Download JEE Main 2023 Question Paper (15 Apr - Shift 1) Mathematics

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06 Apr 2023:Shift-1

06 Apr 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. Let $x = x(y)$ be the solution of the differential equation $2(y+2)\log_e(y+2)dx$+$(x+4-2\log_e(y+2))dy$=0, $y>-1$ with $x(e^4-2)$=1. Then $x(e^9-2)$ is equal to
   
   1. $\frac{10}{3}$
   2. $\frac{32}{9}$
   3. $\frac{4}{9}$
   4. 3
2. If $\int \limits_{0}^{1}\frac{1}{(5+2x-2x^2)(1+e^{(2-4x)})}dx$=$\frac{1}{\alpha}\log_e\left(\frac{1+\alpha}{\beta}\right)$, then $\alpha^4-\beta^4$ is equal to
   
    
   1. 21
   2. 0
   3. 19
   4. -21
3.  Let $[x]$ denote the greatest integer function and $f(x)$ = max {1 +$ x + [x]$, 2 + $x$, $x + 2[x]$}, $0\leq x \leq 2$. Let $m$ be the number of points in [0,2], where $f$ is not continuous and $n$ be the number of points in (0,2) where $f$ is not differentiable. Then $(m + n)^2$+2 is equal to
   
    
   1. 3
   2. 6
   3. 2
   4. 11
4.  The total number of three digit numbers, divisible by 3, which can be formed using the digits 1, 3, 5, 8 , if repetition of digits is allowed , is
   
    
   1. 20
   2. 22
   3. 21
   4. 18
5.  Let $A_1$ and $A_2$ be two arithmetic means and $G_1$, $G_2$, $G_3$ be three geometric means of two distinct positive numbers. Then $G_1^4$ + $G_2^4$ + $G_3^4$ + $G_1^2G_3^2$ is equal to
   
    
   1. $2(A_1 + A_2) G_1^2G_3^2$
   2. $(A_1 + A_2)^2 G_1 G_3$
   3. $2(A_1 + A_2) G_1 G_3$
   4. $(A_1 + A_2) G_1^2 G_3^2$
6.  A bag contains 6 white and 4 black balls. A die is rolled once and the number of balls equal to the number obtained on the die are drawn from the bag at random. The probability that all the balls drawn are white is
   
    
   1. $\frac{11}{50}$
   2. $\frac{1}{4}$
   3. $\frac{9}{50}$
   4. $\frac{1}{5}$
7.  If the domain of the function $f(x)$ = $\log_e(4x^2 + 11x + 6)$ + $sin^{–1}(4x + 3)$+$\cos^{-1}\left(\frac{10x+6}{3}\right)$ is $(\alpha, \beta]$, then $36|\alpha+\beta|$ is equal to
   
    
   1. 45
   2. 54
   3. 63
   4. 72
8.  Let $ABCD$ be a quadrilateral. If $E$ and $F$ are the mid points of the diagonals $AC$ and $BD$ respectively and $(\vec{AB}-\vec{BC})$+$(\vec{AD}-\vec{DC})$=$k\vec{FE}$, then $k$ is equal to
   
    
   1. 4
   2. -4
   3. -2
   4. 2
9.  Let $(a+bx+cx^2)^{10}$=$\sum \limits_{i=0}^{20}p_ix^i$, $a, b, c \in N$. If $p_1$=20 and $p_2$=210, then $2(a+b+c)$ is equal to
   
    
   1. 6
   2. 12
   3. 15
   4. 8
10.  If the set $\left\{Re\left(\frac{z-\bar{z}+z\bar{z}}{2-3z+5\bar{z}}\right):z \in C, Re(z)=3\right\}$ is equal to the interval $(\alpha, \beta]$, then 24$(\beta-\alpha)$ is equal to
    
     
    1. 27
    2. 36
    3. 42
    4. 30
11.  Let the system of linear equations
    $– x$ + $2y$$ – 9z$ = 7
    $– x$ + $3y$ + $7z$ = 9
    $– 2x$ + $y$ + $5z$ = 8
    $– 3x$ + $y$ + $13z$ = $\lambda$
    has a unique solutions $x = \alpha$, $y = \beta$, $z = \gamma$. Then the distance of the point $(\alpha, \beta, \gamma)$ from the plane]
    
     
    1. 9
    2. 11
    3. 13
    4. 7
12.  The mean and standard deviation of 10 observation are 20 and 8 respectively. Later on, it was observed that one observation was recorded as 50 instead of 40. Then the correct variance is
    
     
    1. 12
    2. 13
    3. 11
    4. 14
13.  Let $S$ be the set of all values of $\lambda$, for which the shortest distance between the line $\frac{x-\lambda}{0}$=$\frac{y-3}{4}$=$\frac{z+6}{1}$ and $\frac{x+\lambda}{3}$=$\frac{y}{-4}$=$\frac{z-6}{0}$ is 13. Then 8$|\sum \limits_{\lambda \in S}\lambda|$ is equal to
    
     
    1. 306
    2. 304
    3. 302
    4. 308
14.  The number of common tangent, to the circle $x^2$ + $y^2$$ – 18x$$ – 15y$ + 131 = 0 and $x^2$ + $y^2$$ – 6x $$– 6y $– 7 = 0, is
    
