Download JEE Main 2024 Question Paper (08 Apr - Shift 1) Mathematics

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09 Apr 2024:Shift-1

09 Apr 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. The value of $k \in N$ for which the integral $I_n$=$\int \limits_{0}^{1}(1-x^k)^ndx$, $n \in N$, satisfies 147 $I_{20}$=148$I_{21}$ is:
   
      
   1. 10
   2. 8
   3. 14
   4. 7
2. The sum of all the solutions of the equation $(8)^{2x}$– $16•(8)^x$+ 48 = 0 is :
   
      
   1. $1+\log_6(8)$
   2. $\log_8(6)$
   3. $1+\log_8(6)$
   4. $\log_8(4)$
3. Let the circles $C_1$ : $(x – \alpha)^2$ + $(y – \beta)^2$=$r_1^2$ and $C_2$ : $(x – 8)^2$+$\left(y-\frac{15}{2}\right)^2$=$r_2^2$ touch each otherexternally at the point (6, 6). If the point (6, 6) divides the line segment joining the centres of the circles $C_1$ and $C_2$ internally in the ratio 2 : 1, then $(\alpha + \beta)$ + 4$(r_1^2+r_2^2)$ equals
   
      
   1. 110
   2. 130
   3. 125
   4. 145
4. Let $P(x, y, z)$ be a point in the first octant, whose projection in the xy-plane is the point $Q$. Let $OP$ = $\gamma$;  the angle between $OQ$ and the positive $x-$axis be $\theta$; and the angle between $OP$ and the positive $z-$axis be $\phi$, where $O$ is the origin. Then the distance of $P$ from the $x-$axis is :
   
      
   1. $\gamma \sqrt{1-\sin^2\phi \cos^2\theta}$
   2. $\gamma \sqrt{1+\cos^2\theta \sin^2\phi}$
   3. $\gamma \sqrt{1-\sin^2\theta \cos^2\phi}$
   4. $\gamma \sqrt{1+\cos^2\phi \sin^2\theta}$
5. The number of critical points of the function $f(x)$ = $(x – 2)^{2/3} (2x + 1)$ is :
   
      
   1. 2
   2. 0
   3. 1
   4. 3
6. Let $f(x)$ be a positive function such that the area bounded by $y = f(x)$, $y$ = 0 from $x$ = 0 to $x = a$ > 0 is $e^{–a}$+ $4a^2$ + $a – 1$. Then the differential equation, whose general solution is $y = c_1f(x)$ + $c_2$, where $c_1$ and $c_2$ are arbitrary constants, is :
   
      
   1. $8(e^x-1)\frac{d^2y}{dx^2}$+$\frac{dy}{dx}$=0
   2. $8(e^x+1)\frac{d^2y}{dx^2}$$-\frac{dy}{dx}$=0
   3. $8(e^x+1)\frac{d^2y}{dx^2}$+$\frac{dy}{dx}$=0
   4. $8(e^x-1)\frac{d^2y}{dx^2}$$-\frac{dy}{dx}$=0
7. Let $f(x)$ = $4\cos^3x$ + $3 \sqrt{3} \cos^2 x$ – 10. The number of points of local maxima of $f$ in interval $(0, 2\pi)$ is:
   
      
   1. 1
   2. 2
   3. 3
   4. 4
8. Let $\begin{equation*}A=\begin{bmatrix}2 & a & 0\\1 & 3 & 1\\ 0 & 5 & b\end{bmatrix}\end{equation*}$. If $A^3$=$4A^2-A-21I$, where $I$ is the identity matrix of order 3 × 3, then $2a + 3b$ is equal to :
   
      
   1. -10
   2. -13
   3. -9
   4. -12
9.  If the shortest distance between the lines
   $L_1$:$\vec{r}$=$(2+\lambda)\hat{i}$+$(1-3\lambda)\hat{j}$+$(3+4\lambda)\hat{k}$, $\lambda \in R$
   $L_2$:$\vec{r}$=$2(1+\mu)\hat{i}$+$3(1+\mu)\hat{j}$+$(5+\mu)\hat{k}$, $\mu \in R$
   is $\frac{m}{\sqrt{n}}$, where $gcd (m, n)$ = 1, then the value of $m + n$ equals.
   
