JEE Main 6 April 2023 Shift 2 Question Paper

Question 1

Three dice are rolled. If the probability of getting different numbers on the three dice is $\frac{p}{q}$, where $p$ and $q$ are co-prime, then $q-p$ is equal to

Accepted Answer

(A) 4

Answer

(B) 3

Answer

(C) 1

Answer

(D) 2

Question 2

Among the statements: $(S1):2023^{2022}-1999^{2022}$ is divisible by 8. $(S2):13(13)^n-11n-13$ is divisible by 144 for infinitely many $n\in \mathbb{N}$.

Accepted Answer

(C) both (S1) and (S2) are correct

Answer

(A) both (S1) and (S2) are incorrect

Answer

(B) only (S2) is correct

Answer

(D) only (S1) is correct

Question 3

$\lim_{n	o\infty}\left\{\left(2^{\frac{1}{2}}-2^{\frac{1}{3}}ight)\left(2^{\frac{1}{2}}-2^{\frac{1}{5}}ight)\cdots\left(2^{\frac{1}{2}}-2^{\frac{1}{2n+1}}ight)ight\}$ is equal to

Accepted Answer

(D) 0

Answer

(A) $\frac{1}{\sqrt{2}}$

Answer

(B) 1

Answer

(C) $\sqrt{2}$

Question 4

Let the sets $A$ and $B$ denote the domain and range respectively of the function $f(x)=\frac{1}{\sqrt{\lceil x\rceil-x}}$, where $\lceil x\rceil$ denotes the smallest integer greater than or equal to $x$. Then among the statements (S1): $A\cap B=(1,\infty)-\mathbb{N}$ and (S2): $A\cup B=(1,\infty)$

Accepted Answer

(A) only (S1) is true

Answer

(B) both (S1) and (S2) are true

Answer

(C) neither (S1) nor (S2) is true

Answer

(D) only (S2) is true

Question 5

If the solution curve $f(x, y) = 0$ of the differential equation $(1+\log_e x)\frac{dx}{dy}-x\log_e x=e^y$, $x>0$, passes through the points $(1,0)$ and $(\alpha,2)$ then $\alpha^\alpha$ is equal to

Accepted Answer

(D) $e^{2e^2}$

Answer

(A) $e^{2e^{\sqrt{2}}}$

Answer

(B) $e^{\sqrt{2}e^2}$

Answer

(C) $e^{e^2}$

Question 6

The sum of all values of $\alpha$, for which the points whose position vectors $\hat{i}-2\hat{j}+3\hat{k}$, $2\hat{i}-3\hat{j}+4\hat{k}$, $(\alpha+1)\hat{i}+2\hat{k}$ and $9\hat{i}+(\alpha-8)\hat{j}+6\hat{k}$ are coplanar, is equal to

Accepted Answer

(D) 2

Answer

(A) 6

Answer

(B) 4

Answer

(C) $-2$

Question 7

For the system of equations $x+y+z=6$, $x+2y+\alpha z=10$, $x+3y+5z=\beta$, which one of the following is NOT true?

Accepted Answer

(A) System has a unique solution for $\alpha=3,\beta
eq 14$.

Answer

(B) System has no solution for $\alpha=3,\beta=24$.

Answer

(C) System has a unique solution for $\alpha=-3,\beta=14$.

Answer

(D) System has infinitely many solutions for $\alpha=3,\beta=14$.

Question 8

The area bounded by the curves $y=|x-1|+|x-2|$ and $y=3$ is equal to

Accepted Answer

(B) 4

Answer

(A) 3

Answer

(C) 5

Answer

(D) 6

Question 9

Let $P$ be a square matrix such that $P^2=I-P$. For $\alpha,\beta,\gamma,\delta\in \mathbb{N}$, if $P^\alpha+P^\beta=\gamma I-29P$ and $P^\alpha-P^\beta=\delta I-13P$, then $\alpha+\beta+\gamma-\delta$ is equal to

Accepted Answer

(C) 24

Answer

(A) 18

Answer

(B) 40

Answer

(D) 22

Question 10

All the letters of the word PUBLIC are written in all possible orders and these words are written as in a dictionary with serial numbers. Then the serial number of the word PUBLIC is

