Download JEE Main 2023 Question Paper (29 Jan - Shift 2) Mathematics

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30 Jan 2023:Shift-1

30 Jan 2023: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 a function $f:IN \to IR$, satisfying
   $f(1)$+$2f(2)$+$3f(3)$+...+$xf(x)$=$x(x+1)f(x)$; $x \geq 2$ with $f(1)$=1. Then $\frac{1}{f(2022)}$+$\frac{1}{f(2028)}$ is equal to
   
   1. 8200
   2. 8100
   3. 8400
   4. 8000
2.   Let $K$ be the sum of the coefficients of the odd powers of $x$ in the expansion of $(1+x)^{99}$. Let $a$ be the middle term in the expansion of $\left(2+\frac{1}{\sqrt{2}}\right)^{200}$. If $\frac{^{200}C_{99}K}{a}$=$\frac{2^lm}{n}$, where $m$ and $n$ are odd numbers, then the odd pair $(l, n)$ is equal to
   
    
   1. (50, 51)
   2. (51, 99)
   3. (50, 101)
   4. (51, 101)
3.   The value of the integral $\int \limits_1^2 \left( \frac{t^4+1}{t^6+1} \right)dt$ is
   
    
   1. $\tan^{-1} \frac{1}{2}$+$\frac{1}{3} \tan^{-1}8-$$\frac{\pi}{3}$
   2. $\tan^{-1} \frac{1}{2}-$$\frac{1}{3} \tan^{-1}8$+$\frac{\pi}{3}$
   3. $\tan^{-1} 2-$$\frac{1}{3} \tan^{-1}8$+$\frac{\pi}{3}$
   4. $\tan^{-1} 2$+$\frac{1}{3} \tan^{-1}8-$$\frac{\pi}{3}$
4.   The set of all values of $t \in R$ , for which the matrix
   $\begin{equation*} \begin{bmatrix} e^t & e^{-t}(sin t-2cos t) & e^{-t}(-2sin t-cos t) \\ e^t & e^{-t}(2sin t+cos t) & e^{-t}(sin t-2cos t) \\ e^t & e^{-t}cost & e^{-t} sint \end{bmatrix} \end{equation*}$ is invertible, is
   
    
   1. $R$
   2. $\left \{(2k+1) \frac{\pi}{2}, k \in Z \right \}$
   3. $\left \{k\pi + \frac{\pi}{4}, k \in Z \right \}$
   4. $\left \{ k \pi, k \in Z \right \}$
5.   If the tangent at a point $P$ on the parabola $y^2=3x$ is a parallel to the line $x$+$2y$=1 and the tangents at the points $Q$ and $R$ on the ellipse $\frac{x^2}{4}$+$\frac{y^2}{1}$ are perpendicular to the line $x-y$=2, then the area of the triangle $PQR$ is :
   
    
   1. $\frac{9}{\sqrt{5}}$
   2. $\frac{3}{2} \sqrt{5}$
   3. $3 \sqrt{5}$
   4. $5 \sqrt{3}$
6.   Let $R$ be a relation defined on $N$ as $a R b$ if $2a + 3b$ is a multiple of $5,a,b \in N$ . Then $R$ is
   
    
   1. not reflexive
   2. an equivalence relation
   3. symmetric but not transitive
   4. transitive but not symmetric
7.   The shortest distance between the lines $\frac{x-1}{2}$=$\frac{y+8}{-7}$=$\frac{z-4}{5}$ and $\frac{x-1}{2}$=$\frac{y-2}{1}$=$\frac{z-6}{-3}$ is
   
    
   1. $3 \sqrt{3}$
   2. $4 \sqrt{3}$
   3. $2 \sqrt{3}$
   4. $5 \sqrt{3}$
8. The set of all values of $\lambda$ for which the equation $cos^2x-$$2sin^4x-$$2cos^2x$=$\lambda$ has a real solution $x$, is
   
    
   1. $\left[-2, -\frac{3}{2} \right]$
   2. $\left[-1, -\frac{1}{2} \right]$
   3. $\left[-2, -1 \right]$
   4. $\left[-\frac{3}{2}, -1\right]$
9.   The value of the integral $\int \limits_{\frac{1}{2}}^{2} \frac{\tan^{-1}x}{x}dx$ is equal to
   
