Download JEE Main 2024 Question Paper (29 Jan - Shift 2) Mathematics

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30 Jan 2024:Shift-1

30 Jan 2024: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 $A$=$\begin{equation*} \begin{bmatrix} 2 & 1 & 2 \\ 6 & 2 & 11 \\ 3 & 3 & 2 \end{bmatrix} \end{equation*}$ and $P$=$\begin{equation*} \begin{bmatrix} 1 & 2 & 0 \\ 5 & 0 & 2 \\ 7 & 1 & 5\end{bmatrix}\end{equation*}$. The sum of the prime factors of $|P^{-1} AP - 2I|$ is equal to
   
      
   1. 26
   2. 27
   3. 66
   4. 23
2. Number of ways of arranging 8 identical books into 4 identical shelves where any number of shelves may remain empty is equal to
   
      
   1. 18
   2. 16
   3. 12
   4. 15
3. Let $P(3, 2, 3)$, $Q(4, 6, 2)$ and $R(7, 3, 2)$ be the vertices of $\Delta PQR$. Then, the angle $\angle{QPR}$ is
   
      
   1. $\frac{\pi}{6}$
   2. $\cos^{-1}\left(\frac{7}{18}\right)$
   3. $\cos^{-1}\left(\frac{1}{18}\right)$
   4. $\frac{\pi}{3}$
4. If the mean and variance of five observations are $\frac{24}{5}$ and $\frac{194}{25}$ respectively and the mean of first four observations is $\frac{7}{2}$, then the variance of the first four observations in equal to
   
      
   1. $\frac{4}{5}$
   2. $\frac{77}{12}$
   3. $\frac{5}{4}$
   4. $\frac{105}{4}$
5. The function $f(x)$ = $2x+3(x)^{\frac{2}{3}}$, $x \in R$, has
   
      
   1. exactly one point of local minima and no point of local maxima
   2. exactly one point of local maxima and no point of local minima
   3. exactly one point of local maxima and exactly one point of local minima
   4. exactly two points of local maxima and exactly one point of local minima
6. Let $r$ and $\theta$ respectively be the modulus and amplitude of the complex number $z$=$2-i\left(2\tan\frac{5\pi}{8}\right)$, then $(r, \theta)$is equal to
   
      
   1. $\left(2\sec \frac{3\pi}{8}, \frac{3\pi}{8}\right)$
   2. $\left(2\sec \frac{3\pi}{8}, \frac{5\pi}{8}\right)$
   3. $\left(2\sec \frac{5\pi}{8}, \frac{3\pi}{8}\right)$
   4. $\left(2\sec \frac{11\pi}{8}, \frac{11\pi}{8}\right)$
7. The sum of the solutions $x \in R$ of the equation $\frac{3\cos 2x+\cos^32x}{\cos^6x -\sin^6x}$=$x^3-x^2$+6 is
   
      
   1. 0
   2. 1
   3. -1
   4. 3
8. Let $\vec{OA}$=$\vec{a}$, $\vec{OB}$=$12\vec{a}+4\vec{b}$ and $\vec{OC}$=$\vec{b}$, where $O$ is the origin. If $S$ is the parallelogram with adjacent sides $OA$ and $OC$, then $\frac{\text{area of the quadrilateral OABC}}{\text{area of S}}$ is equal to........
   
      
   1. 6
   2. 10
   3. 7
   4. 8
9. If $\log_e a$, $\log_e b$, $\log_e c$ are in an A.P. and $\log_e a –\log_e2b$, $\log_e2b – log_e3c$, $log_e3c – log_e a$ are also in an A.P, then $a : b : c$ is equal to
   
      
   1. 9:6:4
   2. 16:4:1
   3. 25:10:4
   4. 6:3:2
10. If $\int \frac{\sin^{\frac{3}{2}}x+\cos^{\frac{3}{2}}x}{\sqrt{\sin^3x\cos^3x\sin(x-\theta)}}dx$=$A\sqrt{\cos \theta \tan x-\sin \theta}$+$B\sqrt{\cos\theta-\sin \theta \cot x}$+C, where $C$ is the integration constant, then $AB$ is equal to
    
