Download JEE Main 2023 Question Paper (24 Jan - Shift 2) Mathematics

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25 Jan 2023:Shift-1

25 Jan 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 $y=y(x)$ be the solution of the differential equation $(x^2-3y^2)dx$+$3xydy$=0, $y(1)=1$. Then $6y^2(e)$ is equal to
   
   1. $\frac{3}{2}e^2$
   2. $3e^2$
   3. $e^2$
   4. $2e^2$
2. The number of real solutions of the equation $3\left(x^2+\frac{1}{x^2}\right)$$-2\left(x+\frac{1}{x}\right)$+5=0, is
   
   1. 4
   2. 0
   3. 2
   4. 3
3. Let $f(x)$ be a function such that $f(x+y)$=$f(x)•f(y)$ for all $x, y\in N$. If $f(1)=3$ and $\sum \limits_{k=1}^{n}f(k)$=3279, then the value of $n$ is
   
   1. 8
   2. 7
   3. 9
   4. 6
4.  If the system of equations
   $x$+$2y$+$3z$=3
   $4x$+$3y-4z$=4
   $8x$+$4y-\lambda z$=9+$\mu$
   has infinitely many solutions, then the ordered pair $(\lambda, \mu)$ is equal to :
   
   1. $\left(-\frac{72}{5}, -\frac{21}{5}\right)$
   2. $\left(\frac{72}{5}, -\frac{21}{5}\right)$
   3. $\left(\frac{72}{5}, \frac{21}{5}\right)$
   4. $\left(-\frac{72}{5}, \frac{21}{5}\right)$
5. The number of integers, greater than 7000 that can be formed, using the digits 3,5,6,7,8 without repetition, is
   
   1. 48
   2. 168
   3. 120
   4. 220
6. The locus of the mid points of the chords of the circle $C_1$:$(x-4)^2$+$(y-5)^2$=4 which subtend an angle $\theta_1$ at the centre of the circle $C_1$, is a circle of radius $r_1$. If $\theta_1$=$\frac{\pi}{3}$, $\theta_3$=$\frac{2\pi}{3}$ and $r_1^2$=$r_2^2$+$r_3^2$, then $\theta_2$ is equal to.......
   
   1. $\frac{3\pi}{4}$
   2. $\frac{\pi}{6}$
   3. $\frac{\pi}{4}$
   4. $\frac{\pi}{2}$
7.  If $f(x)$=$x^3-$$x^2f'(1)$+$xf''(2)-$$f'''(3)$, $x \in R$ , then
   
   1. 2$f(0)-$$f(1)$+$f(3)$=$f(2)$
   2. $f(3)-f(2)$=$f(1)$
   3. $3f(1)$+$f(2)$=$f(3)$
   4. $f(1)$+$f(2)$+$f(3)$=$f(0)$
8. If $(^{30}C_1)^2$+$2(^{30}C_2)^2$+$3(^{30}C_3)^2$+....+$30(^{30}C_30)^2$=$\frac{\alpha 60!}{(30!)^2}$ then $\alpha$ is equal to
   
   1. 15
   2. 10
   3. 60
   4. 30
9.  Let the plane containing the line of intersection of the planes $P1$: $x$+$(\lambda+4)y$+$z$=1 and $P2$ : $2x$+$y$+$z$=2 pass through the points $(0,1,0)$ and $(1,0,1)$. Then the distance of the point $(2\lambda, \lambda, -\lambda)$ from the plane $P2$ is
   
   1. $2\sqrt{6}$
   2. $5\sqrt{6}$
   3. $3\sqrt{6}$
   4. $4\sqrt{6}$
10.  Let the six numbers $a_1$, $a_2$, $a_3$, $a_4$, $a_5$, $a_6$, be in A.P. and $a_1$+$a_3$=10. If the mean of these six numbers is $\frac{19}{2}$ and their variance is $\sigma^2$, then $8\sigma^2$ is equal to
    
    1. 105
    2. 220
    3. 210
    4. 200
11.  Let p and q be two statements. Then ~ (p∧(p=>~ q)) is equivalent to
    
    1. p∨(p∧q)
    2. p∨(p∧(~q))
    3. (~p)∨q
    4. p∨((~p)∧q)
12. The set of all values of $a$ for which $\lim \limits_{x \to a}([x-5]-[2x+2])$=0, where $[\alpha]$ denotes the greatest integer less than or equal to $\alpha$ is equal to
    
    1. (-7.5, -6.5]
    2. [-7.5, -6.5]
    3. (-7.5, -6.5)
    4. [-7.5, -6.5)$
13. If $f(x)$=$\frac{2^{2x}}{2^{2x}+2}$, $x \in R$, then $f\left(\frac{1}{2023}\right)$+$f\left(\frac{2}{2023}\right)$+...+$f\left(\frac{2022}{2023}\right)$ is equal to
    
    1. 1010
    2. 1011
    3. 2011
    4. 2010
14.  The equations of the sides $AB$ and $AC$ of a triangle $ABC$ are $(\lambda+1)x$+$\lambda y$=4 and $\lambda x$+$(1-\lambda y)y$+$\lambda$=0 respectively. Its vertex $A$ is on the $y-$axis and its orthocentre is $(1,2)$ . The length of the tangent from the point $C$ to the part of the parabola $y^2$=$6x$ in the first quadrant is :
    
