NTA JEE Main 28th June 2022 Shift 1

NTA JEE Main 28th June 2022 Shift 1 - Question 1

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Assertion A: Product of Pressure ($$P$$) and time ($$t$$) has the same dimension as that of coefficient of viscosity.
Reason: Coefficient of viscosity = $$\frac{\text{Force}}{\text{Velocity gradient}}$$

A

Both A and R true, and R is correct explanation of A.

B

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

C

A is true but R is false.

D

A is false but R is true.

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NTA JEE Main 28th June 2022 Shift 1 - Question 2

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Motion of a particle in $$x-y$$ plane is described by a set of following equations $$x = 4\sin\left(\frac{\pi}{2} - \omega t\right)$$ m and $$y = 4\sin(\omega t)$$ m. The path of the particle will be

A

circular

B

helical

C

parabolic

D

elliptical

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NTA JEE Main 28th June 2022 Shift 1 - Question 3

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A particle of mass $$m$$ is moving in a circular path of constant radius $$r$$ such that its centripetal acceleration $$a_c$$ is varying with time $$t$$ as $$a_c = k^2rt^2$$, where $$k$$ is a constant. The power delivered to the particle by the force acting on it is

A

Zero

B

$$2mk^2r^2t$$

C

$$mk^2r^2t$$

D

$$2mk^2rt$$

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NTA JEE Main 28th June 2022 Shift 1 - Question 4

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Match List-I with List-II

List-I	List-II
(A) M.I. of solid sphere of radius R about any tangent.	(I) $$\frac{5}{3}MR^2$$
(B) M.I. of hollow sphere of radius R about any tangent.	(II) $$\frac{7}{5}MR^2$$
(C) M.I. of circular ring of radius R about its diameter.	(III) $$\frac{1}{4}MR^2$$
(D) M.I. of circular disc of radius R about any diameter.	(IV) $$\frac{1}{2}MR^2$$

A

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

B

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

C

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

D

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

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NTA JEE Main 28th June 2022 Shift 1 - Question 5

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Two planets $$A$$ and $$B$$ of equal mass are having their period of revolutions $$T_A$$ and $$T_B$$ such that $$T_A = 2T_B$$. These planets are revolving in the circular orbits of radii $$r_A$$ and $$r_B$$ respectively. Which out of the following would be the correct relationship of their orbits?

A

$$2r_A^2 = r_B^3$$

B

$$r_A^3 = 2r_B^3$$

C

$$r_A^3 = 4r_B^3$$

D

$$T_A^2 - T_B^2 = \frac{\pi^2}{GM}(r_B^3 - 4r_A^3)$$

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NTA JEE Main 28th June 2022 Shift 1 - Question 6

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A water drop of diameter $$2$$ cm is broken into $$64$$ equal droplets. The surface tension of water is $$0.075$$ N m$$^{-1}$$. In this process the gain in surface energy will be

A

$$2.8 \times 10^{-4}$$ J

B

$$1.5 \times 10^{-3}$$ J

C

$$1.9 \times 10^{-4}$$ J

D

$$9.4 \times 10^{-5}$$ J

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NTA JEE Main 28th June 2022 Shift 1 - Question 7

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Statement - I : When $$\mu$$ amount of an ideal gas undergoes adiabatic change from state $$(P_1, V_1, T_1)$$ to state $$(P_2, V_2, T_2)$$, then work done is $$W = \frac{\mu R(T_2 - T_1)}{1-\gamma}$$, where $$\gamma = \frac{C_p}{C_v}$$ and $$R$$ = universal gas constant.
Statement - II : In the above case, when work is done on the gas, the temperature of the gas would rise.

A

Both statement-I and statement-II are true.

B

Both statement-I and statement-II are false.

C

Statement-I is true but statement-II is false.

D

Statement-I is false but statement-II is true.

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NTA JEE Main 28th June 2022 Shift 1 - Question 8

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A radar sends an electromagnetic signal of electric field $$(E_0) = 2.25$$ V m$$^{-1}$$ and magnetic field $$(B_0) = 1.5 \times 10^{-8}$$ T which strikes a target on line of sight at a distance of $$3$$ km in a medium. After that, a part of signal (echo) reflects back towards the radar with same velocity and by same path. If the signal was transmitted at time $$t = 0$$ from radar, then after how much time echo will reach to the radar?

