Download JEE Main 2023 Question Paper (12 Apr - Shift 1) Mathematics

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13 Apr 2023:Shift-1

13 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 number of five digit numbers, greater than 40000 and divisible by 5, which can be formed using the digits 0, 1, 3, 5, 7 and 9 without repetition, is equal to
   
      
   1. 120
   2. 132
   3. 72
   4. 96
2. Let $\alpha$, $\beta$ be the roots of the quadratic equation $x^2$+$\sqrt{6}x$+3=0. Then $\frac{\alpha^{23}+\beta^{23}+\alpha^{14}+\beta^{14}}{\alpha^{15}+\beta^{15}+\alpha^{10}+\beta^{10}}$ is equal to
   
      
   1. 729
   2. 72
   3. 81
   4. 9
3. Let <$a_n$> be a sequence such that $a_1$+$a_2$+........+$a_n$=$\frac{n^2+3n}{(n+1)(b+2)}$. If $28\sum \limits_{k=1}^{10}\frac{1}{a_k}$=$p_1$$p_2$$p_3$........$p_m$, where $p_1$, $p_2$,.......,$p_m$ are the first $m$ prime numbers, then $m$ is equal to.........
   
      
   1. 7
   2. 6
   3. 5
   4. 8
4. Let the lines $l_1$:$\frac{x+5}{3}$=$\frac{y+4}{1}$=$\frac{z-\alpha}{-2}$ and $l_2$:$3x$+$2y$+$z-2$=0=$x-3$+$2z-13$ be coplanar If the point $P(a, b, c)$ on $l_1$ is nearest to the point $Q(-4, -3, 2)$, then $|a| + |b|+|c|$ is equal to
   
      
   1. 12
   2. 14
   3. 10
   4. 8
5. Let $P\left(\frac{2\sqrt{3}}{\sqrt{7}},\frac{6}{\sqrt{7}}\right)$, $Q$, $R$ and $S$ be four points on the ellipse $9x^2$+$4y^2$+36. Let $PQ$ and $RS$ be mutually perpendicular and pass through the origin. If $\frac{1}{(PQ)^2}$+$\frac{1}{(RS)^2}$=$\frac{p}{q}$, where $p$ and $q$ are coprime, then $p + q$ is equal to
   
      
   1. 143
   2. 137
   3. 157
   4. 147
6. Let $a$, $b$, $c$ be three distinct real numbers, none equal to one. If the vectors $a\hat{i}$+$\hat{j}$+$\hat{k}$, $\hat{i}$+$b\hat{j}$+$\hat{k}$ and $\hat{i}$+$\hat{j}$+$c\hat{k}$ are coplanar, then $\frac{1}{1-a}$+$\frac{1}{1-b}$+$\frac{1}{1-c}$ is equal to
   
      
   1. 1
   2. $-1$
   3. $-2$
   4. 2
7. If the local maximum value of the function $f(x)$=$\left(\frac{\sqrt{3e}}{2\sin x}\right)^{\sin^2x}$, $x \in \left(0, \frac{\pi}{2}\right)$, is $\frac{k}{e}$, then $\left(\frac{k}{e}\right)^{8}$+$\frac{k^8}{e^5}$+$k^8$ is equal to
   
      
   1. $e^5$+$e^6$+$e^{11}$
   2. $e^3$+$e^5$+$e^{11}$
   3. $e^3$+$e^6$+$e^{11}$
   4. $e^3$+$e^6$+$e^{10}$
8. Let $D$ be the domain of the function $f(x)$=$\sin^{-1}\left(\log_{3x}\left(\frac{6+2\log_3x}{-5x}\right)\right)$. If the range of the function $g:D\to R$ defined by $g(x)$=$x-[x]$, ($[x]$ is the greatest integer function), is $(\alpha, \beta)$, then $\alpha^2+\frac{5}{\beta}$ is equal to
   
