Download JEE Main 2023 Question Paper (31 Jan - Shift 1) Mathematics

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01 Feb 2023:Shift-1

01 Feb 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.  For all $z \in C$ on the curve $C_1$:$|z|$=4, let the locus of the point $z+\frac{1}{z}$ be the curve $C_2$. Then :
    
   1. the curves $C_1$ and $C_2$ intersect at 4 points
   2. the curve $C_1$ lies inside $C_2$
   3. the curves $C_1$ and $C_2$ intersect at 2 points
   4. the curve $C_2$ lies inside $C_1$
2. The value of $\int \limits_{\frac{\pi}{3}}^{\frac{\pi}{2}}\frac{(2+3\sin x)}{\sin x(1+\cos x)}dx$ is equal to
    
   1. -2+3$\sqrt{3}$+$\log_e\sqrt{3}$
   2. $\frac{10}{3}-$$\sqrt{3}$+$\log_e\sqrt{3}$
   3. $\frac{10}{3}-$$\sqrt{3}$$-\log_e\sqrt{3}$
   4. $\frac{7}{2}-$$\sqrt{3}-$$\log_e\sqrt{3}$
3.  Let a differentiable function $f$ satisfy $f(x)$+$\int \limits_{3}^{x}\frac{f(t)}{dt}$=$\sqrt{x+1}$, $x \geq 3$. Then 12$f(8)$ is equal to :
    
   1. 34
   2. 1
   3. 19
   4. 17
4.  If the domain of the function $f(x)$=$\frac{[x]}{1+x^2}$, where $[x]$ is greatest integer $\leq x$, is $[2, 6)$, then it's range is:
   1. $\left(\frac{5}{37}\right.,\left. \frac{2}{5}\right]$$-\left\{\frac{9}{29}, \frac{27}{109}, \frac{18}{89}, \frac{9}{53}\right\}$
   2. $\left(\frac{5}{37}\right.,\left. \frac{2}{5}\right]$
   3. $\left(\frac{5}{26}\right.,\left. \frac{2}{5}\right]$
   4. $\left(\frac{5}{26}\right.,\left. \frac{2}{5}\right]$$-\left\{\frac{9}{29}, \frac{27}{109}, \frac{18}{89}, \frac{9}{53}\right\}$
5.  Let $R$ be a relation on $N×N$ defined by $(a, b)R(c, d)$ if any only if $ad(b-c)$=$bc(a-d)$. Then $R$ is
    
   1. reflexive and symmetric but not transitive
   2. transitive but neither reflexive nor symmetric
   3. symmetric and transitive but not reflexive
   4. symmetric but neither reflexive nor transitive
6. Let $y$=$f(x)$ represent a parabola with focus $\left(-\frac{1}{2}, 0\right)$ and directrix $y=-\frac{1}{2}$. Then $S$=$\left\{x \in R:\tan^{-1}\left(\sqrt{f(x)}\right)\right.$$\left. \sin^{-1}\left(\sqrt{f(x)+1}\right)=\frac{\pi}{2}\right\}$:
    
   1. contains exactly one element
   2. is an infinite set
   3. contains exactly two elements
   4. is an empty set
7. Let $y=f(x)$=$\sin^3\left(\frac{\pi}{3}\left(\cos\left(\frac{\pi}{3\sqrt{2}}(-4x^3+5x^2+1)^{\frac{3}{2}}\right)\right)\right)$. Then, at $x$=1
    
   1. $2y'$+$3\pi^2y$=0
   2. $\sqrt{2}y'$$-3\pi^2y$=0
   3. $y'$+$3\pi^2y$=0
   4. $\sqrt{2}y'$+$\sqrt{3}\pi^2y$=0
8.  If the sum and product of four positive consecutive terms of a G.P., are 126 and 1296, respectively, then the sum of common ratios of all such GPs is
    
   1. $\frac{9}{2}$
   2. 3
   3. 7
   4. 14
9.  For the system of linear equations
   $x$+$y$+$z$=6
   $\alpha x$+$\beta y$+$7z$=3 $x$+$2y$+$3z$=14
   which of the following is NOT true?
    
