JEE Main 13 April 2023 Shift 2 Question Paper

Question 1

If the system of equations $2x + y - z = 5$, $2x - 5y + \lambda z = \mu$, $x + 2y - 5z = 7$ has infinitely many solutions, then $(\lambda + \mu)^2 + (\lambda - \mu)^2$ is equal to

Accepted Answer

(A) 916

Answer

(B) 912

Answer

(C) 920

Answer

(D) 904

Question 2

The coefficient of $x^5$ in the expansion of $\left(2x^3 - \dfrac{1}{3x^2}\right)^5$ is

Accepted Answer

(C) $\dfrac{80}{9}$

Answer

(A) 8

Answer

(B) 9

Answer

(D) $\dfrac{26}{3}$

Question 3

The plane, passing through the points $(0, -1, 2)$ and $(-1, 2, 1)$ and parallel to the line passing through $(5, 1, -7)$ and $(1, -1, -1)$, also passes through the point

Accepted Answer

(C) $(-2, 5, 0)$

Answer

(A) $(1, -2, 1)$

Answer

(B) $(0, 5, -2)$

Answer

(D) $(2, 0, 1)$

Question 4

Let $\alpha, \beta$ be the roots of the equation $x^2 - \sqrt{2}x + 2 = 0$. Then $\alpha^{14} + \beta^{14}$ is equal to

Accepted Answer

(D) $-128$

Answer

(A) $-64\sqrt{2}$

Answer

(B) $-128\sqrt{2}$

Answer

(C) $-64$

Question 5

Let $a_1, a_2, a_3, \ldots$ be a G.P. of increasing positive numbers. Let the sum of its $6^{th}$ and $8^{th}$ terms be 2 and the product of its $3^{rd}$ and $5^{th}$ terms be $\dfrac{1}{9}$. Then $6(a_2 + a_4)(a_4 + a_6)$ is equal to

Accepted Answer

(D) $3$

Answer

(A) $2\sqrt{2}$

Answer

(B) $2$

Answer

(C) $3\sqrt{3}$

Question 6

Let $(\alpha, \beta)$ be the centroid of the triangle formed by the lines $15x - y = 82$, $6x - 5y = -4$ and $9x + 4y = 17$. Then $\alpha + \beta$ and $9\alpha - \beta$ are the roots of the equation

Accepted Answer

(B) $x^2 - 13x + 42 = 0$

Answer

(A) $x^2 - 7x + 12 = 0$

Answer

(C) $x^2 - 14x + 48 = 0$

Answer

(D) $x^2 - 10x + 25 = 0$

Question 7

Let $|\vec{a}| = 2$, $|\vec{b}| = 3$ and the angle between the vectors $\vec{a}$ and $\vec{b}$ be $\dfrac{\pi}{4}$. Then $\left|(\vec{a} + 2\vec{b}) 	imes (2\vec{a} - 3\vec{b})ight|^2$ is equal to

Accepted Answer

(D) 882

Answer

(A) 482

Answer

(B) 441

Answer

(C) 841

Question 8

Let N be the foot of perpendicular from the point $P(1, -2, 3)$ on the line passing through the points $(4, 5, 8)$ and $(1, -7, 5)$. Then the distance of N from the plane $2x - 2y + z + 5 = 0$ is

Accepted Answer

(C) 7

Answer

(A) 6

Answer

(B) 9

Answer

(D) 8

Question 9

If $\lim\limits_{x 	o 0} \dfrac{e^{ax} - \cos(bx) - \dfrac{cxe^{-cx}}{2}}{1 - \cos(2x)} = 17$, then $5a^2 + b^2$ is

Accepted Answer

(C) 68

Answer

(A) 72

Answer

(B) 76

Answer

(D) 64

Question 10

Let the centre of a circle C be $(\alpha, \beta)$ and its radius $r < 8$. Let $3x + 4y = 24$ and $3x - 4y = 32$ be two tangents and $4x + 3y = 1$ be a normal to C. Then $(\alpha - \beta + r)$ is equal to

