Download JEE Main 2023 Question Paper (29 Jan - Shift 1) 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. Let $\alpha$ and $\beta$ be real numbers. Consider a 3 x 3 matrix $A$ such that $A^2$=$3A$+$\alpha I$. If $A^4$=$21A$+$\beta I$, then
   
   1. $\alpha$=1
   2. $\alpha$=4
   3. $\beta=-8$=
   4. $\beta$=8
2. Let the tangents at the points $A(4, -11)$ and $B(8, -5)$ on the circle $x^2$+$y^2-$$3x$+$10y-$15=0, intersect at the point $C$. Then the radius of the circle, whose centre is $C$ and the line joining $A$ and $B$ is its tangent, is equal to
   
    
   1. $\frac{3\sqrt{3}}{2}$
   2. $\frac{2\sqrt{13}}{3}$
   3. $\sqrt{13}$
   4. $2\sqrt{13}$
3. Let $y$=$f(x)$ be the solution of the differential equation $y(x+1)dx-$$x^2dy$=0, $y(1)=e$. Then $\lim \limits_{x \to 0^+} f(x)$ is equal to
   
    
   1. $\frac{1}{e}$
   2. $e^2$
   3. $\frac{1}{e^2}$
   4. 0
4. If the vectors $\vec{a}$=$\lambda \hat{i}$+$\mu \hat{j}$+$4 \hat{k}$, $\vec{b}$=$-2 \hat{i}$+$4 \hat{j}$$-2 \hat{k}$ and $\vec{c}$=$2 \hat{i}$+$3 \hat{j}$+$\hat{k}$ are coplanar and the projection of $\vec{a}$ on the vector $\vec{b}$ is $\sqrt{54}$ units, then the sum of all possible values of $\lambda$+$\mu$ is equal to
   
    
   1. 0
   2. 6
   3. 18
   4. 24
5. Let $A$=$\left \{(x,y) \in R^2 \right.$ : $\left. y \geq 0, 2x \leq y \leq \sqrt{4-(x-1)^2} \right \}$ and $B$=$\left \{(x,y) \in R×R \right.$ : $\left. 0 \leq y \leq min \left \{2x, \sqrt{4-(x-1)^2} \right \} \right \}$. Then the ratio of the area of $A$ to the area of $B$ is
   
    
   1. $\frac{\pi+1}{\pi-1}$
   2. $\frac{\pi-1}{\pi+1}$
   3. $\frac{\pi}{\pi+1}$
   4. $\frac{\pi}{\pi-1}$
6. Consider the following system of equations
   $\alpha x$+$2y$+$z$=1
   $2\alpha x$+$3y$+$z$=1
   $3 x$+$\alpha y$+$2z$=$\beta$
   for some $\alpha, \beta \in R$ . Then which of the following is NOT correct.
   
    
   1. It has no solution for $\alpha$=3 and for all $\beta \neq$2
   2. It has no solution for $\alpha=-1$ and for all $\beta \in R$
   3. It has no solution for $\alpha \neq -1$ and $\beta$=2
   4. It has no solution if $\alpha=-1$ and $\beta \neq$2
7. The domain of $f(x)$=$\frac{log_{x+1}(x-2)}{e^{2 log_e x}-(2x+3)}$, $x \in R$ is
   
   1. $(2, \infty)-${3}
   2. $R-${-1, 3}
   3. $(-1, \infty)-${3}
   4. $R-${3}
8. Three rotten apples are mixed accidently with seven good apples and four apples are drawn one by one without replacement. Let the random variable $X$ denote the number of rotten apples. If $\mu$ and $\sigma^2$ represent mean and variance of $X$, respectively, then $10(\mu^2+\sigma^2)$ is equal to
   
    
   1. 30
   2. 20
   3. 25
   4. 250
9. Let $f(\theta)$=3 $\left(sin^4 \left(\frac{3 \pi}{2} - \theta \right)+sin^4(3 \pi + \theta) \right)-$$2(1-sin^2 2 \theta)$ and $S$= $\left \{ \theta \in [0, \pi] : f'( \theta )=- \frac{\sqrt{3}}{2} \right \}$. If $4 \beta$= $\sum \limits_{\theta \in S} \theta$, then $f(\beta)$ is equal to
   
