Download JEE Main 2024 Question Paper (01 Feb - Shift 1) Mathematics

Please wait! Loading...

 Download as PDF

27 Jan 2024:Shift-1

27 Jan 2024:Shift-2

All the Papers

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. A bag contains 8 balls, whose colours are either white or black. 4 balls are drawn at random without replacement and it was found that 2 balls are white and other 2 balls are black. The probability that the bag contains equal number of white and black balls is:
   
      
   1. $\frac{2}{5}$
   2. $\frac{2}{7}$
   3. $\frac{1}{7}$
   4. $\frac{1}{5}$
2. The value of the integral $\int \limits_{0}^{\frac{\pi}{4}}\frac{x dx}{\sin^4(2x)+\cos^4(2x)}$ equals
   
      
   1. $\frac{\sqrt{2}\pi^2}{8}$
   2. $\frac{\sqrt{2}\pi^2}{16}$
   3. $\frac{\sqrt{2}\pi^2}{32}$
   4. $\frac{\sqrt{2}\pi^2}{64}$
3. If $A$=$\begin{equation*}\begin{bmatrix} \sqrt{2} & 1 \\ -1 & \sqrt{2}\end{bmatrix}\end{equation*}$, $B$=$\begin{equation*}\begin{bmatrix} 1 & 0 \\ 1 & 1\end{bmatrix}\end{equation*}$, $C$=$ABA^T$ and $X$=$A^TC^2A$, then $det X$ is equal to:
   
      
   1. 243
   2. 729
   3. 27
   4. 891
4. If $\tan A$=$\frac{1}{\sqrt{x(x^2+x+1)}}$, $\tan B$=$\frac{\sqrt{x}}{\sqrt{x^2+x+1}}$ and $\tan C$=$(x^{-3}+x^{-2}+x^{-1})^{\frac{1}{2}}$, $0 < A, B, C < \frac{\pi}{2}$, then $A+B$ is equal to :
   
      
   1. $C$
   2. $\pi - C$
   3. $2\pi - C$
   4. $\frac{\pi}{2} - C$
5. If $n$ is the number of ways five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero, then $n$ is equal to:
   
      
   1. 47
   2. 53
   3. 51
   4. 43
6. Let $S$={$z \in C:|z-1|=1$ and $(\sqrt{2}-1)(z+\bar{z})$$-i(z-\bar{z})$=$2\sqrt{2}$}. Let $z_1$, $z_2 \in S$ be such that $|z_1|$=$\max \limits_{z \in S}|z|$ and $|z_2|$=$\min \limits_{z \in S}|z|$. Then $|\sqrt{2}z_1-z_2|^2$ equals to
   
      
   1. 1
   2. 4
   3. 3
   4. 2
7. Let the median and the mean deviation about the median of 7 observation 170, 125, 230, 190, 210, $a$, $b$ be 170 and $\frac{205}{7}$ respectively. Then the mean deviation about the mean of these 7 observations is :
   
      
   1. 31
   2. 28
   3. 30
   4. 32
8. Let $\vec{a}$=$-5\hat{i}$+$\hat{j}$$-3\hat{k}$, $\vec{b}$=$\hat{i}$+$2\hat{j}$$-4\hat{k}$ and $\vec{c}$=$((\vec{a}×\vec{b})×\hat{i})×\hat{i})×\hat{i}$. Then $\vec{c}•(-\hat{i}+\hat{j}+\hat{k})$ is equal to
   
      
   1. -12
   2. -10
   3. -13
   4. -15
9. Let $S$={$x \in R:(\sqrt{3}+\sqrt{2})^x$+$(\sqrt{3}-\sqrt{2})^x$=10}. Then the number of elements in $S$ is
   
      
   1. 4
   2. 0
   3. 2
   4. 1
10. The area enclosed by the curves $xy + 4y$ = 16 and $x + y$ = 6 is equal to :
    
       
    1. $28-30\log_e2$
    2. $30-28\log_e2$
    3. $30-32\log_e2$
    4. $32-30\log_e2$
11. Let $f:R \to R$ and $g:R \to R$ be defined as $f(x)$=$\left\{\begin{array}{cc} \log_e x &, x > 0 \\ e^{-x} &, x \leq 0\end{array} \right.$ and $g(x)$=$\left\{\begin{array}{cc} x &, x \geq 0 \\ e^x &, x < 0 \end{array} \right.$. Then, $gof:R \to R$ is:
    