       
    1. $4 cosec(2\theta)$
    2. $4 \sec \theta$
    3. $2 \sec \theta$
    4. $8 cosec(2\theta)$
11. The distance of the point (2, 3) from the line $2x –3y$ + 28 = 0, measured parallel to the line $\sqrt{3}x-y$+1=0 is equal to
    
       
    1. $4\sqrt{2}$
    2. $6\sqrt{3}$
    3. $3+4\sqrt{2}$
    4. $4+6\sqrt{3}$
12. If $\sin\left(\frac{y}{x}\right)$=$\log_e|x|$+$\frac{\alpha}{2}$ is the solution of the differential equation $x \cos\left(\frac{y}{x}\right)\frac{dy}{dx}$=$y \cos\left(\frac{y}{x}\right)+x$ and $y(1)$=$\frac{\pi}{3}$, then $\alpha^2$ is equal to
    
       
    1. 3
    2. 12
    3. 4
    4. 9
13. If each term of a geometric progression $a_1$, $a_2$, $a_3$,… with $a_1$=$\frac{1}{8}$ and $a_2 \neq a_1$, is the arithmetic mean of the next two terms and $S_n$=$a_1$+$a_2$+....+$a_n$, then $S_{20}$ – $S_{18}$ is equal to
    
       
    1. $2^{15}$
    2. $-2^{18}$
    3. $2^{18}$
    4. $-2^{15}$
14. Let $A$ be the point of intersection of the lines $3x + 2y$ = 14, $5x – y$ = 6 and $B$ be the point of intersection of the lines $4x + 3y$ = 8, $6x + y$ = 5. The distance of the point $P(5, –2)$ from the line $AB$ is
    
       
    1. $\frac{13}{2}$
    2. 8
    3. $\frac{5}{2}$
    4. 6
15. Let $x$ = $\frac{m}{n}$ ($m, n$ are co-prime natural numbers) be a solution of the equation $\cos (2sin^{-1} x$=$\frac{1}{9}$ and let $\alpha, \beta$ be the roots of the equation $mx^2$– $nx$ –$m + n$ = 0. Then the point $( \alpha, \beta)$ lies on the line
    
       
    1. $3x+2y=2$
    2. $5x-8y=-9$
    3. $3x-2y=-2$
    4. $5x+8y=9$
16. The function $f(x)$=$\frac{x}{x^2-6x-16}$, $x \in R-${$-2$, 8}
    
       
    1. decreases in (–2, 8) and increases in $(-\infty , 2) \cup (8, \infty)$
    2. decreases in $(-\infty, -2) \cup ( -2,8) \cup (8, \infty)$
    3. decreases in $(-\infty , -2)$ and increases in $(8, \infty)$
    4. increases in $(-\infty , -2) \cup ( -2,8) \cup (8, \infty)$
17. Let $y$=$\log_e\left(\frac{1-x^2}{1+x^2}\right)$, $-1 < x < 1$. Then at $x=\frac{1}{2}$, the value of 225$(y ' - y")$ is equal to
    
       
    1. 732
    2. 746
    3. 742
    4. 736
18. If $R$ is the smallest equivalence relation on the set {1, 2, 3, 4} such that {(1,2), (1,3)} $\subset R$, then the number of elements in $R$ is ______
    
       
    1. 10
    2. 12
    3. 8
    4. 15
19.  An integer is chosen at random from the integers 1, 2, 3, …, 50. The probability that the chosen integer is a multiple of atleast one of 4, 6 and 7 is
    
