Download JEE Main 2024 Question Paper (31 Jan - Shift 1) Mathematics

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01 Feb 2024:Shift-1

01 Feb 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. For $0 < c < b < a$, let $(a + b – 2c)x^2$ + $(b + c – 2a)x$+ $(c + a – 2b)$ = 0 and $\alpha \neq 1$ be one of its root. Then, among the two statements
   (I) If $\alpha \in (-1, 0)$, then $b$ cannot be the geometric mean of $a$ and $c$
   (II) If $\alpha \in(0,1)$, then $b$ may be the geometric mean of $a$ and $c$
   
      
   1. Both (I) and (II) are true
   2. Neither (I) nor (II) is true
   3. Only (II) is true
   4. Only (I) is true
2. Let a be the sum of all coefficients in the expansion of $(1 – 2x + 2x^2)^{2024}$$(3 – 4x^2+2x^3)^{2024}$ and $b$=$\lim \limits_{x \to 0}\left(\frac{\int \limits_{0}^{x}\frac{\log(1+t)}{t^{2024}+1}dt}{x^2}\right)$. If the equations $cx^2 + dx + e$ = 0 and $2bx^2+ ax + 4$ = 0 have a common root, where $c, d, e \in R$, then $d : c : e$equals
   
      
   1. 2 : 1 : 4
   2. 4 : 1 : 4
   3. 1 : 2 : 4
   4. 1 : 1 : 4
3. If the foci of a hyperbola are same as that of the ellipse $\frac{x^2}{9}$+$\frac{y^2}{25}$=1 and the eccentricity of the hyperbola is $\frac{15}{8}$ times the eccentricity of the ellipse, then the smaller focal distance of the point $\left(\sqrt{2}, \frac{14}{3}\sqrt{\frac{2}{5}}\right)$ on the hyperbola, is equal to
   
      
   1. $7\sqrt{\frac{2}{5}}-\frac{8}{3}$
   2. $14\sqrt{\frac{2}{5}}-\frac{4}{3}$
   3. $14\sqrt{\frac{2}{5}}-\frac{16}{3}$
   4. $7\sqrt{\frac{2}{5}}+\frac{8}{3}$
4. If one of the diameters of the circle $x^2$ + $y^2$ – $10x$ + $4y$ + 13 = 0 is a chord of another circle $C$, whose center is the point of intersection of the lines $2x$ + $3y$ = 12 and $3x – 2y$ = 5, then the radius of the circle $C$ is
   
      
   1. $\sqrt{20}$
   2. 4
   3. 6
   4. $3\sqrt{2}$
5. The area of the region
   $\left\{(x, y):y^2\leq 4x, x < 4, \frac{xy(x-1)(x-2)}{(x-3)(x-4)} > 0, x \neq 0)\right\}$ is
   
      
   1. $\frac{16}{3}$
   2. $\frac{64}{3}$
   3. $\frac{8}{3}$
   4. $\frac{32}{3}$
6. If $f(x)$=$\frac{4x+3}{6x-4}$, $x \neq \frac{2}{3}$ and $(fof) (x)$ = $g(x)$, where $g:R-\frac{2}{3}\to R-\frac{2}{3}$, then $(gogog) (4)$ is equal to
   
      
   1. $-\frac{19}{20}$
   2. $\frac{19}{20}$
   3. $-4$
   4. 4
7. $\lim \limits_{x \to 0}\frac{e^{2|\sin x|}-2|\sin x|-1}{x^2}$
   
      
   1. is equal to – 1
   2. does not exist
   3. is equal to 1
   4. is equal to 2
8. If the system of linear equations
   $x - 2y + z=-4$
   $2x+\alpha y+3z= 5$
   $3x - y + \beta z = 3$
   has infinitely many solutions, then $12\alpha + 13\beta$ is equal to
   
      
   1. 8
   2. 9
   3. 6
   4. 7
9. The solution curve of the differential equation $y\frac{dx}{dy}$=$x(\log_ex-\log_ey+1)$, $ x > 0, y > 0$ passing through the point $(e, 1)$ is
   
      
   1. $\left|\log_e\frac{y}{x}\right|=x$
   2. $\left|\log_e\frac{y}{x}\right|=y^2$
   3. $\left|\log_e\frac{x}{y}\right|=y$
   4. $\left|\log_e\frac{x}{y}\right|=y+1$
10. Let $\alpha, \beta, \gamma, \delta \in Z$ and let $A (\alpha, \beta)$, $B (1, 0)$, $C (\gamma, \delta)$and $D (1, 2)$ be the vertices of a parallelogram $ABCD$. If $AB$ = 10and the points $A$ and $C$ lie on the line $3y = 2x + 1$, then $2 (\alpha+\beta+\delta+\gamma)$ is equal to
    
