Download JEE Main 2023 Question Paper (08 Apr - Shift 2) Mathematics

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10 Apr 2023:Shift-1

10 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. The area of the quadrilateral $ABCD$ with vertices $A (2, 1, 1)$, $B(1, 2, 5)$, $C(–2, –3, 5)$ and $D(1, –6, –7)$ is equal to
   
   1. 8$\sqrt{38}$
   2. 48
   3. 54
   4. 9$\sqrt{38}$
2.  The value of 36 (4 $cos^29°$ – 1)(4 $cos^227°$ – 1)(4 $cos^2 81°$ – 1) (4 $cos^2$243° –1) is
   
    
   1. 27
   2. 18
   3. 54
   4. 36
3.  Let $A$={$\theta \in (0, 2\pi):\frac{1+2i\sin\theta}{1-i\sin \theta}$ is purely imaginary}. The sum of elements in $A$ is
   
    
   1. $\pi$
   2. $2\pi$
   3. $3 \pi$
   4. $4 \pi$
4.  If the probability that the random variable $X$ take values $x$ is given by $P(X = x)$ = $k (x + 1)3^{–x}$, $x$ = 0, 1, 2, 3, ……, where $k$ is a constant, then $P(X ≥ 2)$ is
   
    
   1. $\frac{20}{27}$
   2. $\frac{7}{18}$
   3. $\frac{7}{27}$
   4. $\frac{11}{18}$
5.  For $a, b \in Z$ and $a – b ≤ 10$, let the angle between the plane $P : ax + y – z = b$ and the line $l : x – 1$ = $a – y$ = $z + 1$ be $\cos^{-1}\left(\frac{1}{3}\right)$ if the distance of the point (6, –6, 4) from the plane $P$ is $3\sqrt{6}$, then $a^4+b^2$ is equal to
   
    
   1. 32
   2. 25
   3. 48
   4. 85
6.  Let $a_n$ be the $n^{th}$ term of the series 5 + 8 + 14 + 23 + 35 + 50 + ……. And $s_n$=$\sum \limits_{k=1}^{n}a_k$. Then $S_{30}-a_{40}$ is equal to
   
    
   1. 11280
   2. 11290
   3. 11310
   4. 11260
7.  Let $A$ = {1, 2, 3, 4, 5, 6, 7}. Then the relation $R$ = {$(x, y) \in A × A$ : $x + y$ = 7} is
   
    
   1. reflexive but neither symmetric nor transitive
   2. transitive but neither symmetric nor reflexive
   3. symmetric but neither reflexive nor transitive
   4. an equivalence relation
8.  The absolute difference of the coefficients of $x^{10}$ and $x^7$in the expansion of $\left(2x^2+\frac{1}{x}\right)^{11}$ is equal to
   
    
   1. $10^3-10$
   2. $13^3-13$
   3. $11^3-11$
   4. $12^3-12$
9. The integral $\int \left(\left(\frac{x}{2}\right)^{x}+\left(\frac{2}{x}\right)^{x}\right)\log_2xdx$ is equal to
   
   1. $\left(\frac{x}{2}\right)^{x}-\left(\frac{2}{x}\right)^{x}+C$
   2. $\left(\frac{x}{2}\right)^{x}\log_2\left(\frac{2}{x}\right)^{x}+C$
   3. $\left(\frac{x}{2}\right)^{x}+\left(\frac{2}{x}\right)^{x}$
   4. $\left(\frac{x}{2}\right)^{x}\log_2\left(\frac{2}{x}\right)^{x}$
10. If $A$=$\begin{equation*}\begin{bmatrix} 1 & 5 \\ \lambda & 10 \end{bmatrix} \end{equation*}$, $A^{-1}$=$\alpha A$+$\beta I$ and $\alpha+\beta$=-2, then $4 \alpha^2$+$\beta^2$+$\lambda^2$ is equal to..........
    