Accepted Answer

(B) 582

Answer

(A) 580

Answer

(C) 578

Answer

(D) 576

Question 11

Let the line $L$ pass through the point $(0,1,2)$, intersect the line $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and be parallel to the plane $2x+y-3z=4$. Then the distance of the point $P(1,-9,2)$ from the line $L$ is

Accepted Answer

(D) $\sqrt{74}$

Answer

(A) 9

Answer

(B) $\sqrt{54}$

Answer

(C) $\sqrt{69}$

Question 12

A plane $P$ contains the line of intersection of the plane $\vec{r}\cdot(\hat{i}+\hat{j}+\hat{k})=6$ and $\vec{r}\cdot(2\hat{i}+3\hat{j}+4\hat{k})=-5$. If $P$ passes through the point $(0,2,-2)$, then the square of distance of the point $(12,12,18)$ from the plane $P$ is

Accepted Answer

(A) 620

Answer

(B) 156

Answer

(C) 310

Answer

(D) 144

Question 13

Let $f(x)$ be a function satisfying $f(x)+f(\pi-x)=\pi^2$, $\forall x\in \mathbb{R}$. Then $\int_0^\pi f(x)\sin x\,dx$ is equal to

Accepted Answer

(D) $\pi^2$

Answer

(A) $\frac{\pi^2}{4}$

Answer

(B) $\frac{\pi^2}{2}$

Answer

(C) $2\pi^2$

Question 14

If the coefficients of $x^7$ in $\left(ax^2+\frac{1}{2bx}\right)^{11}$ and $x^{-7}$ in $\left(ax-\frac{1}{3bx^2}\right)^{11}$ are equal, then

Accepted Answer

(B) $729ab=32$

Answer

(A) $64ab=243$

Answer

(C) $243ab=64$

Answer

(D) $32ab=729$

Question 15

Among the statements: (S1): $(p\Rightarrow q)\lor(\sim p\land q)$ is a tautology (S2): $(q\Rightarrow p)\Rightarrow(\sim p\land q)$ is a contradiction

Accepted Answer

(A) neither (S1) nor (S2) is True

Answer

(B) only (S1) is True

Answer

(C) only (S2) is True

Answer

(D) both (S1) and (S2) are True

Question 16

If the tangents at the points $P$ and $Q$ on the circle $x^2+y^2-2x+y=5$ meet at the point $R\left(\frac{9}{4},2\right)$, then the area of the triangle $PQR$ is

Accepted Answer

(D) $\frac{5}{8}$

Answer

(A) $\frac{13}{4}$

Answer

(B) $\frac{13}{8}$

Answer

(C) $\frac{5}{4}$

Question 17

Let the vectors $\vec{a},\vec{b},\vec{c}$ represent three coterminous edges of a parallelepiped of volume $V$. Then the volume of the parallelepiped, whose coterminous edges are represented by $\vec{a},\ \vec{b}+\vec{c}$ and $\vec{a}+2\vec{b}+3\vec{c}$ is equal to

Accepted Answer

(C) V

Answer

(A) 3V

Answer

(B) 6V

Answer

(D) 2V

Question 18

If $\gcd(m,n)=1$ and $1^2-2^2+3^2-4^2+\cdots+(2021)^2-(2022)^2+(2023)^2=1012\,m^2n$ then $m^2-n^2$ is equal to

Accepted Answer

(B) 240

Answer

(A) 200

Answer

(C) 220

Answer

(D) 180

Question 19

In a group of 100 persons 75 speak English and 40 speak Hindi. Each person speaks at least one of the two languages. If the number of persons, who speak only English is $a$ and the number of persons who speak only Hindi is $b$, then the eccentricity of the ellipse $25\left(\frac{x^2}{b^2}+\frac{y^2}{a^2}\right)=1$ is equal to

