Download JEE Main 2023 Question Paper (13 Apr - Shift 1) Mathematics

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15 Apr 2023:Shift-1

06 Apr 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 $y = y_1(x)$ and $y= y_2(x)$ be the solution curves of the differential equation $\frac{dy}{dx}$=$y$+7 with initial conditions $y_1(0)$ = 0 and $y_2(0)$=1 respectively. Then the curves $y = y_1(x)$ and $y=y_2(x)$ intersect a
   
    
   1. one point
   2. no point
   3. infinite number of points
   4. two points
2.  Let the tangent and normal at the point $(3\sqrt{3}, 1)$ on the ellipse $\frac{x^2}{36}$+$\frac{y^2}{4}$ meet the $y-$axis at the points $A$ and $B$ respectively. Let the circle $C$ be drawn taking $AB$ as a diameter and the line $x = 2 \sqrt{5}$ intersect $C$ at the points $P$ and $Q$ . If the tangents at the points $P$ and $Q$ on the circle intersect at the point $(\alpha,\beta)$, then $(\alpha^2-\beta^2)$ is equal to
   
    
   1. 60
   2. $\frac{304}{5}$
   3. $\frac{314}{5}$
   4. 61
3.  The negation of the statement ((A ∧ ( B ∨ C))=> (A ∨ B))=> A is
   
    
   1. a fallacy
   2. equivalent to B ∨ ~C
   3. equivalent to ~C
   4. equivalent to ~A
4.  For the system of linear equations
   $2x$ + $4y$ + $2az$ = $b$ $x$ + $2y$ + $3z$ = 4 $2x$$ – 5y $+ $2z$ =8
   Which of following is NOT correct?
   
    
   1. It has unique solution if $a$ = $b$ = 6
   2. It has infinitely many solutions if $a$ = 3, $b$ = 8
   3. It has unique solution if $a$ = $b$ = 8
   4. It has infinitely many solutions if $a$ = 3, $b$ = 6
5. The distance of the point (–1,2,3) from the plane $\vec{r}$•$(\hat{i}-2\hat{j}+3\hat{k}$)=10 parallel to the line of the shortest distance between the lines $\vec{r}$=$(\hat{i}-\hat{j})$+$\lambda(2\hat{i}+\hat{k})$ and $\vec{r}$=$(2\hat{i}-\hat{j})$+$\mu(\hat{i}-\hat{j}+\hat{k})$ is
   
    
   1. $3\sqrt{6}$
   2. $3\sqrt{5}$
   3. $2\sqrt{5}$
   4. $2\sqrt{6}$
6.  Among
   (S1):$\lim \limits_{n \to \infty} \frac{1}{n^2}(2+4+6+....+2n)$=1
   (S2):$\lim \limits_{n \to \infty} \frac{1}{n^{16}}(1^{16}+2^{16}+3^{16}+....+n^{16})$=$\frac{1}{16}$
   
    
   1. Both (S1) and (S2) are false
   2. Only (S2) is true
   3. Both (S1) and (S2) are true
   4. Only (S1) is true
7.  A coin is biased so that the head is 3 times as likely to occur as tail. This coin is tossed until a head or three tails occur. If $X$ denotes the number of tosses of the coin, then the mean of $X$ is
   
    
   1. $\frac{81}{64}$
   2. $\frac{15}{16}$
   3. $\frac{37}{16}$
   4. $\frac{21}{16}$
8. Let $B$=$\begin{equation}\begin{bmatrix} 1 & 3 & \alpha \\ 1 & 2 & 3 \\ \alpha & \alpha & 4 \end{bmatrix}\end{equation}$, $\alpha > 2$ be the adjoint of a matrix $A$ and $|A|$ =2 . Then $\begin{equation}\begin{bmatrix} \alpha & -2\alpha & \alpha \end{bmatrix}\end{equation}$$\begin{equation}\begin{bmatrix} \alpha \\ -2\alpha \\ \alpha \end{bmatrix}\end{equation}$ is equal to
   
