Download JEE Main 2023 Question Paper (25 Jan - Shift 2) Mathematics

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29 Jan 2023:Shift-1

29 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. Let $f(x)$=$2x^n$+$\lambda$, $\lambda \in R$, $n \in N$ and $f(4)$=133, $f(5)$=255. Then the sum of all the positive integer divisors of $(f(3)-f(2))$ is
   1. 60
   2. 61
   3. 59
   4. 58
2. Let $A$=$\begin{equation*} \begin{bmatrix} \frac{1}{\sqrt{10}} & \frac{3}{\sqrt{10}} \\ \frac{-3}{\sqrt{10}} & \frac{1}{\sqrt{10}} \end{bmatrix} \end{equation*}$ and $B$=$\begin{equation*} \begin{bmatrix} 1 & -i \\ 0 & 1 \end{bmatrix} \end{equation*}$, where $i$=$\sqrt{-1}$.If $M$=$A^TBA$ , then the inverse of the matrix $AM^{2023}A^T$ is
    
   1. $\begin{equation*} \begin{bmatrix} 1 & 0 \\ -2023i & 1 \end{bmatrix} \end{equation*}$
   2. $\begin{equation*} \begin{bmatrix} 1 & 2023i \\ 0 & 1 \end{bmatrix} \end{equation*}$
   3. $\begin{equation*} \begin{bmatrix} 1 & -2023i \\ 0 & 1 \end{bmatrix} \end{equation*}$
   4. $\begin{equation*} \begin{bmatrix} 1 & 0 \\ 2023i & 1 \end{bmatrix} \end{equation*}$
3. Let $A$, $B$, $C$ be 3 x 3 matrices such that $A$ is symmetric and $B$ and $C$ are skew-symmetric. Consider the statements
   (S1) $A^{13}B^{26}-B^{26}A^{13}$ is symmetric
   (S2) $A^{26}C^{13}-C^{13}A^{26}$ is symmetric
    
   1. Both S1 and S2 are false
   2. Both S1 and S2 are true
   3. Only S1 is true
   4. Only S2 is true
4. The number of numbers, strictly between 5000 and 10000 can be formed using the digits 1,3,5,7,9 without repetition, is
    
   1. 72
   2. 120
   3. 6
   4. 12
5. Let $f: R\to R$ be a function defined by $f(x)$=$\log_{\sqrt{m}}{\sqrt{2}(\sin x - \cos x)+m - 2}$, for some $m$, such that the range of $f$ is [0, 2]. Then the value of $m$ is ........
    
   1. 3
   2. 5
   3. 2
   4. 4
6. The number of functions $f$ : {1,2,3,4} $\to $${a \in Z, |a| \leq 8}$ satisfying $f(n)$+$\frac{1}{n}f(n+1)$=1, $\forall n \in${1,2,3} is
    
   1. 2
   2. 1
   3. 3
   4. 4
7. Let $y=y(t)$ be a solution of the differential equation $\frac{dy}{dt}$+$\alpha y$=$\gamma e^{- \beta t}$, where $\alpha > 0$, $\beta > 0$ and $\gamma > 0$. Then $\lim \limits_{ t \to \infty} y(t)$
    
   1. is 0
   2. is - 1
   3. is 1
   4. does not exist
8. The integral of 16 $\int \limits_1^2 \frac{dx}{x^3(x^2+2)^2}$ is equal to
    
   1. $\frac{11}{12}-$$log_e4$
   2. $\frac{11}{6}-$$log_e4$
   3. $\frac{11}{6}$+$log_e4$
   4. $\frac{11}{12}$+$log_e4$
9. If the four points, whose position vectors are $3 \hat{i}-4 \hat{j}+2 \hat{k}$, $ \hat{i}+ \hat{j}- \hat{k}$, $-2 \hat{i}- \hat{j}+3 \hat{k}$, and $5 \hat{i}-2 \alpha \hat{j}+4 \hat{k}$ are coplanar, then $\alpha$ is equal to
    
