Download JEE Main 2023 Question Paper (31 Jan - Shift 2) Mathematics

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01 Feb 2023:Shift-1

01 Feb 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. If $\phi(x)$=$\frac{1}{\sqrt{x}}\int \limits_{\frac{\pi}{4}}^{x}(4\sqrt{2}\sin t-3\phi '(t))dt$, $x>0$, then $\phi '\left(\frac{\pi}{4}\right)$ is equal to :
   
   1. $\frac{8}{\sqrt{\pi}}$
   2. $\frac{4}{6+\sqrt{\pi}}$
   3. $\frac{4}{6-\sqrt{\pi}}$
   4. $\frac{8}{6+\sqrt{\pi}}$
2.  Let the plane $P$:$8x$+$a_1y$+$a_2z$+12=0 be parallel to the line $L:$$\frac{x+2}{2}$=$\frac{y-3}{3}$=$\frac{z+4}{5}$. If the intercept of $P$ on the $y-$ axis is 1, then the distance between $P$ and $L$ is:
   
    
   1. $\sqrt{\frac{7}{2}}$
   2. $\sqrt{\frac{2}{7}}$
   3. $\frac{6}{\sqrt{14}}$
   4. $\sqrt{14}$
3.  $\lim \limits_{x \to \infty} \frac{(\sqrt{3x+1}+\sqrt{3x-1})^{6}+(\sqrt{3x+1}-\sqrt{3x-1})^{6}}{(x+\sqrt{x^2-1})^{6}+(x-\sqrt{x^2-1})^{6}}$ 
   
    
   1. is equal to 27
   2. is equal to $\frac{27}{2}$
   3. does not exist
   4. is equal to 9
4.  Let $f:R-{2, 6} \to R$ be real valued function defined as $f(x)$=$\frac{x^2+2x+1}{x^2-8x+12}$. Then range of $f$ is
   
    
   1. $\left(-\infty, -\frac{21}{4}\right] \cup [0, \infty)$
   2. $\left(-\infty, -\frac{21}{4}\right) \cup (0, \infty)$
   3. $\left(-\infty, -\frac{21}{4}\right] \cup [1, \infty)$
   4. $\left(-\infty, -\frac{21}{4}\right] \cup \left[\frac{21}{4}, \infty \right)$
5.  Let $\alpha >0$. If $\int \limits_{0}^{\alpha} \frac{x}{\sqrt{x+\alpha}-\sqrt{x}}dx$=$\frac{16+20\sqrt{2}}{15}$, then $\alpha$ is equal to :
   
    
   1. 2
   2. 4
   3. $\sqrt{2}$
   4. $2\sqrt{2}$
6.  Let $H$ be the hyperbola, whose foci are $(1±\sqrt{2}, 0)$ and eccentricity is $\sqrt{2}$. Then the length of its latus rectum is………
   
    
   1. 3
   2. $\frac{5}{2}$
   3. 2
   4. $\frac{3}{2}$
7.  The foot of perpendicular from the origin $O$ to a plane $P$ which meets the co-ordinate axes at the points $A$, $B$, $C$ is $(2,a,4)$, $a \in N$. If the volume of the tetrahedron $OABC$ is 144 $unit^3$, then Which of the following points is NOT on $P$?
   
    
   1. (3, 0, 4)
   2. (0, 4, 4)
   3. (0, 6, 3)
   4. (2, 2, 4)
8. The number of values of {$r \in p, q, ~p, ~q}$ for which $((p∧q)=>(r∨q))∧((p∧r)=>q)$ is a tautology is:
   
    
   1. 2
   2. 1
   3. 3
   4. 4
9.  Let the mean and standard deviation of marks of class $A$ of 100 students be respectively 40 and $\alpha$(>0), and the mean and standard deviation of marks of class $B$ of $n$ students be respectively 55 and $30-\alpha$. If the mean and variance of the marks of the combined class of 100+$n$ students are respectively 50 and 350, then the sum of variances of classes $A$ and $B$ is :
   
    
   1. 500
   2. 650
   3. 900
   4. 450
10.  Let $a_1$, $a_2$, $a_3$...... be an A.P. If $a_7$=3, the product $a_1$$a_4$ is minimum and the sum of its first $n$ terms is zero, then $n!-4a_{n(n+2)}$ is equal to :
    
     
    1. $\frac{381}{4}$
    2. 9
    3. $\frac{33}{4}$
    4. 24
11.  Among the relations
    $S$=$\left\{(a, b):a, b \in R-{0}, 2+\frac{a}{b} >0 \right\}$ and $T$=$\left\{(a, b):a, b \in R, a^2-b^2 \in Z\right\}$
    
     
    1. both S and T are symmetric
    2. neither S nor T is transitive
    3. S is transitive but T is not
    4. T is symmetric but S is not
12.  The complex number $z$=$\frac{i-1}{\cos \frac{\pi}{3}+i\sin{\pi}{3}}$ is equal to:
    
     
    1. $\sqrt{2}i\left(\cos\frac{5\pi}{12}-i\sin\frac{5\pi}{12}\right)$
    2. $\cos\frac{\pi}{12}-i\sin\frac{\pi}{12}$
    3. $\sqrt{2}\left(\cos\frac{5\pi}{12}+i\sin\frac{5\pi}{12}\right)$
    4. $\sqrt{2}\left(\cos\frac{\pi}{12}+i\sin\frac{\pi}{12}\right)$
13. If a point $P(\alpha, \beta, \gamma)$ satisfying
    $\begin{equation*} \begin{pmatrix} \alpha & \beta & \gamma \end{pmatrix} \begin{pmatrix} 2 & 10 & 8 \\ 9 & 3 & 8 \\ 8 & 4 & 8 \end{pmatrix} \end{equation*}$=$\begin{equation*} \begin{pmatrix} 0 & 0 & 0 \end{pmatrix} \end{equation*}$ lies on the plane $2x$+$4y$+$3z$=5, then $6\alpha$+$9\beta$+$7\gamma$ is equal to:
    