    
   1. 16
   2. 0
   3. -16
   4. 32
9.  The set of all $\alpha \in R$ for which the equation $x|x – 1|$ + $|x + 2|$ + $\alpha$ = 0 has exactly one real root, is
   
    
   1. $(-\infty, \infty)$
   2. $(-6, -3)$
   3. $(-6, \infty)$
   4. $(-\infty, -3)$
10.  $\int \limits_{0}^{\infty}\frac{6}{e^{3x}+6e^{2x}+11e^{x}+6}dx$=
    
     
    1. $\log_e\left(\frac{256}{81}\right)$
    2. $\log_e\left(\frac{64}{27}\right)$
    3. $\log_e\left(\frac{512}{81}\right)$
    4. $\log_e\left(\frac{32}{27}\right)$
11.  For $x\in R$, two real valued functions $f(x)$ and $g(x)$ are such that, $g(x)$= $x +1$ and $fog(x)$=$x+3– x$. Then $f(0)$ is equal
    
     
    1. 1
    2. 0
    3. 5
    4. -3
12.  The area of the region enclosed by the curve $f(x)$ = $max {sinx, cosx}$, $–\pi \leq x \leq x$ and the $x-$axis is
    
     
    1. $2(\sqrt{2}+1)$
    2. $4(\sqrt{2})$
    3. 4
    4. $2\sqrt{2}(\sqrt{2}+1)$
13.  Let $PQ$ be a focal chord of the parabola $y^2$= $36x$ of length 100, making an acute angle with the positive $x-$axis. Let the ordinate of $P$ be positive and $M$ be the point on the line segment $PQ$ such that $PM:MQ$ = 3:1. Then which of the following points does NOT lie on the line passing through $M$ and perpendicular to the line $PQ$?
    
     
    1. (3, 33)
    2. (6, 29)
    3. (-3, 43)
    4. (-6, 45)
14.  Fractional part of the number $\frac{4^{2022}}{15}$ is equal to
    
     
    1. $\frac{4}{15}$
    2. $\frac{1}{15}$
    3. $\frac{14}{15}$
    4. $\frac{8}{15}$
15.  Let the equation of plane passing through the line of intersection of the planes $x$+$2y$+$az$=2 and $x–y$+$z$ = 3 be $5x–11y$ + $bz$=$6a–1$. For $c\in Z$, if the distance of this plane from the point $(a, –c, c)$ is $\frac{2}{\sqrt{a}}$then $\frac{a+b}{c}$ is equal to
    
     
    1. 4
    2. -4
    3. 2
    4. -2
16. For the differentiable function $f: R –{0} \to R$, let $3f(x)+$$2f\left(\frac{1}{x}\right)$=$\frac{1}{x}-$10 then $\left|f(3)+f'\left(\frac{1}{4}\right)\right|$ is equal to
    
     
    1. 13
    2. 7
    3. $\frac{29}{25}$
    4. $\frac{33}{5}$
17.  Let $s_1$, $s_2$, $s_3$, …….. $s_{10}$ respectively be the sum to 12 terms of 10 A.P. s whose first terms are 1, 2, 3, ……..,10 and the common differences are 1, 3, 5,………, 19 respectively. Then $\sum \limits_{i=1}^{10}s_i$ is equal to
    
     
    1. 7360
    2. 7380
    3. 7260
    4. 7220
18.  Let $\vec{a}$=$\hat{i}$+$4\hat{j}$+$2\hat{k}$, $\vec{b}$=$3\hat{i}-$$2\hat{j}$+$7\hat{k}$ and $\vec{c}$=$2\hat{i}-$$\hat{j}$+$4\hat{k}$. If a vector $\vec{d}$ satisfies $\vec{d}×\vec{b}$=$\vec{c}×\vec{b}$ and $\vec{d}•\vec{a}$=24 then $|\vec{d}|^2$ is equal to
    
     
    1. 323
    2. 423
    3. 313
    4. 413
19.  The number of symmetric matrices of order 3, with all the entries from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, is
    
