Download JEE Main 2023 Question Paper (25 Jan - Shift 1) 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 $\vec{a}$, $\vec{b}$ and $\vec{c}$ be three non zero vectors such that $\vec{b}$.$\vec{c}$=0 and $\vec{a}$×$\left(\vec{b}×\vec{c}\right)$=$\frac{\vec{b}-\vec{c}}{2}$. If $\vec{d}$ be a vector such that $\vec{b}$.$\vec{d}$=$\vec{a}$.$\vec{b}$, then $\left(\vec{a}×\vec{b}\right)$.$\left(\vec{c}×\vec{d}\right)$ is equal to
   1. $\frac{3}{4}$
   2. $\frac{1}{4}$
   3. $\frac{1}{2}$
   4. $-\frac{1}{4}$
2. The mean and variance of the marks obtained by the students in a test are 10 and 4 respectively. Later, the marks of one of the students is increased from 8 to 12. If the new mean of the marks is 10.2, then their new varience is equal to
   1. 4.08
   2. 4.04
   3. 3.92
   4. 3.96
3. The value of $\lim \limits_{n \to \infty}$ $\frac{1+2-3+4+5-6+...+(3n-2)+(3n-1)-3n}{\sqrt{2n^4+4n+3}-\sqrt{n^4+5n+4}}$ is :
   1. $\frac{3}{2}$$\left(\sqrt{2}+1\right)$
   2. $\frac{3}{2\sqrt{2}}$
   3. $\frac{\sqrt{2}+1}{2}$
   4. $3\left(\sqrt{2}+1\right)$
4. Consider the lines $L_1$ and $L_2$ given by
   $L_1$: $\frac{x-1}{2}$=$\frac{y-3}{1}$=$\frac{z-2}{2}$
   $L_2$: $\frac{x-2}{1}$=$\frac{y-2}{2}$=$\frac{z-3}{3}$
   A line $L_3$ having direction ratios 1, -1, -2, intersects $L_1$ and $L_2$ at the points $P$ and $Q$ respectively. Then the length of line segment  $PQ$ is
   1. 4
   2. 2$\sqrt{6}$
   3. 3 $\sqrt{2}$
   4. 4$\sqrt{3}$
5. The vector $\vec{a}$=$-\hat{i}$+$2\hat{j}$+$\hat{k}$ is rotated through a right angle, passing through the $y-$ axis in its way and the resulting vector is $\vec{b}$. Then the projection of $3\vec{a}$+$\sqrt{2}\vec{b}$ on $\vec{c}$=$5 \hat{i}$+$4\hat{j}$+$3\hat{k}$ is:
   1. $2\sqrt{3}$
   2. $\sqrt{6}$
   3. $3\sqrt{2}$
   4. 1
6. Let $f(0, 1) \to R$ be a function defined by $f(x)$=$\frac{1}{1-e^{-x}}$ and $g(x)$=$(f(-x)-f(x))$. Consider two statements
   (I) $g$ is an increasing function in (0, 1)
   (II)$g$ is one-one in (0, 1)
   Then,
   1. Neither (I) nor (II) is true
   2. Both (I) and (II) are true
   3. Only (II) is true
   4. Only (I) is true
7. Let $z_1$=2+3$i$ and $z_2$=3+4$i$. The set $S$={$z \in C:$$|z-z_1|^2-$$|z-z_2|^2$=$|z_1-z_2|^2$}represents a
   1. straight line with the sum of its intercepts on the coordinate axes equals -18
   2. hyperbola with the length of the transverse axis 7
   3. hyperbola with eccentricity 2
   4. straight line with the sum of its intercepts on the coordinate axes equals 14
8. Let $x$, $y$, $z$>1 and $A$=\begin{equation*} \begin{bmatrix} 1 & log_xy & log_xz \\ log_xy & 2 & log_yz \\ log_xz & log_xy & 3 \end{bmatrix} \end{equation*}. Then $|adj(adj A^2)|$ is equal to
   1. $4^8$
   2. $2^4$
   3. $6^4$
   4. $2^8$
