Download JEE Main 2024 Question Paper (01 Feb - Shift 2) Mathematics

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27 Jan 2024:Shift-1

27 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 $f(x)$ = $|2x^2+5|x|–3|$, $x \in R$. If $m$ and $n$ denote the number of points where $f$ is not continuous and not differentiable respectively, then $m + n$ is equal to :
   
      
   1. 5
   2. 2
   3. 0
   4. 3
2. Let $\alpha$ and $\beta$ be the roots of the equation $px^2$+$qx-r$=0, where $p \neq 0$. If $p$, $q$ and $r$ be the consecutive terms of a non-constant $G.P$ and $\frac{1}{\alpha}$+$\frac{1}{\beta}$=$\frac{3}{4}$, then the value of $(\alpha - \beta)^2$ is
   
      
   1. $\frac{80}{9}$
   2. 9
   3. $\frac{20}{3}$
   4. 8
3. The number of solutions of the equation $4 \sin^2 x – 4 \cos^3x + 9 – 4 \cos x$ = 0; $x \in [ –2\pi, 2\pi]$ is :
   
      
   1. 1
   2. 3
   3. 2
   4. 0
4. The value of $\int \limits_{0}^{1}(2x^3-3x^2-x+1)^{\frac{1}{3}}dx$ is equal to :
   
      
   1. 0
   2. 1
   3. 2
   4. -1
5. Let $P$ be a point on the ellipse $\frac{x^2}{9}+\frac{y^2}{4}$=1. Let the line passing through $P$ and parallel to $y-$axis meet the circle $x^2$+ $y^2$ = 9 at point $Q$ such that $P$ and $Q$are on the same side of the $x-$axis. Then, the eccentricity of the locus of the point $R$ on $PQ$ such that $PR$ : $RQ$ = 4 : 3 as $P$ moves on the ellipse, is :
   
      
   1. $\frac{11}{19}$
   2. $\frac{13}{21}$
   3. $\frac{\sqrt{139}}{23}$
   4. $\frac{\sqrt{13}}{7}$
6. Let $m$ and $n$ be the coefficients of seventh and thirteenth terms respectively in the expansion of $\left(\frac{1}{3}x^{\frac{1}{3}}+\frac{1}{2x^{\frac{2}{3}}}\right)^{18}$. Then $\left(\frac{n}{m}\right)^{\frac{1}{3}}$ is
   
      
   1. $\frac{4}{9}$
   2. $\frac{1}{9}$
   3. $\frac{1}{4}$
   4. $\frac{9}{4}$
7. Let $\alpha$ be a non-zero real number. Suppose $f : R \to R$ is a differentiable function such that $f (0)$ = 2 and $\lim \limits_{x \to - \infty} f(x)$=1. If $f '(x)$ = $\alpha f(x)$ +3, for all $x \in R$, then $f (–log_e2)$ is equal to____.
   
      
   1. 3
   2. 5
   3. 9
   4. 7
8. Let $P$ and $Q$ be the points on the line $\frac{x+3}{8}$=$\frac{y-4}{2}$=$\frac{z+1}{2}$ which are at a distance of 6 units from the point $R (1,2,3)$. If the centroid of the triangle $PQR$ is $(\alpha, \beta, \gamma)$, then $\alpha^2+\beta^2+\gamma^2$ is:
   
      
   1. 26
   2. 36
   3. 18
   4. 24
9. Consider a $\Delta ABC$ where $A(1,3,2)$, $B(–2,8,0)$ and $C(3,6,7)$. If the angle bisector of $\Delta BAC$ meets the line $BC$ at $D$, then the length of the projection of the vector $\vec{AD}$ on the vector $\vec{AC}$ is:
   
