Download JEE Main 2023 Question Paper (11 Apr - Shift 1) Mathematics

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12 Apr 2023:Shift-1

13 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.  If equation of the plane that contains the point $(–2, 3, 5)$ and is perpendicular to each of the planes $2x$ + $4y$ + $5z$ = 8 and $3x –$$ 2y$ + $3z$ = 5 is $\alpha x$ + $\beta y$ + $\gamma z$ + 97 = 0 then $\alpha$ +$\beta$ + $\gamma$=
   
   1. 15
   2. 16
   3. 17
   4. 18
2.  Let $(\alpha, \beta, \gamma)$ be the image of the point $P(2, 3, 5)$ in the plane $2x$ + $y –$$ 3z$ = 6. Then $\alpha$+$\beta$+$\gamma$ is equal to
   
      
   1. 10
   2. 12
   3. 9
   4. 5
3.  Let $y = y(x)$ be a solution curve of the differential equation. $(1 – x^2y^2)dx$ = $ydx + xdy$. If the line $x$ = 1 intersects the curve $y = y(x)$ at $y$ = 2 and the line $x$ = 2 intersects the curve $y = y(x)$ at $y = \alpha$, then a value of $\alpha$ is
   
      
   1. $2\frac{1-3e^2}{2(3e^2+1)}$
   2. $2\frac{3e^2}{2(3e^2-1)}$
   3. $2\frac{3e^2}{2(3e^2+1)}$
   4. $2\frac{1+3e^2}{2(3e^2-1)}$
4.  Let $A$ be a 2 × 2 matrix with real entries such that $A'$ = $\alpha A$ + $I$ where $a \in R – {–1, 1}$. If $det(A^2 – A)$ = 4, then the sum of all possible values of $\alpha$ is
   
      
   1. $\frac{3}{2}$
   2. 0
   3. $\frac{5}{2}$
   4. $2$
5. Let $f : [2, 4] => R$ be a differentiable function such that $(x log_e x) f'(x)$ + $(log x) f(x)$ +$f(x)\geq 1$, $x \in [2, 4]$ with $f(2)$ = $\frac{1}{2}$ and $f(4)$ = $\frac{1}{4}$. Consider the following two statements:
   (A) : $f (x) \leq1$, for all $x \in [2, 4]$
   (B) : $f (x) \geq \frac{1}{8}$, for all $x \in [2, 4]$ Then,
   
      
   1. Neither statement (A) nor statement (B) is true
   2. Both the statements (A) and (B) are true
   3. Only statement (B) is true
   4. Only statement (A) is true
6.  for any vector $\vec{a}$= $a_1 \hat{i}$+$a_2 \hat{j}$+$a_3 \hat{k}$, with $10|a_i|$ < 1, $i$ = 1, 2, 3, consider the following statements: (A) : max{$|a_1|$, $|a_2|$, $|a_3|$} $\leq | \vec{a} |$
   (B) : $| \vec{a} | \leq 3$ max{$|a_1|$, $|a_2|$, $|a_3|$}
   
   
      
   1. Neither (A) nor (B) is true
   2. Only (A) is true
   3. Both (A) and (B) are true
   4. Only (B) is true
7. The value of the integral $\int \limits_{-\log_e2}^{\log_e2}e^x(\log_e(e^x+\sqrt{1+e^{2x}}))dx$ is equal to
   
   1. $\log_e\left(\frac{2(2+\sqrt{5})}{\sqrt{1+\sqrt{5}}}\right)-\frac{\sqrt{5}}{2}$
   2. $\log_e\left(\frac{\sqrt{2}(3-\sqrt{5})^2}{\sqrt{1+\sqrt{5}}}\right)+\frac{\sqrt{5}}{2}$
   3. $\log_e\left(\frac{\sqrt{2}(2+\sqrt{5})^2}{\sqrt{1+\sqrt{5}}}\right)-\frac{\sqrt{5}}{2}$
   4. $\log_e\left(\frac{(2+\sqrt{5})^2}{\sqrt{1+\sqrt{5}}}\right)+\frac{\sqrt{5}}{2}$
8.  Let $S$ = {$M = [a_{ij}]$, $a_{ij} \in {0, 1, 2}$, $1 \leq i, j \leq 2$} be a sample space and $A$ = {$M \in S$ : $M$ is invertible} be an event. Then $P(A)$ is equal to
   