     
    1. $6^{10}$
    2. $10^9$
    3. $10^6$
    4. $9^{10}$
20.  $\max \limits_{0 \leq x \leq \pi}\left\{x-2\sin x \cos x +\frac{1}{3}\sin 3x\right\}$=
    
    1. $\pi$
    2. $\frac{5\pi+2+3\sqrt{3}}{6}$
    3. $\frac{\pi+2-3\sqrt{3}}{6}$
    4. 0

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. Let for $x \in R$, $S_0(x)=x$, $S_k(x)$=$C_k(x)$+$k\int \limits_{0}^{x}S_{k-1}(t)dt$, where $C_0$=1, $C_k$=$1-\int \limits_{0}^{1} S_{k-1}(x)dx$, $k$=1, 2, 3,………then $S_2(3)$ + $6C_3$ is equal to _______.
2.  Let $m_1$ and $m_2$ be the slopes of the tangents drawn from the point $P(4, 1)$ to the hyperbola $H:$$\frac{y^2}{25}-\frac{x^2}{16}$=1 . If $Q$ is the point from which the tangents drawn to $H$ have slopes $|m_1|$ and $|m_2|$ and they make positive intercepts $\alpha$ and $\beta$ on the $x-$axis, then $\frac{(PQ)^2}{\alpha \beta}$ is equal to
3.  The sum to 20 terms of the series $2.2^2$ – $3^2$ + $2.4^2$ – $5^2$ + $2.6^2$ – …… is equal _____ .
4.  If $S$=$\left\{x \in R:\sin^{-1}\left(\frac{x+1}{\sqrt{x^2+2x+2}}\right)\right.$$-\left. \sin^{-1}\left(\frac{x}{\sqrt{x^2+1}}\right)=\frac{\pi}{4}\right\}$, then $\sum \limits_{x \in S}\left(\sin \left((x^2+x+5)\frac{\pi}{2}\right)-\cos ((x^2+x+5)\pi)\right)$ is equal to......
5.  Let $\vec{a}$=$3\hat{i}$+$\hat{j}$$-\hat{k}$ and $\vec{c}$=$2\hat{i}-3\hat{j}$+$3\hat{k}$. If $\vec{b}$ is a vector such that $\vec{a}$=$\vec{b}×\vec{c}$ and $|\vec{b}|^2$=50, then $|72-|\vec{b}+\vec{c}|^2|$ is equal to.......
6.  The number of seven digit positive integers formed using the digits 1,2,3 and 4 only and sum of the digits equal to 12 is _____ .
7.  Let the image of the point $\left(\frac{5}{3}, \frac{5}{3}, \frac{8}{3}\right)$ in the plane $x-2y$+$z-$2=0 be $P$. If the distance of the point $Q(6, –2, \alpha)$, $\alpha$ > 0, from P is 13, then $\alpha$ is equal to _____ .
8.  Let $w$ = $z\bar{z}$ +$k_1z$+$k_2iz$+$\lambda(1+i)$, $k_1$, $k_2 \in R$. Let $Re(w)$ = 0 be the circle $C$ of radius 1 in the first quadrant touching the line $y$ =1 and the $y-$axis. If the curve $Im(w)$ = 0 intersects $C$ at $A$ and $B$, then $30(AB)^2$ is equal to ____________ .
9.  Let the mean of the data
   be 5. If $m$ and $\sigma^2$ are respectively the mean deviation about the mean and the variance of the data, then $\frac{3\alpha}{m+\sigma^2}$ is equal to ..........
10. Let $\alpha$ be the constant term in the binomial expansion of $\left(\sqrt{x}-\frac{6}{x^{\frac{3}{2}}}\right)^n$, $n \leq 15$. If the sum of the coefficients of the remaining terms in the expansion is 649 and the coefficient of $x^{-n}$ is $\lambda \alpha$, then $\lambda$ is equal to ______

13 Apr 2023:Shift-2

13 Apr 2023:Ans-Key

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