A

$$2.0 \times 10^{-5}$$ s

B

$$4.0 \times 10^{-5}$$ s

C

$$1.0 \times 10^{-5}$$ s

D

$$8.0 \times 10^{-5}$$ s

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NTA JEE Main 28th June 2022 Shift 1 - Question 9

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The velocity of sound in a gas, in which two wavelengths $$4.08$$ m and $$4.16$$ m produce $$40$$ beats in $$12$$ s, will be

A

$$282.8$$ m s$$^{-1}$$

B

$$175.5$$ m s$$^{-1}$$

C

$$353.6$$ m s$$^{-1}$$

D

$$707.2$$ m s$$^{-1}$$

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NTA JEE Main 28th June 2022 Shift 1 - Question 10

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Statement-I : A point charge is brought in an electric field. The value of electric field at a point near to the charge may increase if the charge is positive.
Statement-II : An electric dipole is placed in a uniform electric field. The net electric force on the dipole will not be zero.

A

Both statement-I and statement-II are true.

B

Both statement-I and statement-II are false.

C

Statement-I is true but statement-II is false.

D

Statement-I is false but statement-II is true.

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NTA JEE Main 28th June 2022 Shift 1 - Question 11

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The three charges $$\frac{q}{2}, q$$ and $$\frac{q}{2}$$ are placed at the corners $$A, D$$ and $$C$$ of a square of side $$a$$ as shown in figure. The magnitude of electric field ($$E$$) at the corner $$B$$ of the square, is

A

$$\frac{q}{4\pi\varepsilon_0 a^2}\left(\frac{1}{\sqrt{2}} + \frac{1}{2}\right)$$

B

$$\frac{q}{4\pi\varepsilon_0 a^2}\left(1 + \frac{1}{\sqrt{2}}\right)$$

C

$$\frac{q}{4\pi\varepsilon_0 a^2}\left(1 - \frac{1}{\sqrt{2}}\right)$$

D

$$\frac{q}{4\pi\varepsilon_0 a^2}\left(\frac{1}{\sqrt{2}} - \frac{1}{2}\right)$$

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NTA JEE Main 28th June 2022 Shift 1 - Question 12

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An infinitely long hollow conducting cylinder with radius $$R$$ carries a uniform current along its surface. Choose the correct representation of magnetic field $$(B)$$ as a function of radial distance $$(r)$$ from the axis of cylinder.

A

B

C

D

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NTA JEE Main 28th June 2022 Shift 1 - Question 13

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The refracting angle of a prism is $$A$$ and refractive index of the material of the prism is $$\cot\left(\frac{A}{2}\right)$$. Then the angle of minimum deviation will be

A

$$180° - 2A$$

B

$$90° - A$$

C

$$180° + 2A$$

D

$$180° - 3A$$

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NTA JEE Main 28th June 2022 Shift 1 - Question 14

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The aperture of the objective is $$24.4$$ cm. The resolving power of this telescope, if a light of wavelength $$2440$$ Å is used to see the object will be

A

$$8.1 \times 10^6$$

B

$$10.0 \times 10^7$$

C

$$8.2 \times 10^5$$

D

$$1.0 \times 10^{-8}$$

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NTA JEE Main 28th June 2022 Shift 1 - Question 15

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The de Broglie wavelengths for an electron and a photon are $$\lambda_e$$ and $$\lambda_p$$ respectively. For the same kinetic energy of electron and photon, which of the following presents the correct relation between the de Broglie wavelengths of two?

A

$$\lambda_p \propto \lambda_e^2$$

B

$$\lambda_p \propto \lambda_e$$

C

$$\lambda_p \propto \sqrt{\lambda_e}$$

D

$$\lambda_p \propto \sqrt{\frac{1}{\lambda_e}}$$

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NTA JEE Main 28th June 2022 Shift 1 - Question 16

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The $$Q$$-value of a nuclear reaction and kinetic energy of the projectile particle, $$K_p$$ are related as

A

$$Q = K_p$$

B

$$(K_p + Q) < 0$$

C

$$Q < K_p$$

D

$$(K_p + Q) > 0$$

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NTA JEE Main 28th June 2022 Shift 1 - Question 17

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In the following circuit, the correct relation between output ($$Y$$) and inputs $$A$$ and $$B$$ will be

A

$$Y = A \cdot B$$

B

$$Y = A + B$$

C

$$Y = \overline{A \cdot B}$$

D

$$Y = \overline{A + B}$$

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NTA JEE Main 28th June 2022 Shift 1 - Question 18

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For using a multimeter to identify diode from electrical components, choose the correct statement out of the following about the diode

A

It is two terminal device which conducts current in both directions.

