Download JEE Main 2023 Question Paper (01 Feb - Shift 2) Mathematics

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24 Jan 2023:Shift-1

24 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.  The sum of the absolute maximum and minimum values of the function $f(x)$=$|x^2-5x+6|$$-3x$+2 in the interval $[-1, 3]$ is equal to:
   
   1. 10
   2. 12
   3. 13
   4. 24
2.  The area of the region given by $\left\{(x, y):xy \leq 8, 1 \leq y \leq x^2\right\}$ is:
   
    
   1. $8\log_e2+\frac{7}{6}$
   2. $8\log_e2-\frac{13}{3}$
   3. $16\log_e2-\frac{14}{3}$
   4. $16\log_e2+\frac{7}{3}$
3.  Let $f:R-{0, 1} \to R$ be a function such that $f(x)$+$f\left(\frac{1}{1-x}\right)$=$1+x$. Then $f(2)$ is equal to
   
    
   1. $\frac{7}{4}$
   2. $\frac{7}{3}$
   3. $\frac{9}{4}$
   4. $\frac{9}{2}$
4. The number of integral values of $k$, for which one root of the equation $2x^2-8x+k$=0 lies in the interval (1, 2) and its other root lies in the interval (2, 3), is:
   
    
   1. 0
   2. 1
   3. 2
   4. 3
5.  If $y(x)$=$x^x, x >0$, then $y"(2)-2y'(2)$ is equal to :
   
    
   1. $8 \log_e2-2$
   2. $4 \log_e2+2$
   3. $4( \log_e2)^2+2$
   4. $4( \log_e2)^2-2$
6.  Let $\vec{a}$=$2\hat{i}-7\hat{j}+5\hat{k}$, $\vec{b}$=$\hat{i}+\hat{k}$ and $\vec{c}$=$\hat{i}+2\hat{j}-3\hat{k}$ be three given vectors. If $\vec{r}$ is vector such that $\vec{r}×\vec{a}$=$\vec{c}×\vec{a}$ and $\vec{r}$ is vector such that $\vec{r}$×$\vec{a}$=$\vec{c}×\vec{a}$ and $\vec{r}•\vec{b}$=0, then $|\vec{r}|$ is equal to:
   
    
   1. $\frac{\sqrt{914}}{7}$
   2. $\frac{11}{7}$
   3. $\frac{11}{5}\sqrt{2}$
   4. $\frac{11}{7}\sqrt{2}$
7.  Let $S$=$\left\{x \in R:0< x <1\right.$ and $\left.2\tan^{-1}\left(\frac{1-x}{1+x}\right)=\cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)\right\}$. If $n(S)$ denotes the number of elements in $S$ then :
   
    
   1. $n(S)$=1 and the element in $S$ is less than $\frac{1}{2}$
   2. $n(S)$=1 and the element in $S$ is more than $\frac{1}{2}$
   3. $n(S)$=0
   4. $n(S)$=2 and only one element in $S$ is less than $\frac{1}{2}$
8.  Let $P(S)$ denote the power set of $S$={1, 2, 3,....., 10}. Define the relations $R_1$ and $R_2$ on $P(S)$ as $AR_1B$ if $(A \cap B^{c}) \cup (B \cap A^c)$=$\phi$ and $AR_2B$ if $A \cup B^c$=$B \cup A^c$, $\forall A, B \in P(S).$ Then:
   
    
   1. both $R_1$ and $R_2$ are not equivalence relations
   2. only $R_1$is an equivalence relation
   3. both $R_1$ and $R_2$ are equivalence relations
   4. only $R_2$is an equivalence relation
9.  Let 9=$x_1$<$x_2$<.....<$x_7$ be in an A.P. with common difference $d$. If the standard deviation  of $x_1$, $x_2$, ........, $x_7$ is 4 and the mean is $\bar{x}$, then $\bar{x}+x_6$ is equal to:
   
