Download JEE Main 2023 Question Paper (13 Apr - Shift 2) Mathematics

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15 Apr 2023:Shift-1

06 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.  Let $(\alpha, \beta)$ be the centroid of the triangle formed by the lines $15x –y$ = 82, $6x –5y$ = -4 and $9x$ + $4y$ = 17. Then $\alpha + 2\beta$ and $2\alpha–\beta$ are the roots of the equation
   
   1. $x^2-14x$+$48$=0
   2. $x^2-10x$+$25$=0
   3. $x^2-13x$+$42$=0
   4. $x^2-7x$+$12$=0
2. Let for $A$=$\begin{equation}\begin{bmatrix}1 & 2 & 3 \\ \alpha & 3 & 1 \\ 1 & 1 & 2 \end{bmatrix}\end{equation}$, $|A|$=2. If $|2 adj (2 adj (2A))|$ = $32^n$, then $3n + \alpha$ is equal to
   
    
   1. 11
   2. 9
   3. 10
   4. 12
3.  The statement (p∧(~q))∨((~p)∧q)∨((~p)∧(~q))  is equivalent to _____
   
    
   1. p∨(~q)
   2. (~p)∨q
   3. p∨q
   4. (~p)∨(~q)
4.  The range of $f(x)$=4$\sin^{-1}\left(\frac{x^2}{1+x^2}\right)$ is
   
    
   1. $[0, 2\pi)$
   2. $[0, \pi]$
   3. $[0, \pi)$
   4. $[0, 2\pi]$
5.  The random variable $X$ follows binomial distribution $B (n, p)$, for which the difference of the mean and the variance is 1. If $2P(X = 2)$ = $3P(x = 1)$, then $n^2 P(x > 1)$ is equal to
   
    
   1. 15
   2. 16
   3. 11
   4. 12
6.  Let the centre of a circle $C$ be $(\alpha, \beta)$ and its radius $r$ < 8. Let $3x$ + $4y$ = 24 and $3x – 4y$ = 32 be two tangents and $4x$ + $3y$ = 1 be a normal to $C$. Then $(\alpha –\beta + r)$ is equal to
   
    
   1. 9
   2. 7
   3. 6
   4. 5
7.  Let $a_1$, $a_2$, $a_3$ …….. be a $G.P$ of increasing positive numbers . Let the sum of its $6^{th}$ and $8^{th}$ terms be 2 and the product of its $3^{rd}$and $5^{th}$ terms be $\frac{1}{9}$. Then $6 (a_2 +a_4)$$ (a_4 + a_6)$ is equal to
   
    
   1. $3\sqrt{3}$
   2. 2
   3. $2\sqrt{2}$
   4. 3
8.  The area of the region {$(x, y)$:$x^2 \leq y \leq |x^2-4|$, $y \leq 1$} is
   
    
   1. $\frac{3}{4}(4\sqrt{2}+1)$
   2. $\frac{3}{4}(4\sqrt{2}-1)$
   3. $\frac{4}{3}(4\sqrt{2}-1)$
   4. $\frac{4}{3}(4\sqrt{2}+1)$
9.  The line, that is coplanar to the line $\frac{x+3}{-3}$=$\frac{y-1}{1}$=$\frac{z-5}{5}$ is
   
    
   1. $\frac{x+1}{-1}$=$\frac{y-2}{2}$=$\frac{z-5}{4}$
   2. $\frac{x+1}{-1}$=$\frac{y-2}{2}$=$\frac{z-5}{5}$
   3. $\frac{x-1}{-1}$=$\frac{y-2}{2}$=$\frac{z-5}{5}$
   4. $\frac{x+1}{1}$=$\frac{y-2}{5}$=$\frac{z-5}{5}$
10.  The plane, passing through the points (0, -1, 2) and (-1, 2, 1) and parallel to the line passing through (5,1,-7) and (1,-1,-1), also passes through the point
    
     
    1. (2, 0, 1)
    2. (1, -2, 1)
    3. (0, 5, -2)
    4. (-2, 5, 0)
11.  If $\lim \limits_{x \to 0} \frac{e^{ax}-\cos(bx)-\frac{cxe^{-cx}}{2}}{1-\cos(2x)}$=17, then $5a^2+b^2$ is equal to
    
     
    1. 72
    2. 64
    3. 76
    4. 68
12.  Let for a triangle $ABC$,
    $\vec{AB}$=$-2\hat{i}$+$\hat{j}$+$3\hat{k}$
    $\vec{CB}$=$\alpha \hat{i}$+$\beta \hat{j}$+$\gamma \hat{k}$
    $\vec{CA}$=$4\hat{i}$+$3\hat{j}$+$\delta \hat{k}$
    if $\delta$ > 0 and the area of the triangle $ABC$ is $5\sqrt{6}$, then $\vec{CB}.\vec{CA}$ is equal to
    
     
    1. 120
    2. 54
    3. 60
    4. 108
13.  If the system of equations
    $2x$+$y$$– z$ = 5
    $2x$$ – 5y$ + $\lambda z$ = $\mu$
    $x$ + $2y $$– 5z $= 7
    has infinitely many solutions, then $(\alpha +\mu)^2$+$(\alpha – \mu)^2$ is equal to
    