B

It is two terminal device which conducts current in one direction only

C

It does not conduct current gives an initial deflection which decays to zero.

D

It is three terminal device which conducts current in one direction only between central terminal and either of the remaining two terminals

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NTA JEE Main 28th June 2022 Shift 1 - Question 19

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Given below are two statements : One is labelled as Assertion A and the other is labelled as Reason R.
Assertion A : $$n - p - n$$ transistor permits more current than a $$p - n - p$$ transistor.
Reason R : Electrons have greater mobility as a charge carrier.

A

Both A and R true, and R is correct explanation of A.

B

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

C

A is true but R is false.

D

A is false but R is true.

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NTA JEE Main 28th June 2022 Shift 1 - Question 20

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Match List-I with List-II

List-I	List-II
(A) Television signal	(I) 03 KHz
(B) Radio signal	(II) 20 KHz
(C) High Quality Music	(III) 02 MHz
(D) Human speech	(IV) 06 MHz

A

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

B

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

C

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

D

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

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NTA JEE Main 28th June 2022 Shift 1 - Question 21

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A hanging mass $$M$$ is connected to a four times bigger mass by using a string pulley arrangement, as shown in the figure. The bigger mass is placed on a horizontal ice-slab and being pulled by $$2Mg$$ force. In this situation, tension in the string is $$\frac{x}{5}Mg$$ for $$x =$$ ______. Neglect mass of the string and friction of the block (bigger mass) with ice slab.

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NTA JEE Main 28th June 2022 Shift 1 - Question 22

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A pendulum is suspended by a string of length $$250$$ cm. The mass of the bob of the pendulum is $$200$$ g. The bob is pulled aside until the string is at $$60°$$ with vertical as shown in the figure. After releasing the bob, the maximum velocity attained by the bob will be ______ m s$$^{-1}$$. (if $$g = 10$$ m s$$^{-2}$$)

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NTA JEE Main 28th June 2022 Shift 1 - Question 23

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A man of $$60$$ kg is running on the road and suddenly jumps into a stationary trolly car of mass $$120$$ kg. Then the trolly car starts moving with velocity $$2$$ m s$$^{-1}$$. The velocity of the running man was ______ m s$$^{-1}$$, when he jumps into the car.

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NTA JEE Main 28th June 2022 Shift 1 - Question 24

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The position vector of $$1$$ kg object is $$\vec{r} = (3\hat{i} - \hat{j})$$ m and its velocity $$\vec{v} = (3\hat{j} + \hat{k})$$ m s$$^{-1}$$. The magnitude of its angular momentum is $$\sqrt{x}$$ N m s, where $$x$$ is ______

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NTA JEE Main 28th June 2022 Shift 1 - Question 25

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The total internal energy of two mole monoatomic ideal gas at temperature $$T = 300$$ K will be ______ J. (Given $$R = 8.31$$ J mol$$^{-1}$$ K$$^{-1}$$)

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NTA JEE Main 28th June 2022 Shift 1 - Question 26

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A meter bridge setup is shown in the figure. It is used to determine an unknown resistance $$R$$ using a given resistor of $$15$$ $$\Omega$$. The galvanometer $$(G)$$ shows null deflection when tapping key is at $$43$$ cm mark from end $$A$$. If the end correction for end $$A$$ is $$2$$ cm, then the determined value of $$R$$ will be ______ $$\Omega$$.

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NTA JEE Main 28th June 2022 Shift 1 - Question 27

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Current measured by the ammeter $$(A)$$ in the reported circuit when no current flows through $$10\Omega$$ resistance, will be ______ A.

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NTA JEE Main 28th June 2022 Shift 1 - Question 28

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A singly ionized magnesium atom $$(A = 24)$$ ion is accelerated to kinetic energy $$5$$ keV, and is projected perpendicularly into a magnetic field $$B$$ of the magnitude $$0.5$$ T. The radius of path formed will be ______ cm.

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NTA JEE Main 28th June 2022 Shift 1 - Question 29

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An AC source is connected to an inductance of $$100$$ mH, a capacitance of $$100$$ $$\mu$$F and a resistance of $$120$$ $$\Omega$$ as shown in figure. The time in which the resistance having a thermal capacity $$2$$ J °C$$^{-1}$$ will get heated by $$16°$$C is ______ s.