Accepted Answer

(C) $\frac{\sqrt{119}}{12}$

Answer

(A) $\frac{3\sqrt{15}}{12}$

Answer

(B) $\frac{\sqrt{117}}{12}$

Answer

(D) $\frac{\sqrt{129}}{12}$

Question 20

Let $f(x)=\frac{x}{\left(1+x^night)^{\frac{1}{n}}}$, $x\in \mathbb{R}-\{-1\}$, $n\in \mathbb{N}$, $n>2$. If $f^n(x)=(\underbrace{f\circ f\circ \cdots \circ f}_{n	ext{ times}})(x)$, then $\lim_{n	o\infty}\int x^{n-2}\left(f^n(x)ight)dx$ is equal to

Accepted Answer

0

Question 21

The value of $	an 9^\circ-	an 27^\circ-	an 63^\circ+	an 81^\circ$ is

Accepted Answer

4

Question 22

If the lines $\frac{x-1}{2}=\frac{2-y}{-3}=\frac{z-3}{\alpha}$ and $\frac{x-4}{5}=\frac{y-1}{2}=\frac{z}{\beta}$ intersect, then the magnitude of the minimum value of $8\alpha\beta$ is

Accepted Answer

18

Question 23

If $(20)^{19}+2(21)(20)^{18}+3(21)^2(20)^{17}+\cdots+20(21)^{19}=k(20)^{18}$ then $k$ is equal to

Accepted Answer

400

Question 24

The number of 4-letter words, with or without meaning, each consisting of 2 vowels and 2 consonants, which can be formed from the letters of the word UNIVERSE without repetition is

Accepted Answer

432

Question 25

The number of points, where the curve $y=x^5-20x^3+50x+2$ crosses the x-axis, is

Accepted Answer

5

Question 26

For $\alpha,\beta,z\in \mathbb{C}$ and $\lambda>1$, if $\sqrt{\lambda-1}$ is the radius of the circle $|z-\alpha|^2+|z-\beta|^2=2\lambda$, then $|\alpha-\beta|$ is equal to

Accepted Answer

2

Question 27

Let a curve $y=f(x)$, $x\in(0,\infty)$ pass through the points $P\left(1,\frac{3}{2}\right)$ and $Q\left(a,\frac{1}{2}\right)$. If the tangent at any point $R(b,f(b))$ to the given curve cuts the y-axis at the point $S(0,c)$ such that $bc=3$, then $(PQ)^2$ is equal to

Accepted Answer

5

Question 28

Let the eccentricity of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ be reciprocal to that of the hyperbola $2x^2-2y^2=1$. If the ellipse intersects the hyperbola at right angles, then square of length of the latus-rectum of the ellipse is

Accepted Answer

2

Question 29

If the mean and variance of the frequency distribution $x_i = 2,4,6,8,10,12,14,16$ with frequencies $f_i = 4,4,\alpha,15,8,\beta,4,5$ are 9 and 15.08 respectively, then the value of $\alpha^2+\beta^2-\alpha\beta$ is

Accepted Answer

25

Question 30

A 2 metre long scale with least count of 0.2 cm is used to measure the locations of objects on an optical bench. While measuring the focal length of a convex lens, the object pin and the convex lens are placed at 80 cm mark and 1 m mark, respectively. The image of the object pin on the other side of lens coincides with image pin that is kept at 180 cm mark. The percentage error in the estimation of focal length is:

Accepted Answer

(C) $1.70$

Answer

(A) $1.02$

Answer

(B) $0.85$

Answer

(D) $0.51$

Question 31

A capacitor of capacitance 150.0 $\mu F$ is connected to an alternating source of emf given by $E = 36\sin(120\pi t)$ V. The maximum value of current in the circuit is approximately equal to:

Accepted Answer

(A) $2\,A$

Answer

(B) $\frac{1}{\sqrt{2}}\,A$

Answer

(C) $\sqrt{2}\,A$

Answer

(D) $2\sqrt{2}\,A$

Question 32

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A: When you squeeze one end of a tube to get toothpaste out from the other end, Pascal's principle is observed.
Reason R: A change in the pressure applied to an enclosed incompressible fluid is transmitted undiminished to every portion of the fluid and to the walls of its container.
In the light of the above statements, choose the most appropriate answer from the options given below

Accepted Answer

(C) Both A and R are correct and R is the correct explanation of A

Answer

(A) A is not correct but R is correct

Answer

(B) A is correct but R is not correct

Answer

(D) Both A and R are correct but R is NOT the correct explanation of A

Question 33

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A: The phase difference of two light waves change if they travel through different media having same thickness, but different indices of refraction.
Reason R: The wavelengths of waves are different in different media.
In the light of the above statements, choose the most appropriate answer from the options given below

