Download JEE Main 2023 Question Paper (10 Apr - Shift 1) Mathematics

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11 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. For the system of linear equation
   $2x$$ – y$ + $3z$ = 5
   $3x$ + $2y – z$ = 7
   $4x$ + $5y$ + $\alpha z$ = $\beta$,
   which of the following is NOT correct?
   
   1. The system has infinitely many solutions for $\alpha$ = –6 and $\beta$= 9
   2. The system is inconsistent for $\alpha$=$-5$ and $\beta$=$8$
   3. The system has infinitely many solutions for $\alpha$=$-5$ and $\beta$=9
   4. The system has a unique solution for $\alpha \neq 5$ and $\beta$=8
2.  Let the first term a and the common ratio r of a geometric progression be positive integers. If the sum of squares of its first three terms is 33033, then the sum of these three terms is equal to
   
    
   1. 241
   2. 220
   3. 210
   4. 231
3.  A line segment $AB$ of length $\lambda$ moves such that the points $A$ and $B$ remain on the periphery of a circle of radius $\lambda$. Then the locus of the point, that divides the line segment $AB$ in the ratio 2:3, is a circle of radius
   
    
   1. $\frac{\sqrt{19}}{7}\lambda$
   2. $\frac{2}{3}\lambda$
   3. $\frac{\sqrt{19}}{5}\lambda$
   4. $\frac{3}{5}\lambda$
4.  Let two vertices of a triangle $ABC$ be (2, 4, 6) and (0, –2, –5), and its centroid be (2, 1, –1). If the image of the third vertex in the plane $x$ + $2y$ + $4z$ = 11 is $(\alpha, \beta, \gamma)$, then $\alpha \beta $+ $\beta \gamma$ + $\gamma \alpha$ is equal to
   
    
   1. 72
   2. 76
   3. 74
   4. 70
5.  Let $N$ denote the sum of the numbers obtained when two dice are rolled. If the $m$ probability that $2^N < N!$ is $\frac{m}{n}$, where $m$ and $n$ are co-prime, then $4m-3n$ is equal to
   
    
   1. 8
   2. 6
   3. 12
   4. 10
6.  If $f(x)$=$\frac{(\tan1°)x+\log_e(123)}{x\log_e(1234)-(\tan1°)}$, $x >0$, then the least value of $f(f(x))$+$f\left(f\left(\frac{4}{x}\right)\right)$ is
   
    
   1. 8
   2. 2
   3. 4
   4. 0
7.  Let $f$ be a differentiable function such that $x^2f(x)$$-x$=$4\int \limits_{0}^{x}tf(t)dt$, $f(1)$=$\frac{2}{3}$. Then $18f(3)$ is equal to
   
    
   1. 210
   2. 160
   3. 180
   4. 150
8. The shortest distance between the lines $\frac{x+2}{1}$=$\frac{y}{-2}$=$\frac{z-5}{2}$ and $\frac{x-4}{1}$=$\frac{y-1}{2}$=$\frac{z+3}{0}$ is
   
    
   1. 7
   2. 6
   3. 9
   4. 8
9.  Let the ellipse $E$: $x^2$ + $9y^2$ = 9 intersect the positive $x-$ and $y-$axes at the points $A$ and $B$ respectively. Let the major axis of $E$ be a diameter of the circle $C$. Let the line passing through $A$ and $B$ meet the circle $C$ at the point $P$. If the area of the triangle with vertices $A$, $P$ and the origin $O$ is$\frac{m}{n}$, where $m$ and $n$ are co-prime, then $m – n$ is equal to
   
    
   1. 15
   2. 18
   3. 16
   4. 17
10.  If $A$ is a 3 × 3 matrix and $| A |$ = 2, then $3 adj (|3A|A^2)$ is equal to
    
     
    1. $3^{11}•6^{10}$
    2. $3^{12}•6^{11}$
    3. $3^{12}•6^{10}$
    4. $3^{10}•6^{11}$
11.  Let the complex number $z$ = $x + iy$ be such that $\frac{2z-3i}{2z+i}$ is purely imaginary. If $x+y^2$=0, then $y^4+y^2-y$ is equal to :
    
     
    1. $\frac{2}{3}$
    2. $\frac{3}{2}$
    3. $\frac{4}{3}$
    4. $\frac{3}{4}$
12.  96$\cos\frac{\pi}{33}$$\cos\frac{2\pi}{33}$$\cos\frac{4\pi}{33}$$\cos\frac{8\pi}{33}$$\cos\frac{16\pi}{33}$
    
     
    1. 3
    2. 1
    3. 2
    4. 4
13. If $I(x)$=$\int e^{\sin^2x}(\cos x \sin 2x -\sin x)dx$ and $I(0)$=1, then $I\left(\frac{\pi}{3}\right)$ is equal to
    
     
    1. $e^{\frac{3}{4}}$
    2. $-e^{\frac{3}{4}}$
    3. $-\frac{1}{2}e^{\frac{3}{4}}$
    4. $\frac{1}{2}e^{\frac{3}{4}}$
14.  A square piece of tin of side 30 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. If the volume of the box is maximum, then its surface area (in cm²) is equal to
    