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NTA JEE Main 28th June 2022 Shift 1 - Question 30

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A telegraph line of length $$100$$ km has a capacity of $$0.01$$ $$\mu$$F km$$^{-1}$$ and it carries an alternating current at $$0.5$$ kilo cycle per second. If minimum impedance is required, then the value of the inductance that needs to be introduced in series is ______ mH. (If $$\pi = \sqrt{10}$$)

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NTA JEE Main 28th June 2022 Shift 1 - Question 31

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Element "E" belongs to the period 4 and group 16 of the periodic table. The valence shell electron configuration of the element, which is just above "E" in the group is

A

$$2s^2 2p^4$$

B

$$3d^{10}, 4s^2, 4p^4$$

C

$$3s^2 3p^4$$

D

$$4d^{10}, 5s^2, 5p^4$$

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NTA JEE Main 28th June 2022 Shift 1 - Question 32

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Which one of the following techniques is not used to spot components of a mixture separated on thin layer chromatographic plate?

A

$$I_2$$ (Solid)

B

U.V. Light

C

Visualisation agent as a component of mobile phase

D

Spraying of an appropriate reagent

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NTA JEE Main 28th June 2022 Shift 1 - Question 33

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Which of the following structures are aromatic in nature?

A

A, B, C & D

B

Only A, B & C

C

Only A & C

D

Only B, C & D

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NTA JEE Main 28th June 2022 Shift 1 - Question 34

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The formula of the purple colour formed in Lassaigne's test for sulphur using sodium nitroprusside is

A

$$Na_4[Fe(CN)_5(NOS)]$$

B

$$NaFe[Fe(CN)_6]$$

C

$$Na[Cr(NH_3)_2(NCS)_4]$$

D

$$Na_2[Fe(CN)_5(NO)]$$

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NTA JEE Main 28th June 2022 Shift 1 - Question 35

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Which amongst the following is not a pesticide?

A

DDT

B

Dieldrin

C

Organophosphate

D

Sodium Arsenite

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NTA JEE Main 28th June 2022 Shift 1 - Question 36

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The incorrect statement about the imperfections in solids is

A

Schottky defect decreases the density of the substance.

B

Frenkel defect does not alter the density of the substance.

C

Interstitial defect increases the density of the substance.

D

Vacancy defect increases the density of the substance.

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NTA JEE Main 28th June 2022 Shift 1 - Question 37

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The Zeta potential is related to which property of colloids?

A

Colour

B

Brownian movement

C

Charge on surface of colloidal particle

D

Tyndall effect

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NTA JEE Main 28th June 2022 Shift 1 - Question 38

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Given are two statements one is labelled as Assertion and other is labelled as Reason.
Assertion: Magnesium can reduce $$Al_2O_3$$ at a temperature below $$1350°$$C, while above $$1350°$$C aluminium can reduce MgO.
Reason: The melting and boiling points of magnesium are lower than those of aluminium.

A

Both Assertion and Reason are correct, and Reason is correct explanation of Assertion.

B

Both Assertion and Reason are correct, but Reason is NOT the correct explanation of Assertion.

C

Assertion is correct Reason is not correct.

D

Assertion is not correct, Reason is correct.

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NTA JEE Main 28th June 2022 Shift 1 - Question 39

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Nitrogen gas is obtained by thermal decomposition of:

A

$$NaNO_2$$

B

$$NaNO_3$$

C

$$Ba(N_3)_2$$

D

$$Ba(NO_3)_2$$

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NTA JEE Main 28th June 2022 Shift 1 - Question 40

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Given below are two statements :
Statement I: The pentavalent oxide of group-15 element, $$E_2O_5$$, is less acidic than trivalent oxide, $$E_2O_3$$, of the same element.
Statement II : The acidic character of trivalent oxide of group 15 elements, $$E_2O_3$$, decreases down the group.

A

Both Statement I and Statement II are true.

B

Both Statement I and Statement II are false.

C

Statement I is false but statement II is true.

D

Statement I true, but Statement II is false.

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NTA JEE Main 28th June 2022 Shift 1 - Question 41

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Dihydrogen reacts with CuO to give

A

$$Cu(OH)_2$$

B

$$Cu(s)$$

C

$$Cu_2O$$

D

$$CuH_2$$

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NTA JEE Main 28th June 2022 Shift 1 - Question 42

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Which one of the lanthanoids given below is the most stable in divalent form?