Accepted Answer

(C) Both A and R are correct and R is the correct explanation of A

Answer

(A) Both A and R are correct but R is NOT the correct explanation of A

Answer

(B) A is correct but R is not correct

Answer

(D) A is not correct but R is correct

Question 34

Figure shows a part of an electric circuit. The potentials at points $a$, $b$ and $c$ are 30 V, 12 V and 2 V respectively. The current through the 20 $\Omega$ resistor will be

Accepted Answer

(A) $0.4\,A$

Answer

(B) $0.2\,A$

Answer

(C) $0.6\,A$

Answer

(D) $1.0\,A$

Question 35

As shown in the figure, a particle is moving with constant speed $\pi$ m/s. Considering its motion from A to B, the magnitude of the average velocity is:

Accepted Answer

(D) $1.5\sqrt{3}$ m/s

Answer

(A) $\pi$ m/s

Answer

(B) $\sqrt{3}$ m/s

Answer

(C) $2\sqrt{3}$ m/s

Question 36

The work functions of Aluminium and Gold are 4.1 eV and 5.1 eV respectively. The ratio of the slope of the stopping potential versus frequency plot for Gold to that of Aluminium is

Accepted Answer

(C) $1$

Answer

(A) $1.24$

Answer

(B) $2$

Answer

(D) $1.5$

Question 37

The ratio of speed of sound in hydrogen gas to the speed of sound in oxygen gas at the same temperature is:

Accepted Answer

(A) $4 : 1$

Answer

(B) $1 : 2$

Answer

(C) $1 : 4$

Answer

(D) $1 : 1$

Question 38

A child of mass 5 kg is going round a merry-go-round that makes 1 rotation in 3.14 s. The radius of the merry-go-round is 2 m. The centrifugal force on the child will be

Accepted Answer

(D) $40\ N$

Answer

(A) $80\ N$

Answer

(B) $50\ N$

Answer

(C) $100\ N$

Question 39

A particle starts with an initial velocity of $10.0\,	ext{ms}^{-1}$ along x-direction and accelerates uniformly at the rate of $2.0\,	ext{ms}^{-2}$. The time taken by the particle to reach the velocity of $60.0\,	ext{ms}^{-1}$ is

Accepted Answer

(D) $25\,s$

Answer

(A) $6\,s$

Answer

(B) $3\,s$

Answer

(C) $30\,s$

Question 40

Choose the incorrect statement from the following:

Accepted Answer

(D) The linear speed of a planet revolving around the sun remains constant.

Answer

(A) The speed of satellite in a given circular orbit remains constant.

Answer

(B) For a planet revolving around the sun in an elliptical orbit, the total energy of the planet remains constant.

Answer

(C) When a body falls towards earth, the displacement of earth towards the body is negligible.

Question 41

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A: Diffusion current in a p-n junction is greater than the drift current in magnitude if the junction is forward biased.
Reason R: Diffusion current in a p-n junction is from the n-side to the p-side if the junction is forward biased.
In the light of the above statements, choose the most appropriate answer from the options given below

Accepted Answer

(C) A is correct but R is not correct

Answer

(A) Both A and R are correct and R is the correct explanation of A

Answer

(B) Both A and R are correct but R is NOT the correct explanation of A

Answer

(D) A is not correct but R is correct

Question 42

A dipole comprises of two charged particles of identical magnitude $q$ and opposite in nature. The mass '$m$' of the positive charged particle is half of the mass of the negative charged particle. The two charges are separated by a distance '$l$'. If the dipole is placed in a uniform electric field '$\vec{E}$' in such a way that dipole axis makes a very small angle with the electric field '$\vec{E}$'. The angular frequency of the oscillations of the dipole when released is given by:

Accepted Answer

(C) $\sqrt{\dfrac{4qE}{3ml}}$

Answer

(A) $\sqrt{\dfrac{8qE}{3ml}}$

Answer

(B) $\sqrt{\dfrac{4qE}{ml}}$

Answer

(D) $\sqrt{\dfrac{8qE}{ml}}$

Question 43

The energy density associated with electric field $\vec{E}$ and magnetic field $\vec{B}$ of an electromagnetic wave in free space is given by ($\varepsilon_0$ - permittivity of free space, $\mu_0$ - permeability of free space)

Accepted Answer

(D) $U_E=\dfrac{\varepsilon_0 E^2}{2},\ U_B=\dfrac{B^2}{2\mu_0}$

Answer

(A) $U_E=\dfrac{E^2}{2\,\varepsilon_0},\ U_B=\dfrac{B^2}{2\mu_0}$

Answer

(B) $U_E=\dfrac{E^2}{2\,\varepsilon_0},\ U_B=\dfrac{\mu_0 B^2}{2}$

Answer

(C) $U_E=\dfrac{\varepsilon_0 E^2}{2},\ U_B=\dfrac{\mu_0 B^2}{2}$

Question 44

The temperature of an ideal gas is increased from 200 K to 800 K. If r.m.s. speed of gas at 200 K is $v_0$, then r.m.s. speed of the gas at 800 K will be:

Accepted Answer

(D) $2v_0$

Answer

(A) $v_0$

Answer

(B) $4v_0$

Answer

(C) $\dfrac{v_0}{4}$

Question 45

A student is provided with a variable voltage source $V$, a test resistor $R_T=10\,\Omega$, two identical galvanometers $G_1$ and $G_2$ and two additional resistors, $R_1 = 10\,M\Omega$ and $R_2 = 0.001\,\Omega$. For conducting an experiment to verify Ohm's law, the most suitable circuit is:

Accepted Answer

(C) (3)

Answer

(A) (1)

Answer

(B) (2)

Answer

(D) (4)

Question 46

A body cools in 7 minutes from 60 °C to 40 °C. The temperature of the surrounding is 10 °C. The temperature of the body after the next 7 minutes will be:

Accepted Answer

(C) $28\,°C$

Answer

(A) $32\,°C$

Answer

(B) $30\,°C$

Answer

(D) $34\,°C$

Question 47

A small particle of mass $m$ moves in such a way that its potential energy $U = \dfrac{1}{2}m\omega^2 r^2$ where $\omega$ is constant and $r$ is the distance of the particle from origin. Assuming Bohr's quantization of momentum and circular orbit, the radius of $n^{th}$ orbit will be proportional to:

Accepted Answer

(A) $\sqrt{n}$

Answer

(B) $n$

Answer

(C) $n^2$

Answer

(D) $\dfrac{1}{n}$

Question 48

For an amplitude modulated wave the minimum amplitude is 3 V, while the modulation index is 60%. The maximum amplitude of the modulated wave is:

Accepted Answer

(B) $12\,V$

Answer

(A) $15\,V$

Answer

(C) $10\,V$

Answer

(D) $5\,V$

Question 49

The weight of a body on the surface of the earth is 100 N. The gravitational force on it when taken at a height, from the surface of earth, equal to one-fourth the radius of the earth is:

Accepted Answer

(B) $64\ N$

Answer

(A) $100\ N$

Answer

(C) $50\ N$

Answer

(D) $25\ N$

Question 50

As shown in the figure, the voltmeter reads 2 V across a 5 $\Omega$ resistor. The resistance of the voltmeter is

Accepted Answer

20

Question 51

A metal block of mass $m$ is suspended from a rigid support through a metal wire of diameter 14 mm. The tensile stress developed in the wire under equilibrium state is $7	imes10^5$ Nm$^{-2}$. The value of mass $m$ is ____ kg. (Take $g = 9.8$ ms$^{-2}$ and $\pi = \dfrac{22}{7}$)

Accepted Answer

11

Question 52

As shown in the figure, two parallel plate capacitors having equal plate area of 200 cm$^2$ are joined together. The equivalent capacitance of the combination is $x\,\varepsilon_0$ F. The value of $x$ is

Accepted Answer

5

Question 53

A ring and a solid sphere rotating about an axis passing through their centers have same radii of gyration. The axis of rotation is perpendicular to plane of ring. The ratio of radius of ring to that of sphere is $\sqrt{\dfrac{2}{x}}$. The value of $x$ is