     
    1. 912
    2. 916
    3. 904
    4. 920
14.  Let $S$={$z \in C$:$\bar{z}=i(z^2+Re(\bar{z}))$}. Then $\sum \limits_{z \in S}|z|^2$ is equal to
    
     
    1. 4
    2. $\frac{7}{2}$
    3. $\frac{5}{2}$
    4. 3
15.  The coefficient of $x^5$ in the expansion of $\left(2x^3-\frac{1}{3x^2}\right)^5$ is
    
     
    1. $\frac{26}{3}$
    2. $\frac{80}{9}$
    3. 8
    4. 9
16. The value of $\frac{e^{-\frac{\pi}{4}}+\int \limits_{0}^{\frac{\pi}{4}}e^{-x}(\tan^{50}x)dx}{\int \limits_0^{\frac{\pi}{4}}e^{-x}(\tan^{49}x+\tan^{51}x)dx}$ is
    
     
    1. 25
    2. 49
    3. 50
    4. 51
17.  Let $N$ be the foot of perpendicular from the point $P(1, –2, 3)$ on the line passing through the point (4,5,8) and (1, –7, 5). Then the distance of $N$ from the plane $2x $$–2y$ + $z$ + 5 = 0 is
    
     
    1. 6
    2. 9
    3. 8
    4. 7
18.  Let $|\vec{a}|$=2, $|\vec{b}|$=3 and the angle between the vectors $\vec{a}$ and $\vec{b}$ be $\frac{\pi}{4}$. Then $|(\vec{a}+2\vec{b}$)×(2\vec{a}-3\vec{b})|^2$ is equal to
    
     
    1. 441
    2. 882
    3. 841
    4. 482
19.  Let $\alpha$, $\beta$ be the roots of the equation $x^2-$$\sqrt{2}x$+2=0. Then $\alpha^{14}$+$\beta^{14}$ is equal to
    
     
    1. -64
    2. -128
    3. -128$\sqrt{2}$
    4. -64$\sqrt{2}$
20.  All words, with or without meaning, are made using all the letters of the word "MONDAY" . These words are written as in a dictionary with serial numbers. The serial number of the word "MONDAY" is
    
     
    1. 324
    2. 326
    3. 327
    4. 328

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.  The remainder, when $7^{103}$ is divided by 17, is
2.  Let $[\alpha]$ denote the greatest integer $\leq \alpha$. Then $[\sqrt{1}]$+$[\sqrt{2}]$+$[\sqrt{3}]$+ - - - +$[\sqrt{120}]$ is equal to
3.  The mean and standard deviation of the marks of 10 students were found to be 50 and 12 respectively. Later, it was observed that two marks 20 and 25 were wrongly read as 45 and 50 respectively. Then the correct variance is ______ .
4.  If $y = y(x)$ is the solution of the differential equation $\frac{dy}{dx}$+$\frac{4x}{(x^2-1)}y$$-\frac{x+2}{(x^2-1)^{\frac{5}{2}}$, $x>1$ such that $y(2)$=$\frac{2}{9}\log_e(2+\sqrt{3})$ and $y(\sqrt{2})$=$\alpha \log_e(\sqrt{\alpha}+\beta)$+$\beta-\sqrt{\gamma}$, $\alpha$, $\beta$, $\gamma \in N$, then $\alpha \beta \gamma$ is equal to ............
5.  Let $f_n$=$\int \limits_{0}^{\frac{\pi}{2}}\left(\sum \limits_{k=1}^{n}\sin^{k-1}x \right)\left(\sum \limits_{k=1}^{n}(2k-1)\sin^{k-1}x \right)\cos x dx$, $n \in N$. Then $f_{21}$-$f_{20}$ is equal to
6.  Let $A$ = {–4, –3, –2, 0, 1, 3, 4} and $R$ = {$(a, b) \in A × A$ : $b = |a|$ or $b^2$= $a + 1$ } be a relation on $A$. then the minimum number of elements, that must be added to the relation $R$ so that it becomes reflexive and symmetric, is
7.  For $x \in (–1, 1]$, then number of solution of the equation $sin^{–1}x$ = 2 $tan^{–1}x$ is equal to
8.  Let $f(x)$=$\sum \limits_{k=1}^{10}kx^k$, $x \in R$. If $2f(2)$+$f'(2)$=119(2)^n$+1 then $n$ is equal to
9.  Total numbers of 3-digit numbers that are divisible by 6 and can be formed by using the digits 1,2,3,4, 5 with repetition, is
10.  The foci of hyperbola are $(± 2, 0)$ and its eccentricity is $\frac{3}{2}$. A tangent, perpendicular to the line $2x$ + $3y$ = 6, is drawn at a point in the first quadrant on the hyperbola. If the intercepts made by the tangent on the $x –$ and $y –$ axes are $a$ and $b$ respectively, then $|6a|$ + $|5b|$ is equal to _______.

13 Apr 2023:Shift-1

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