Download JEE Main 2023 Question Paper (11 Apr - Shift 2) Mathematics

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12 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.  The angle of elevation of the top $P$ of a tower from the feet of one person standing due South of the tower is 45° and from the feet of another person standing due west of the tower is 30°. If the height of the tower is 5 meters, then the distance ( in meters) between the two persons is equal to
   
      
   1. 10
   2. $5\sqrt{5}$
   3. $\frac{5}{2}\sqrt{5}$
   4. 5
2. Let $a$, $b$, $c$ and $d$ be positive real numbers such that $a$ + $b$ + $c$ + $d$ = 11. If the maximum value of $a^5 b^3c^2 d$ is $3750\beta$, then the value of $\beta$ is
   
      
   1. 55
   2. 108
   3. 90
   4. 110
3. If $f : R \to R$be a continuous function satisfying $\int \limits_{0}^{\frac{\pi}{2}}f(\sin 2x)\sin x dx$+$\alpha \int \limits_{0}^{\frac{\pi}{4}} f (\cos 2x ) \cos x dx$=0, then the value of $\alpha$ is
   
      
   1. $-\sqrt{3}$
   2. $\sqrt{3}$
   3. $-\sqrt{2}$
   4. $\sqrt{2}$
4. Let $f$ and $g$ be two functions defined by $f(x)$=$\left\{\begin{array}{cc} x+1, & x < 0 \\ |x-1|, &  x \geq 0 \end{array}\right.$ and $g(x)$=$\left\{\begin{array}{cc} x+1, & x < 0 \\ 1, &  x \geq 0 \end{array}\right.$. Then (gof ) $(x)$ is
   
      
   1. continuous everywhere but not differentiable at $x$ = 1
   2. continuous everywhere but not differentiable exactly at one point
   3. differentiable everywhere
   4. not continuous at $x = –1$
5. If the radius of the largest circle with centre (2, 0) inscribed in the ellipse $x^2$ + $4y^2$ = 36 is $r$, then $12r^2$is equal to
   
      
   1. 69
   2. 72
   3. 115
   4. 92
6. Let the mean of 6 observations 1, 2, 4, 5, $x$ and $y$ be 5 and their variance be 10. Then their mean deviation about the mean is equal to
   
      
   1. $\frac{7}{3}$
   2. $\frac{10}{3}$
   3. $\frac{8}{3}$
   4. 3
7. Let $A$ = {1, 3, 4, 6, 9} and $B$ = {2, 4, 5, 8, 10}. Let $R$ be a relation defined on $A×B$ such that $R$ ={($(a_1, b_1)$, $(a_2,b_2)$): $a_1 \leq b_2$ and $b_1 \leq a_2$}. Then the number of elements in the set $R$ is
   
      
   1. 52
   2. 160
   3. 26
   4. 180
8. Let $P$ be the plane passing through the points (5, 3, 0), (13, 3, –2) and (1, 6, 2). For $\alpha \in N$, if the distances of the points $A(3, 4, \alpha)$ and $B(2, \beta, a)$ from the plane $P$ are 2 and 3 respectively, then the positive value of $a$ is
   
      
   1. 5
   2. 6
   3. 4
   4. 3
9. If the letters of the word MATHS are permuted and all possible words so formed are arranged as in a dictionary with serial number, then the serial number of the word THAMS is
   
      
   1. 102
   2. 103
   3. 101
   4. 104
10. If four distinct points with position vectors $\vec{a}$, $\vec{b}$, $\vec{c}$and $\vec{d}$are coplanar, then $[\vec{a} \vec{b} \vec{c}]$is equal to
    