       
    1. $\frac{8}{25}$
    2. $\frac{21}{50}$
    3. $\frac{9}{50}$
    4. $\frac{14}{25}$
20. Let a unit vector $\hat{u}$=$x\hat{i}$+$y\hat{j}$+$z\hat{k}$ make angles $\frac{\pi}{2}$, $\frac{\pi}{3}$ and $\frac{2\pi}{3}$ with the vectors $\frac{1}{\sqrt{2}}\hat{i}$+$\frac{1}{\sqrt{2}}\hat{k}$, $\frac{1}{\sqrt{2}}\hat{j}$+$\frac{1}{\sqrt{2}}\hat{k}$ and $\frac{1}{\sqrt{2}}\hat{i}$+$\frac{1}{\sqrt{2}}\hat{j}$ respectively. If $\vec{v}$=$\frac{1}{\sqrt{2}}\hat{i}$+$\frac{1}{\sqrt{2}}\hat{j}$+$\frac{1}{\sqrt{2}}\hat{k}$, then $|\hat{u}-\vec{v}|^2$ is equal to
    
       
    1. $\frac{11}{2}$
    2. $\frac{5}{2}$
    3. 9
    4. 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. Let $\alpha$, $\beta$ be the roots of the equation $x^2-\sqrt{6}x+3$=0 such that $Im (\alpha)$ > $Im (\beta)$. Let $a$, $b$ be integers not divisible by 3 and $n$ be a natural number such that $\frac{\alpha^{99}}{\beta}$+$\alpha^{98}$=$3^n(a+ib)$, $i$=$\sqrt{-1}$. Then $n + a + b$ is equal to _______.
2. Let for any three distinct consecutive terms $a$, $b$, $c$ of an $A.P$, the lines $ax$ + $by$ + $c$ = 0 be concurrent at the point $P$ and $Q(\alpha, \beta)$ be a point such that the system of equations
   $x$ + $y$ + $z$ = 6,
   $2x$ + $5y$ + $\alpha z$ = $\beta$ and
   $x$ + $2y$ + $3z$ = 4, has infinitely many solutions. Then $(PQ)^2$ is equal to ________.
3. Let $P(\alpha ,\beta)$ be a point on the parabola $y^2= 4x$. If $P$ also lies on the chord of the parabola $x^2= 8y$ whose mid point is $\left(1, \frac{5}{4}\right)$. Then $(\alpha-28)$$(\beta-8)$ is equal to...........
4. If $\int \limits_{\frac{\pi}{6}}^{\frac{\pi}{3}}\sqrt{1-\sin 2x}dx$=$\alpha$+$\beta\sqrt{2}$+$\gamma\sqrt{3}$, where $\alpha$, $\beta$and $\gamma$ are rational numbers, then $3\alpha+4\beta- \gamma$ is equal to _______.
5. Let the area of the region {$(x, y): 0 \leq x \leq 3$, $0 \leq y \leq min{x^2+ 2, 2x + 2}$} be $A$. Then $12A$ is equal to _____.
6. Let $O$ be the origin, and M and N be the points on the lines $\frac{x-5}{4}$=$\frac{y-4}{1}$=$\frac{z-5}{3}$ and $\frac{x+8}{12}$=$\frac{y+2}{5}$=$\frac{z+11}{9}$ respectively such that MN is the shortest distance between the given lines. Then $\vec{OM}$•$\vec{ON}$ is equal to ...................
7. Let $f(x)$=$\sqrt{\lim \limits_{r \to x}\left\{\frac{2r^2[(f(r))^2-f(x)f(r)]}{r^2-x^2}-r^3e^{\frac{f(r)}{r}}\right\}}$ be differentiable in $(-\infty, 0) \cup (0, \infty)$ and $f(1)$ = 1. Then the value of $ea$, such that $f(a)$ = 0, is equal to _____.
8. Remainder when ${64^{32}}^{32}$is divided by 9 is equal to…………….
9. Let the set $C$={$(x, y) | x^2 -2^y=2024,$ $x, y \in N$. Then $\sum \limits_{(x,y)\in C}(x + y)$is equal to _______.
10. Let the slope of the line $45x + 5y$ + 3 = 0 be $27r_1$+$\frac{9r_2}{2}$ for some $r_1$, $r_2 \in R$. Then $\operatorname{Lim}\limits_{x \rightarrow 3}\left(\int_3^x \frac{8 t^2}{\frac{3 r_2 x}{2}-r_2 x^2-r_1 x^3-3 x} d t\right)$ is equal to ____.

29 Jan 2024:Shift-1

29 Jan 2024:Ans-Keys

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