   1. $\frac{73}{17}$
   2. $-\frac{107}{17}$
   3. $-\frac{73}{17}$
   4. $\frac{107}{17}$
10. Let $T$ And $C$ respectively by the transverse and conjugate axes of the hyperbola $16x^2-$$y^2$+$64x$+$4y$+44=0. Then the area of the region above the parabola $x^2$=$y+4$, below the transverse axis $T$ and on the right of the conjugate axis $C$ is :
     
    1. $4 \sqrt{6} - \frac{28}{3}$
    2. $4 \sqrt{6} + \frac{44}{3}$
    3. $4 \sqrt{6} - \frac{44}{3}$
    4. $4 \sqrt{6} + \frac{28}{3}$
11. $\sum_{k=0}^6{ }^{51-k} C_3$ is equal to
     
    1. $^{52} C_4 - ^{45}C_4$
    2. $^{52} C_3 - ^{45}C_3$
    3. $^{51} C_3 - ^{45}C_3$
    4. $^{51} C_4 - ^{45}C_4$
12. Let $\vec{a}$=$- \hat{i} - \hat{j} + \hat{k}$, $\vec{a}• \vec{b}$=1 and $\vec{a} × \vec{b}$=$\hat{i} - \hat{j}$. Then $\vec{a} - 6 \vec{b}$ is equal to
     
    1. $3(\hat{i} + \hat{j} + \hat{k})$
    2. $3(\hat{i} - \hat{j} + \hat{k})$
    3. $3(\hat{i} + \hat{j} - \hat{k})$
    4. $3(\hat{i} - \hat{j} - \hat{k})$
13. Let the function $f(x)$=$2x^3$+$(2p-7)x^2$+$3(2p-9)x$$-6$ have a maxima for some value of $x < 0$ and a minima for some value of $x > 0$. Then, the set of all values of $p$ is
     
    1. $\left(-\frac{9}{2}, \frac{9}{2} \right)$
    2. $\left(\frac{9}{2}, \infty \right)$
    3. $\left(-\infty, \frac{9}{2} \right)$
    4. $\left(0, \frac{9}{2} \right)$
14. Let $z$ be a complex number such that $\left| \frac{z-2i}{z+i} \right|$=2, $z \neq i$. Then $z$ lies on the circle of radius 2 and centre
     
    1. (0, 2)
    2. (2, 0)
    3. (0, 0)
    4. (0, -2)
15. The equations of two sides of a variable triangle are $x$ = 0 and $y$ = 3, and its third side is a tangent to the parabola $y^2$=$6x$ . The locus of its circumcentre is :
     
    1. $4y^2-$$18y-$$3x$+18=0
    2. $4y^2-$$18y$+$3x$+18=0
    3. $4y^2$+$18y$+$3x$+18=0
    4. $4y^2-$$18y-$$3x-$18=0
16. If the function $f(x)$=$\left\{\begin{array}{cc}(1+|\cos x|) \frac{\lambda}{|\cos x|} & , 0 < x < \frac{\pi}{2} \\ \mu & , \quad x=\frac{\pi}{2} \\ e^{\frac{\cot 6 x}{\cot 4 x}} & , \frac{\pi}{2} < x < \pi\end{array}\right.$ is continuous at $x=\frac{\pi}{2}$, then $9\lambda$+$6\log_e\mu$+$\mu^6-$$e^{6\lambda}$ is equal to
     
    1. 10
    2. 11
    3. $2e^4+8$
    4. 8
17. The shortest distance between the lines $x$+1=$2y$=$-12z$ and $x$= $y$+2=$6z-$6 is
     
    1. 3
    2. 2
    3. $\frac{5}{2}$
    4. $\frac{3}{2}$
18. Let $N$ be the sum of the numbers appeared when two fair dice are rolled and let the probability that $N-2$, $\sqrt{3N}$, $N$+2 are in geometric progression be $\frac{k}{48}$. Then the value of $k$ is
     