Accepted Answer

(A) 7

Answer

(B) 9

Answer

(C) 5

Answer

(D) 6

Question 11

All words, with or without meaning, are made using all the letters of the word MONDAY. These words are written as in a dictionary with serial numbers. The serial number of the word MONDAY is

Accepted Answer

(A) 327

Answer

(B) 326

Answer

(C) 328

Answer

(D) 324

Question 12

The range of $f(x) = 4\sin^{-1}\left(\dfrac{x^2}{x^2+1}\right)$ is

Accepted Answer

(B) $[0, 2\pi)$

Answer

(A) $[0, \pi]$

Answer

(C) $[0, \pi)$

Answer

(D) $[0, 2\pi]$

Question 13

The statement $(\sim(p \Leftrightarrow \sim q)) \wedge q$ is equivalent to

Accepted Answer

(A) $(\sim p) \wedge (\sim q)$

Answer

(B) $p \wedge (\sim q)$

Answer

(C) $(\sim p) \vee q$

Answer

(D) $p \vee q$

Question 14

The random variable X follows binomial distribution $B(n, p)$ for which the difference of the mean and the variance is 1. If $2P(X = 2) = 3P(X = 1)$, then $n^2 P(X > 1)$ is equal to

Accepted Answer

(C) 11

Answer

(A) 12

Answer

(B) 15

Answer

(D) 16

Question 15

Let for $A = \begin{bmatrix} 1 & 2 & 3 \ \alpha & 3 & 1 \ 1 & 1 & 2 \end{bmatrix}$, $|A| = 2$. If $\left|2\,	ext{adj}\,(2\,	ext{adj}\,(2A))ight| = 32^n$, then $3n + \alpha$ is equal to

Accepted Answer

(D) 11

Answer

(A) 10

Answer

(B) 9

Answer

(C) 12

Question 16

Let $S = \left\{ z \in \mathbb{C} : \bar{z} = i(z^2 + 	ext{Re}(\bar{z})) ight\}$. Then $\sum\limits_{z \in S} |z|^2$ is equal to

Accepted Answer

(B) $4$

Answer

(A) $\dfrac{7}{2}$

Answer

(C) $\dfrac{5}{2}$

Answer

(D) $3$

Question 17

The area of the region $\left\{(x, y) : x^2 \le y \le \left|x^2 - 4\right|,\ y \ge 1\right\}$ is

Accepted Answer

(B) $\dfrac{4}{3}\left(4\sqrt{2} - 1\right)$

Answer

(A) $\dfrac{3}{4}\left(4\sqrt{2} - 1\right)$

Answer

(C) $\dfrac{4}{3}\left(4\sqrt{2} + 1\right)$

Answer

(D) $\dfrac{3}{4}\left(4\sqrt{2} + 1\right)$

Question 18

Let for a triangle ABC, $\overrightarrow{AB} = -2\hat{i} + \hat{j} + 3\hat{k}$, $\overrightarrow{CB} = \alpha\hat{i} + \beta\hat{j} + \gamma\hat{k}$, $\overrightarrow{CA} = 4\hat{i} + 3\hat{j} + \delta\hat{k}$. If $\delta > 0$ and the area of the triangle ABC is $5\sqrt{6}$, then $\overrightarrow{CB} \cdot \overrightarrow{CA}$ is equal to

Accepted Answer

(A) 60

Answer

(B) 120

Answer

(C) 108

Answer

(D) 54

Question 19

The line, that is coplanar to the line $\dfrac{x+3}{-3} = \dfrac{y-1}{1} = \dfrac{z-5}{5}$, is

Accepted Answer

(B) $\dfrac{x+1}{-1} = \dfrac{y-2}{2} = \dfrac{z-5}{5}$

Answer

(A) $\dfrac{x+1}{1} = \dfrac{y-2}{2} = \dfrac{z-5}{5}$

Answer

(C) $\dfrac{x+1}{-1} = \dfrac{y-2}{2} = \dfrac{z-5}{4}$

Answer

(D) $\dfrac{x-1}{-1} = \dfrac{y-2}{2} = \dfrac{z-5}{5}$

Question 20

The value of $\dfrac{\displaystyle\int_{0}^{\frac{\pi}{4}} e^{-x} + \int_{0}^{\frac{\pi}{4}} e^{-x}	an^{50}x\,dx}{\displaystyle\int_{0}^{\frac{\pi}{4}} e^{-x}\left(	an^{49}x + 	an^{51}xight)dx}$ is

