Download JEE Main 2024 Question Paper (30 Jan - Shift 2) Mathematics

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31 Jan 2024:Shift-1

31 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. Consider the system of linear equations
   $x + y + z$ = 5,
   $x + 2y +\lambda^2 z$ = 9, $x + 3y +\lambda z$ = $\mu$, where $\lambda, \mu \in R$. Then, which of the following statement is NOT correct?
   
      
   1. System has infinite number of solution if $\lambda$=1 and $\mu$ =13
   2. System is inconsistent if $\lambda = 1$ and $\mu \neq$13
   3. System is consistent if $\lambda \neq$=1 and  $\mu$=13
   4. System has unique solution if $lambda \neq 1 and $\mu \neq$ 13
2. For $\alpha, \beta \in \left(0, \frac{\pi}{2}\right)$, let $3 \sin(\alpha+\beta)$=$2\sin(\alpha-\beta)$ and a real number $k$ be such that $\tan \alpha$=$k \tan \beta$. Then the value of $k$ is equal to :
   
      
   1. $-\frac{2}{3}$
   2. $-5$
   3. $\frac{2}{3}$
   4. 5
3. Let $A(\alpha, 0)$ and $B(0, \beta)$ be the points on the line $5x + 7y$ = 50. Let the point $P$ divide the line segment $AB$ internally in the ratio 7 : 3. Let $3x – 25$ = 0 be a directrix of the ellipse $E:\frac{x^2}{a^2}+\frac{y^2}{b^2}$=1 and the corresponding focus be $S$. If from $S$, the perpendicular on the $x-$axis passes through $P$, then the length of the latus rectum of $E$ is equal to
   
      
   1. $\frac{25}{3}$
   2. $\frac{32}{9}$
   3. $\frac{25}{9}$
   4. $\frac{32}{5}$
4. Let $\vec{a}$=$\hat{i}$+$\alpha\hat{j}$+$\beta \hat{k}$, $\alpha$, $\beta \in R$. Let a vector $\vec{b}$ be such that the angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{4}$ and $|\vec{b}|^2$=6. If $\vec{a}•\vec{b}$=$3\sqrt{2}$, then the value of $(\alpha^2+\beta^2)|\vec{a}×\vec{b}|^2$ is equal to
   
      
   1. 90
   2. 75
   3. 95
   4. 85
5. Let $f(x)$=$(x+3)^2$$(x-2)^3$, $x \in [-4, 4]$. If $M$ and $m$ are the maximum and minimum values of $f$, respectively in [–4, 4], then the value of $M – m$ is :
   
      
   1. 600
   2. 392
   3. 608
   4. 108
6. Let $a$ and $b$ be be two distinct positive real numbers. Let $11^{th}$ term of a GP, whose first term is $a$ and third term is $b$, is equal to pth term of another $GP$, whose first term is $a$ and fifth term is $b$. Then $p$ is equal to
   
      
   1. 20
   2. 25
   3. 21
   4. 24
7. If $x^2– y^2$ + $2hxy$ + $2gx$ + $2fy$ + $c$ = 0 is the locus of a point, which moves such that it is always equidistant from the lines $x + 2y$ + 7 = 0 and $2x – y$+ 8 = 0, then the value of $g + c + h – f$ equals
   
      
   1. 14
   2. 6
   3. 8
   4. 29
8. Let $\vec{a}$, and $\vec{b}$ be two vectors such that $|\vec{b}|$=1 and $|(\vec{b}×\vec{a}-\vec{b}|^2$ is equal to
   
      
   1. 3
   2. 5
   3. 1
   4. 4
9. Let $y$=$f(x)$ be a thrice differentiable function in (–5, 5). Let the tangents to the curve $y=f(x)$ at $(1, f(1))$ and $(3, f(3))$ make angles $\frac{\pi}{6}$ and $\frac{\pi}{4}$, respectively with positive $x-$ axis. If $27\int \limits_{1}^{3}((f'(t))^2+1)f"(t)dt$=$\alpha + \beta\sqrt{3}$ where $\alpha$, $\beta$ are integers, then the value of $\alpha$+$\beta$ equals
   