    
   1. $\pi log_e 2$
   2. $\frac{\pi}{2} log_e 2$
   3. $\frac{\pi}{4} log_e 2$
   4. $\frac{1}{2} log_e 2$
10. Let $S$={$w_1$, $w_2$, ....} be the sample space associated to a random experiment. Let $P(W_n)$=$\frac{P(W_{n-1}}{2}$, $n \geq 2$. Let $A$=${2k+3l : k, l \in N}$ and $B$=${W_n : n \in A}$. Then $P(B)$ is equal to
    
     
    1. $\frac{1}{32}$
    2. $\frac{3}{64}$
    3. $\frac{1}{16}$
    4. $\frac{3}{32}$
11.   Let $y$=$y(x)$ be the solution of the differential equation $xlog_ex \frac{dy}{dx}$+$y$=$x^2log_ex$, $(x>1)$. $y(2)=2$, then $y(e)$ is equal to
    
     
    1. $\frac{1+e^2}{4}$
    2. $\frac{1+e^2}{2}$
    3. $\frac{2+e^2}{2}$
    4. $\frac{4+e^2}{4}$
12.   Let $f$ and $g$ be twice differentiable functions on $R$ such that
    $f"(x)$=$g"(x)$+$6x$
    $f'(1)$=$4g'(1)-$3=9
    $f'(1)$=$4g'(1)-$3=9
    Then which of the following is NOT true?
    
     
    1. There exists $x_0 \in (1, 3/2)$  such that f(x_0)$=g(x_0)$
    2. $g(-2)-$$f(-2)$=20
    3. $|f'(x)-g'(x)|$ < 6 => $-1 < x < 1$
    4. If $-1 < x < 2$, then $|f(x)-g'(x)|$ < 8
13. The letters of the word OUGHT are written in all possible ways and these words are arranged as in a dictionary, in a series. Then the serial number of the word TOUGH is
    
     
    1. 86
    2. 89
    3. 79
    4. 84
14. The statement B => ((~A) v B) is equivalent to
    
     
    1. B => ((~A) => B)
    2. A => ((~A) => B)
    3. A => (A <=> B)
    4. B => (A => B)
15. The plane $2x - y + z$= 4 intersects the line segment joining the points $A (a, -2,4)$ and $B (2,b, -3)$at the point $C$ in the ratio 2 : 1 and the distance of the point $C$ from the origin is $\sqrt{5}$ . If $ab$ < 0  and $P$ is the point $(a - b, b, 2b - a)$ then $CP^2$ is equal to
    
     
    1. $\frac{17}{3}$
    2. $\frac{97}{3}$
    3. $\frac{16}{3}$
    4. $\frac{73}{3}$
16. If $\vec{a}$=$\hat{i}+2\hat{k}$, $\vec{b}$=$\hat{i}+\hat{j}+\hat{k}$, $\vec{c}$=$7\hat{i}-3\hat{j}+4\hat{k}$, $\vec{r}×\vec{b}$+$\vec{b}×\vec{c}$=$\vec{0}$ and $\vec{r}•\vec{a}$=0. Then $\vec{r}•\vec{c}$ is equal to
    
     
    1. 30
    2. 32
    3. 36
    4. 34
17. If the lines $\frac{x-1}{1}$=$\frac{y-2}{2}$=$\frac{z+3}{3}$ and $\frac{x-a}{2}$=$\frac{y+2}{3}$=$\frac{z-3}{1}$ intersect at the point $P$, then the distance of the point $P$ from the plane $z=a$ is
    
     
    1. 16
    2. 22
    3. 10
    4. 28
18. Let $\vec{a}$=$4\hat{i}$+$3\hat{j}$ and $\vec{b}$=$3\hat{i}-4\hat{j}+5\hat{k}$. If $\vec{c}$ is a vector such that $\vec{c}•(\vec{a}×\vec{b})$+25=0, $\vec{c}•(\hat{i}+\hat{j}+\hat{k})$=4 and projection of $\vec{c}$ on $\vec{a}$ is 1, then the projection of $c$ on $\vec{b}$ equals
    