    
   1. 2$\left(9+\frac{8}{\sqrt{7}}\right)$
   2. 18$\left(1+\frac{1}{\sqrt{3}}\right)$
   3. 25
   4. 34
10.  Let $P(x_0, y_0)$ be the point on the hyperbola $3x^2-4y^2$=36, which is nearest to the line $3x$+$2y$=1. Then $\sqrt{2}(y_0-x_0)$ is equal to :
    
     
    1. 9
    2. -3
    3. 3
    4. -9
11. If $A=\frac{1}{2} \begin{equation*}\begin{bmatrix} 1 & \sqrt{3} \\ -\sqrt{3} & 1 \end{bmatrix} \end{equation*}$, then:
    
     
    1. $A^{30}$+A^{25}$$-A$=$I$
    2. $A^{30}-A^{25}$=$2I$
    3. $A^{30}$=$A^{25}$
    4. $A^{30}$+A^{25}$+$A$=$I$
12.  The sum $\sum \limits_{n=1}^{\infty}\frac{2n^2+3n+4}{(2n)!}$ is equal to :
    
     
    1. $\frac{11e}{2}+\frac{7}{2e}$
    2. $\frac{11e}{2}+\frac{7}{2e}-4$
    3. $\frac{13e}{4}+\frac{5}{4e}$
    4. $\frac{13e}{4}+\frac{5}{4e}-4$
13.  Which of the following statements is a tautology?
    
     
    1. (p∧(p->q))->∼q
    2. p∨(p∧q)
    3. (p∧q)->(∼(p)->q)
    4. p->(p∧(p->q))
14.  Two dice are thrown independently. Let $A$ be the event that the number appeared on the $1^{st}$ die is less than the number appeared on the $2^{nd}$ die, $B$ be the even that the number appeared on the $1^{st}$die is even and that on the second die is odd, and $C$ be the event that the number appeared on the $1^{st}$ die is odd and that on the $2^{nd}$ is even. Then :
    
     
    1. the number of favourable cases of the events $A$, $B$ and $C$ are 15, 6 and 6 respectively
    2. the number of favourable cases of the event $(A \cup B) \cap C$ is 6
    3. $A$ and $B$ are mutually exclusive
    4. $B$ and $C$ are independent
15. Let $a$, $b$, be two real numbers such that $ab$ < 0. If the complex number $\frac{1+ai}{b+i}$ is of unit modulus and $a+ib$ lies on the circle $|z-1|$=$|2z|$, then a possible value of $\frac{1+[a]}{4b}$, where $[t]$ is greatest integer function, is:
    
     
    1. $\frac{1}{2}$
    2. $-1$
    3. $-\frac{1}{2}$
    4. 1
16. The value of the integral $\int \limits_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\frac{x+\frac{\pi}{4}}{2-\cos 2x}dx$ is
    
     
    1. $\frac{\pi^2}{6\sqrt{3}}$
    2. $\frac{\pi^2}{3\sqrt{3}}$
    3. $\frac{\pi^2}{6}$
    4. $\frac{\pi^2}{12\sqrt{3}}$
17.  Let $\alpha x$=$exp(x^{\beta}y^{\gamma})$ be the solution of the differential equation $2x^2ydy-$$(1-xy^2)dx$=0, $x>0$, $y(2)$=$\sqrt{\log_e2}$. Then $\alpha+\beta-\gamma$ equals:
    
     
    1. -1
    2. 3
    3. 1
    4. 0
18.  For the system of linear equations $\alpha x$+$y$+$z$=1, $x$+$\alpha y$+$z$=1, $x$+$y$+$\alpha z$=$\beta$, which one of the following statements is NOT correct?
    
     
    1. It has infinitely many solutions if $\alpha$=2 and $\beta$=-1
    2. It has no solution if $\alpha$=-2 and $\beta$=1
    3. $x$+$y$+$z$=$\frac{3}{4}$ if $\alpha$=2 and $\beta$=1
    4. It has infinitely many solutions if $\alpha=1$ and $\beta$=1
19.  Let $\vec{a}$=$5\hat{i}$$-\hat{j}$$-3\hat{k}$ and $\vec{b}$=$\hat{i}$+$3\hat{j}$+$5\hat{k}$ be two vectors. Then which one of the following statements is TRUE?
    