    1. $\frac{11}{12}-$$log_e4$
    2. $\frac{11}{6}-$$log_e4$
    3. $\frac{11}{6}$+$log_e4$
    4. $\frac{11}{12}$+$log_e4$
19. Let $\Delta$, $\nabla \in {\bigwedge, \bigvee}$ be such that $(p\to q)$$\Delta (p \nabla q)$ is a tautology. Then
     
    1. $\Delta$=$\bigwedge$, $\nabla$=$\bigvee$
    2. $\Delta$=$\bigwedge$, $\nabla$=$\bigwedge$
    3. $\Delta$=$\bigvee$, $\nabla$=$\bigwedge$
    4. $\Delta$=$\bigvee$, $\nabla$=$\bigvee$
20. The foot of perpendicular of the point (2, 0, 5) on the line $\frac{x+1}{2}$=$\frac{y-1}{5}$=$\frac{z+1}{-1}$ is $(\alpha, \beta, \gamma)$. Then, which of the following is NOT correct?
     
    1. $\frac{\alpha \beta}{\gamma}$=$\frac{4}{15}$
    2. $\frac{\alpha}{\beta}=-8$
    3. $\frac{\gamma}{\alpha}$=$\frac{5}{8}$
    4. $\frac{\beta}{\gamma}=-5$

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 $a \in R$ and let $\alpha$, $\beta$ be the roots of the equation $x^2$+$60^{\frac{1}{4}}x$+$a$=0 . If $\alpha^4$+$\beta^4$=$-30$, then the product of all possible values of $a$ is…………
2. The remainder when $(2023)^{2023}$ is divisible by 35 is ...........
3. 25% of the population are smokers. A smoker has 27 times more chances to develop lung cancer than a non smoker. A person is diagnosed with lung cancer and the probability that this person is a smoker is $\frac{k}{10}$. Then the value of $k$ is……..
4. A triangle is formed by $X-$axis, $Y-$axis and the line $3x$+$4y$=60. Then the number of points $P(a,b)$ which lie strictly inside the triangle, where $a$ is an integer and $b$ is a multiplier of $a$, is………
5. For the two positive numbers $a$, $b$, if $a$, $b$ and $\frac{1}{18}$ are in a geometric progression, while $\frac{1}{a}$,10and $\frac{1}{b}$ are in an arithmetic progression, then $16a$+$12b$ is equal to…….
6. If $m$ and $n$ respectively are the numbers of positive and negative values of $\theta$ in the interval $[-\pi,\pi]$ that satisfy the equation $cos 2 \theta cos \frac{\theta}{2}$=$cos 3 \theta cos \frac{9 \theta}{2}m$ then $mn$ is equal to……..
7. If $\int \limits_{\frac{1}{3}}^{3} | log_e x |dx$=$\frac{m}{n}log_e \left(\frac{n^2}{e}\right)$, where $m$, $n$ are coprime natural numbers, then $m^2+n^2-$5 is equal to  ...
8. Suppose Anil‘s mother wants to give 5 whole fruits to Anil from a basket of 7 red apples, 5 white apples and 8 oranges. If in the selected 5 fruits, at least 2 oranges, at least one red apple and at least one white apple must be given, then the number of ways, Anil’s mother can offer 5 fruits to Anil is……….
9. If the shortest distance between the line joining the points $(1,2,3)$ and $(2,3,4)$, and the line $\frac{x-1}{2}$=$\frac{y+1}{-1}$=$\frac{z-2}{0}$ is $\alpha$, then $28 \alpha^2$ is equal to ....
10. Points $P(-3,2)$, $Q(9,10 )$ and $R(\alpha , 4)$ lie on a circle $C$ with $PR$ as its diameter. The tangents to $C$ at the points $Q$ and $R$ intersect at the point $S$. If $S$ lies on the line $2x-$$ ky$= 1, then $k$ is equal to…………

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