   1. For every point $(\alpha, \beta)$ $\neq (7, 7)$ on the line $x-2y$+7=0, the system has infinitely many solutions
   2. If $\alpha$=$\beta$=7, then the system has no solution
   3. There is a unique point $(\alpha, \beta)$ on the line $x$+$2y$+18=0 for which the system has many solutions
   4. If $\alpha=\beta$ and $\alpha \neq 7$, then the system has a unique solution
10.  Let $\alpha \in (0, 1)$ and $\beta=\log_e(1-\alpha)$. Let $P_n(x)$=$x$+$\frac{x^2}{2}$+$\frac{x^3}{3}$+...+$\frac{x^n}{n}$, $x \in (0, 1)$. Then the integral $\int \limits_{0}^{\alpha}\frac{t^{50}}{1-t}dt$ is equal to
     
    1. $P_{50}(\alpha)-\beta$
    2. $\beta-P_{50}(\alpha)$
    3. $\beta+P_{50}(\alpha)$
    4. -($\beta+P_{50}(\alpha)$)
11. If the maximum distance of normal to the ellipse $\frac{x^2}{4}$+$\frac{y^2}{b^2}$=1, $b$<2, from the origin is 1, then the eccentricity of the ellipse is :
     
    1. $\frac{1}{\sqrt{2}}$
    2. $\frac{\sqrt{3}}{2}$
    3. $\frac{1}{2}$
    4. $\frac{\sqrt{3}}{4}$
12.  A bag contains 6 balls. Two balls are drawn from it at random and both are found to be black. The probability that the bag contains at least 5 black balls is
     
    1. $\frac{2}{7}$
    2. $\frac{5}{7}$
    3. $\frac{5}{6}$
    4. $\frac{3}{7}$
13. If $\sin^{-1}\frac{\alpha}{17}$+$\cos^{-1}\frac{4}{5}$+$\tan^{-1}\frac{77}{36}$=0, $0 < \alpha < 13$, then $\sin^{-1}(\sin \alpha)$+$\cos^{-1}(\cos \alpha)$ is equal to
     
    1. $\pi$
    2. $16-5\pi$
    3. 16
    4. 0
14.  A wire of length $20 m$ is to be cut into two pieces. A piece of length $l_1$ is bent to make a square of area $A_1$ and the other piece of length $l_2$ is made into a circle of area $A_2$. If $2A_1$+ $3A_2$ is minimum then $(\pi l_1):l_2$ is equal to :
     
    1. 3:1
    2. 4:1
    3. 1:6
    4. 6:1
15. Let $A=\begin{equation*} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 4 & -1 \\ 0 & 12 & -3 \end{pmatrix} \end{equation*}$. Then the sum of the diagonal elements of the matrix $(A+I)^{11}$ is equal to :
     
    1. 4097
    2. 4094
    3. 2050
    4. 6144
16.  Let a circle $C_1$ be obtained on rolling the circle $x^2$+$y^2$$-4x$$-6y$+11=0 upwards 4 units on the tangent $T$ to it at the point (3, 2). Let $C_2$ be the image of $C_1$ in $T$. Let $A$ and $B$ be the centres of circles $C_1$ and $C_2$ respectively, and $M$ and $N$ be respectively the feet of perpendiculars drawn from $A$ and $B$ on the $x-$axis. Then the area of the trapezium $AMNB$ is:
     
    1. $4(1+\sqrt{2})$
    2. $3+2\sqrt{2}$
    3. $2(1+\sqrt{2})$
    4. $2(2+\sqrt{2})$
17.  Let $\vec{a}$=$2\hat{i}$+$\hat{j}$+$\hat{k}$ and $vec{b}$ and $\vec{c}$ be two nonzero vectors such that $|vec{a}+\vec{b}+\vec{c}|$=$|\vec{a}+\vec{b}-\vec{c}|$ and $\vec{b}.\vec{c}$=0. Consider the following two statements:
    $|\vec{a}+\lambda \vec{c}|$$ \geq |\vec{a}|$ for all $\lambda \in R$
    $\vec{a}$ and $\vec{c}$ are always parallel.
    Then,
     