     
    1. Projection of $\vec{a}$ on $\vec{b}$ is $\frac{17}{\sqrt{35}}$ and the direction of the projection vector is opposite to the direction of $\vec{b}$
    2. Projection of $\vec{a}$ on $\vec{b}$ is $\frac{-17}{\sqrt{35}}$ and the direction of the projection vector is same as of $\vec{b}$
    3. Projection of $\vec{a}$ on $\vec{b}$ is $\vec{-17}{\sqrt{35}}$ and the direction of the projection vector is opposite to the direction of $\vec{b}$
    4. Projection of $\vec{a}$ on $\vec{b}$ is $\frac{17}{\sqrt{35}}$ and the direction of the projection vector is same as of $\vec{b}$
20.  Let the plane $P$ pass through the intersection of the planes $2x$+$3y$$-z$=2 and $x$+$2y$+$3z$=6, and be perpendicular to the plane $2x$+$y$$-z$+1=0. If $d$ is the distance of $P$ from the point $(-7, 1, 1)$, then $d^2$ is equal to :
    
    1. $\frac{250}{83}$
    2. $\frac{15}{53}$
    3. $\frac{25}{83}$
    4. $\frac{250}{82}$

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 $\int \limits_{0}^{\pi}\frac{5^{\cos x}(1+\cos x \cos 3x+\cos^2 x+\cos^3x \cos 3x)dx}{1+5^{\cos x}}$=$\frac{k\pi}{16}$, then $k$ is equal to ............
2.  Let $\alpha x$+$\beta y$+$\gamma z$=1 be the equation of a plane passing through the point (3, -2, 5) and perpendicular to the line joining the points (1, 2, 3) and (-2, 3, 5). Then the value of $\alpha \beta \gamma$ is equal to.........
3.  Number of integral solutions to the equation $x$+$y$+$z$=21, where $x \geq 1$, $y \geq 3$, $z \geq 4$, is equal to............
4.  The total number of six digit numbers, formed using the digits 4, 5, 9 only and divisible by 6, is…………
5.  The point of intersection $C$ of the plane $8x$+$y$+$2z$=0 and the line joining the points $A(-3, -6, 1)$ and $B(2, 4, -3)$ divides the line segment $AB$ internally in the ratio $k:1$. If $a$, $b$, $c$, $(|a|, |b|, |c|)$ are coprime) are the directions ratios of the perpendicular from the point $C$ on the line $\frac{1-x}{1}$=$\frac{y+4}{2}$=$\frac{z+2}{3}$, then $|a+b+c|$ is equal to.............
6.  The sum of the common terms of the following three arithmetic progressions
   3,7,11,15,.......,399 ,
   2,5,8,11,.....,359 and
   2,7,12,17,.....,197 , is equal to…………
7.  If the $x-$intercept of a focal chord of the parabola $y^2$=$8x$+$4y$+4 is 3, then the length of this chord is equal to..........
8. Let the sixth term in the binomial expansion of $\left(\sqrt{2^{\log_2(10-3^x)}}+\sqrt{5}{2^{(x-2)\log_23}}\right)^m$ in the increasing powers of $2^{(x-2)\log_23}$, be 21. If the binomial coefficients of the second, third and fourth terms in the expansion are respectively the first, third and fifth terms of $A.P.$, then the sum of the squares of all possible values of x is………..
9.  The line $x$ = 8 is the directrix of the ellipse is $E$:$\frac{x^2}{a^2}+\frac{y^2}{b^2}$=1 with the corresponding focus (2, 0). If the tangent to $E$ at the point $P$ in the first quadrant passes through the point $(0, 4\sqrt{3})$ and intersects the $x-axis$ at $Q$, then $(3PQ)^2$ is equal to ..............
10. If the term without $x$ in the expansion of $\left(x^{\frac{2}{3}}+\frac{\alpha}{x^3}\right)^{22}$ is 7315, then $|\alpha|$ is equal to...............

01 Feb 2023:Shift-1

01 Feb 2023:Ans-Key

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