      
   1. $\frac{47}{81}$
   2. $\frac{50}{81}$
   3. $\frac{16}{27}$
   4. $\frac{49}{81}$
9.  Consider ellipse $E_k$ : $kx^2$+$k^2y^2}$ = 1, $k$ = 1, 2, ….., 20. Let $C_k$ be the circle which touches the four chords joining the end points (one on minor axis and another on major axis) of the ellipse $E_k$. If $r_k$ is the radius of the circle $C_k$, then the value of $\sum \limits_{k=1}^{20}\frac{1}{r_k^2}$ is
   
      
   1. 3320
   2. 3210
   3. 3080
   4. 2870
10.  The number of integral solutions $x$ of $\log_{\left(x+\frac{7}{2}\right)}\left(\frac{x-7}{2x-3}\right)^2 \geq 0$ is
    
       
    1. 7
    2. 8
    3. 5
    4. 6
11.  An organization awarded 48 medals in event $'A'$, 25 in event $'B'$ and 18 in event $'C'$. If these medals went to total 60 men and only five men got medals in all the three events, then, how many received medals in exactly two of three events?
    
       
    1. 15
    2. 10
    3. 9
    4. 21
12.  The number of triplets $(x, y, z)$, where $x, y, z$ are distinct non negative integers satisfying $x + y + z$ = 15, is
    
       
    1. 114
    2. 80
    3. 136
    4. 92
13.  Let $x_1$, $x_2$, …….., $x_{100}$ be in an arithmetic progression, with $x_1$ = 2 and their mean equal to 200. If $y_i = i(x_i – i)$, $1 \leq i \leq 100$, then the mean of $y_1$, $y_2$, ….., $y_{100}$ is
    
       
    1. 10049.50
    2. 10101.50
    3. 10051.50
    4. 10100
14. Area of the region {$(x, y)$ : $x^2$+ $(y – 2)^2 \leq 4$, $x^2 \geq 2y$} is
    
       
    1. $\pi-\frac{8}{3}$
    2. $2\pi-16\frac{16}{3}$
    3. $2\pi+16\frac{16}{3}$
    4. $\pi+\frac{8}{3}$
15.  Let $w_1$ be the point obtained by the rotation of $z_1$ = 5 + $4i$ about the origin through a right angle in the anticlockwise direction, and $w_2$ be the point obtained by the rotation of $z_2$ = 3 + $5i$ about the origin through a right angle in the clockwise direction. Then the principal argument of $w_1$ – $w_2$ is equal to
    
       
    1. $-\pi+\tan^{-1}\frac{33}{5}$
    2. $-\pi+\tan^{-1}\frac{8}{9}$
    3. $\pi-\tan^{-1}\frac{8}{9}$
    4. $-\pi-\tan^{-1}\frac{33}{5}$
16.  Let $R$ be a rectangle given by the lines $x = 0$, $x = 2$, $y = 0$ and $y = 5$. Let $A(\alpha, 0)$ and $B(0,\beta)$, $\alpha  \in [0, 2]$and $\beta \in [0, 5]$, be such that the line segment $AB$ divides the area of the rectangle $R$ in the ratio 4 : 1. Then, the mid-point of $AB$ lies on a
    
       
    1. circle
    2. parabola
    3. straight line
    4. hyperbola
17.  Let $f(x)$ = $[x^2 – x]$ + $|–x + [x]|$, where $x \in R$ and $[t]$ denotes the greatest integer less than or equal to $t$, Then $f$ is
    
       
    1. continuous at $x$ = 0, but not continuous at $x$ = 1
    2. continuous at $x$ = 0 and $x$ = 1
    3. continuous at $x$ = 1, but not continuous at $x$ = 0
    4. not continuous at $x = 0$ and $x = 1$
18.  The number of elements in the set $S$ = {$\theta \in [0, 2\pi]$ : $3 cos^4 \theta$– $5 cos^2 \theta$ – $2 sin^6 \theta$ + 2 = 0} is
    
       
    1. 10
    2. 12
    3. 9
    4. 8
19.  Let sets $A$ and $B$ have 5 elements each. Let the mean of the elements in sets $A$ and $B$ be 5 and 8 respectively and the variance of the elements in sets $A$ and $B$ be 12 and 20 respectively. A new set $C$ of 10 elements is formed by subtracting 3 from each element of $A$ and adding 2 to each element of $B$. Then the sum of the mean and variance of the elements of $C$ is ______.
    