      
   1. $-14$
   2. 26
   3. $-16$
   4. 36
10. Let $P$ be a point on the hyperbola $H$:$\frac{x^2}{9}-\frac{y^2}{4}$=1, in the first quadrant such that the area of triangle formed by P and the two foci of $H$ is $2 \sqrt{13}$. Then, the square of the distance of $P$ from the origin is
    
       
    1. 18
    2. 26
    3. 22
    4. 20
11. Bag A contains 3 white, 7 red balls and bag B contains 3 white, 2 red balls. One bag is selected at random and a ball is drawn from it. The probability of drawing the ball from the bag A, if the ball drawn in white, is :
    
       
    1. $\frac{1}{4}$
    2. $\frac{1}{9}$
    3. $\frac{1}{3}$
    4. $\frac{3}{10}$
12. Let $f : R \to R$ be defined $f(x)$=$a e^{2x}$+$be^x$+$cx$. If $f(0)=-1$, $f'(log_e 2)$=21 and $\int \limits_{0}^{\log_e4}(f(x)-cx)dx$=$\frac{39}{2}$, then the value of $|a+b+c|$equals :
    
       
    1. 16
    2. 10
    3. 12
    4. 8
13. Let $L_1$:$\vec{r}$=$(\hat{i}-\hat{j}+2\hat{k})$+$\lambda(\hat{i}-\hat{j}+2\hat{k})$, $\lambda \in R$
    $L_2$:$\vec{r}$=$(\hat{j}-\hat{k})$+$\mu(3\hat{i}+\hat{j}+p\hat{k})$, $\mu \in R$
    $L_3$:$\vec{r}$=$\delta(l\hat{i}+m\hat{j}+n\hat{k})$, $\delta \in R$
    Be three lines such that $L_1$ is perpendicular to $L_2$and $L_3$ is perpendicular to both $L_1$ and $L_2$. Then the point which lies on $L_3$ is
    
       
    1. (–1, 7, 4)
    2. (–1, –7, 4)
    3. (1, 7, –4)
    4. (1, –7, 4)
14. Let $a$ and $b$ be real constants such that the function $f$ defined by $f(x)$=$\left\{\begin{array}{cc} x^3+3x+a &, x \leq 1\\bx+2 &, x > 1 \end{array} \right.$ be differentiable on $R$. Then, the value of $\int \limits_{-2}^{2}f(x)dx$ equals
    
       
    1. $\frac{15}{6}$
    2. $\frac{19}{6}$
    3. 21
    4. 17
15. Let $f : R-{0} \to R$ be a function satisfying $\left(\frac{x}{y}\right)$=$\frac{f(x)}{f(y)}$ for all $x$, $y$, $f(y) \neq 0$. $If f'(1)$ =2024, then
    
       
    1. $xf' (x)$ – $2024 f(x)$ = 0
    2. $xf'(x)$ + $2024f(x)$ = 0
    3. $xf'(x)$ + $f(x)$ = 2024
    4. $xf'(x) –2024f(x)$ = 0
16. If $z$ is a complex number, then the number of common roots of the equation $z^{1985}$+$z^{100}$+1=0 and $z^3$+$2z^2$+$2z$+1=0, is equal to:
    
       
    1. 1
    2. 2
    3. 0
    4. 3
17. Suppose $2 – p$, $p$, $2 – q$ are the coefficient of four consecutive terms in the expansion of $(1+x)^n$. Then the value of $p^2-\alpha^2$+$6\alpha$+2p$ equals
    
       
    1. 4
    2. 10
    3. 8
    4. 6
18. If the domain of the function $f(x)$ = $\log_e \left(\frac{2x+3}{4x^2+x-3}\right)$+$\cos^{-1}\left(\frac{2x-1}{x+2}\right)$ is $(\alpha, \beta]$, then the value of $5\beta - 4\alpha$ is equal to:
    