Accepted Answer

5

Question 54

A simple pendulum with length 100 cm and bob of mass 250 g is executing S.H.M. of amplitude 10 cm. The maximum tension in the string is found to be $\dfrac{x}{40}$ N. The value of $x$ is

Accepted Answer

99

Question 55

Two concentric circular coils with radii 1 cm and 1000 cm, and number of turns 10 and 200 respectively are placed coaxially with centers coinciding. The mutual inductance of this arrangement will be ____ $	imes 10^{-8}$ H. (Take $\pi^2=10$)

Accepted Answer

4

Question 56

A beam of light consisting of two wavelengths 7000 Å and 5500 Å is used to obtain interference pattern in Young's double slit experiment. The distance between the slits is 2.5 mm and the distance between the plane of slits and the screen is 150 cm. The least distance from the central fringe, where the bright fringes coincide, will be $n 	imes 10^{-5}$ m. The value of $n$ is

Accepted Answer

462

Question 57

A body is dropped on ground from a height '$h_1$' and after hitting the ground, it rebounds to a height '$h_2$'. If the ratio of velocities of the body just before and after hitting ground is 4, then percentage loss in kinetic energy of the body is $\dfrac{x}{4}$. The value of $x$ is

Accepted Answer

375

Question 58

Experimentally it is found that 12.8 eV energy is required to separate a hydrogen atom into a proton and an electron. The orbital radius of the electron in a hydrogen atom is $\dfrac{9}{x}	imes10^{-10}$ m. The value of $x$ is ($\dfrac{1}{4\pi\varepsilon_0}=9	imes10^9$ Nm$^2$/C$^2$ and electronic charge $=1.6	imes10^{-19}$ C)

Accepted Answer

16

Question 59

A proton with a kinetic energy of 2.0 eV moves into a region of uniform magnetic field of magnitude $\dfrac{\pi}{2}	imes10^{-3}$ T. The angle between the direction of magnetic field and velocity of proton is 60°. The pitch of the helical path taken by the proton is ____ cm. (Take mass of proton $=1.6	imes10^{-27}$ kg and charge on proton $=1.6	imes10^{-19}$ C)

Accepted Answer

40

Question 60

Ion having highest hydration enthalpy among the given alkaline earth metal ions is:

Accepted Answer

(A) $Be^{2+}$

Answer

(B) $Ba^{2+}$

Answer

(C) $Sr^{2+}$

Answer

(D) $Ca^{2+}$

Question 61

The IUPAC name of $K_3[Co(C_2O_4)_3]$ is:

Accepted Answer

(A) Potassium trioxalatocobaltate(III)

Answer

(B) Potassium trisoxalato cobalt(III)

Answer

(C) Potassium trisoxalato cobaltate(III)

Answer

(D) Potassium trioxalatocobalt(III)

Question 62

Match List I (Natural Amino acid) with List II (One Letter Code). Choose the correct answer from the options given below:

Accepted Answer

(D) A-IV, B-I, C-II, D-III

Answer

(A) A-IV, B-I, C-III, D-II

Answer

(B) A-I, B-III, C-IV, D-II

Answer

(C) A-III, B-I, C-II, D-IV

Question 63

Element not present in Nessler's reagent is:

Accepted Answer

(D) N

Answer

(A) Hg

Answer

(B) I

Answer

(C) K

Question 64

Structure of $BeCl_2$ in solid state, vapour phase and at very high temperature respectively are:

Accepted Answer

(B) Polymeric, Dimeric, Monomeric

Answer

(A) Dimeric, Polymeric, Monomeric

Answer

(C) Monomeric, Dimeric, Polymeric

Answer

(D) Polymeric, Monomeric, Dimeric

List I (Natural Amino acid)	List II (One Letter Code)
A.Arginine	I.D
B.Aspartic acid	II.N
C.Asparagine	III.A
D.Alanine	IV.R

JEE Main 6 April 2023 Shift 2 Question Paper with Answers

Physics — JEE Main 6 April 2023 Shift 2

Chemistry — JEE Main 6 April 2023 Shift 2

Mathematics — JEE Main 6 April 2023 Shift 2

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