A

Yb (Atomic Number 70)

B

Sm (Atomic Number 62)

C

Eu (Atomic Number 63)

D

Ce (Atomic Number 58)

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NTA JEE Main 28th June 2022 Shift 1 - Question 43

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Given below are two statements :
Statement I: $$[Ni(CN)_4]^{2-}$$ is square planar and diamagnetic complex, with $$dsp^2$$ hybridization for Ni but $$[Ni(CO)_4]$$ is tetrahedral, paramagnetic and with $$sp^3$$ hybridication for Ni.
Statement II : $$[NiCl_4]^{2-}$$ and $$[Ni(CO)_4]$$ both have same d-electron configuration, have same geometry and are paramagnetic.
In light the above statements, choose the correct answer form the options given below

A

Both Statement I and Statement II are true.

B

Statement I is correct but statement II is false.

C

Statement I is incorrect but statement II is true.

D

Both Statement I and Statement II are false.

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NTA JEE Main 28th June 2022 Shift 1 - Question 44

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The major product (P) in the reaction

A

B

C

D

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NTA JEE Main 28th June 2022 Shift 1 - Question 45

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Which one of the following compounds is inactive towards $$S_N1$$ reaction?

A

B

C

D

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NTA JEE Main 28th June 2022 Shift 1 - Question 46

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The correct structure of product 'A' formed in the following reaction,

A

B

C

D

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NTA JEE Main 28th June 2022 Shift 1 - Question 47

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Identify the major product formed in the following sequence of reactions :

A

B

C

D

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NTA JEE Main 28th June 2022 Shift 1 - Question 48

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A primary aliphatic amine on reaction with nitrous acid in cold (273 K) and there after raising temperature of reaction mixture to room temperature (298 K), gives a/an

A

Alcohol

B

nitrile

C

diazonium salt

D

secondary amine

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NTA JEE Main 28th June 2022 Shift 1 - Question 49

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Which one of the following is NOT a copolymer?

A

Neoprene

B

PHBV

C

Buna-S

D

Butadiene-styrene

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NTA JEE Main 28th June 2022 Shift 1 - Question 50

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Stability of $$\alpha$$-Helix structure of proteins depends upon

A

H-bonding interaction

B

dipolar interaction

C

van der Waals forces

D

$$\pi$$-stacking interaction

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NTA JEE Main 28th June 2022 Shift 1 - Question 51

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If the work function of a metal is $$6.63 \times 10^{-19}$$ J, the maximum wavelength of the photon required to remove a photoelectron from the metal is ______ nm. Nearest integer
[Given : $$h = 6.63 \times 10^{-34}$$ J s, and $$c = 3 \times 10^{8}$$ m s$$^{-1}$$]

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NTA JEE Main 28th June 2022 Shift 1 - Question 52

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The hybridization of P exhibited in $$PF_5$$ is $$sp^x d^y$$. The value of $$y$$ is ______

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NTA JEE Main 28th June 2022 Shift 1 - Question 53

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$$4.0$$ L of an ideal gas is allowed to expand isothermally into vacuum until the total volume is $$20$$ L. The amount of heat absorbed in this expansion is ______ L atm.

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NTA JEE Main 28th June 2022 Shift 1 - Question 54

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A $$2.0$$ g sample containing $$MnO_2$$ is treated with HCl liberating $$Cl_2$$. The $$Cl_2$$ gas is passed into a solution of KI and $$60.0$$ mL of $$0.1$$ M $$Na_2S_2O_3$$ is required to titrate the liberated iodine. The percentage of $$MnO_2$$ in the sample is ______ Nearest integer
[Atomic masses (in u) Mn = 55; Cl = 35.5; O = 16, I = 127, Na = 23, K = 39, S = 32]

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NTA JEE Main 28th June 2022 Shift 1 - Question 55

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In the estimation of bromine, $$0.5$$ g of an organic compound gave $$0.40$$ g of silver bromide. The percentage of bromine in the given compound is ______ % (nearest integer)
(Relative atomic masses of Ag and Br are 108u and 80u, respectively).