Accepted Answer

(A) 50

Answer

(B) 49

Answer

(C) 51

Answer

(D) 52

Question 21

The mean and standard deviation of the marks of 10 students were found to be 50 and 12 respectively. Later, it was observed that two marks 20 and 25 were wrongly read as 45 and 50 respectively. Then the correct variance is

Accepted Answer

269

Question 22

Let $A = \{-4, -3, -2, 0, 1, 3, 4\}$ and $R = \{(a, b) \in A 	imes A : b = |a| 	ext{ or } b^2 = a + 1\}$ be a relation on A. Then the minimum number of elements, that must be added to the relation R so that it becomes reflexive and symmetric, is

Accepted Answer

7

Question 23

Let $f(x) = \sum\limits_{k=1}^{10} k x^k$, $x \in \mathbb{R}$. If $2f(2) + f'(2) = 119(2)^k + 1$ then $k$ is equal to

Accepted Answer

10

Question 24

Total number of 3-digit numbers that are divisible by 6 and can be formed by using the digits 1, 2, 3, 4, 5 with repetition, is

Accepted Answer

16

Question 25

Let $[\alpha]$ denote the greatest integer $\le \alpha$. Then $\left[\sqrt{1}\right] + \left[\sqrt{2}\right] + \left[\sqrt{3}\right] + \ldots + \left[\sqrt{120}\right]$ is equal to

Accepted Answer

825

Question 26

For $x \in (-1, 1]$, the number of solutions of the equation $\sin^{-1}x = 2	an^{-1}x$ is equal to

Accepted Answer

2

Question 27

If $y = y(x)$ is the solution of the differential equation $\dfrac{dy}{dx} + \dfrac{4x}{(x^2-1)}y = \dfrac{x+2}{(x^2-1)^{\frac{5}{2}}}$, $x > 1$ such that $y(2) = \dfrac{2}{9}\log_e\left(2 + \sqrt{3}\right)$ and $y\left(\sqrt{2}\right) = \alpha\log_e\left(\sqrt{\alpha} + \beta\right) + \beta - \sqrt{\gamma}$, $\alpha, \beta, \gamma \in \mathbb{N}$, then $\alpha\beta\gamma$ is equal to

Accepted Answer

6

Question 28

The foci of a hyperbola are $(\pm 2, 0)$ and its eccentricity is $\dfrac{3}{2}$. A tangent, perpendicular to the line $2x + 3y = 6$, is drawn at a point in the first quadrant on the hyperbola. If the intercepts made by the tangent on the x- and y-axes are $a$ and $b$ respectively, then $|6a| + |5b|$ is equal to

Accepted Answer

12

Question 29

Let $f_n = \int_{0}^{\frac{\pi}{2}} \left(\sum\limits_{k=1}^{n} \sin^{k-1}x\right)\left(\sum\limits_{k=1}^{n} (2k-1)\sin^{k-1}x\right)\cos x\, dx$, $n \in \mathbb{N}$. Then $f_{21} - f_{20}$ is equal to

Accepted Answer

41

Question 30

The remainder, when $7^{103}$ is divided by 17, is

Accepted Answer

12

Question 31

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A: The binding energy per nucleon is practically independent of the atomic number for nuclei of mass number in the range 30 to 170.
Reason R: Nuclear force is short ranged.
In the light of the above statements, choose the correct answer from the options given below

Accepted Answer

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

Answer

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

Answer

(B) A is true but R is false

Answer

(C) A is false but R is true

Question 32

The output from a NAND gate having inputs A and B given below will be:

Accepted Answer

(A) (1)

Answer

(B) (2)

Answer

(C) (3)

Answer

(D) (4)

Question 33

In the network shown below, the charge accumulated in the capacitor in steady state will be:

Accepted Answer

(A) 7.2 $\mu$C

Answer

(B) 4.8 $\mu$C

Answer

(C) 10.3 $\mu$C

Answer

(D) 12 $\mu$C

Question 34

Given below are two statements:
Statement I: For a planet, if the ratio of mass of the planet to its radius increases, the escape velocity from the planet also increases.
Statement II: Escape velocity is independent of the radius of the planet. In the light of above statements, choose the most appropriate answer from the options given below

Accepted Answer

(B) Statement I is correct but Statement II is incorrect

Answer

(A) Both Statement I and Statement II are incorrect

Answer

(C) Statement I is incorrect but Statement II is correct

Answer

(D) Both Statement I and Statement II are correct

Question 35

A particle executes SHM of amplitude A. The distance from the mean position when its kinetic energy becomes equal to its potential energy is:

Accepted Answer

(C) $\dfrac{1}{\sqrt{2}}A$

Answer

(A) $\sqrt{2}A$

Answer

(B) $2A$

Answer

(D) $\dfrac{1}{2}A$

Question 36

A passenger sitting in a train A moving at 90 km/h observes another train B moving in the opposite direction for 8 s. If the velocity of the train B is 54 km/h, then length of train B is:

Accepted Answer

(D) 320 m

Answer

(A) 80 m

Answer

(B) 200 m

Answer

(C) 120 m

Question 37

The initial pressure and volume of an ideal gas are $P_0$ and $V_0$. The final pressure of the gas when the gas is suddenly compressed to volume $\dfrac{V_0}{4}$ will be: (Given $\gamma$ = ratio of specific heats at constant pressure and at constant volume)

Accepted Answer

(B) $P_0(4)^{\gamma}$

Answer

(A) $P_0(4)^{1-\gamma}$

Answer

(C) $P_0$

Answer

(D) $4P_0$

Question 38

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A: A spherical body of radius $(5 \pm 0.1)$ mm having a particular density is falling through a liquid of constant density. The percentage error in the calculation of its terminal velocity is 4%.
Reason R: The terminal velocity of the spherical body falling through the liquid is inversely proportional to its radius.
In the light of the above statements, choose the correct answer from the options given below

Accepted Answer

(D) A is true but R is false

Answer

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

Answer

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

Answer

(C) A is false but R is true

Question 39

In an electromagnetic wave, at an instant and at a particular position, the electric field is along the negative z-axis and magnetic field is along the positive x-axis. Then the direction of propagation of electromagnetic wave is:

Accepted Answer

(B) negative y-axis

Answer

(A) at 45$^\circ$ angle from positive y-axis

Answer

(C) positive z-axis

Answer

(D) positive y-axis

Question 40

The distance travelled by an object in time t is given by $s = (2.5)t^2$. The instantaneous speed of the object at $t = 5$ s will be:

Accepted Answer

(D) 25 m s$^{-1}$

Answer

(A) 12.5 m s$^{-1}$

Answer

(B) 62.5 m s$^{-1}$

Answer

(C) 5 m s$^{-1}$

Question 41

An electron is moving along the positive x-axis. If the uniform magnetic field is applied parallel to the negative z-axis, then A. The electron will experience magnetic force along positive y-axis B. The electron will experience magnetic force along negative y-axis C. The electron will not experience any force in magnetic field D. The electron will continue to move along the positive x-axis E. The electron will move along circular path in magnetic field. Choose the correct answer from the options given below:

Accepted Answer

(A) B and E only

Answer

(B) A and E only

Answer

(C) C and D only

Answer

(D) B and D only

Question 42

Two planets A and B of radii R and 1.5 R have densities $\rho$ and $\rho/2$ respectively. The ratio of acceleration due to gravity at the surface of B to A is:

Accepted Answer

(C) 3 : 4

Answer

(A) 2 : 3

Answer

(B) 2 : 1

Answer

(D) 4 : 3

Question 43

Given below are two statements:
Statement I: An AC circuit undergoes electrical resonance if it contains either a capacitor or an inductor.
Statement II: An AC circuit containing a pure capacitor or a pure inductor consumes high power due to its non-zero power factor. In the light of above statements, choose the correct answer from the options given below