       
    1. 10
    2. 12
    3. 11
    4. 9
19.  Let $f : R \to R$ be a function defined $f(x)$=$\frac{x}{(1+x^4)^{1/4}}$ and $g(x)$=$f(f(f(f(x))))$ then $18\int \limits_{0}^{\sqrt{2\sqrt{5}}}x^2g(x)dx$
    
       
    1. 33
    2. 36
    3. 42
    4. 39
20. Let $R$=$\begin{equation*}\begin{pmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{pmatrix} \end{equation*}$ be a non-zero 3×3 matrix, where $x \sin \theta$=$y \sin\left(\theta +\frac{2\pi}{3}\right)$=$z \sin\left(\theta +\frac{4\pi}{3}\right) \neq 0$, $\theta \in (0, 2\pi)$. For a square matrix $M$, let trace $(M)$ denote the sum of all the diagonal entries of $M$. Then, among the statements:
    (I) Trace $(R)$ = 0
    (II) If trace $(adj(adj(R))$ = 0, then $R$ has exactly one non-zero entry.
    
       
    1. Both (I) and (II) are true
    2. Neither (I) nor (II) is true
    3. Only (II) is true
    4. Only (I) is true

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 $Y = Y(X)$ be a curve lying in the first quadrant such that the area enclosed by the line $Y – y$ = $Y'(x) (X – x)$ and the co-ordinate axes, where $(x, y)$ is any point on the curve, is always $\frac{-y^2}{2Y'(x)}+1$, $Y'(x) \neq 0$. If $Y(1)$ = 1, then $12Y(2)$ equals ______ .
2. Let a line passing through the point (–1, 2, 3) intersect the lines $L_1$:$\frac{x-1}{3}$=$\frac{y-2}{2}$=$\frac{z+1}{-2}$ at $M(\alpha, \beta, \gamma)$ and $L_2$:$\frac{x+2}{-3}$=$\frac{y-2}{-2}$=$\frac{z-1}{4}$ at $N(a, b, c)$. Then the value of $\frac{(\alpha+\beta+\gamma)^2}{(a+b+c)^2}$ equals.........
3. Consider two circles $C_1$ : $x^2$ + $y^2$= 25 and $C_2$ : $(x – \alpha)^2$ + $y^2$ = 16, where $\alpha \in (5, 9)$. Let the angle between the two radii (one to each circle) drawn from one of the intersection points of $C_1$ and $C_2$ be $\sin^{-1}\left(\frac{\sqrt{63}}{8}\right)$. If the length of common chord of $C_1$ and $C_2$ is $\beta$, then the value of $(\alpha \beta)^2$ equals _____ .
4. Let $\alpha$=$\sum \limits_{k=0}^{n}\left(\frac{({}^nC_k)^2}{k+1}\right)$ and $\beta$=$\sum \limits_{k=0}^{n-1}\left(\frac{{}^nC_k{}^nC_{k+1}}{k+2}\right)$. If $5\alpha$=$6\beta$, then $n$ equals..............
5. Let $S_n$ be the sum to $n-$terms of an arithmetic progression 3, 7, 11, ...... .
   If 40 < $\left(\frac{6}{n(n+1)}\sum \limits_{k=1}^{n}S_k\right)$ < 42, then $n$ equals ..............
6. In an examination of Mathematics paper, there are 20 questions of equal marks and the question paper is divided into three sections : A, B and C . A student is required to attempt total 15 questions taking at least 4 questions from each section. If section A has 8 questions, section B has 6 questions and section C has 6 questions, then the total number of ways a student can select 15 questions is _______ .
7. The number of symmetric relations defined on the set {1, 2, 3, 4} which are not reflexive is _____ .
8. The number of real solutions of the equation $x(x^2+3|x|+5|x-1|+6|x-2|)$=0 is.......
9. The area of the region enclosed by the parabola $(y – 2)^2$ = $x – 1$, the line $x – 2y$ + 4 = 0 and the positive coordinate axes is ______.
10. The variance $\sigma^2$of the data
    

    $x_i$	0	1	5	6	10	12	17
    $f_i$	3	2	3	2	6	3	3

    
    
    
    is............

30 Jan 2024:Shift-1

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