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NTA JEE Main 28th June 2022 Shift 1 - Question 56

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The vapour pressures of two volatile liquids A and B at $$25°$$C are $$50$$ Torr and $$100$$ Torr, respectively. If the liquid mixture contains $$0.3$$ mole fraction of A, then the mole fraction of liquid B in the vapour phase is $$\frac{x}{17}$$. The value of $$x$$ is ______

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NTA JEE Main 28th June 2022 Shift 1 - Question 57

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The solubility product of a sparingly soluble salt $$A_2X_3$$ is $$1.1 \times 10^{-23}$$. If specific conductance of the solution is $$x \times 10^{-3}$$ S m$$^2$$ mol$$^{-1}$$. The value of $$x$$ is ______

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NTA JEE Main 28th June 2022 Shift 1 - Question 58

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The quantity of electricity in Faraday needed to reduce $$1$$ mol of $$Cr_2O_7^{2-}$$ to $$Cr^{3+}$$ is ______

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NTA JEE Main 28th June 2022 Shift 1 - Question 59

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For a first order reaction A $$\to$$ B, the rate constant, $$k = 5.5 \times 10^{-14}$$ s$$^{-1}$$. The time required for $$67\%$$ completion of reaction is $$x \times 10^{-1}$$ times the half life of reaction. The value of $$x$$ is ______ Nearest integer) (Given : $$\log 3 = 0.4771$$)

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NTA JEE Main 28th June 2022 Shift 1 - Question 60

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Number of complexes which will exhibit synergic bonding amongst, $$[Cr(CO)_6], [Mn(CO)_5]$$ and $$[Mn_2(CO)_{10}]$$ is ______

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NTA JEE Main 28th June 2022 Shift 1 - Question 61

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The total number of 5-digit numbers, formed by using the digits $$1, 2, 3, 5, 6, 7$$ without repetition, which are multiple of $$6$$, is

A

$$72$$

B

$$48$$

C

$$24$$

D

$$60$$

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NTA JEE Main 28th June 2022 Shift 1 - Question 62

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Let $$A_1, A_2, A_3, \ldots$$ be an increasing geometric progression of positive real numbers. If $$A_1 A_3 A_5 A_7 = \frac{1}{1296}$$ and $$A_2 + A_4 = \frac{7}{36}$$, then the value of $$A_6 + A_8 + A_{10}$$ is equal to

A

$$43$$

B

$$33$$

C

$$37$$

D

$$48$$

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NTA JEE Main 28th June 2022 Shift 1 - Question 63

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If $$\sum_{k=1}^{31} \left(^{31}C_{k}\right) \left(^{31}C_{k-1} \right) - \sum_{k=1}^{30} \left(^{30}C_{k}\right) \left(^{30}C_{k-1} \right)= \frac{\alpha(60!)}{(30!)(31!)}$$, where $$\alpha \in R$$, then the value of $$16\alpha$$ is equal to

A

$$1411$$

B

$$1320$$

C

$$1615$$

D

$$1855$$

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NTA JEE Main 28th June 2022 Shift 1 - Question 64

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If the tangents drawn at the point $$O(0,0)$$ and $$P(1+\sqrt{5}, 2)$$ on the circle $$x^2 + y^2 - 2x - 4y = 0$$ intersect at the point $$Q$$, then the area of the triangle $$OPQ$$ is equal to

A

$$\frac{3+\sqrt{5}}{2}$$

B

$$\frac{4+2\sqrt{5}}{2}$$

C

$$\frac{5+3\sqrt{5}}{2}$$

D

$$\frac{7+3\sqrt{5}}{2}$$

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NTA JEE Main 28th June 2022 Shift 1 - Question 65

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Let the eccentricity of the hyperbola $$H : \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ be $$\sqrt{\frac{5}{2}}$$ and length of its latus rectum be $$6\sqrt{2}$$. If $$y = 2x + c$$ is a tangent to the hyperbola $$H$$, then the value of $$c^2$$ is equal to

A

$$18$$

B

$$20$$

C

$$24$$

D

$$32$$

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NTA JEE Main 28th June 2022 Shift 1 - Question 66

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Let $$p, q, r$$ be three logical statements. Consider the compound statements
$$S_1 : ((\sim p) \vee q) \vee ((\sim p) \vee r)$$ and $$S_2 : p \to (q \vee r)$$
Then, which of the following is NOT true?