    
   1. $\frac{3}{2}$
   2. $\frac{5}{4}$
   3. $\frac{9}{8}$
   4. $\frac{11}{8}$
10. Let $B$ and $C$ be the two points on the line $y+x$=0 such that $B$ and $C$ are symmetric with respect to the origin. Suppose $A$ is a point on $y-2x$=2 such that $\Delta ABC$ is an equilateral triangle. Then, the area of the $\Delta ABC$ is
    
     
    1. $3 \sqrt{3}$
    2. $\frac{8}{\sqrt{3}}$
    3. $2 \sqrt{3}$
    4. $\frac{10}{\sqrt{3}}$
11. Fifteen football players of a club-team are given 15 T-shirts with their names written on the backside. If the players pick up the T-shirts randomly, then the probability that at least 3 players pick the correct T-shirt is
    
     
    1. $\frac{5}{24}$
    2. $\frac{2}{15}$
    3. $\frac{5}{36}$
    4. $\frac{1}{6}$
12. Let $f(x)$=$x$+$\frac{a}{\pi^2-4} sin x$ + $\frac{b}{\pi^2-4} cos x, x \in R$ be a function which satisfies $f(x)$=$x$+$\int \limits_{0}^{\pi/2} sin(x+y) f(y) dy$. Then $(a+b)$ is equal to
    
     
    1. $-\pi(\pi+2)$
    2. $-\pi(\pi-2)$
    3. $-2 \pi(\pi-2)$
    4. $-2 \pi(\pi+2)$
13. A light ray emits from the origin making an angle 30° with the positive $x-$axis. After getting reflected by the line $x+y$= 1, if this ray intersects $x-$axis at $Q$, then the abscissa of $Q$ is
    
     
    1. $\frac{2}{(\sqrt{3}-1)}$
    2. $\frac{2}{(3+\sqrt{3})}$
    3. $\frac{\sqrt{3}}{2(\sqrt{3}+1)}$
    4. $\frac{2}{(3-\sqrt{3})}$
14. Let $\Delta$ be the area of the region {$(x, y) \in R^2 $ : $x^2+y^2 \leq 21$, $y^2 \leq 4x$, $x \geq 1$}. Then $\frac{1}{2} \left(\Delta-21 sin^{-1} \frac{2}{\sqrt{7}} \right)$ is equal to
    
     
    1. $\sqrt{3}-\frac{2}{3}$
    2. $\sqrt{3}-\frac{4}{3}$
    3. $2\sqrt{3}-\frac{2}{3}$
    4. $2\sqrt{3}-\frac{1}{3}$
15. If $p$, $q$ and $r$ are three propositions, then which of the following combination of truth values of $p$, $q$ and $r$ makes the logical expression {$(p \bigvee q) \bigwedge {((\text{~} p) \bigvee r}) \to ((\text{~ }q) \bigvee r )$ false?
    
    1. $p$=$T$, $q$=$T$, $r$=$F$
    2. $p$=$F$, $q$=$T$, $r$=$F$
    3. $p$=$T$, $q$=$F$, $r$=$T$
    4. $p$=$T$, $q$=$F$, $r$=$F$
16. Let $f : R \to R$ be a function such that $f(x)$=$\frac{x^2+2x+1}{x^2+1}$. Then
    
     
    1. $f(x)$ is many-one in $(-\infty, -1)$
    2. $f(x)$ is many-one in $(1, \infty)$
    3. $f(x)$ is one-one in $(-\infty, \infty)$
    4. $f(x)$ is one-one in $[1, \infty)$ but not in $(-\infty, \infty)$
17. Let $[x]$ denote the greatest integer $\leq x$ . Consider the function $f(x)$=max $\left \{x^2, 1+[x] \right \}$. Then the value of the integral $\int \limits_0^2 f(x) dx$is
    
     
    1. $\frac{8+4 \sqrt{2}}{3}$
    2. $\frac{4+5 \sqrt{2}}{3}$
    3. $\frac{1+5 \sqrt{2}}{3}$
    4. $\frac{5+4 \sqrt{2}}{3}$
18. Let $\lambda \neq 0$ be a real number. Let $\alpha, \beta$ be the roots of the equation $14 x^2 -$$31 x$+$3 \lambda$=0 and $\alpha, \gamma$ be the roots of the equation $35 x^2 -$$53 x$+$4 \lambda$=0. Then $\frac{3 \alpha}{\beta}$ and $\frac{4 \alpha}{\gamma}$ are the roots of the equation
    