     
    1. 10
    2. 19
    3. 14
    4. 12
11.  The negation of (p ∧(~q)) ∨ ( ~ p) is equivalent to
    
     
    1. p∨(q∨(~p))
    2. p∧q
    3. p∧(q∧(~p))
    4. p∧(~p)
12. If $\alpha$ > $\beta$ > 0 are the roots of the equation $ax^2$ + $bx$ + 1 = 0, and $\lim \limits_{x \to \frac{1}{\alpha}}\left(\frac{1-\cos(x^2+bx+a)}{2(1-\alpha x)^2}\right)^{\frac{1}{2}}$=$\frac{1}{k}\left(\frac{1}{\beta}-\frac{1}{\alpha}\right)$, then $k$ is equal to
    
     
    1. $2 \alpha$
    2. $2 \beta$
    3. $\beta$
    4. $\alpha$
13.  If the number of words, with or without meaning, which can be made using all the letters of the word MATHEMATICS in which C and S do not come together, is (6!)k, then k is equal to
    
     
    1. 1890
    2. 2835
    3. 5670
    4. 945
14.  Let $S$ be the set of all values of $\theta \in [–\pi, \pi]$ for which the system of linear equations $x$ + $y$ + $3z$ = 0
    $–x$ + $(\tan \theta) y$ + $7z$ = 0
    $x$ + $y$ + $(tan \theta) z$ = 0
    has non-trivial solution. Then $\frac{120}{\pi}\sum \limits_{\theta \in S} \theta$ is equal to
    
     
    1. 20
    2. 30
    3. 40
    4. 10
15.  Let P be the plane passing through the line $\frac{x-1}{1}$=$\frac{y-2}{-3}$=$\frac{z+5}{7}$ and the point (2, 4, –3). If the image of the point (–1, 3, 4) in the plane $P$ is $(\alpha, \beta, \lambda)$ then $\alpha+\beta+\lambda$ is equal to
    
     
    1. 12
    2. 9
    3. 10
    4. 11
16.  Let $O$ be the origin and $OP$ and $OQ$ be the tangents to the circle $x^2$ + $y^2$$ – 6x$ + $4y$ + 8 = 0 at the points $P$ and $Q$ on it. If the circumcircle of the triangle $OPQ$ passes through the point $\left(\alpha, \frac{1}{2}\right)$ then a value of $\alpha$ is
    
     
    1. $\frac{3}{2}$
    2. $-\frac{1}{2}$
    3. 1
    4. $\frac{5}{2}$
17. Let the mean and variance of 12 observations be$\frac{9}{2}$ and 4 respectively. Later on, it was observed that two observations were considered as 9 and 10 instead of 7 and 14 respectively. If the correct variance is$\frac{m}{n}$, where $m$ and $n$ are coprime, then $m + n$ is equal to
    
     
    1. 316
    2. 317
    3. 314
    4. 315
18. Let the vectors $\vec{u_1}$=$\hat{i}$+$\hat{j}$+$a\hat{k}$, $\vec{u_2}$=$\hat{i}$+$b\hat{j}$+$\hat{k}$ and $u_3$=$c\hat{i}$+$\hat{j}$+$\hat{k}$ be coplanar. If the vectors $\vec{v_1}$=$(a+b)\hat{i}$+$c\hat{j}$+$c\hat{k}$, $\vec{v_2}$=$a\hat{i}$+$(b+c)\hat{j}$+$a\hat{k}$ and $\vec{v_3}$=$b\hat{i}$+$b\hat{j}$+$(c+a)\hat{k}$ are also coplanar, then $6(a+b+c)$ is equal to
    
     
    1. 4
    2. 0
    3. 12
    4. 6
19.  Let $A (0, 1)$, $B(1, 1)$ and $C(1, 0)$ be the mid-points of the sides of a triangle with incentre at the point $D$. If the focus of the parabola $y^2$ = $4ax$ passing through $D$ is $(\alpha + \beta \sqrt{2} , 0)$, where $\alpha$ and $\beta$ are rational numbers, then$\frac{\alpha}{\beta^2}$ is equal to
    