     
    1. -1
    2. 11
    3. $\frac{11}{5}$
    4. $\frac{5}{4}$
14.  The set of all values of $a^2$ for which the line $x+y$=0 bisects two distinct chords drawn from a point $P\left(\frac{1+a}{2}, \frac{1-a}{2}\right)$ on the circle $2x^2$+$2y^2$$-(1+a)x-(1-a)y$=0, is equal to:
    
     
    1. $(8, \infty)$
    2. $(0, 4]$
    3. $(4, \infty)$
    4. $(2, 12]$
15.  Let $P$ be the plane, passing through the point $(1, -1, -5)$ and perpendicular to the line joining the points $(4, 1, -3)$ and $(2, 4, 3)$. Then the distance of $P$ from the point $(3, -2, 2)$ is
    
     
    1. 7
    2. 4
    3. 6
    4. 5
16.  Let $y=y(x)$ be the solution of the differential equation $(3y^2-5x^2)ydx$+$2x(x^2-y^2)dy$=0 such that $y(1)=1$. Then $|(y(2))^3-12y(2)|$ is equal to:
    
     
    1. $32\sqrt{2}$
    2. 32
    3. 64
    4. $16\sqrt{2}$
17.  Let : $\vec{a}=\hat{i}+2\hat{j}+3\hat{k}$, $\vec{b}$=$\hat{i}-\hat{j}+2\hat{k}$ and $\vec{c}$=$5\hat{i}-3\hat{j}+3\hat{k}$ be three vectors. If $\vec{r}$ is a vector such that, $\vec{r}×\vec{b}=\vec{c}×\vec{b}$ and $\vec{r}•\vec{a}$=0, then 25$|\vec{r}|^2$ is equal to
    
     
    1. 339
    2. 560
    3. 449
    4. 336
18. Let $(a, b)⊂(0, 2\pi)$ be the largest interval for which $\sin^{-1}(\sin \theta)-\cos^{-1}(\sin \theta)>0$, $\theta \in (0, 2\pi)$, holds. If $\alpha x^2$+$\beta x$+$\sin^{-1}(x^2-6x+10)$=0 and $\alpha-\beta$=$b-a$, then $\alpha$ is equal to:
    
     
    1. $\frac{\pi}{8}$
    2. $\frac{\pi}{16}$
    3. $\frac{\pi}{48}$
    4. $\frac{\pi}{12}$
19.  The absolute minimum value, of the function $f(x)$=$|x^2-x+1|$+$[x^2-x+1]$, where $[t]$ denotes the greatest integer function, in the interval $[-1, 2]$ is:
    
     
    1. $\frac{3}{2}$
    2. $\frac{3}{4}$
    3. $\frac{5}{4}$
    4. $\frac{1}{4}$
20.  The equation $e^{4x}$+$8e^{3x}$+$13e^{2x}-8e^x+1$=0, $x \in R$ has:
    
    1. no solution
    2. four solutions two of which are negative
    3. two solutions and both are negative
    4. two solutions and only one of them is negative

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 $\vec{a}$, $\vec{b}$, $\vec{c}$ be three vectors such that $|\vec{a}|$=$\sqrt{31}$, 4$|\vec{b}|$=$|\vec{c}|$=2 and 2$(\vec{a}×\vec{b})$=$(\vec{c}×\vec{a}).$ If the angle between $\vec{b}$ and $\vec{c}$ is $\frac{2\pi}{3}$, then $\left(\frac{\vec{a}×\vec{c}}{\vec{a}•\vec{b}}\right)^2$ is equal to:
2.  Let $A$ be the event that the absolute difference between two randomly choosen real numbers in the sample space $[0, 60]$ is less than or equal to $a$. If $P(A)$=$\frac{11}{36}$, then $a$ is equal to .........
3.  If $^{2n+1}P_{n-1}:^{2n-1}P_n$=11:21, then $n^2+n$+15 is equal to :
4. If the constant term in the binomial expansion of $\left(\frac{x^{\frac{5}{2}}}{2}-\frac{4}{x^l}\right)^{9}$ is $-84$ and the coefficient of $x^{-3l}$ is $2^{\alpha}\beta$, where $\beta$<0 is an odd number, then $|\alpha l-\beta|$ is equal to ........
5. Let $A$ be a $n×n$ matrix such that $|A|$=2. If the determinant of the matrix $Adj(2•Adj(2A^{-1}))$ is $2^{84}$, then $n$ is equal to..........
6.  Let $S$ be the set of all $a \in N$ such that the area of the triangle formed by the tangent at the point $P(b, c), b, c \in N$, on the parabola $y^2=2ax$ and the lines $x=b$, $y=0$ is 16 $unit^2$, then $\sum \limits_{a \in S} a$ is equal to............
7.  The sum $1^2-2.3^2$+$3.5^2-4.7^2$+$5.9^2-......$+$15.29^2$ is.........
8.  The coefficient of $x^{-6}$, in the expansion of $\left(\frac{4x}{5}+\frac{5}{2x^2}\right)^9$, is............
9.  Let the area of the region $\left\{(x,y):|2x-1|\leq y\leq|x^2-x|, 0 \leq x \leq 1 \right\}$ be $A$. Then $(6A+11)^2$ is equal to........
10.  Let $A=[a_{ij}]$, $a_{ij} \in Z \cap [0, 4]$, $1 \leq i, j \leq 2$. The number of matrices $A$ such that the sum of all entries is a prime number $p \in (2, 13)$ is.........

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