    1. both (i) and (ii) are correct
    2. only (ii) is correct
    3. neither (i) nor (ii) is correct
    4. only (i) is correct
18.  (S1) (p=>q)∨(p∧(~ q)) is a tautology
    (S2) ((~ p)=>(~ q)) ∧((~ p)∨q)) is a contradiction. Then
     
    1. only (S1) is correct
    2. both (S1) and (S2) are correct
    3. both (S1) and (S2) are wrong
    4. only (S2) is correct
19.  The number of real roots of the equation $\sqrt{x^2-4x+3}$+$\sqrt{x^2-9}$=$\sqrt{4x^2-14x+6}$, is:
     
    1. 0
    2. 3
    3. 1
    4. 2
20.  Let the shortest distance between the lines $L: \frac{x-5}{-2}$=$\frac{y-\lambda}{0}$=$\frac{z+\lambda}{1}$, $\lambda \geq 0$ and $L_1$:$x+1$=$y-1$=$4-z$ be $2\sqrt{6}$. If $(\alpha, \beta, \gamma)$ lies on $L$, then which of the following is NOT possible?
    1. $\alpha+2\gamma$=24
    2. $2\alpha-\gamma$=9
    3. $\alpha-2\gamma$=19
    4. $2\alpha +\gamma$=7

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 variance of the frequency distribution
   

   $x_i$	2	3	4	5	6	7	8
   frequency, $f_i$	3	6	16	$\alpha$	9	5	6

   
   is 3, then $\alpha$ is equal to.........
2.  Number of 4-digit numbers that are less than or equal to 2800 and either divisible by 3 or by 11, is equal to………
3. Let for $x \in R$, $f(x)$=$\frac{x+|x|}{2}$ and $g(x)$=$\left\{\begin{array}{cc} x & , x < 0 \\ x^2 &, x \geq 0, \end{array}\right.$. Then area bounded by the curve $y=(fog)(x)$ and the lines $y=0$, $2y-x$=15 is equal to……….
4.  Let 5 digit numbers be constructed using the digits 0, 2, 3, 4, 7, 9 with repetition allowed, and are arranged in ascending order with serial numbers. Then the serial number of the number 42923 is……..
5.  Let the line $L:$$\frac{x-1}{2}$=$\frac{y+1}{-1}$=$\frac{z-3}{1}$ intersect the plane $2x$+$y$+$3z$=16  at the point $P$. Let the point $Q$ be the foot of perpendicular from the point $R$ $(1, -1, -3)$ on the line L. If $\alpha$ is the area of triangle $PQR$, then $\alpha^2$ is equal to……..
6. Let $\theta$ be the angle between the planes $P_1$:$\vec{r}•(\hat{i}+\hat{j}+2\hat{k})$=9 and $P_2$:$\vec{r}$•$(2\hat{i}-\hat{j}+\hat{k})$=15. Let $L$ be the line that meets $P_2$ at the point (4, -2, 5) and makes an angle $\theta$ with the normal of $P_2$. If $\alpha$ is the angle between $L$ and $P_2$, then $(\tan^2\theta)(\cot^2\alpha$)$ is equal to .............
7.  Let $\alpha$ >0, be the smallest number such that the expansion of $\left(x^{2/3}+\frac{2}{x^3}\right)^{30}$ has a term  $\beta x^{-\alpha}$, $\beta \in N$. Then $\alpha$ is equal to...........
8.  The remainder on dividing $5^{99}$ by 11 is...........
9.  Let $a_1$, $a_2$,....... , $a_n$ be in A.P. If $a_5=2a_7$ and $a_{11}$=18, then $12 \left(\frac{1}{\sqrt{a_{10}}}+\sqrt{a_{11}}+\frac{1}{\sqrt{a_{11}}}+\sqrt{a_{12}}+......+\frac{1}{\sqrt{a_{17}}}+\sqrt{a_{18}}\right)$ is equal to ...........
10.  Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a}|$=$\sqrt{14}$, $\vec{b}|$=$\sqrt{6}$ and $|\vec{a}×\vec{b}|$=$\sqrt{48}$. Then $(\vec{a}•\vec{b})^2$ is equal to..........

31 Jan 2023:Shift-2

31 Jan 2023:Ans-Key

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