Download JEE Main 2023 Question Paper (08 Apr - Shift 1) Mathematics

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10 April 2023:Shift-1

10 April 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 the number of elements in sets $A$ and $B$ be five and two respectively. Then the number of subsets of A ×B each having at least 3 and at most 6
   
   1. 782
   2. 792
   3. 752
   4. 772
2.  The number of arrangements of the letters of the word "INDEPENDENCE" in which all the vowels always occur together is
   
    
   1. 18000
   2. 33600
   3. 16800
   4. 14800
3.  The number of ways, in which 5 girls and 7 boys can be seated at a round so that no two girls sit together, is
   
    
   1. $7(360)^2$
   2. $7(720)^2$
   3. $126(5!)^2$
   4. 720
4.  If the equation of the plane containing the line $x$ + 2$y$ + 3$z$ – 4 = 0 = 2$x$ + $y-z$ +5 and perpendicular to the plane $\vec{r}$=$(\hat{i}-\hat{j})$+$\lambda (\hat{i}+\hat{j}+\hat{k})$+$\mu(\hat{i}-2\hat{j}+3\hat{k})$ is $ax$+$by$+$cz$=4, than $(a - b + c)$ is equal
   
    
   1. 20
   2. 22
   3. 21
   4. 18
5.  If the points with position vectors $\alpha \hat{i}$+$10 \hat{j}$+$13 \hat{k}$, $6 \hat{i}$+$11\hat{j}$+$11\hat{k}$, $\frac{9}{2} \hat{i}$+$\beta \hat{j}$$-8\hat{k}$ are collinear, then $(19\alpha-6\beta)^2$ is equal to............
   
    
   1. 25
   2. 16
   3. 49
   4. 36
6.  Let $S_k$$\frac{1+2+..... +K}{k}$ and $\sum \limits_{j=1}^{n}S_j^2$=$\frac{n}{A}(Bn^2+Cn+D)$, where $A$, $B$, $C$, $D \in N$ and $A$ has least value. Then,
   
    
   1. $A$ + $B$ =5$(D-C)$
   2. $A$ + $C$+ $D$ is not divisible by B
   3. $A$ +$B$ + $C$ +$D$ is divisible by 5
   4. $A$ + $B$ is divisible by $D$
7.  Negation of (p=>q)=>(q=>p) is
   
    
   1. q∧(~p)
   2. q∧(~p)
   3. (~q)∧p
   4. (~q)∧p
8.  If for $z$ = $\alpha$ + $i\beta$$|z + 2|$ = $z + 4 (1 + i)$, then $\alpha + \beta$ and $\alpha \beta$ are the roots of the
   
    
   1. $x^2$+$3x-4$=0
   2. $x^2+x-$12=0
   3. $x^2+2x-$3=0
   4. $x^2+7x$+12=0
9.  If the coefficients of three consecutive terms in the expansion of $(1+x)^n$ are in the ratio 1:5:20, then the coefficient of the fourth term is
   
    
   1. 3654
   2. 2436
   3. 1827
   4. 5481
10. Let $A$=$\begin{equation*} \begin{bmatrix} 2 & 1 & 0 \\ 1 & 2 & -1 \\ 0 & -1 & 2 \end{bmatrix} \end{equation*}$. If $|adj(adj(adi2A))|$=$(16)^n$, then $n$ is equal to
    
     
    1. 9
    2. 12
    3. 8
    4. 10
11.  Let $\alpha, \beta, \gamma$ be the three roots of the equation $x^3$+ $bx$ + $c$ =0. If $\beta \gamma$ = 1 = $–\alpha$, then $b^3$ + $2c^3$ – $3\alpha^3$ – $6\beta^3$ – $8\gamma^3$ is equal to
    
     
    1. 21
    2. 19
    3. $\frac{155}{2}$
    4. $\frac{169}{8}$
12.  In a bolt factory, machines $A$, $B$ and $C$, manufacture respectively 20%, 30% and 50% of the total bolts, Of their output , 3, 4 and 2 percent are respectively defective bolts, $A$ blot is drawn at random from the product. If the bolt drawn is found the defective, then the probability that is manufactured by the machine $C$is
    
     
    1. $\frac{9}{28}$
    2. $\frac{5}{14}$
    3. $\frac{2}{7}$
    4. $\frac{3}{7}$
13. Let $f(x)$=$\frac{\sin x-\cos x-\sqrt{2}}{\sin x -\cos x}$, $x \in [0, \pi]-\left\{\frac{\pi}{4}\right\}$. Then $f\left(\frac{7\pi}{12}\right)$$f''\left(\frac{7\pi}{12}\right)$ is equal to
    
     
    1. $\frac{-1}{3\sqrt{3}}$
    2. $\frac{2}{9}$
    3. $\frac{2}{3\sqrt{3}}$
    4. $\frac{-2}{3}$
14.  The shortest distance between the lines $\frac{x-4}{4}$=$\frac{y+2}{5}$=$\frac{z+3}{3}$ and $\frac{x-1}{3}$=$\frac{y-3}{4}$=$\frac{z-4}{2}$ is
    