       
    1. one-one but not onto
    2. neither one-one nor onto
    3. onto but not one-one
    4. both one-one and onto
12. If the system of equations
    $2x + 3y – z$ = 5
    $x + \alpha y + 3z$ = –4
    $3x – y + \beta z$ = 7
    has infinitely many solutions, then $13 \alpha \beta$ is equal to
    
       
    1. 1110
    2. 1120
    3. 1210
    4. 1220
13. For $0 < \theta < \pi/2$, if the eccentricity of the hyperbola $x^2– y^2 cosec^2\theta$ = 5 is $\sqrt{7}$ times eccentricity of the ellipse $x^2 cosec^2\theta + y^2$= 5, then the value of $\theta$ is :
    
       
    1. $\frac{\pi}{6}$
    2. $\frac{5\pi}{12}$
    3. $\frac{\pi}{3}$
    4. $\frac{\pi}{4}$
14. Let $y = y(x)$ be the solution of the differential equation $\frac{dy}{dx}$=$2x(x+y)^3$$-x(x+y)$$-1$, $y(0)$=1. Then, $\left(\frac{1}{\sqrt{2}}+y\left(\frac{1}{\sqrt{2}}\right)\right)^2$ equals :
    
       
    1. $\frac{4}{4+\sqrt{e}}$
    2. $\frac{3}{3-\sqrt{e}}$
    3. $\frac{2}{1+\sqrt{e}}$
    4. $\frac{1}{2-\sqrt{e}}$
15. Let $f : R \to R$ be defined as
    $\left\{\begin{array}{cc} \frac{a-b \cos 2x}{x^2} &; x < 0 \\ x^2+cx+2 &; 0 \leq x \leq 1 \\ 2x+1 &; x > 1\end{array} \right.$
    If $f$ is continuous everywhere in $R$ and $m$ is the number of points where $f$ is NOT differential then $m + a + b + c$ equals :
    
       
    1. 1
    2. 4
    3. 3
    4. 2
16. Let $\frac{x^2}{a^2}$+$\frac{y^2}{b^2}$=1, $a > b$ be an ellipse, whose eccentricity is $\frac{1}{\sqrt{2}}$ and the length of the latus rectum is 14. Then the square of the eccentricity of $\frac{x^2}{a^2}-\frac{y^2}{b^2}$=1 is:
    
       
    1. 3
    2. 7/2
    3. 3/2
    4. 5/2
17. Let 3, $a$, $b$, $c$ be in $A.P.$ and 3, $a – 1$, $b + l$, $c + 9$ be in G.P. Then, the arithmetic mean of $a$, $b$ and $c$ is :
    
       
    1. -4
    2. -1
    3. 13
    4. 11
18. Let $C : x^2 + y^2$ = 4 and $C’ : x^2 + y^2- 4\lambda x$ + 9 = 0 be two circles. If the set of all values of $\lambda$ so that the circles $C$ and $C’$ intersect at two distinct points, is $R– [a, b]$, then the point $(8a + 12, 16b – 20)$ lies on the curve :
    
       
    1. $x^2+ 2y^2 – 5x + 6y$ = 3
    2. $5x^2– y = – 11$
    3. $x^2– 4y^2$ = 7
    4. $6x^2+ y^2$= 42
19.  If $5f(x)$+$4f\left(\frac{1}{x}\right)$=$x^2-2$, $\forall x \neq 0$ and $y$=$9x^2f(x)$, then $y$ is strictly increasing in :
    
       
    1. $\left(0, \frac{1}{\sqrt{5}}\right) \cup \left(\frac{1}{\sqrt{5}}, \infty\right)$
    2. $\left(-\frac{1}{\sqrt{5}}, 0\right) \cup \left(\frac{1}{\sqrt{5}}, \infty\right)$
    3. $\left(-\frac{1}{\sqrt{5}}, 0\right) \cup \left(0, \frac{1}{\sqrt{5}}\right)$
    4. $\left(-\infty, \frac{1}{\sqrt{5}}\right) \cup \left(0, \frac{1}{\sqrt{5}}\right)$
20. If the shortest distance between the lines $\frac{x - \lambda}{-2}$=$\frac{y - 2}{1}$=$\frac{z - 1}{1}$ and $\frac{x - \sqrt{3}}{1}$=$\frac{y - 1}{-2}$=$\frac{z - 2}{1}$ is 1, then the sum of all possible values of $\lambda$ is :
    