      
   1. 384
   2. 387
   3. 377
   4. 390
10. Let the sum of two positive integers be 24. If the probability, that their product is not less than $\frac{3}{4}$ times their greatest positive product, is $\frac{m}{n}$, where $gcd(m, n)$ = 1, then $n – m$ equals :
    
       
    1. 9
    2. 11
    3. 8
    4. 10
11. If $\sin x$=$-\frac{3}{5}$, where $\pi$ < $x$ < $\frac{3\pi}{2}$, then $80(\tan^2x-\cos x)$ is equal to:
    
       
    1. 109
    2. 108
    3. 18
    4. 19
12. Let $I(x)$=$\int \frac{6}{\sin^2 x(1-\cot x)^2}dx$. If $I(0)$=3, then $I\left(\frac{\pi}{12}\right)$ is equal to:
    
       
    1. $\sqrt{3}$
    2. $3\sqrt{3}$
    3. $6\sqrt{3}$
    4. $2\sqrt{3}$
13. The equations of two sides $AB$ and $AC$ of a triangle $ABC$ are $4x + y$ = 14 and $3x – 2y$ = 5, respectively. The point $\left(2, -\frac{4}{3}\right)$ divides the third side $BC$ internally in the ratio 2 : 1. The equation of the side $BC$ is :
    
       
    1. $x – 6y$ – 10 = 0
    2. $x – 3y$ – 6 = 0
    3. $x + 3y$ + 2 = 0
    4. $x + 6y$ + 6 = 0
14. Let $[t]$ be the greatest integer less than or equal to $t$. Let $A$ be the set of al prime factors of 2310 and $f : A \to Z$ be the function $f(x)$=$\left[\log_2\left(x^2+\left[\frac{x^3}{5}\right]\right)\right]$. The number of one-to-one functions from $A$ to the range of $f$ is :
    
       
    1. 20
    2. 120
    3. 25
    4. 24
15. Let $z$ be a complex number such that $|z + 2|$ = 1 and $lm\left(\frac{z+1}{z+2}\right)$=$\frac{1}{5}$. Then the value of $|Re(\bar{z+2})|$ is:
    
       
    1. $\frac{\sqrt{6}{5}}$
    2. $\frac{1+\sqrt{6}}{5}$
    3. $\frac{24}{5}$
    4. $\frac{2\sqrt{6}}{5}$
16. If the set $R$={$(a, b); a+5b$=42, $a, b \in N$} has $m$ elements and $\sum \limits_{n=1}^{m}(1+i^{n!})$=$x+iy$, where $I$ = $\sqrt{-1}$, then the value of $m + x + y$ is :
    
       
    1. 8
    2. 12
    3. 4
    4. 5
17. For the function $f(x)$ = $(cosx) – x$ + 1, $x \in R$,between the following two statements
    (S1) $f(x)$ = 0 for only one value of $x$ is $[0, \pi]$.
    (S2) $f(x)$ is decreasing in $\left[0, \frac{pi}{2}\right]$ and increasing in $\left[\frac{\pi}{2}, \pi\right]$.
    
       
    1. Both (S1) and (S2) are correct
    2. Only (S1) is correct
    3. Both (S1) and (S2) are incorrect
    4. Only (S2) is correct
18. The set of all $\alpha$, for which the vector $\vec{a}$=$\alpha t\hat{i}$+$6\hat{j}$$-3\hat{k}$ and $\vec{b}$=$t\hat{i}$$-2\hat{j}$$-2\alpha t \hat{k}$ are inclined at an obtuse angle for all $t \in R$ is :
    
       
    1. [0, 1)
    2. (-2, 0]
    3. (-4/3, 0]
    4. (-4/3, 1)
19.  Let $y = y(x)$ be the solution of the differential equation $(1 + y^2)$$e^{\tan x}dx$ + $\cos^2x(1 + e^{2\tan x)}dy$ = 0, $y(0)$ = 1. Then $y\left(\frac{\pi}{4}\right)$ is equal to:
    
