Download JEE Main 2024 Question Paper (27 Jan - Shift 1) Mathematics

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29 Jan 2024:Shift-1

29 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. ${}^{n-1}C_r$=$(k^2-8){}^nC_{r+1}$ if only if
   
      
   1. $2\sqrt{2} < k \leq 3$
   2. $2\sqrt{3} < k \leq 3\sqrt{2}$
   3. $2\sqrt{3} < k < 3\sqrt{3}$
   4. $2\sqrt{2} < k < 2\sqrt{3}$
2. The distance, of the point (7, –2, 11) from the line $\frac{x-6}{1}$=$\frac{y-4}{0}$=$\frac{z-8}{3}$ along the line $\frac{x-5}{2}$=$\frac{y-1}{-3}$=$\frac{z-5}{6}$, is:
   
      
   1. 12
   2. 14
   3. 18
   4. 21
3. Let $x = x(t)$ and $y = y(t)$ be solutions of the differential equations $\frac{dx}{dt}$+$ax$=0 and $\frac{dy}{dt}$+$by$=0 respectively, $a$, $b \in R$. Given that $x(0)$ = 2; $y(0)$ = 1 and $3y(1)$ = $2x(1)$, the value of $t$, for which $x(t)$ = $y(t)$, is :
   
      
   1. $\log_{\frac{2}{3}}2$
   2. $\log_43$
   3. $\log_34$
   4. $\log_{\frac{4}{3}}2$
4. If $(a, b)$ be the orthocentre of the triangle whose vertices are (1, 2), (2, 3) and (3, 1), and $I_1$=$\int \limits_{a}^{b} x \sin(4x-x^2)dx$, $I_2$=$\int \limits_{a}^{b}\sin(4x-x^2)dx$, then $36\frac{I_1}{I_2}$ is equal to:
   
      
   1. 72
   2. 88
   3. 80
   4. 66
5. If $A$ denotes the sum of all the coefficients in the expansion of $(1 – 3x + 10x^2 )^n$ and $B$ denotes the sum of all the coefficients in the expansion of $(1 + x^2)^n$, then :
   
      
   1. $A$=$B^3$
   2. $3A$=$B$
   3. $B$=$A^3$
   4. $A$=$3B$
6. The number of common terms in the progressions 4, 9, 14, 19, ...... , up to $25^{th}$ term and 3, 6, 9, 12, ......., up to $37^{th}$ term is :
   
      
   1. 9
   2. 5
   3. 7
   4. 8
7. If the shortest distance of the parabola $y^2= 4x$ from the centre of the circle $x^2+ y^2$$– 4x – 16y$ + 64 = 0 is $d$, then $d^2$is equal to :
   
      
   1. 16
   2. 24
   3. 20
   4. 36
8. If the shortest distance between the lines $\frac{x-4}{1}$=$\frac{y+1}{2}$=$\frac{z}{-3}$ and $\frac{x-\lambda}{2}$=$\frac{y+1}{4}$=$\frac{z-2}{-5}$ is $\frac{6}{\sqrt{5}}$, then the sum of all possible values of $\lambda$ is :
   
      
   1. 5
   2. 8
   3. 7
   4. 10
9. If $\int \limits_{0}^{1}\frac{1}{\sqrt{3+x}+\sqrt{1+x}}dx$=$a+b\sqrt{2}$+$c\sqrt{3}$, where $a$, $b$, $c$ are rational numbers, then $2a + 3b – 4c$ is equal to :
   
      
   1. 4
   2. 10
   3. 7
   4. 8
10. Let $S$ = {l, 2, 3, ... , 10}. Suppose $M$ is the set of all the subsets of $S$, then the relation $R$ = {$(A, B)$: $A \cap B \neq \phi$; $A, B \in M$} is :
    
       
    1. symmetric and reflexive only
    2. reflexive only
    3. symmetric and transitive only
    4. symmetric only
11. If $S$ = {$z \in C : |z – i|$ = $|z + i| = |z–1|$}, then, $n(S)$ is:
    
       
    1. 1
    2. 0
    3. 3
    4. 2
12. Four distinct points $(2k, 3k)$, (1, 0), (0, 1) and (0, 0) lie on a circle for $k$ equal to :
    
       
    1. $\frac{2}{13}$
    2. $\frac{3}{13}$
    3. $\frac{5}{13}$
    4. $\frac{1}{13}$
13. Consider the function
    $f(x)$=$\left\{\begin{array}{cc}\frac{\mathrm{a}\left(7 \mathrm{x}-12-x^2\right)}{\mathrm{b}\left|\mathrm{x}^2-7 \mathrm{x}+12\right|} & ,\mathrm{x} < 3\\ \begin{array}{cc}2^{\frac{\sin (x-3)}{x-[x]}} & ,\mathrm{x} > 3 \\ \mathrm{b} &, x=3\end{array} \\ & \end{array}\right.$
    Where $[x]$ denotes the greatest integer less than or equal to $x$. If $S$ denotes the set of all ordered pairs $(a, b)$ such that $f(x)$ is continuous at $x = 3$, then the number of elements in $S$ is :
    
       
    1. 2
    2. infinitely many
    3. 4
    4. 1
14. Let $a_1$, $a_2$, ….. $a_{10}$ be 10 observations such that $\sum \limits_{k=1}^{10}a_k$=50 and $\sum \limits_{\forall k < j}a_k•a_j$=1000. Then the standard deviation of $a_1$, $a_2$, .., $a_{10}$ is equal to :
    
       
    1. 5
    2. $\sqrt{5}$
    3. 10
    4. $\sqrt{115}$
15. The length of the chord of the ellipse $\frac{x^2}{25}$+$\frac{y^2}{16}$=1, whose midpoint is $\left(1, \frac{2}{5}\right)$, is equal to :
    