       
    1. 10
    2. 5
    3. 12
    4. 8
11. Let $y = y(x)$ be the solution of the differential equation $\frac{dy}{dx}$=$\frac{(\tan x)+y}{\sin x(\sec x-\sin x \tan x)}$, $x \in \left(0, \frac{\pi}{2}\right)$ satisfying the condition $y\left(\frac{\pi}{4}\right)$=2. Then $y\left(\frac{\pi}{3}\right)$ is
    
       
    1. $\sqrt{3}(2+\log_e\sqrt{3})$
    2. $\frac{3}{2}(2+\log_e3)$
    3. $\sqrt{3}(1+2\log_e3)$
    4. $\sqrt{3}(2+\log_e 3)$
12. Let $\vec{a}$=$3\hat{i}+\hat{j}-2\hat{k}$, $\vec{b}$=$4\hat{i}+\hat{j}+7\hat{k}$ and $\vec{c}$=$\hat{i}-3\hat{j}+4\hat{k}$ be three vectors. If a vectors $\vec{p}$satisfies $\vec{p}×\vec{b}$=$\vec{c}×\vec{b}$ and $\vec{p}•\vec{a}$=0, then $\vec{p}•(\hat{i}-\hat{j}-\hat{k})$ is equal to
    
       
    1. 24
    2. 36
    3. 28
    4. 32
13. The sum of the series $\frac{1}{1-3•1^2+1^4}$+$\frac{2}{1-3•2^2+2^4}$+$\frac{3}{1-3•^2+3^4}$+.....upto 10 terms is
    
       
    1. $\frac{45}{109}$
    2. $-\frac{45}{109}$
    3. $\frac{55}{109}$
    4. $-\frac{55}{109}$
14. The distance of the point $Q(0, 2, –2)$ form the line passing through the point $P(5, –4, 3)$ and perpendicular to the lines $\vec{r}$=$(-3\hat{i}+2\hat{k})$+$\lambda(2\hat{i}+3\hat{j}+5\hat{k})$, $\lambda \in R$ and $\vec{r}$=$(\hat{i}-2\hat{j}+\hat{k})$+$\mu(-\hat{i}+3\hat{j}+2\hat{k})$, $\mu \in R$ is
    
       
    1. $\sqrt{86}$
    2. $\sqrt{20}$
    3. $\sqrt{54}$
    4. $\sqrt{74}$
15. For $\alpha, \beta, \gamma \neq 0$. If  $\sin^{-1}\alpha$+$\sin^{-1}\beta$+$\sin^{-1}\gamma$=$\pi$ and $(\alpha+\beta+\gamma)$$(\alpha-\gamma+\beta)$=$3\alpha \beta)$, then $\gamma$ equal to
    
       
    1. $\frac{\sqrt{3}}{2}$
    2. $\frac{1}{\sqrt{2}}$
    3. $\frac{\sqrt{3}-1}{2\sqrt{2}}$
    4. $\sqrt{3}$
16. Two marbles are drawn in succession from a box containing 10 red, 30 white, 20 blue and 15 orange marbles, with replacement being made after each drawing. Then the probability, that first drawn marble is red and second drawn marble is white, is
    
       
    1. $\frac{2}{25}$
    2. $\frac{4}{25}$
    3. $\frac{2}{3}$
    4. $\frac{4}{75}$
17. Let $g(x)$ be a linear function and
    $f(x)$=$\left\{\begin{array} {cc} g(x) &, x \leq 0\\ \left(\frac{1+x}{2+x}\right)^{\frac{1}{x}} &, x > 0 \end{array} \right.$, is continuous at $x = 0$. If $(f'(1)$=$f(-1)$, then the value of $g(3)$ is
    
       
    1. $\frac{1}{3}\log_e\left(\frac{4}{9e^{1/3}}\right)$
    2. $\frac{1}{3}\log_e\left(\frac{4}{9}\right)+1$
    3. $\log_e\left(\frac{4}{9}\right)-1$
    4. $\log_e\left(\frac{4}{9e^{1/3}}\right)$
18. If $f(x)$=$\begin{vmatrix} x^3 & 2x^2+1 & 1+3x \\ 3x^2+2 & 2x & x^3+6 \\ x^3-x & 4 & x^2-2 \end{vmatrix}$ for all $x \in R$, then $2f(0)$ + $f'(0)$ is equal to
    