9. Let $M$ be the maximum value of the product of two positive integers when their sum is 66. Let the sample space $S$=$\left\{x \in Z : x(66-x) \geq \frac{5}{9}M \right.$and the event $A$={$x \in S : x$ is a multiple of 3}. Then $P(A)$ is equal to
   1. $\frac{15}{44}$
   2. $\frac{1}{3}$
   3. $\frac{1}{5}$
   4. $\frac{7}{22}$
10. Let $f(x)$=$\int \frac{2x}{(x^2+1)(x^2+3)}dx$. If $f(3)$=$\frac{1}{2}(log_e5-log_e6)$, then $f(4)$ is equal to
    1. $log_e19-log_e20$
    2. $\frac{1}{2}log_e19-log_e17$
    3. $log_e17-log_e18$
    4. $\frac{1}{2}log_e17-log_e19$
11. Let $x$ = 2 be a local minima of the function $f(x)$=$2x^4$-18$x^2$+8$x$+12, $x \in (-4, 4)$. If $M$ is local maximum value of the function $f$ in (-4, 4), then $M$ =
    1. $12\sqrt{6}-\frac{31}{2}$
    2. $18\sqrt{6}-\frac{31}{2}$
    3. $12\sqrt{6}-\frac{33}{2}$
    4. $18\sqrt{6}-\frac{33}{2}$
12. Let $y$=$y(x)$ be the solution curve of the differential equation $\frac{dy}{dx}$=$\frac{y}{x}$(1+$xy^2$$(1+log_ex)$), $x>0$, $y(1)$=3. Then $\frac{y^2(x)}{9}$ is equal to :
    1. $\frac{x^2}{2x^3(2+log_ex^3)-3}$
    2. $\frac{x^2}{7-3x^3(2+log_ex^2)}$
    3. $\frac{x^2}{5-2x^3(2+log_ex^3)}$
    4. $\frac{x^2}{3x^3(1+log_ex^2)-2}$
13. The points of intersection of the line $ax$+$ by$= 0, $(a \neq b)$ and the circle $x^2$+$y^2-$$ 2 x$= 0 are $A(\alpha ,0)$ and $B(1, \beta)$. The image of the circle with $AB$ as a diameter in the line $x$+$ y$+ 2 =0 is :
    1. $x^2$+$y^2$+$3x$+$3y$+4=0
    2. $x^2$+$y^2$+$3x$+$5y$+8=0
    3. $x^2$+$y^2$$-5x$$-5y$+12=0
    4. $x^2$+$y^2$+$5x$+$5y$+12=0
14. Let $S_1$and $S_2$ be respectively the sets of all $a \in R - {0}$ for which the system of linear equations
    $ax$+$ 2ay-$$ 3az$= 1
    $(2a+1) x$ +$(2a+ 3) y$+ $(a+ 1) z $=2
    $(3a+ 5) x$+$( a +5) y$+$( a + 2) z $=3
    has unique solution and infinitely many solutions. Then
    1. $S_1$is an infinite set and $n (S _2)$= 2
    2. $S_1$=$\phi$ and $S_2$=$ R - {0}$
    3. $S_1$=$ R - {0}$ and $S_2$=$\phi$
    4. $n (S_1)$ = 2 and $S_2$ is an infinite set
15. If $a_r$is the coefficient of $x^{10-r}$in the Binomial expansion of $(1+x)^{10}$, then $\sum \limits_{r=1}^{10} r^3\left(\frac{a_r}{a_{r-1}}\right)^2$ is equal to
    1. 5445
    2. 3025
    3. 4895
    4. 1210
16. The distance of the point $P(4, 6, -2)$ from the line passing through the point $(-3, 2, 3)$ and parallel to a line with direction ratios $3,3, -1$ is equal to :
    1. $2 \sqrt{3}$
    2. $3$
    3. $\sqrt{14}$
    4. $\sqrt{6}$
17. The distance of the point $(6, -2 \sqrt{ 2 })$ from the common tangent $y$=$ mx$+$ c$, $m$ > 0 of the curve $x$= $2y^2$ and $x$= 1 + $y^2$ is:
    1. $\frac{1}{3}$
    2. $5$
    3. $5 \sqrt{3}$
    4. $\frac{14}{3}$
18. The minimum value of the function $f(x)$=$\int_0^2 e^{|x-t|}dt$ is
    1. $2(e-1)$
    2. $e(e-1)$
    3. $2e-1$
    4. $2$
19. Let $y(x)$=$(1+x)$$(1+x^2)$$(1+x^4)$$(1+x^8)$$(1+x^{16})$ . Then $y' - y''$ at $x=- 1$ is equal to :