      
   1. 46
   2. 135
   3. 136
   4. 45
     
9.  Let $y = y(x)$, $y$ > 0, be a solution curve of the differential equation (1 + $x^2$) $dy$ = $y (x – y) dx$. If $y (0)$ = 1 and $y(2 \sqrt{2})$, = $\beta$,then
   
      
   1. $e^{3\beta^{-1}}$=$e(3+2\sqrt{2})$
   2. $e^{\beta^{-1}}$=$e^{-2}(5+\sqrt{2})$
   3. $e^{\beta^{-1}}$=$e^{-2}(3+2\sqrt{2})$
   4. $e^{3\beta^{-1}}$=$e(5+\sqrt{2})$
10. Among the two statements
    (S1) : $(p=>q) \bigwedge (q \bigwedge(~q))$ is a contradiction and
    (S2) : $(p \bigwedge q) \bigvee ((\text{~}p) \bigwedge q) \bigvee (p \bigwedge(\text{~}q)) \bigvee ((\text{~}p) \bigwedge (\text{~}q))$ is a tautology
    
    
       
    1. only (S2) is true
    2. only (S1) is true
    3. both are false
    4. both are true
11. If $\lambda \in Z$, $\vec{a}$=$\lambda \hat{i}$+$\hat{j}$$-\hat{k}$ and $\vec{b}$=$3\hat{i}$$-\hat{j}$+2$\hat{k}$. Let $\vec{c}$ be a vector such that $(\vec{a}+\vec{b}+\vec{c})$×$\vec{c}$=0, $\vec{a}$•$\vec{c}$=$-17$ and $\vec{b}$•$\vec{c}$=$-20$. Then $|\vec{c}×(\lambda \hat{i}+\hat{j}+\hat{k})|^2$ is equal to
    
       
    1. 62
    2. 46
    3. 53
    4. 49
12. The sum, of the coefficients of the first 50 terms in the binomial expansion of $(1 – x)^{100}$, is equal to
    
       
    1. $-{}^{101} C_{50}$
    2. ${}^{99} C_{49}$
    3. $-{}^{99} C_{49}$
    4. ${}^{101} C_{50}$
13. The area of the region enclosed by the curve $y = x^3$and its tangent at the point (-1, -1) is
    
       
    1. $\frac{27}{4}$
    2. $\frac{19}{4}$
    3. $\frac{23}{4}$
    4. $\frac{31}{4}$
14. Let $A$ = $\begin{equation*} \begin{bmatrix} 1 & \frac{1}{51} \\ 0 & 1 \end{bmatrix} \end{equation*}$. If $B$=$\begin{equation*} \begin{bmatrix} 1 & 2 \\ -1 & -1\end{bmatrix} \begin{bmatrix} -1 & -2 \\ 1 & 1\end{bmatrix} \end{equation*}$, then the sum of all the elements of the matrix $\sum \limits_{n=1}^{50} B^n$ is equal to
    
       
    1. 100
    2. 50
    3. 75
    4. 125
15. Let the plane $P$ : $4x$$ – y$ + $z$ = 10 be rotated by an angle $\frac{\pi}{2}$ about its line of intersection with the plane $x$ + $y $$– z$ = 4. If $\alpha$ is the distance of the point (2, 3, − 4) from the new position of the plane $P$, then 35$\alpha$ is
    
       
    1. 90
    2. 85
    3. 105
    4. 126
16. If $\frac{1}{n+1}{}^n C_n$+$\frac{1}{n}{}^n C_{n-1}$+....+$\frac{1}{2}{}^n C_1$+${}^n C_0$=$\frac{1023}{10}$, then $n$ is equal to
    
       
    1. 6
    2. 9
    3. 8
    4. 7
17. Let $C$ be the circle in the complex plane with centre $z_0$=$\frac{1}{2}(1+3i)$and radius $r$ = 1. Let $z_1$ = $1+ i$ and the complex number $z_2$ be outside the circle $C$ such that $|z_1 – z_0| $$|z_2 – z_0|$ = 1. If $z_0$, $z_1$ and $z_2$ are collinear, then the smaller value of $|z_2|^2$is equal to
    