    1. $6 \sqrt{3}$
    2. $3 \sqrt{6}$
    3. $6 \sqrt{2}$
    4. $2\sqrt{6}$
15.  Let $C (\alpha, \beta)$ be the circumcenter of the triangle formed by the lines
    $4x$+$3y$ = 69,
    $4y-$$3x$=17, and
    $x$+$7y$=61.
    Then $(\alpha– \beta)^2$ + $\alpha$ + $\beta$ is equal to
    
     
    1. 18
    2. 15
    3. 16
    4. 17
16.  The area of the region {$(x, y):x^2 \leq y \leq 8-x^2$, $y \leq 7$} is
    
     
    1. 18
    2. 20
    3. 21
    4. 24
17. Let $P$=$\begin{equation*}\begin{bmatrix} \frac{3}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{\sqrt{3}}{2} \end{bmatrix} \end{equation*}$, $A$=$\begin{equation*}\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} \end{equation*}$ and $Q$=$PAP^T$. If $P^TQ^{2007}P$=$\begin{equation*}\begin{bmatrix} a & b \\ c & d \end{bmatrix} \end{equation*}$, then $2a$ + $b $$-3c$$ –4d$ equal to
    
     
    1. 2005
    2. 2004
    3. 2007
    4. 2006
18.  Let $I(x)$=$\int \frac{(x+1)}{x(1+xe^x)^2}dx$, $x>0$. If $\lim \limits_{x \to \infty} I(x)=0$, then $I(1)$ is equal to
    
     
    1. $\frac{e+1}{e+2}-\log_e(e+1)$
    2. $\frac{e+1}{e+2}+\log_e(e+1)$
    3. $\frac{e+2}{e+1}-\log_e(e+1)$
    4. $\frac{e+2}{e+1}+\log_e(e+1)$
19.  Let $R$ be the focus of the parabola $y^2$= $20x$ and the line $y$ = $mx$ + $c$ intersect the parabola at two points $P$ and $Q$. Let the point $G (10,10)$ be the centroid of the triangle $PQR$. If $c-m$=6, then $(PQ)^2$ is
    
     
    1. 325
    2. 296
    3. 317
    4. 316
20. $\lim \limits_{x \to 0}\left(\left(\frac{(1-\cos ^2(3x))}{\cos^3(4x)}\right)\right.$$\left. \left(\frac{\sin^3(4x)}{(\log_e(2x+1))^5}\right)\right)$ is equal to.........
    
     
    1. 15
    2. 9
    3. 24
    4. 18

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 $a_n$ is the greatest term in the sequence $a_n$=$\frac{n^3}{n^4+147}$, $n=1, 2, 3, ........ , then $a$ is equal to..........
2.  Let the mean and variance of 8 number $x$ , $y$,10,12,6,12,4,8 be 9 and 9.25 respectively, If $x>y$, then $3x – 2y$ is equal to_____.
3.  Let $[t] $denote the greatest integer $\leq t$. If the constant term in the expansion of $\left(3x^2-\frac{1}{2x^5}\right)^7$ is $a$, then $[ a ]$ is equal to _____.
4.  Let $\lambda_1, \lambda_2$ be the values of $\lambda$ for which the points $\left(\frac{5}{2}, 1, \lambda \right)$ and $(-2, 0, 1)$ are at equal distance from the plane $2x$ + $3y$$ -6z$ +7=0. If $\lambda_1$ > $\lambda_2$, then the distance of the point $(\lambda_1– \lambda_2, \lambda_2, \lambda_1)$ from the line $\frac{x-5}{1}$=$\frac{y-1}{2}$=$\frac{z+7}{2}$ is...........
5.  The largest natural number $n$ such that $3^n$ divides 66! is ________.
6.  Consider a circle $C_1$: $x^2$+ $y^2$ $– 4x$ $– 2y$=$\alpha$–5, Let its mirror image in the line $y$=$2x$+1 be another circle $C_2$:$5x^2$+ $5y^2$–$10f x$$-10gy$+36=0. Let $r$ be the radius of $C_2$, then $\alpha + r$ is equal to ___________
7.  If the solution curve of the differential equation $(y – 2logex)dx$ + $(xlog_ex^2)dy$ =0, $x>1$ passes through the points $\left(e, \frac{4}{3}\right)$ and $(e^4, a)$ then $a$ is equal to..........
8. Let $[t]$ denote the greatest integer $\leq t$, Then $\frac{2}{\pi} \int \limits_{\pi/6}^{5\pi/6}(8[cosec x]-5[\cot x])dx$ is equal to.............
9.  Let $A$ = {0, 3, 4, 6, 7, 8, 9, 10} and $R$ be the relation defined on $A$ such that $R$ = {$(x, y) \in A × A:x-y$ is odd positive integer or $x - y$ = 2} . The minimum number of elements that must be added to the relation $R$, so that it is a symmetric relation , is equal to ______.
10.  Let $\vec{a}$=$6\hat{i}$+$9\hat{j}$+$12\hat{k}$, $\vec{b}$=$\alpha \hat{i}$+$11\hat{j}-2\hat{k}$ and $\vec{c}$ be vectors such that $\vec{a}×\vec{c}$=$\vec{a}×\vec{b}$. If $\vec{a}•\vec{c}$=-12, $\vec{c}•(\hat{i}-2\hat{j}+\hat{k})$=5, then $\vec{c}•(\hat{i}+\hat{j}+\hat{k})$ is equal to ..............

08 Apr 2023:Shift-2

08 Apr 2023:Ans-Key

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