     
    1. $\frac{5}{\sqrt{2}}$
    2. $\frac{1}{\sqrt{2}}$
    3. $\frac{3}{\sqrt{2}}$
    4. $\frac{1}{5}$
19.   The number of 3 digit numbers, that are divisible by either 3 or 4 but not divisible by 48, is
    
     
    1. 400
    2. 472
    3. 432
    4. 507
20. The area of the region $A$={$(x, y)$ : $|cos x - sin x|$ $\leq y \leq sin x$, $0 \leq x \leq \frac{\pi}{2}$} is
    
    1. $\frac{3}{\sqrt{5}}-$$\frac{3}{\sqrt{2}}$+1
    2. $1-\frac{3}{\sqrt{2}}+\frac{4}{\sqrt{5}}$
    3. $\sqrt{5}+2\sqrt{2}-4.5$
    4. $\sqrt{5}-2\sqrt{2}+1$

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. Let $\alpha$=8-14$i$, $A$=$\left\{z \in c : \frac{\alpha z- \bar{\alpha} \bar{z}}{z^2-(\bar{z})^2-112i}=1 \right\}$ and $B$={$z \in C : |z+3i|=4$}. Then $\sum \limits_{z \in A \cap B}(Re z-Im z)$ is equal to .......
2. A triangle is formed by the tangents at the point (2, 2) on the curves $y^2$=$2x$ and $x^2$+$y^2$=$4x$ and the line $x$+$y$+2=0. If $r$ is the radius of its circumcircle, then $r^2$ is equal to……..
3. Let $A$ be a symmetric matrix such that $|A|$=2 and $\begin{equation*} \begin{bmatrix} 2 & 1 \\ 3 & \frac{3}{2} \end{bmatrix} A \end{equation*}$=$\begin{equation*} \begin{bmatrix} 1 & 2 \\ \alpha & \beta \end{bmatrix} \end{equation*}$. If the sum of the diagonal elements of $A$ is $s$, then $\frac{\beta s}{\alpha ^2}$ is equal to .........
4. Let ${a_k}$ and ${b_k}$, $k \in N$ be two G.P.s with common ratios $r_1$ and $r_2$ respectively such that $a_1$=$b_1$=4 and $r_1$ < $r_2$. Let $c_k$=$a_k$+$b_k$, $k \in N$. If $c_2$=5 and $c_3$=$\frac{13}{4}$ then $\sum \limits_{k=1}^{\infty} c_k-(12a_6+8b_4)$ is equal to......
5. The total number of 4-digit numbers whose greatest common divisor with 54 is 2, is…….
6.   A circle with centre (2, 3) and radius 4 intersects the line $x$+$y$=3 at the points $P$ and $Q$. If the tangents at $P$ and $Q$ intersect at the point $S(\alpha, \beta)$, then $4\alpha-7\beta$ is equal to……..
7. Let $X$={11, 12, 13, ... , 40, 41} and $Y$={61, 62, 63, ... , 90, 91} be the two sets of observations. If $\bar{x}$ and $\bar{y}$ are their respective means and $\sigma^2$ is the variance of all the observations in $X \cup Y$, then $|\bar{x}+\bar{y}-\sigma^2|$ is equal to……….
8. If the equation of the normal to the curve $y$=$\frac{x-a}{(x+b)(x-2)}$ at the point $(1, -3)$ is $x-4y$=13, then the value of a + b is equal to…….
9.   Let $\alpha_1$, $\alpha_2$, ... , $\alpha_7$ be the roots of the equation $x^7$+$3x^5$$-13x^3$$-15x$=0 and $|\alpha_1| \geq |\alpha_2|$$\geq .... \geq|\alpha_7|$. Then $\alpha_1 \alpha_2 - \alpha_3 \alpha_4$+$\alpha_5 \alpha_6$ is equal to.....
10. Let $a_1$, $b_1$=1 and $a_n$=$a_{n-1}+(n-1)$, $b_n$=$b_{n-1}+a_{n-1}$,  $\forall n \geq 2$. If $S$=$\sum \limits_{n=1}{10} \frac{b_n}{2^n}$ and $T$=$\sum \limits_{n=1}^{8} \frac{n}{2^{n-1}}$, these $2^7(2S-T)$ is equal to.....

29 Jan 2023:Shift-1

29 Jan 2023:Ans-Key

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