Accepted Answer

(A) Both Statement I and Statement II are false

Answer

(B) Statement I is true but Statement II is false

Answer

(C) Both Statement I and Statement II are true

Answer

(D) Statement I is false but Statement II is true

Question 44

A vehicle of mass 200 kg is moving along a levelled curved road of radius 70 m with angular velocity of 0.2 rad/s. The centripetal force acting on the vehicle is:

Accepted Answer

(A) 560 N

Answer

(B) 2800 N

Answer

(C) 14 N

Answer

(D) 2240 N

Question 45

To radiate EM signal of wavelength $\lambda$ with high efficiency, the antennas should have a minimum size equal to:

Accepted Answer

(B) $\dfrac{\lambda}{4}$

Answer

(A) $\dfrac{\lambda}{2}$

Answer

(C) $2\lambda$

Answer

(D) $4\lambda$

Question 46

Given below are two statements:
Statement I: Out of microwaves, infrared rays and ultraviolet rays, ultraviolet rays are the most effective for the emission of electrons from a metallic surface.
Statement II: Above the threshold frequency, the maximum kinetic energy of photoelectrons is inversely proportional to the frequency of the incident light. In the light of above statements, choose the correct answer from the options given below

Accepted Answer

(A) Statement I is true but Statement II is false

Answer

(B) Both Statement I and Statement II are true

Answer

(C) Statement I is false but Statement II is true

Answer

(D) Both Statement I and Statement II are false

Question 47

A 10 $\mu$C charge is divided into two parts and placed at 1 cm distance so that the repulsive force between them is maximum. The charges of the two parts are:

Accepted Answer

(B) 5 $\mu$C, 5 $\mu$C

Answer

(A) 9 $\mu$C, 1 $\mu$C

Answer

(C) 7 $\mu$C, 3 $\mu$C

Answer

(D) 8 $\mu$C, 2 $\mu$C

Question 48

In the equation $\left[X + \dfrac{a}{Y^2}\right][Y - b] = RT$, $X$ is pressure, $Y$ is volume, R is universal gas constant and $T$ is temperature. The physical quantity equivalent to the ratio $\dfrac{a}{b}$ is:

Accepted Answer

(A) Energy

Answer

(B) Impulse

Answer

(C) Pressure gradient

Answer

(D) Coefficient of viscosity

Question 49

In a Young's double slit experiment, the ratio of amplitude of light coming from slits is 2:1. The ratio of the maximum to minimum intensity in the interference pattern is:

Accepted Answer

(B) 9 : 1

Answer

(A) 9 : 4

Answer

(C) 2 : 1

Answer

(D) 25 : 9

Question 50

The mean free path of molecules of a certain gas at STP is 1500d, where d is the diameter of the gas molecules. While maintaining the standard pressure, the mean free path of the molecules at 373 K is approximately:

Accepted Answer

(B) 2049d

Answer

(A) 1098d

Answer

(C) 750d

Answer

(D) 1500d

Question 51

A biconvex lens of focal length 10 cm is cut in two identical parts along a plane perpendicular to the principal axis. The power of each lens after cut is ____ D.

Accepted Answer

5

Question 52

An atom absorbs a photon of wavelength 500 nm and emits another photon of wavelength 600 nm. The net energy absorbed by the atom in this process is $n 	imes 10^{-4}$ eV. The value of n is ____. [Assume the atom to be stationary during the absorption and emission process] (Take $h = 6.6 	imes 10^{-34}$ Js and $c = 3 	imes 10^8$ m/s)

Accepted Answer

4125

Question 53

Three point charges $q$, $-2q$ and $2q$ are placed on x-axis at a distance $x = 0$, $x = \dfrac{3}{4}R$ and $x = R$ respectively from origin as shown. If $q = 2 	imes 10^{-6}$ C and R = 2 cm, the magnitude of net force experienced by the charge $-2q$ is ____ N.

Accepted Answer

5440

Question 54

In the circuit shown, the energy stored in the capacitor is $n$ $\mu$J. The value of n is ____.