       
    1. 0
    2. $2\sqrt{3}$
    3. $3\sqrt{3}$
    4. $-2\sqrt{3}$

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 $x = x(t)$ is the solution of the differential equation $(t + 1)dx$ = $(2x + (t + 1)^4 ) dt$, $x(0) = 2$, then, $x(1)$ equals ___________.
2. The number of elements in the set $S$ = {$(x, y, z)$ : $x, y, z \in Z$, $x + 2y + 3z$ = 42, $x, y, z \geq 0$} equals __________.
3. If the Coefficient of $x^{30}$ in the expansion of $\left(1+\frac{1}{x}\right)^6$$(1+x^2)^7$$(1-x^3)^8$; $x \neq 0$ is $\alpha$, then $|\alpha|$ equals............
4. Let 3, 7, 11, 15, ...., 403 and 2, 5, 8, 11, . . ., 404 be two arithmetic progressions. Then the sum, of the common terms in them, is equal to _________.
5. Let ${x}$ denote the fractional part of $x$ and $f(x)$ = $\frac{\cos^{-1}(1-\text{{x}}^2)\sin^{-1}(1-\text{{x}})}{\text{{x}}-\text{{x}}^3}$, $x \neq 0$. If $L$and $R$ respectively denotes the left hand limit and the right hand limit of $f(x)$ at $x = 0$, then $\frac{32}{\pi^2}{(L^2+R^2)}$ is equal to ....................
6. Let the line $L : \sqrt{2}x + y$ = $\alpha$ pass through the point of the intersection $P$ (in the first quadrant) of the circle $x^2$ + $y^2$ = 3 and the parabola $x^2 = 2y$. Let the line $L$touch two circles $C_1$ and $C_2$ of equal radius $2 \sqrt{3}$. If the centres $Q_1$ and $Q_2$ of the circles $C_1$ and $C_2$ lie on the $y-$axis, then the square of the area of the triangle $PQ_1Q_2$ is equal to ___________.
7. Let $P$ = {$z \in C : |z + 2 – 3i | \leq 1$} and $Q$ = {$z \in C : z (1 + i) + \bar{z} (1 – i) \leq –8$} . Let in $P \cap Q$, $|z – 3 + 2i|$ be maximum and minimum at $z_1$ and $z_2$ respectively. If $|z_1|^2 + 2|z|^2$= $\alpha + \beta\sqrt{2}$, where $\alpha$, $\beta$ are integers, then $\alpha$ + $\beta$ equals ________.
8. If $\int \limits_{-\pi/2}^{\pi/2}\frac{8\sqrt{2} \cos x dx}{(1+e^{\sin x})(1+\sin^4 x)}$=$\alpha \pi$+$\beta \log_e(3+2\sqrt{2})$, where $\alpha, \beta$ are integers, then $\alpha^2+\beta^2$ equals..........
9. Let the line of the shortest distance between the lines $L_1$ : $\vec{r}$=$(\hat{i}+2\hat{j}+3\hat{k})$+$\lambda(\hat{i}-\hat{j}+\hat{k})$ and $L_2$ : $\vec{r}$=$(4\hat{i}$+$5\hat{j}$+$6\hat{k})$+$\mu(\hat{i}+\hat{j}-\hat{k})$ intersect $L_1$ and $L_2$ at $P$ and $Q$ respectively. If $(\alpha, \beta, \gamma)$ is the midpoint of the line segment $PQ$, then $2(\alpha + \beta + \gamma)$ is equal to ___________.
10. Let $A$= {1, 2, 3, . . 20}. Let $R_1$ and $R_2$ two relation on $A$ such that
    $R_1$ = {$(a, b) : b$ is divisible by $a$}
    $R_2$ = {$(a, b)$ : a is an integral multiple of $b$}.
    Then, number of elements in $R_1 – R_2$ is equal to ________.

01 Feb 2024:Shift-2

01 Feb 2024:Ans-Keys

All the Papers Ans-Keys

 Download as PDF