       
    1. $\frac{13}{2}$
    2. $\frac{5}{2}$
    3. $\frac{3}{2}$
    4. $\frac{7}{2}$
18. If the point $\left(\alpha, \frac{7\sqrt{3}}{3}\right)$ lies on the curve traced by the mid-points of the line segments of the lines  $x \cos\theta $+$y \sin \theta $=7, $\theta \in \left(0, \frac{\pi}{2}\right)$ between the co- ordinates axes, then $\alpha$ is equal to
    
       
    1. 7
    2. $-7$
    3. $-7\sqrt{3}$
    4. $7\sqrt{3}$
19.   Two dice $A$ and $B$ are rolled, Let the numbers obtained on $A$ and $B$ be $\alpha$ and $\beta$ respectively. If the variance of $\alpha – \beta$ is $\frac{p}{q}$ where $p$ and $q$ are co-prime, then the sum of the positive divisors of $p$ is equal to
    
       
    1. 36
    2. 48
    3. 31
    4. 72
20. In a triangle $ABC$, if $cos A$ + 2 $cos B$ + $cos C$ = 2 and the lengths of the sides opposite to the angles $A$ and $C$ are 3 and 7 respectively, then $cos A – cos C$ is equal to
    
       
    1. $\frac{3}{7}$
    2. $\frac{9}{7}$
    3. $\frac{10}{7}$
    4. $\frac{5}{7}$

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. A fair $n (n > 1)$ faces die is rolled repeatedly until a number less than $n$ appears. If the mean of the number of tosses required is $\frac{n}{9}$, then $n$ is equal to __________.
2. Let the digits $a$, $b$, $c$ be in $A.P.$ Nine-digit numbers are to be formed using each of these three digits thrice such that three consecutive digits are in $A.P.$ at least once. How many such numbers can be formed?
3. Let $[x]$ be the greatest integer $\leq x$. Then the number of points in the interval (–2,1), where the function $f(x)$ = $| [x] |$ + $\sqrt{x – [x]}$ is discontinuous is________.
4. Let the plane $x$ + $3y – 2z$ + 6 = 0 meet the co-ordinate axes at the points $A$,$B$,$C$. If the orthocentre of the triangle $ABC$ is $\left(\alpha, \beta, \frac{6}{7}\right)$, then $98 (\alpha + \beta)^2$is equal to______.
5. Let $I(x)$=$\int \sqrt{\frac{x+7}{x}}dx$ and $I(9)$=12+$7\log_e7$. If $I(1)$=$\alpha$+7$\log_e(1+2\sqrt{2})$, then $\alpha^4$ is equal to.........
6. Let $D_k$=$\begin{equation*} \begin{vmatrix} 1 & 2k & 2k-1 \\ n & n^2+n+2 & n^2 \\ n & n^2+n & n^2+n+2 \end{vmatrix} \end{equation*}$. If $\sum \limits_{k=1}^{n} D_k$=96, then $n$ is equal to........
7. Let the positive numbers $a_1$, $a_2$, $a_3$, $a_4$ and $a_5$ be in a $G.P.$ Let their mean and variance be $\frac{31}{10}$ and $m$and $n$ respectively, where $m$ and $n$ are co-prime. If the mean of their reciprocals is $\frac{31}{40}$and $a_3$ + $a_4$ + $a_5$ = 14, then $m + n$ is equal to________.
8. The number of relations, on the set {1,2,3} containing (1,2) and (2,3), which are reflexive and transitive but not symmetric, is_____
9. If $\int \limits_{-0.15}^{0.15}|100x^2-1|dx$=$\frac{k}{3000}$, then $k$ is equal to........
10. Two circles in the first quadrant of radii $r_1$ and $r_2$touch the coordinate axes. Each of them cuts off an intercept of 2 units with the line $x$ + $y$ = 2. Then $r_1^2$+$r_2^2$–$r_1r_2$ is equal to________.

13 Apr 2023:Shift-2

12 Apr 2023:Ans-Keys

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