       
    1. $[\vec{d} \vec{c} \vec{a}]$+$[\vec{b} \vec{d} \vec{a}]$+$[\vec{c} \vec{d} \vec{b}]$
    2. $[\vec{d} \vec{b} \vec{a}]$+$[\vec{a} \vec{c} \vec{d}]$+$[\vec{d} \vec{b} \vec{c}]$
    3. $[\vec{a} \vec{d} \vec{b}]$+$[\vec{d} \vec{c} \vec{a}]$+$[\vec{d} \vec{b} \vec{c}]$
    4. $[\vec{b} \vec{c} \vec{d}]$+$[\vec{d} \vec{a} \vec{c}]$+$[\vec{d} \vec{b} \vec{a}]$
11.   The sum of the coefficients of three consecutive terms in the binomial expansion of $(1 + x)^{n+2}$, which are in the ratio 1 : 3 : 5, is equal to
    
       
    1. 63
    2. 92
    3. 25
    4. 41
12. Let $y$ = $y (x)$ be the solution of the differential equation $\frac{dy}{dx}$+$\frac{5}{x(x^5+1)}y$=$\frac{(x^5+1)^2}{x^7}$, $x$ > 0. If $y(1)$ = 2, then $y(2)$ is equal to
    
       
    1. $\frac{693}{128}$
    2. $\frac{637}{128}$
    3. $\frac{697}{128}$
    4. $\frac{679}{128}$
13. The converse of $((~ p) \bigwedge q)=> r $ is
    
       
    1. $(p \bigvee (~ q))=>( ~ r)$
    2. $((~ p) \bigvee q)=> r$
    3. $(~ r) =>((~ p) \bigwedge q)$
    4. $(~ r)=> p \bigwedge q$
14. If the $1011^{th}$ term from the end in the binominal expansion of $\left(\frac{4x}{5}-\frac{5}{2x}\right)^{2022}$ is 1024 times $1011^{th}$ term from the beginning, the $|x|$ is equal to
    
       
    1. 8
    2. 10
    3. 12
    4. 15
15. If the system of linear equations
    $7x$ + $11y$ + $\alpha z$ = 13
    $5x$ + $4y$ + $7z$ = $\beta$
    $175x$ + $194y$ + $57z$ = 361
    has infinitely many solutions, then $\alpha$ + $\beta$ + 2 is equal to :
    
       
    1. 3
    2. 6
    3. 5
    4. 4
16. Let the line passing through the point $P (2, –1, 2)$ and $Q (5, 3, 4)$ meet the plane $x – y$ +$z$ = 4 at the point $T$. Then the distance of the point R from the plane $x$ + $2y$ + $3z$ +2 = 0 measured parallel to the line $\frac{x-7}{2}$=$\frac{y+3}{2}$=$\frac{z-2}{1}$ is equal to
    
       
    1. 3
    2. $\sqrt{61}$
    3. $\sqrt{31}$
    4. $\sqrt{189}$
17.  Let the function $f : [0, 2] \to R$ be defined as
    $f(x)$=$\left\{\begin{array}{cc} e^{\text{min }{x^2, x-[x]}}, & x \in [0, 1) \\ e^{[x-\log_ex]}, & x \in [1, 2) \end{array}\right.$
    where $[t]$ denotes the greatest integer less than or equal to $t$. Then the value of the integral $\int \limits_{0}^{2}xf(x)dx$ is
    
       
    1. $(e-1)\left(e^2+\frac{1}{2}\right)$
    2. $1+\frac{3e}{2}$
    3. $2e-\frac{1}{2}$
    4. $2e-1$
18. For $a \in C$, let $A$ ={$z \in C:Re (a +\bar{z})$ > $Im (\bar{a}+z)$} and $B$ = {$z \in C:Re (a + \bar{z}$ < $Im (\bar{a} + z)$}. The among the two statements:
    (S1) : If $Re (a)$, $Im (a)$ >0, then the set $A$ contains all the real numbers
    (S2) : If $Re (a)$, $Im (a)$ < 0, then the set $B$ contains all the real numbers,
    
    
       
    1. only (S1) is true
    2. both are false
    3. only (S2) is true
    4. both are true
19. If $\begin{vmatrix} x+1 & x & x \\ x & x+\lambda & x \\ x & x & x+\lambda^2 \end{vmatrix}$=$\frac{9}{8}(103x+81)$, then $\lambda$, $\frac{\lambda}{3}$ are the roots of the equation
    