    1. 496
    2. 464
    3. 944
    4. 976
20. The statement ($p$ ^ (~ $q$)) => ($p$ => (~ $q$)) is
    1. a contradiction
    2. equivalent to (~ $p$) v ( ~ $q$)
    3. equivalent to $p$ v $q$
    4. a tautology

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 the sum of all the solutions of $\tan^{-1} \left(\frac{2x}{1-x^2}\right)$+$\cot^{-1} \left(\frac{1-x^2}{2x}\right)$=$\frac{\pi}{3}$, -1 < $x$ < 1, $x \neq 0$, is $\alpha -$$\frac{4}{\sqrt{3}}$, then $\alpha$ is equal to
2. Let $S$={ 1,2,3,5,7,10,11}. The number of non-empty subsets of $S$ that have the sum of all elements a multiple of 3, is………
3. Let the equation of the plane passing through the line
   $x$ - $2y$ - $z$ - 5 = 0 = $x$ + $y$ + $3z$ - 5 and parallel to the line
   $x$+ $y$ + $2z$ - 7 = 0 = $2x$ + $3y$ + $z$ - 2 be $ax$ + $by$ + $cz$ = 65. Then the distance of the point $(a,b,c)$ from the plane $2x$ + $2y$ - $z$ + 16 = 0 is…….
4. Let $A_1$ ,$A_2$ ,$A_3$ be the three $A.P.$ with the same common difference $d$ and having their first terms as $A$,$A+1$,$A+2$, respectively. Let $a,b,c$ be the $7^{th}$, $9^{th}$, $17^{th}$ terms of $A_1$ ,$A_2$ ,$A_3$, respectively such that
   $\begin{equation*} \begin{vmatrix}a & 7 & 1 \\ 2b & 17 & 1 \\ c & 17 & 1 \end{vmatrix}+70=0\end{equation*}$
   If $a$ = 29, then the sum of first 20 terms of an $AP$ whose first term is $c-a-b$ and common difference is $\frac{d}{12}$, is equal to………..
5. Let $S$= $\left\{ \alpha : log_2(9^{2 $\alpha -4} +13) \right.$$\left. - log_2 \left(\frac{5}{2}, 3^{2 \alpha -4} +1 \right) =2 \right\}$. Then the maximum value of $\beta$ for which the equation $x^2-$$2\left(\sum \limits_{\alpha \in S} \alpha \right)^2 x$+ $\sum \limits_{\alpha \in S} (\alpha+1)^2 \beta$ =0 has real roots, is .........
6. For some $a,b,c \in N$, let $f(x)$=$ax-3$ and $g(x)$=$x^b+c$, $x \in R$. If $(fog)^{-1}(x)$=$\left(\frac{x-7}{2}\right)^{1/3}$, then $(fog)(ac)$ + $(gof) (b)$ is equal to…..
7. The constant term in the expansion of $\left(2x+\frac{1}{x^7}+3x^2\right)^5$ is ........
8. It the area enclosed by the parabola $P_1$ : $2y$= $5x^2$ and $P_2$ : $x^2-$$y$ + 6 = 0 is equal to the area enclosed by $P_1$ and $y$=$\alpha x$, $\alpha$ > 0, then $\alpha^3$ is equal to…..
9. The vertices of a hyperbola $H$ are $(±6, 0)$ and its eccentricity is $\frac{\sqrt{5}}{2}$. Let $N$ be the normal to $H$ at a point in the first quadrant and parallel to the line $\sqrt{2}x$ + $y$ = $2 \sqrt{2}$ . If $d$ is the length of the line segment of $N$ between $H$ and the $y-$axis then $d^2$ is equal to……….
10. Let $x$ and $y$ be distinct integers where $1 \leq x \leq 25$ and $1 \leq y \leq 25$. Then, the number of ways of choosing $x$ and $y$, such that $x + y$ is divisible by 5, is………..

25 Jan 2023:Shift-2

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