    1. $2\sqrt{2}$
    2. 2
    3. $\sqrt{6}$
    4. 4
15.  The value of $\left(\frac{1+\sin\frac{2\pi}{9}+i\cos\frac{2\pi}{9}}{1+\sin\frac{2\pi}{9}-i\cos\frac{2\pi}{9}}\right)^3$ is
    
    1. $\frac{1}{2}(1-i\sqrt{3})$
    2. $-\frac{1}{2}(1-i\sqrt{3})$
    3. $\frac{1}{2}(\sqrt{3}+i)$
    4. $-\frac{1}{2}(\sqrt{3}-i)$
16.  Let $\vec{\alpha}$=$4\hat{i}$+$3\hat{j}$+$5\hat{k}$ and $\vec{\beta}$=$\hat{i}$+$2\hat{j}$$-4\hat{k}$. Let $\vec{\beta_1}$ be parallel to $\vec{\alpha}$ and $\vec{\beta_2}$ be perpendicular to $\vec{\alpha}$. If $\vec{\beta}$=$\vec{\beta_1}+\vec{\beta_2}$, then the value of $5\vec{\beta_2}•(\hat{i}+\hat{j}+\hat{k})$ is
    
    1. 7
    2. 9
    3. 6
    4. 11
17.  $\int \limits_{\frac{3\sqrt{2}}{4}}^{\frac{3\sqrt{3}}{4}}\frac{48}{\sqrt{9-4x^2}}dx$ is equal to
    
    1. $\frac{\pi}{3}$
    2. $2\pi$
    3. $\frac{\pi}{2}$
    4. $\frac{\pi}{6}$
18.  Let $A$ be a 3 x 3 matrix such that $|adj(adj(adj A))|$=$12^4$. Then $|A^{-1} adjA|$ is equal to
    
    1. $2\sqrt{3}$
    2. 1
    3. $\sqrt{6}$
    4. 12
19.   If the foot of the perpendicular drawn from $(1,9,7)$ to the line passing through the point $(3,2,1)$ and parallel to the planes $x$+$2y$+$z$=0 and $3y-z$=3 is $(\alpha, \beta, \gamma)$, then $\alpha+\beta+\gamma$ is equal to
    
    1. 1
    2. 3
    3. 5
    4. -1
20. The number of square matrices of order 5 with entries from the set {0,1} , such that the sum of all the elements in each row is 1 and the sum of all the elements in each column is also 1, is
    
    1. 120
    2. 125
    3. 225
    4. 150

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 $S$={$\theta \in [0, 2\pi):\tan(\pi \cos \theta)+\tan (\pi \sin \theta)$=0}. Then $\sum \limits_{\theta \in S} \sin^2\left(\theta +\frac{\pi}{4}\right)$ is equal to..........
2.  The minimum number of elements that must be added to the relation $R$= {$(a,b)$, $(b,c)$, $(b,d)$ on the set ${a,b,c,d}$ so that it is an equivalence relation, is…….
3.  The equations of the sides $AB$, $BC$ and $CA$ of a triangle $ABC$ are : $2x+y$=0, $x+py$=$21a$, $a\neq 0$ and $x-y$=3 respectively. Let $P(2,a)$ be the centroid of $ABC$ . Then $(BC)^2$ is equal to
4.  If $\frac{1^3+2^3+3^3+....\text{up to n terms}}{1•3+2•5+3•7+....\text{up to n terms}}$=$\frac{9}{5}$, , then the value of $n$ is
5. Let $f$ be a differentiable function defined on $[0, \frac{\pi}{2}]$ such that $f(x)>0$ and $f(x)$+$\int \limits_{0}^{x}f(t)\sqrt{1-(\log_ef(t))^2}dt$=$e$, $\forall x \in [0, \frac{\pi}{2}]$. Then $\left(6\log_ef\left(\frac{\pi}{6}\right)\right)^2$ is equal to..........
6.  If the shortest between the lines
   $\frac{x+\sqrt{6}}{2}$=$\frac{y-\sqrt{6}}{3}$=$\frac{z-\sqrt{6}}{4}$ and $\frac{x-\lambda}{3}$=$\frac{y-2\sqrt{6}}{4}$=$\frac{z+2\sqrt{6}}{5}$ is 6, then the square of sum of all possible values of $\lambda$ is
7.  If the area of the region bounded by the curves $y^2-2y$=$-x$, $x+y$=0 is $A$, then $8A$ is equal to
8.  Three urns $A$, $B$ and $C$ contain 4 red, 6 black; 5 red, 5 black; and $\lambda$ red, 4 black balls respectively. One of the urns is selected at random and a ball is drawn. If the ball drawn is red and the probability that it is drawn from urn $C$ is 0.4 then the square of the length of the side of the largest equilateral triangle, inscribed in the parabola $\lambda^2$=$\lambda x$ with one vertex at the vertex of the parabola, is
9.  Let the sum of the coefficients of the first three terms in the expansion of $\left(x-\frac{3}{x^2}\right)^n$, $x \neq 0$, $n \in N$, be 376. Then the coefficient of $x^4$ is..........
10.  Let $\vec{a}$=$\hat{i}$+$2\hat{j}$+$\lambda \hat{k}$, $\vec{b}$=$3 \hat{i}$$-5\hat{j}$$-\lambda \hat{k}$, $\vec{a}$•$\vec{c}$=7, 2$\vec{b}$•$\vec{c}$+43=0, $\vec{a}$×$\vec{c}$=$\vec{b}$×$\vec{c}$. Then $|\vec{a}•\vec{b}|$ is equal to

24 Jan 2023: Shift-1

24 Jan 2023:Ans-Keys

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