A

If $$S_2$$ is True, then $$S_1$$ is True

B

If $$S_2$$ is False, then $$S_1$$ is False

C

If $$S_2$$ is False, then $$S_1$$ is True

D

If $$S_1$$ is False, then $$S_2$$ is False

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NTA JEE Main 28th June 2022 Shift 1 - Question 67

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Let $$AB$$ and $$PQ$$ be two vertical poles, $$160$$ m apart from each other. Let $$C$$ be the middle point of $$B$$ and $$Q$$, which are feet of these two poles. Let $$\frac{\pi}{8}$$ and $$\theta$$ be the angles of elevation from $$C$$ to $$P$$ and $$A$$, respectively. If the height of pole $$PQ$$ is twice the height of pole $$AB$$, then $$\tan^2\theta$$ is equal to

A

$$\frac{3-2\sqrt{2}}{2}$$

B

$$\frac{3+\sqrt{2}}{2}$$

C

$$\frac{3-2\sqrt{2}}{4}$$

D

$$\frac{3-\sqrt{2}}{4}$$

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NTA JEE Main 28th June 2022 Shift 1 - Question 68

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Let $$A$$ be a matrix of order $$3 \times 3$$ and $$\det(A) = 2$$. Then $$\det(\det(A) \text{ adj}(5 \text{ adj}(A^3)))$$ is equal to

A

$$256 \times 10^6$$

B

$$1024 \times 10^6$$

C

$$512 \times 10^6$$

D

$$256 \times 10^{11}$$

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NTA JEE Main 28th June 2022 Shift 1 - Question 69

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If the system of linear equations
$$2x + 3y - z = -2$$
$$x + y + z = 4$$
$$x - y + |\lambda|z = 4\lambda - 4$$ where $$\lambda \in \mathbb{R}$$,
has no solution, then

A

$$\lambda = 7$$

B

$$\lambda = -7$$

C

$$\lambda = 8$$

D

$$\lambda^2 = 1$$

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NTA JEE Main 28th June 2022 Shift 1 - Question 70

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Let a function $$f : \mathbb{N} \to \mathbb{N}$$ be defined by
$$f(n) = \begin{cases} 2n, & n = 2, 4, 6, 8, \ldots \\ n-1, & n = 3, 7, 11, 15, \ldots \\ \frac{n+1}{2}, & n = 1, 5, 9, 13, \ldots \end{cases}$$
then, $$f$$ is

A

One-one and onto

B

One-one but not onto

C

Onto but not one-one

D

Neither one-one nor onto

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NTA JEE Main 28th June 2022 Shift 1 - Question 71

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Let $$f : \mathbb{R} \to \mathbb{R}$$ be defined as $$f(x) = \begin{cases} [e^x], & x < 0 \\ ae^x + [x-1], & 0 \leq x < 1 \\ b + [\sin(\pi x)], & 1 \leq x < 2 \\ [e^{-x}] - c, & x \geq 2 \end{cases}$$
where $$a, b, c \in \mathbb{R}$$ and $$[t]$$ denotes greatest integer less than or equal to $$t$$. Then, which of the following statements is true?

A

There exists $$a, b, c \in \mathbb{R}$$ such that $$f$$ is continuous on $$\mathbb{R}$$.

B

If $$f$$ is discontinuous at exactly one point, then $$a+b+c = 1$$

C

If $$f$$ is discontinuous at exactly one point, then $$a+b+c \neq 1$$.

D

$$f$$ is discontinuous at atleast two points, for any values of $$a, b$$ and $$c$$.

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NTA JEE Main 28th June 2022 Shift 1 - Question 72

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The number of real solutions of $$x^7 + 5x^3 + 3x + 1 = 0$$ is equal to ______

A

$$0$$

B

$$1$$

C

$$3$$

D

$$5$$

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NTA JEE Main 28th June 2022 Shift 1 - Question 73

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Let $$[t]$$ denote the greatest integer less than or equal to $$t$$. Then the value of the integral $$\int_0^1 [-8x^2 + 6x - 1] dx$$ is equal to

A

$$-1$$

B

$$-\frac{5}{4}$$

C

$$\frac{\sqrt{17}-13}{8}$$

D

$$\frac{\sqrt{17}-16}{8}$$

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NTA JEE Main 28th June 2022 Shift 1 - Question 74

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The area of the region $$S = \{(x,y) : y^2 \leq 8x, y \geq \sqrt{2}x, x \geq 1\}$$ is

