}{2}$=$\frac{y+4}{-3}$=$\frac{z}{2}$ is :
    
     
    1. $\frac{19}{3}$
    2. 7
    3. 9
    4. $\frac{13}{3}$
16.  The coefficient of $x^{301}$ in $(1+x)^{500}$+$x(1+x)^{499}$+$x^2(1+x)^{498}$+...+$x^{500}$ is:
    
     
    1. $^{500}C_{301}$
    2. $^{501}C_{302}$
    3. $^{500}C_{300}$
    4. $^{501}C_{200}$
17.  Suppose $f:R \to R(0, \infty)$ be a differentiable function such that $5f(x+y)$=$f(x).f(y)$, $\forall x, y \in R$. If $f(3)$=320, then $\sum \limits_{n=0}^{5}f(n)$ is equal to :
    
     
    1. 6825
    2. 6525
    3. 6875
    4. 6575
18.  If the solution of the equation $\log_{\cos x}\cot x+4\log_{\sin x}\tan x$=1, $x \in \left(0, \frac{\pi}{2}\right)$, is $\sin^{-1}\left(\frac{\alpha+\sqrt{\beta}}{2}\right)$, where $\alpha, \beta$ are integers, then $\alpha+\beta$ is equal to :
    
     
    1. 4
    2. 3
    3. 6
    4. 5
19. Let $\begin{equation*} A=\begin{bmatrix} m & n \\ p & q \end{bmatrix} \end{equation*}$, $d=|A| \neq 0$ and $|A-d(Adj A)|$=0. Then
    
     
    1. $1+d^2$=$m^2+q^2$
    2. $1+d^2$=$(m+q)^2$
    3. $(1+d)^2$=$m^2+q^2$
    4. $(1+d)^2$=$(m+q)^2$
20.  A straight line cuts off the intercepts $OA=a$ and $OB=b$ on the positive directions of $x-$axis and $y-$axis respectively. If the perpendicular from origin $O$ to this line makes an angle of $\frac{\pi}{6}$with positive direction of $y-$axis and the area of $\Delta OAB$ is $\frac{98}{3}\sqrt{3}$, then $a^2-b^2$ is equal to :
    
     
    1. 196
    2. $\frac{392}{3}$
    3. $\frac{196}{3}$
    4. 98

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.  Number of 4-digit numbers (the repeation of digits is allowed) which are made using the digits 1, 2, 3 and 5 and are divisible by 15, is equal to……..
2.  The mean and variance of 7 observations are 8 and 16 respectively. If one observation 14 is omitted and a and b are respectively mean and variance of remaining 6 observation, then $a+3b-5$ is equal to…….
3.   If the equation of the plane passing through the point (1,1, 2) and perpendicular to the line $x-3y+2z-1$=0=$4x-y+z$ is $Ax+By+Cz$=1, then 140$(C-B+A)$ is equal to ...........
4.  Let $f^1 (x)=\frac{3x+2}{2x+3}$, $x \in R-\left\{\frac{-3}{2}\right\}$
   For $n \geq 2$, define $f^n(x)$=$f^1 o f^{n-1}(x)$.
   If $f^5(x)$=$\frac{ax+b}{bx+a}$, $gcd(a, b)$=1, then $a+b$ is equal to............
5.  Let $\sum \limits_{n=0}^{\infty}\frac{n^3((2n!))+(2n-1)(n!)}{(n!)((2n)!)}$=$ae$+$\frac{b}{e}$+$c$, where $a, b, c \in Z$ and $e$=$\sum \limits_{n=0}^{\infty} \frac{1}{n!}$. Then $a^2-b+c$ is equal to..........
6.  Let $\alpha$ be the area of the larger region bounded by the curve $y^2$=$8x$ and the lines $y$=$x$ and $x$=2, which lies in the first quadrant. Then the value of $3 \alpha$ is equal to.......
7.  Let $z=1+i$ and $z_1$=$\frac{1+i \bar{z}}{\bar{z}(1-z)+\frac{1}{z}}$. Then $\frac{12}{\pi} arg(z_1)$ is equal to...........
8. If $\lambda_1$ < $\lambda_2$ are two values of $\lambda$ such that the angle between the planes $P_1$:$\vec{r}•(3\hat{i}-5\hat{j}+\hat{k})$=7 and $P_2$:$\vec{r}•(\lambda \hat{i}+\hat{j}-3\hat{k})$=9 is $sin^{-1}\left(\frac{2\sqrt{6}}{5}\right)$, then the square of the length of the perpendicular from the point $(38 \lambda_1, 10\lambda_2, 2)$ to the plane $P_1$ is..........
9.  Let $S$=${1, 2, 3, 4, 5, 6}$. Then the number of one-one functions $f:S \to P(S)$, where $P(S)$ denote the power set of $S$, such that $f(n) \subset f(m)$ where $n < m$ is...........
10.  $\lim \limits_{x \to 0} \frac{48}{x^4} \int \limits_0^x \frac{t^3}{t^6+1}dt$ is equal to.............

30 Jan 2023:Shift-2

30 Jan 2023:Ans-Key

All the Papers Ans-Keys

Please wait! Loading...

 Download as PDF