       
    1. $\frac{2}{e}$
    2. $\frac{1}{e^2}$
    3. $\frac{1}{e}$
    4. $\frac{2}{e^2}$
20. Let $H$:$\frac{-x^2}{a^2}$+$\frac{y^2}{b^2}$=1 be the hyperbola, whose eccentricity is $\sqrt{3}$and the length of the latus rectum is $4 \sqrt{3}$. Suppose the point $(\alpha, 6)$, $\alpha$ > 0lies on $H$. If $\beta$ is the product of the focal distances of the point $(\alpha, 6)$, then $\alpha^2+\beta$ is equal to :
    
       
    1. 170
    2. 171
    3. 169
    4. 172

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 $\begin{equation*} A=\begin{bmatrix} 2 & -1 \\ 1 & 1 \end{bmatrix}\end{equation*}$. If the sum of the diagonal elements of $A^{13}$ is $3^n$, then $n$ is equal to ______.
2. If the orthocentre of the triangle formed by thelines $2x + 3y$ – 1 = 0, $x + 2y$ – 1 = 0 and $ax + by$ – 1 = 0, is the centroid of another triangle, whose circumecentre and orthocentre respectively are (3, 4) and (–6, –8), then the value of $|a – b|$ is _____.
3. Three balls are drawn at random from a bag containing 5 blue and 4 yellow balls. Let the random variables X and Y respectively denote the number of blue and Yellow balls. If $\bar{X}$ and $\bar{Y}$are the means of X and Y respectively, then $7\bar{X}$ + 4\bar{Y}$is equal to ______.
4. The number of 3-digit numbers, formed using thedigits 2, 3, 4, 5 and 7, when the repetition of digitsis not allowed, and which are not divisible by 3, isequal to ______.
5. Let the positive integers be written in the form :
   
   If the $k^{th}$ row contains exactly $k$ numbers for every natural number $k$, then the row in which the number 5310 will be, is ______.
6. If the range of $f(\theta)$=$\frac{\sin^4\theta+3\cos^2\theta}{\sin^4\theta+\cos^2\theta}$, $\theta \in R$ is $[\alpha, \beta]$,  then the sum of the infinite G.P., whose first term is 64 and the common ratio is $\frac{\alpha}{\beta}$ is equal to...........
7. Let $\alpha =\sum \limits_{r=0}^{n}(4r^2+2r+1){}^nC_r$ and $\beta$=$\left(\sum \limits_{r=0}^{n}\frac{{}^nC_r}{r+1}\right)$+$\frac{1}{n+1}$. If 140 < $\frac{2\alpha}{\beta}$ < 281, then the value of $n$ is...............
8. Let $\vec{a}$=$9\hat{i}-13\hat{j}$+$25\hat{k}$, $\vec{b}$=$3\hat{i}$+$7\hat{j}$$-13\hat{k}$ and $\vec{c}$=$17\hat{i}$$-2\hat{j}$+$\hat{k}$ be three given vectors. If $\vec{r}$ is a vector such that $\vec{r}×\vec{a}$=$(\vec{b}+\vec{c})×\vec{a}$ and $\vec{r}•(\vec{b}-\vec{c})$, then $\frac{593\vec{r}+67\vec{a}|^2}{(593)^2}$ is equal to...........
9. Let the area of the region enclosed by the curve $y$ = $min{\sin x, \cos x}$ and the $x-$axis between $x = –\pi$ to $x = \pi$ be $A$. Then $A^2$ is equal to _______.
10. The value of $\lim \limits_{x\to 0}2\left(\frac{1-\cos x \sqrt{\cos 2x}\sqrt{3}{\cos 3x}.......\sqrt{10}{\cos 10x}}{x^2}\right)$ is..........

08 Apr 2024:Shift-2

08 Apr 2024:Ans-Keys

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