Accepted Answer

75

Question 55

An insulated copper wire of 100 turns is wrapped around a wooden cylindrical core of the cross-sectional area 24 cm$^2$. The two ends of the wire are connected to a resistor. The total resistance in the circuit is 12 $\Omega$. If an externally applied uniform magnetic field in the core along its axis changes from 1.5 T in one direction to 1.5 T in the opposite direction, the charge flowing through a point in the circuit during the change of magnetic field will be ____ mC.

Accepted Answer

60

Question 56

In an experiment with sonometer when a mass of 180 g is attached to the string, it vibrates with fundamental frequency of 30 Hz. When a mass m is attached, the string vibrates with fundamental frequency of 50 Hz. The value of m is ____ g.

Accepted Answer

500

Question 57

A light rope is wound around a hollow cylinder of mass 5 kg and radius 70 cm. The rope is pulled with a force of 52.5 N. The angular acceleration of the cylinder will be ____ rad s$^{-2}$.

Accepted Answer

15

Question 58

A car accelerates from rest to u m/s. The energy spent in this process is E J. The energy required to accelerate the car from u m/s to 2u m/s is nE J. The value of n is ____.

Accepted Answer

3

Question 59

Two plates A and B have thermal conductivities 84 Wm$^{-1}$K$^{-1}$ and 126 Wm$^{-1}$K$^{-1}$ respectively. They have same surface area and same thickness. They are placed in contact along their surfaces. If the temperatures of the outer surfaces of A and B are kept at 100 °C and 0 °C respectively, then the temperature of the surface of contact in steady state is ____ °C.

Accepted Answer

40

Question 60

A straight wire AB of mass 40 g and length 50 cm is suspended by a pair of flexible leads in uniform magnetic field of magnitude 0.40 T as shown in the figure. The magnitude of the current required in the wire to remove the tension in the supporting leads is ____ A. (Take $g = 10$ ms$^{-2}$)

Accepted Answer

2

Question 61

In the wet tests for detection of various cations by precipitation, $Ba^{2+}$ cations are detected by obtaining precipitate of

Accepted Answer

(B) $BaCO_3$

Answer

(A) $Ba(ox)$: Barium oxalate

Answer

(C) $Ba(OAc)_2$

Answer

(D) $BaSO_4$

Question 62

The naturally occurring amino acid that contains only one basic functional group in its chemical structure is

Accepted Answer

(C) asparagine

Answer

(A) arginine

Answer

(B) lysine

Answer

(D) histidine

Question 63

Given below are two statements related to Ellingham diagram.
Statement-I: Ellingham diagrams can be constructed for formation of oxides, sulphides and halides of metals.
Statement-II: It consists of plots of $\Delta H^0$ vs T for formation of oxides of elements.
In the light of the above statements, choose the most appropriate answer from the options given below:

Accepted Answer

(D) Statement I is correct but Statement II is incorrect

Answer

(A) Both Statement I and Statement II are correct

Answer

(B) Statement I is incorrect but Statement II is correct

Answer

(C) Both Statement I and Statement II are incorrect

Question 64

Given below are two statements, one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A: The diameter of colloidal particles in solution should not be much smaller than wavelength of light to show Tyndall effect.
Reason R: The light scatters in all directions when the size of particles is large enough.
In the light of the above statements, choose the correct 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 true but R is false

Answer

(B) A is false but R is true

Answer

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

List I	List II
A.Weak intermolecular forces of attraction	I.Hexamethylenediamine + adipic acid
B.Hydrogen bonding	II. $AlEt_3 + TiCl_4$
C.Heavily branched polymer	III.2-chloro-1,3-butadiene
D.High density polymer	IV.Phenol + formaldehyde

List I (Reagent)	List II (Product)
A. $KOH$ (alc)	I.Nitrile
B. $KCN$ (alc)	II.Ester
C. $AgNO_2$	III.Alkene
D. $H_3CCOOAg$	IV.Nitroalkane

JEE Main 13 April 2023 Shift 2 Question Paper with Answers

Physics — JEE Main 13 April 2023 Shift 2

Chemistry — JEE Main 13 April 2023 Shift 2

Mathematics — JEE Main 13 April 2023 Shift 2

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