     
    1. 2
    2. 1
    3. 4
    4. 3
15.  Let $S$ be the set of all $(\lambda, \mu)$ for which the vectors $\lambda \hat{i}$$-\hat{j}$+$\hat{k}$, $\hat{i}$+$2\hat{j}$+$\mu \hat{k}$ and $3\hat{i}$$-4\hat{j}$+$5\hat{k}$, where $\lambda - \mu$=5, are coplanar, then $\sum \limits_{\lambda \mu \in S}80(\lambda^2+\mu^2)$ is equal to
    
     
    1. 2210
    2. 2370
    3. 2130
    4. 2290
16.  Let the foot of perpendicular of the point $P(3, – 2, – 9)$ on the plane passing through the points (–1, –2, –3), (9, 3, 4), (9, –2, 1) be $Q (\alpha, \beta, \gamma)$ . Then the distance of $Q$ from the origin is
    
     
    1. $\frac{10}{3}$
    2. $\frac{32}{9}$
    3. $\frac{4}{9}$
    4. 3
17. If $(\alpha, \beta)$ is the orthocentre of the triangle $ABC$ with vertices $A(3,–7)$, $B (–1,2)$ and $C(4,5)$, then $9\alpha – 6\beta$ + 60 is equal to
    
     
    1. 35
    2. 30
    3. 25
    4. 40
18. Negation of p∧(q∧~(p∧q)) is
    
     
    1. (~(p∧q))∨p
    2. (~(p∧q))∧q
    3. ~(p∨q)
    4. p∨q
19.  The number of real roots of the equation $x|x| – 5|x + 2|$ + 6 = 0, is
    
     
    1. 4
    2. 3
    3. 6
    4. 5
20.  Let the determinant of a square matrix $A$ of order $m$ be $m – n$ , where $m$ and $n$ satisfy $4m$ + $n$ = 22 and $17m$ + $4n$ = 93 . If det $(n adj(adj(mA)))$ = $3^a5^b6^c$, then $a + b + c$ is equal to
    
     
    1. 84
    2. 96
    3. 109
    4. 101

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.  Consider the triangle with vertices $A(2,1)$, $B (0,0)$ and $C (t,4)$, $t \in [0,4]$. If the maximum and the minimum perimeters of such triangles are obtained at $t = \alpha$ and $t = \beta$ respectively, then $6\alpha + 21\beta$ is equal to
2.  If the area bounded by the curve $2y^2 = 3x$, lines $x$ + $y$ = 3, $y$ = 0 and outside the circle $(x – 3)^2$ + $y^2$= 2 is $A$, then $4(\pi + 4A)$ is equal to
3.  Let an ellipse with centre (1, 0) and latus rectum of length $\frac{1}{2}$ have its major axis along $x-$axis. If its minor axis subtends an angle 60º at the foci, then the square of the sum of the length of its minor and major axes is equal to
4.  A person forgets his 4 digit ATM pin code. But he remembers that in the code all the digits are different, the greatest digit is 7 and the sum of the first two digits is equal to the sum of the last two digit. Then the maximum number of trials necessary to obtain the correct code is
5. Let $f(x)$=$\int \frac{dx}{(3+4x^2)\sqrt{4-3x^2}}$, $|x|$ < $\frac{2}{\sqrt{3}}$. If $f(0)$=0 and $f(1)$=$\frac{1}{\alpha \beta}\tan^{-1}\left(\frac{\alpha}{\beta}\right)$, $\alpha$, $\beta$>0, then $\alpha^2+\beta^2$ is equal to
6.  If the line $x$ = $y$ = $z$ intersects the line $x sinA$ + $y sinB$ + $zsinC$ – 18 = 0 = $xsin2A$ + $ysin2B$ + $zsin2C$ – 9, where $A$, $B$, $C$ are the angles of a triangle $ABC$, then $80\left(\sin \frac{A}{2}\sin \frac{B}{2} \sin \frac{C}{2}\right)$ is equal to
7. If the sum of the series $\left(\frac{1}{2}-\frac{1}{3}\right)$+$\left(\frac{1}{2^2}-\frac{1}{2•3}+\frac{1}{3^2}\right)$+$\left(\frac{1}{2^2}-\frac{1}{2^2•3}+\frac{1}{2•3^2}-\frac{1}{3^3}\right)$+$\left(\frac{1}{2^4}-\frac{1}{2^3•3}+\frac{2^2•3^2}-\frac{1}{2•3^3}+\frac{1}{3^4}\right)$+..... is $\frac{\alpha}{\beta}$, where $\alpha$ and $\beta$ are co-prime, then $\alpha$+$3\beta$ is equal to......
8.  Let $A$ = {1, 2, 3, 4} and $R$ be a relation on the set $A × A$ defined by $R$ = {$((a, b), (c, d))$ : $2a + 3b$ = $4c + 5d$}. Then the number of element in $R$ is _____
9.  The number of element in the set {$n \in N$ : $10 \leq n \leq 100$ and $3n – 3$ is a multiple of 7} is _____
10.  Let the plane $P$ contain the line $2x$ + $y – z$ – 3 = 0 = $5x – 3y$ + $4z$ + 9 and be parallel to the line $\frac{x+2}{2}$=$\frac{3-y}{-4}$=$\frac{z-7}{5}$. Then the distance of the point A (8, –1, – 19) from the plane $P$ measured parallel to the line $\frac{x}{-3}$=$\frac{y-5}{4}$=$\frac{2-z}{-12}$ is equal to........

06 Apr 2023:Shift-2

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