      
   1. $\frac{37}{2\sqrt{38}}$
   2. $\frac{\sqrt{38}}{2}$
   3. $\frac{39}{2\sqrt{38}}$
   4. $\sqrt{19}$
10. Let $S_n$ denote the sum of the first $n$ terms of an arithmetic progression. If $S_{10}$ = 390 and the ratio of the tenth and the fifth terms is 15 : 7, then $S_{15} –S_5$is equal to:
    
       
    1. 800
    2. 890
    3. 790
    4. 690
11. If $\int \limits_{0}^{\frac{\pi}{3}}\cos^4x dx$=$a \pi$+$b\sqrt{3}$, where $a$ and $b$ are rational numbers, then $9a + 8b$ is equal to :
    
       
    1. 2
    2. 1
    3. 3
    4. $\frac{3}{2}$
12. If $z$ is a complex number such that $|z|\leq1$, then the minimum value of $\left|z+\frac{1}{2}(3+4i)\right|$ is:
    
       
    1. $\frac{5}{2}$
    2. 2
    3. 3
    4. $\frac{3}{2}$
13. If the domain of the function $f(x)$=$\frac{\sqrt{x^2-25}}{(4-x^2)}$+$\log_{10} (x^2 + 2x – 15)$ is $(– \infty, \alpha)$ U $[\beta ,\infty)$, then $\alpha^2+ \beta^3$is equal to :
    
       
    1. 140
    2. 175
    3. 150
    4. 125
14. Consider the relations $R_1$ and $R_2$ defined as $aR_1b$<=> $a^2 + b^2$ = 1 for all $a , b, \in R$ and $(a, b) R_2(c, d)$ <=> $a + d = b + c$ for all $(a,b), (c,d) \in N × N$. Then
    
       
    1. Only $R_1$ is an equivalence relation
    2. Only $R_2$ is an equivalence relation
    3. $R_1$ and $R_2$ both are equivalence relations
    4. Neither $R1_$ nor $R_2$ is an equivalence relation
15. If the mirror image of the point $P(3,4,9)$ in the line $\frac{x-1}{3}$=$\frac{y+1}{2}$=$\frac{z-2}{1}$ is $(\alpha, \beta, \gamma)$ is:
    
       
    1. 102
    2. 138
    3. 108
    4. 132
16. Let $f(x)$=$\left\{ \begin{array}{cc} x-1 &, x \text{ is even}, \\ 2x &, x \text{ is odd}, \end{array}\right.$, $x \in N$. If for some $a \in N$, $f(f(f(a)))$=21, then $\lim \limits_{x \to a^-} \left\{\frac{|x|^3}{a}-\left[\frac{x}{a}\right]\right\}$, where $[t]$ denotes the greatest integer less than or equal to $t$, is equal to :
    
       
    1. 121
    2. 144
    3. 169
    4. 225
17. Let the system of equations $x + 2y +3z$ = 5, $2x + 3y + z$ = 9, $4x + 3y + \lambda z$ = $\mu$ have infinite number of solutions. Then $\lambda + 2\mu$ is equal to :
    
       
    1. 28
    2. 17
    3. 22
    4. 15
18. Consider 10 observation $x_1$, $x_2$,…., $x_{10}$. such that $\sum \limits_{i=1}^{10}(x_i-\alpha)$=2 and $\sum \limits_{i=1}^{10}(x_i-\beta)^2$=40, where $\alpha, \beta$ are positive integers. Let the mean and the variance of the observations be $\frac{6}{5}$ and $\frac{84}{25}$ respectively. The $\frac{\beta}{\alpha}$ is equal to:
    
       
    1. 2
    2. $\frac{3}{2}$
    3. $\frac{5}{2}$
    4. 1
19.  Let Ajay will not appear in JEE exam with probability $p$=$\frac{2}{7}$, while both Ajay and Vijay will appear in the exam with probability $q=\frac{1}{5}$. Then the probability, that Ajay will appear in the exam and Vijay will not appear is :
    