     
    1. 800
    2. 1025
    3. 675
    4. 900
15.  The negation of the statement (p∨q)∧(q∨(~r)) is
    
     
    1. ((~p)∨r))∧(~q)
    2. ((~p)∨(~q))∨(~r)
    3. (p∨r)∧(~q)
    4. ((~p)∨(~q))∧(~r)
16. An arc PQ of a circle subtends a right angle at its centre O. The mid-point of the arc PQ is R. If $\vec{OP}$=$\vec{u}$, $\vec{OR}$=$\vec{v}$ and $\vec{OQ}$=$\alpha \vec{u}$+$\beta \vec{v}$, then $\alpha$, $\beta^2$ are the roots of the equation
    
     
    1. $3x^2+2x-1$=0
    2. $x^2+x-2$=0
    3. $3x^2-2x-1$=0
    4. $x^2-x-2$=0
17.  The slope of tangent at any point $(x, y)$ on a curve $y = y(x)$ is $\frac{x^2+y^2}{2xy}$, $x>0$. If $y(2)$=0, then a value of $y(8)$ is
    
     
    1. 4$\sqrt{3}$
    2. 2$\sqrt{3}$
    3. -4$\sqrt{2}$
    4. -2$\sqrt{3}$
18.  Let $O$ be the origin and the position vector of the point $P$ be $-\hat{i}-2\hat{j}+3\hat{k}$. If the position vectors of the points $A$, $B$ and $C$ are $-2\hat{i}+\hat{j}-3\hat{k}$, $2\hat{i}+4\hat{j}-2\hat{k}$ and $-4\hat{i}+2\hat{j}-\hat{k}$ respectively, then the projection of the vector $\vec{OP}$ on a vector perpendicular to the vectors $\vec{AB}$ and $\vec{AC}$ is
    
     
    1. $\frac{8}{3}$
    2. $\frac{10}{3}$
    3. 3
    4. $\frac{7}{3}$
19.  If the coefficient of $x^7$ in $\left(ax-\frac{1}{bx^2}\right)^{13}$ and the coefficient of $x^{-5}$ in $\left(ax+\frac{1}{bx^2}\right)^{13}$ are equal, then $a^4b^4$ is
    
     
    1. 44
    2. 11
    3. 33
    4. 22
20.  Let $P$ be the point of intersection of the line $\frac{x+3}{3}$=$\frac{y+2}{1}$=$\frac{y-z}{2}$ and the plane $x$ + $y$ + $z$ = 2. If the distance of the point $P$ from the plane $3x$$ – 4y$ + $12z$ = 32 is $q$, then $q$ and $2q$ are the roots of the equation
    
     
    1. $x^2-$$18x$$-72$=0
    2. $x^2$+$18x$+72=0
    3. $x^2+18x$$-72$=0
    4. $x^2-18x$+72=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.  If the mean of the frequency distribution

   Class:	0-10	10-20	20-30	30-40	40-50
   Frequency:	2	3	$x$	5	4

   
   is 28, then its variance is –
2.  The number of permutations, of the digits 1, 2, 3, ..., 7 without repetition, which neither contain the string 153 nor the string 2467, is___________
3.  The coefficient of $x^7$ in $(1–x + 2x^3)^{10}$ is _______
4.  The number of elements in the set $\left\{n \in Z:|n^2-10n+19|<6\right\}$ is.........
5.  Some couples participated in a mixed doubles badminton tournament. If the number of matches played, so that no couple played in a match, is 840, then the total numbers of persons, who participated in the tournament, is____________
6.  Let $y = p(x)$ be the parabola passing through the points $(-1, 0)$, $(0, 1)$ and $(1, 0)$. If the area of the region {$(x, y)$:$(x+1)^2$+$(y-1)^2$$\leq 1$, $y \leq p(x)$} is $A$, then $12(\pi-4A)$ is equal to..........
7. Let a common tangent to the curves $y^2$ = $4x$ and $(x – 4)^2$ + $y^2$ = 16 touch the curves at the points $P$ and $Q$. Then $(PQ)^2$ is equal to__________
8.  Let $f:(-2, 2) \to IR$ be defined by
   $f(x)$=$\begin{cases}x[x], -2 < x < 0 \\ (x-1)[x], 0 \leq x < 2 \end{cases}$
   where $[x]$ denotes the greatest integer function. If $m$ and $n$ respectively are the number of points in (–2, 2) at which $y = |f(x)|$ is not continuous and not differentiable, then $m + n$ is equal to_________.
9. Let $a$, $b$, $c$ be three distinct positive real number such that $(2a)^{\log_e a}$=$(bc)^{\log_e b}$ and $b^{\log_e2}$=$a^{\log_ec}$. Then $6a + 5bc$ is equal to _____________
10.  The sum of all those terms, of the arithmetic progression 3, 8, 13, ........ 373, which are not divisible by 3, is equal to _______.

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