     
    1. $7 x^2$+$245 x-$250=0
    2. $49 x^2$+$245 x$+250=0
    3. $7 x^2 -$$245 x$+250=0
    4. $49 x^2 -$$245 x$+250=0
19. Let $x$=2 be a root of the equation $x^2$+$px$+$q$=0 and $f(x)$=$\left\{\begin{array}{cl}\frac{1-\cos \left(x^2-4 p x+q^2+8 q+16\right)}{(x-2 p)^4}, & x \neq 2 p \\ 0, & x=2 p\end{array}\right.$ Then $\lim \limits_{x \to 2p^+}[f(x)]$, where [•] denotes greatest integer function, is
    
     
    1. 0
    2. -1
    3. 2
    4. 1
20. For two non-zero complex numbers $z_1$ and $z_2$ , if $Re(z_1z_2)$=0 and $Re(z_1+z_2)$, then which of the following are possible?
    (i) $Im(z_1)$>0 and $Im(z_2)$>0
    (ii) $Im(z_1)$ < 0 and $Im(z_2)$ > 0
    (iii) $Im(z_1)$ > 0 and $Im(z_2)$ < 0
    (iv) $Im(z_1)$ < 0 and $Im(z_2)$ < 0.
    
    
    1. (ii) and (iii)
    2. (i) and (iii)
    3. (ii) and (iv)
    4. (i) and (ii)

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.   Five digits numbers are formed using the digits 1, 2, 3, 5, 7 with repetitions and are written in descending order with serial numbers. For example, the number 77777 has serial number 1. Then the serial number of 35337 is……
2.   Suppose $f$ is a function satisfying $f(x+y)$=$f(x)$+$f(y)$ for all $x, y \in N$ and $f(1)=\frac{1}{5}$. If $\sum \limits_{n=1}^m \frac{f(n)}{n(n+1)(n+2)}$=$\frac{1}{12}$, then $m$ is equal to
3. Let $f : R \to R$ be a differentiable function that satisfies the relation $f(x+y)$=$f(x)$+$f(y)-$1, $\forall \text{ x, y} \in R$. If $f'(0)$=2, then $|f(-2)|$ is equal to ........
4. Let the equation of the plane $P$ containing the line $x+10$=$\frac{8-y}{2}$=$z$ be $ax$+$by$+$cz$=2$(a+b)$ and the distance of the plane $P$ from the point (1, 27, 7) be $c$. Then $a^2$+$b^2$+$c^2$ is equal to ..........
5.  If the coefficient of $x^9$ in $\left(\alpha x^3+\frac{1}{\beta x} \right)^{11}$ and the coefficient of $x^{-9}$ in $\left(\alpha x-\frac{1}{\beta x^3} \right)^{11}$ are equal, then $(\alpha \beta)^2$ is equal to ........
6.   Let the co-ordinates of one vertex of $\Delta ABC$ be $A(0,2, \alpha)$ and the other two vertices lie on the line $\frac{x+ \alpha}{5}$=$\frac{y-1}{2}$=$\frac{z+4}{3}$. For $\alpha \in Z$, if the area of $\Delta ABC$ is 21 sq. units and the line segment $BC$ has length $2 \sqrt{21}$ units, then $\alpha^2$ is  equal to .......
7.   If all the six digit numbers $x_1$$x_2$$x_3$$x_4$$x_5$$x_6$ with 0 < $x_1$ < $x_2$ < $x_3$ < $x_4$ < $x_5$ < $x_6$ are arranged in the increasing order, then the sum of the digits in the $72^{th}$ number is .......
8. Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be three non-zero non-coplanar vectors. Let the position vectors of four points $A$, $B$, $C$ and $D$ be $\vec{a}-\vec{b}+\vec{c}$, $\lambda \vec{a}-3\vec{b}+4\vec{c}$, $-\vec{a}+2\vec{b}-3\vec{c}$ and $2\vec{a}-4\vec{b}+6\vec{c}$ respectively. If $\vec{AB}$, $\vec{AC}$ and $\vec{AD}$ are coplanar, then $\lambda$ is equal to .........
9. Let $a_1$, $a_2$, $a_3$, ...... be a GP of increasing positive numbers. If the product of fourth and sixth terms is 9 and the sum of fifth and seventh terms is 24, then $a_1a_9$+$a_2a_4a_9$+$a_5$+$a_7$ is equal to .........
10. Let the coefficients of three consecutive terms in the binomial expansion of $(1+2x)^n$ be in the ratio 2 : 5 : 8. Then the coefficient of the term, which is in the middle of these three terms, is ..........

29 Jan 2023:Shift-2

29 Jan 2023:Ans-Key

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