       
    1. $4x^2-24x-27$=0
    2. $4x^2+24x+27$=0
    3. $4x^2-24x+27$=0
    4. $4x^2+24x-27$=0
20. The domain of the function $f(x)$=$\frac{1}{\sqrt{[x]^2-3[x]-10}}$ is (where $[x]$ denotes the greatest integer less than or equal $x$)
    
       
    1. $(-\infty, -3] \cup [6, \infty)$
    2. $(-\infty, -2) \cup (5, \infty)$
    3. $(-\infty, -3] \cup (5, \infty)$
    4. $(-\infty, -2) \cup [6, \infty)$

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 $A$ is the area in the first quadrant enclosed by the curve $C$ : $2x^2$ $– y$ + 1 = 0, the tangent to $C$ at the point (1,3) and the line $x + y$ = 1, then the value of 60 $A$ is ______.
2. Let $A$ = {1, 2, 3, 4, 5} and $B$= {1, 2, 3, 4, 5, 6}. Then the number of functions $f:A \to B$ satisfying $f(1) + f(2)$ = $f(4)–1$ is equal to ______.
3. Let the tangent to the parabola $y^2$ = 12 $x$ at the point $(3, \alpha)$ be perpendicular to the line $2x+2y$ = 3. Then the square of distance of the point (6,–4) from the normal to the hyperbola $\alpha^2 x^2 – 9y^2$ = $9\alpha^2$at its point $(\alpha–1, \alpha + 2)$ is equal to _____.
4. For $k \in N$, if the sum of the series 1+$\frac{4}{k}$+$\frac{8}{k^2}$+$\frac{13}{k^3}$+$\frac{19}{k^4}$+......... is 10, then the value of $k$ is ........
5. Let the line $l$:$x$=$\frac{y-1}{-2}$==$\frac{z-3}{\lambda}$, $\lambda \in R$ meet the plane $P$:$x$+$2y$+$3z$=4 at the point $(\alpha, \beta, \gamma)$. If the angle between the line $l$ and the plane $P$ is $\cos^{-1}\left(\sqrt{\frac{5}{14}}\right)$, then $\alpha$+$2\beta$+$6\gamma$ is equal to......
6. The number of points where the curve $f(x)$ =$e^{8x}$$-e^{6x}$$-3e^{4x}$$-e^{2x}$+1, $x \in R$ cuts $x-$axis, is equal to ________
7. If the line $l_1$ : $3y$$– 2x$ = 3 is the angular bisector of the line $l_2$ : $x –y$ + 1 = 0 and $l_3$ : $\alpha x$ +$\beta y $+ 17, then $\alpha^2$ + $\beta^2$ –$\alpha$ – $\beta$ is equal to ______.
8. Let the probability of getting head for a biased coin be $\frac{1}{4}$. It is tossed repeatedly until a head appears. Let $N$ be the number of tosses required. If the probability that the equation $64x^2$ + $5Nx$ + 1 = 0 has no real root is $\frac{p}{q}$, where $p$ and $q$ are co-prime, then $q – p$ is equal to ______.
9. Let $\vec{a}$=$\hat{i}$+2$\hat{j}$+3$\hat{k}$ and $\vec{b}$=$\hat{i}$+$\hat{j}$$-\hat{k}$. If $\vec {c}$ is a vector such that $\vec{a}•\vec{c}$=11, $\vec{b}•(\vec{a}×\vec{c})$=27 and $\vec{b}•\vec{c}$=$-\sqrt{3}|\vec{b}|$, then $|\vec{a}×\vec{c}|^2$ is equal to........
10. Let $S$=$\left \{z \in C-{i, 2i}:\frac{z^2+8iz-15}{z^2-3iz-2} \in R \right \}$. If $\alpha -\frac{13}{11}i \in S$, $a \in R-{0}$, then 242 $\alpha^2$ is equal to........

11 Apr 2023:Shift-1

11 Apr 2023:Ans-Keys

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