       
    1. $\frac{\sqrt{1691}}{5}$
    2. $\frac{\sqrt{2009}}{5}$
    3. $\frac{\sqrt{1741}}{5}$
    4. $\frac{\sqrt{1541}}{5}$
16. The portion of the line $4x + 5y$ = 20 in the first quadrant is trisected by the lines L1 and $L_2$ passing through the origin. The tangent of an angle between the lines $L_1$ and $L_2$ is :
    
       
    1. $\frac{8}{5}$
    2. $\frac{25}{41}$
    3. $\frac{2}{5}$
    4. $\frac{30}{41}$
17. Let $\vec{a}$=$\hat{i}$+$2\hat{j}$+$\hat{k}$, $\vec{b}$=$3(\hat{i}-\hat{j}+\hat{k})$. Let $\vec{c}$ be the vector such that $\vec{a}×\vec{c}$=$\vec{b}$ and $\vec{a}•\vec{c}$=3. Then $\vec{a}•((\vec{c}×\vec{b})-\vec{b}-\vec{c})$ is equal to:
    
       
    1. 32
    2. 24
    3. 20
    4. 36
18. If $a$=$\lim \limits_{x \to 0}\frac{\sqrt{1+\sqrt{1+x^4}}-\sqrt{2}}{x^4}$ and $b$=$\lim \limits_{x \to 0}\frac{\sin^2x}{\sqrt{2}-\sqrt{1+\cos x}}$, then the value of $ab^3$ is:
    
       
    1. 36
    2. 32
    3. 25
    4. 30
19.  Consider the matrix
    $f(x)$=$\begin{equation*}\begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix} \end{equation*}$
    Given below are two statements :
    Statement I: $f(–x)$ is the inverse of the matrix $f(x)$.
    Statement II: $f(x) f(y)$ = $f(x + y)$.
    In the light of the above statements, choose the correct answer from the options given below
    
       
    1. Statement I is false but Statement II is true
    2. Both Statement I and Statement II are false
    3. Statement I is true but Statement II is false
    4. Both Statement I and Statement II are true
20. The function $f : N – {1} \to N$; defined by $f(n)$ = the highest prime factor of $n$, is :
    
       
    1. both one-one and onto
    2. one-one only
    3. onto only
    4. neither one-one nor onto

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. The least positive integral value of $\alpha$, for which the angle between the vectors $\alpha \hat{i}$$-2\hat{j}$+$2\hat{k}$ and $\alpha \hat{i}$+$2\alpha \hat{j}$$-2\hat{k}$ is acute, is.............
2. Let for a differential function $f:(0, \infty) \to R$, $f(x)-f(y) \geq \log_e\left(\frac{x}{y}\right)$+$x-y$, $\forall x, y \in (0, \infty)$. Then $\sum \limits_{n=1}^{20}f'\left(\frac{1}{n^2}\right)$ is equal to............
3. If the solution of the differential equation $(2x + 3y – 2) dx$ + $(4x + 6y – 7) dy$ = 0, $y(0)$ = 3, is $\alpha x$ + $\beta y$ + $3 log_e |2x + 3y – \gamma|$ = 6, then $\alpha + 2\beta + 3\gamma$is equal to _____.
4. Let the area of the region {$(x, y) : x – 2y + 4 \geq 0$, $x$ + $2y^2\geq 0$, $x + 4y^2 \leq 8$, $y \geq 0$} be $\frac{m}{n}$, where $m$ and $n$ are coprime numbers. Then $m + n$ is equal to _____.
5. If 8=3+$\frac{1}{4}(3+p)$+$\frac{1}{4^2}(3+2p)$+$\frac{1}{4^3}(3+3p)$+...$\infty$, then the value of $p$ is _____.
6. A fair die is tossed repeatedly until a six is obtained. Let $X$ denote the number of tosses required and let $a$ = $P(X = 3)$, $b = P(X = 3)$ and $c$ =$P(X \geq 6 |X > 3)$. Then $\frac{b+c}{a}$ is equal to.........
7. Let the set of all $a \in R$ such that the equation $\cos 2x$ + $a \sin x$= $2a - 7$  has a solution be $[p, q]$ and $r$=$\tan 9° - \tan 27° - $$\frac{1}{\cot 63°}$+$\tan 81°$, then $pqr$ is equal to ..........
8. Let $f(x)$=$x^3$+$x^2f'(1)$+$xf"(2)$+$f"'(3)$, $x \in R$. Then $f '(10)$is equal to _____.
9. Let $A$=$\begin{equation*}\begin{bmatrix}2 & 0 & 1\\ 1 & 1 & 0\\ 1 & 0 & 1 \end{bmatrix} \end{equation*}$, $B$=$\begin{equation*} \begin{bmatrix} B_1 & B_2 & B_3 \end{bmatrix}\end{equation*}$, where $B_1$, $B_2$, and $B_3$ are column matrices, and $AB_1$=$\begin{equation*} \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}\end{equation*}$, $AB_2$=$\begin{equation*} \begin{bmatrix} 2 \\ 3 \\ 0 \end{bmatrix}\end{equation*}$, $AB_3$=$\begin{equation*} \begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix}\end{equation*}$. If $\alpha$ = $|B|$ and $\beta$ is the sum of all the diagonal elements of $B$, then $\alpha^3$+ $\beta^3$ is equal to _____.
10. If $\alpha$ satisfies the equation $x^2$+ $x$ + 1 = 0 and $(1 + \alpha)^7$= $A$ + $B\alpha$ + $C\alpha^2$, $A$, $B$, $C \geq 0$, then $5(3A – 2B – C)$ is equal to _____.

27 Jan 2024:Shift-2

27 Jan 2024:Ans-Keys

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