       
    1. $\frac{9}{35}$
    2. $\frac{18}{35}$
    3. $\frac{24}{35}$
    4. $\frac{3}{35}$
20. Let the locus of the mid points of the chords of circle $x^2+(y–1)^2$=1 drawn from the origin intersect the line $x+y$ = 1 at $P$ and $Q$. Then, the length of $PQ$ is :
    
       
    1. $\frac{1}{\sqrt{2}}$
    2. $\sqrt{2}$
    3. $\frac{1}{2}$
    4. 1

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 three successive terms of a $G.P.$ with common ratio $r(r > 1)$ are the lengths of the sides of a triangle and $[r]$ denotes the greatest integer less than or equal to $r$, then $3[r]$ + $[–r]$ is equal to :
2. Let $A$ = $I_2 – 2MM^T$, where $M$ is real matrix of order 2 × 1 such that the relation $M^T M$ = $I_1$ holds. If $\lambda$ is a real number such that the relation $AX = \lambda X$ holds for some non-zero real matrix $X$ of order 2 × 1, then the sum of squares of all possible values of $\lambda$is equal to :
3. Let $f:(0, \infty) \to R$ and $F(x)$=$\int \limits_{0}^{x}tf(t)dt$. If $F(x^2)$=$x^4$+$x^5$, then $\sum \limits_{r=1}^{12}f(r^2)$ is equal to:
4. If $y$=$\frac{(\sqrt{x}+1)(x^2-\sqrt{x})}{x\sqrt{x}+x+\sqrt{x}}$+$\frac{1}{15}(3\cos^2x - 5)\cos^3x$, then $96y'\left(\frac{\pi}{6}\right)$ is equal to:
5. Let $\vec{a}$=$\hat{i}$+$\hat{j}$+$\hat{k}$, $\vec{b}$=$-\hat{i}$$-8\hat{j}$+$2\hat{k}$ and $\vec{c}$=$4\hat{i}$+$c_2\hat{j}$+$c_3\hat{k}$ be three vectors such that $\vec{b}×\vec{a}$=$\vec{c}×\vec{a}$. If the angle between the vector $\vec{c}$ and the vector $3\hat{i}$+$4\hat{j}$+$\hat{k}$ is $\theta$, then the greatest integer less than or equal to $\tan^2 \theta$ is:
6. The lines $L_1$, $L_2$, …, $L_{20}$ are distinct. For $n$ = 1, 2, 3, …, 10 all the lines $L_{2n–1}$ are parallel to each other and all the lines $L_{2n}$ pass through a given point $P$. The maximum number of points of intersection of pairs of lines from the set {$L_1$, $L_2$, …, $L_{20}$} is equal to :
7. Three points $O(0,0)$, $P(a, a_2)$, $Q(–b, b_2)$, $a > 0$, $b > 0$, are on the parabola $y = x^2$. Let $S_1$ be the area of the region bounded by the line $PQ$ and the parabola, and $S_2$ be the area of the triangle $OPQ$. If the minimum value of $\frac{S_1}{S_2}$ is $\frac{m}{n}$, $gcd(m, n)$=1, then $m+n$ is equal to:
8. The sum of squares of all possible values of $k$, for which area of the region bounded by the parabolas $2y^2$ = $kx$ and $ky^2$ = $2(y – x)$ is maximum, is equal to :
9. If $\frac{dx}{dy}$=$\frac{1+x-y^2}{y}$, $x(1)$=1, then $5x(2)$ is equal to :
10. Let $ABC$ be an isosceles triangle in which $A$ is at (–1, 0), $\angle{A}$=$\frac{2\pi}{3}$, $AB$=$AC$ and $B$ is on the positive $x-$axis. If $BC$=$4\sqrt{4\sqrt{3}}$ and the line $BC$intersects the line $y = x + 3$ at $(\alpha, \beta)$, then $\frac{\beta^4}{\alpha^2}$ is:

01 Feb 2024:Shift-1

01 Feb 2024:Ans-Keys

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