       
    1. 48
    2. 24
    3. 42
    4. 18
19.  Three rotten apples are accidently mixed with fifteen good apples. Assuming the random variable $x$ to be the number of rotten apples in a draw of two apples, the variance of $x$ is
    
       
    1. $\frac{37}{153}$
    2. $\frac{57}{153}$
    3. $\frac{47}{153}$
    4. $\frac{40}{153}$
20. Let $S$ be the set of positive integral values of $a$ for which $\frac{ax^2+2(a+1)x+9a+4}{x^2-8x+32}$ < 0, $\forall x \in R$. Then, the number of elements in $S$ is :
    
       
    1. 1
    2. 0
    3. $\infty$
    4. 3

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 the integral $525\int \limits_{0}^{\frac{\pi}{2}}\sin 2x \cos^{\frac{11}{2}} x \left(1+\cos^{\frac{5}{2}} x\right)^{\frac{1}{2}}dx$ is equal to $(n\sqrt{2}-64)$, then $n$ is equal to ______
2. Let $S$=$(-1, \infty)$ and $f:S \to R$ be defined as $f(x)$=$\int \limits_{-1}^x (e^t-1)^{11}(2t-1)^5$$((t-2)^7(t-3)^{12}(2t-10)^{61}dt$ Let $p$ = Sum of square of the values of $x$, where $f(x)$ attains local maxima on $S$. and $q$ = Sum of the values of $x$, where $f(x)$ attains local minima on $S$. Then, the value of $p^2$+ $2q$ is ________
3. The total number of words (with or without meaning) that can be formed out of the letters of the word ‘DISTRIBUTION’ taken four at a time, is equal to _____
4. Let $Q$ and $R$ be the feet of perpendiculars from the point $P(a, a, a)$ on the lines $x = y$, $z = 1$ and $x = –y$, $z = –1$ respectively. If $\angle{QPR}$ is a right angle, then $12a^2$ is equal to _____
5. In the expansion of
   $(1+x)(1-x^2)$$\left(1+\frac{3}{x}+\frac{3}{x^2}+\frac{1}{x^3}\right)^5$, $x \neq 0$, the sum of the coefficient of $x^3$ and $x^{–13}$ is equal to ___
6. If $\alpha$ denotes the number of solutions of $|1-i|^x$=$2^x$ and $\beta$=$\left(\frac{|z|}{arg(z)}\right)$, where $z$=$\frac{\pi}{4}(1+i)^4\left(\frac{1-\sqrt{\pi}i}{\sqrt{\pi}+i}+\frac{\sqrt{\pi}-i}{1+\sqrt{\pi}i}\right)$, $i$=$\sqrt{-1}$, then the distance of the point $(\alpha, \beta)$ from the line $4x – 3y = 7$ is _____
7. Let the foci and length of the latus rectum of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}$=1, $a$ > $b$ be $(±5, 0)$ and $\sqrt{50}$,  respectively. Then, the square of the eccentricity of the hyperbola $\frac{x^2}{b^2}-\frac{y^2}{a^2b^2}$=1 equals
8. Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a}|$=1, $|\vec{b}|$=4 and $\vec{a}•\vec{b}$=2. If $\vec{c}$=$(2\vec{a}×\vec{b})-3\vec{b}$ and the angle between $\vec{b}$  and $\vec{c}$ is $\alpha$, then $192 \sin^2\alpha$ is equal to............
9. Let $A$ = {1, 2, 3, 4} and $R$ = {(1, 2), (2, 3), (1, 4)} be a relation on $A$. Let $S$ be the equivalence relation on $A$ such that $R \left S$ and the number of elements in $S$ is $n$. Then, the minimum value of $n$is _______
10. Let $f:R \to R$ be a function defined by $f(x)$=$\frac{4^x}{4^x+2}$ and $M$=$\int \limits_{f(a)}^{f(1-a)}x \sin^4(x(1-x))dx$, $N$=$\int \limits_{f(a)}^{f(1-a)} \sin^4(x(1-x))dx$; $a \neq \frac{1}{2}$. If $\alpha M$=$\beta N$, $\alpha, \beta \in N$,, then the least value of $\alpha^2+\beta^2$ is equal to ______

31 Jan 2024:Shift-2

31 Jan 2024:Ans-Keys

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