       
    1. 36
    2. 32
    3. 40
    4. 38
20. Let $\vec{a}$ be a non-zero vector parallel to the line of intersection of the two planes described by $\hat{i} + \hat{j}$, $\hat{i}+\hat{k}$ and $\hat{i} – \hat{j}$, $\hat{j}–\hat{k}$. If $\theta$ is the angle between the vector $\vec{a}$ and the vector $\vec{b}$ = $2 \hat{i} – 2 \hat{j} + \hat{k}$and $\vec{a}•\vec{b}$= 6, then the ordered pair $(\theta, | \vec{a} × \vec{b} |)$ is equal to
    
       
    1. $\left(\frac{\pi}{4}, 6\right)$
    2. $\left(\frac{\pi}{3}, 6\right)$
    3. $\left(\frac{\pi}{3}, 3\sqrt{6}\right)$
    4. $\left(\frac{\pi}{4}, 3\sqrt{6}\right)$

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.  In an examination, 5 students have been allotted their seats as per their roll numbers. The number of ways, in which none of the students sits on the allotted seat, is
2.  Let a line $l$ pass through the origin and be perpendicular to the lines $l_1$:$\vec{r}$=$(\hat{i}-11\hat{j}-7\hat{k})$+$\lambda(\hat{i}+2\hat{j}+3\hat{k})$, $\lambda \in R$ and
   $l_2$:$\vec{r}$=$(-\hat{i}+\hat{k})$+$\mu(2\hat{i}+2\hat{j}+\hat{k})$ $\mu \in R$.
   If $P$ is the point of intersection of $l$ and $l_1$, and $Q (\alpha, \beta, \gamma)$ is the foot of perpendicular from $P$ on $l_2$, then $9(\alpha + \beta + \gamma)$ is equal to ______.
3.  If $a$ and $b$ are the roots of the equation $x^2$$-7x$$-1$=0, then the value of $\frac{a^{21}+b^{21}+a^{17}+b^{17}}{a^{19}+b^{19}}$ is equal to............
4. Let $A$=$\begin{equation}\begin{bmatrix} 0 & 1 & 2 \\ a & 0 & 3 \\ 1 & c & 0 \end{bmatrix}\end{equation}$, where $a, c \in R$. If $A^3$=$A$ and the positive value of $a$ belongs to the interval $(n – 1, n]$, where $n \in N$, then $n$ is equal to _________.
5.  For $m, n$>0, let $\alpha(m, n)$=$\int \limits_{0}^{2}t^m(1+3t)^ndt$. If $11\alpha(10, 6)+18\alpha(11, 5)$=$p(14)^6$, then $p$ is equal to ............
6. Let $S$=109+$\frac{108}{5}$+$\frac{107}{5^2}$+.......+$\frac{2}{5^{107}}$+$\frac{1}{5^{108}}$.Then the value of $(16S – (25)^{–54})$ is equal to ______.
7.  The number of ordered triplets of the truth values of $p$, $q$ and $r$ such that the truth value of the statement $(p ∨ q) ∧ (p ∨ r)$ => $(q ∨ r)$ is True, is equal to _____.
8.  The mean of the coefficients of $x$, $x^2$, …., $x^7$ in the binomial expansion of $(2 + x)^9$ is _______.
9. The number of integral terms in the expansion of $\left(3^{\frac{1}{2}}+5^{\frac{1}{4}}\right)^{680}$ is equal to.........
10.  Let $H_n$:$\frac{x^2}{1+n}-\frac{y^2}{3+n}$=1, $n \in N$. Let $k$ be the smallest even value of $n$ such that the eccentricity of $H_k$is a rational number. If $l$ is the length of the latus rectum of $H_k$, then $21l$ is equal to _____.

11 Apr 2023:Shift-2

11 Apr 2023:Ans-Keys

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