     
    1. 8
    2. 6
    3. $\frac{9}{2}$
    4. 12
20.  $25^{190}$ – $19^{190}$ – $8^{190}$ + $2^{190}$ is divisible by
    
    1. 34 but not by 14
    2. neither 14 nor 34
    3. 14 but not by 34
    4. both 14 and 34

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 the solution curve $x$ = $x(y)$, $0 < y < \frac{\pi}{2}$, of the differential equation $(\log_e\cos y)^2\cos ydx$$-(1+3x\log_e(\cos y))\sin y dy$=0 satisfy $x\left(\frac{\pi}{3}\right)$=$\frac{1}{2\log_e2}$. If $x\left(\frac{\pi}{6}\right)$=$\frac{1}{\log_em-\log_en}$. where $m$ and $n$ are coprime, then $mn$ is equal to
2.  Let $0 < z < y < x$ be three real numbers such that $\frac{1}{x}$, $\frac{1}{y}$, $\frac{1}{z}$ are in an arithmetic progression and $x$, $\sqrt{2}y$, $z$ are in a geometric progression. If $xy$+$yz$+$zx$=$\frac{3}{\sqrt{2}}xyz$, then $3(x+y+z)^2$ is equal to
3. Let $k$ and $m$ be positive real numbers such that the function $f(x)$=$\left\{\begin{array}{cc}3 x^2+k \sqrt{x+1}, & \theta < x < 1 \\ m x^2+k^2, & x \geq 1\end{array}\right.$ is differentiable for all $x >0$. Then $\frac{8f'(8)}{f'\left(\frac{1}{8}\right)}$ is equal to.......
4.  If domain of the function $\log_e\left(\frac{6x^2+5x+1}{2x-1}\right)$+$\cos^{-1}\left(\frac{2x^2-3x+4}{3x-5}\right)$ is $(\alpha, \beta) \cup (\gamma, \delta]$, then $18(\alpha^2+\beta^2+\gamma^2+\delta^2)$ is equal to ................
5.  Let $[t]$ denote the greatest integer function. If $\int \limits_{0}^{2.4}[x^2]dx$=$\alpha$+$\beta \sqrt{2}$+$\gamma \sqrt{3}$+$\delta \sqrt{5}$, then $\alpha$+$\beta$+$\gamma$+$\delta$ is equal to..........
6.  Let $P_1$ be the plane $3x$ – $y$ – $7z$ = 11 and $P_2$ be the plane passing through the points (2, –1, 0), (2, 0, –1), and (5, 1, 1). If the foot of the perpendicular drawn from the point (7, 4, –1) on the line of intersection of the planes $P_1$ and $P_2$ is $(\alpha, \beta, \gamma)$, then $\alpha$ + $\beta$ + $\gamma$ is equal to__________.
7.  The ordinates of the points $P$ and $Q$ on the parabola with focus (3, 0) and directrix $x$ = –3 are in the ratio 3 : 1. If $R (\alpha, \beta)$ is the point of intersection of the tangents to the parabola at $P$ and $Q$, then$\frac{\beta^2}{\alpha}$ is equal to __________
8.  Let $m$ and $n$ be the numbers of real roots of the quadratic equations $x^2$ – $12x$ + $[x]$ + 31 = 0 and $x^2$ – $5 |x + 2 |$ – 4 = 0 respectively, where $[x]$ denotes the greatest integer $≤ x$. Then $m^2$ + $mn$ + $n^2$is equal to __________.
9.  Let $R$ = ${a, b, c, d, e}$ and $S$ = {1, 2, 3, 4}. Total number of onto functions $f : R \to S$ such that $f(a) \neq 1$, is equal to __________.
10.  Let the area enclosed by the lines $x + y$ = 2, $y$ = 0, $x$ = 0 and the curve $f(x)$ = min$\left\{x^2+\frac{3}{4}, 1+[x]\right\}$ where $[x]$denotes the greatest integer $≤ x$, be $A$. Then the value of $12 A$ is __________.

08 Apr 2023:Shift-1

08 Apr 2023:Ans-Key

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