A

$$\frac{5\sqrt{2}}{6}$$

B

$$\frac{19\sqrt{2}}{6}$$

C

$$\frac{13\sqrt{2}}{6}$$

D

$$\frac{11\sqrt{2}}{6}$$

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NTA JEE Main 28th June 2022 Shift 1 - Question 75

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Let the solution curve $$y = y(x)$$ of the differential equation, $$\left[\frac{x}{\sqrt{x^2-y^2}} + e^{y/x}\right]x\frac{dy}{dx} = x + \left[\frac{x}{\sqrt{x^2-y^2}} + e^{y/x}\right]y$$ pass through the points $$(1, 0)$$ and $$(2\alpha, \alpha), \alpha > 0$$. Then $$\alpha$$ is equal to

A

$$\frac{1}{2}\exp\left(\frac{\pi}{6} + \sqrt{e} - 1\right)$$

B

$$\frac{1}{2}\exp\left(\frac{\pi}{3} + \sqrt{e} - 1\right)$$

C

$$\exp\left(\frac{\pi}{6} + \sqrt{e} + 1\right)$$

D

$$2\exp\left(\frac{\pi}{3} + \sqrt{e} - 1\right)$$

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NTA JEE Main 28th June 2022 Shift 1 - Question 76

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Let $$y = y(x)$$ be the solution of the differential equation $$x(1 - x^2)\frac{dy}{dx} + (3x^2y - y - 4x^3) = 0, x > 1$$ with $$y(2) = -2$$. Then $$y(3)$$ is equal to

A

$$-18$$

B

$$-12$$

C

$$-6$$

D

$$-3$$

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NTA JEE Main 28th June 2022 Shift 1 - Question 77

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If two distinct point $$Q$$, $$R$$ lie on the line of intersection of the planes $$-x + 2y - z = 0$$ and $$3x - 5y + 2z = 0$$ and $$PQ = PR = \sqrt{18}$$ where the point $$P$$ is $$(1, -2, 3)$$, then the area of the triangle $$PQR$$ is equal to

A

$$\frac{2}{3}\sqrt{38}$$

B

$$\frac{4}{3}\sqrt{38}$$

C

$$\frac{8}{3}\sqrt{38}$$

D

$$\sqrt{\frac{152}{3}}$$

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NTA JEE Main 28th June 2022 Shift 1 - Question 78

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The acute angle between the planes $$P_1$$ and $$P_2$$, when $$P_1$$ and $$P_2$$ are the planes passing through the intersection of the planes $$5x + 8y + 13z - 29 = 0$$ and $$8x - 7y + z - 20 = 0$$ and the points $$(2, 1, 3)$$ and $$(0, 1, 2)$$, respectively, is

A

$$\frac{\pi}{3}$$

B

$$\frac{\pi}{4}$$

C

$$\frac{\pi}{6}$$

D

$$\frac{\pi}{12}$$

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NTA JEE Main 28th June 2022 Shift 1 - Question 79

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Let the plane $$P : \vec{r} \cdot \vec{a} = d$$ contain the line of intersection of two planes $$\vec{r} \cdot (\hat{i} + 3\hat{j} - \hat{k}) = 6$$ and $$\vec{r} \cdot (-6\hat{i} + 5\hat{j} - \hat{k}) = 7$$. If the plane $$P$$ passes through the point $$(2, 3, \frac{1}{2})$$, then the value of $$\frac{|13\vec{a}|^2}{d^2}$$ is equal to

A

$$90$$

B

$$93$$

C

$$95$$

D

$$97$$

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NTA JEE Main 28th June 2022 Shift 1 - Question 80

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The probability, that in a randomly selected 3-digit number at least two digits are odd, is

A

$$\frac{19}{36}$$

B

$$\frac{16}{36}$$

C

$$\frac{19}{33}$$

D

$$\frac{13}{36}$$

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NTA JEE Main 28th June 2022 Shift 1 - Question 81

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The number of real solutions of the equation $$e^{4x} + 4e^{3x} - 58e^{2x} + 4e^x + 1 = 0$$ is ______

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NTA JEE Main 28th June 2022 Shift 1 - Question 82

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The number of elements in the set $$\{z = a + ib \in C : a, b \in \mathbb{Z}$$ and $$1 < |z - 3 + 2i| < 4\}$$ is ______

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NTA JEE Main 28th June 2022 Shift 1 - Question 83

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The number of positive integers $$k$$ such that the constant term in the binomial expansion of $$\left(2x^3 + \frac{3}{x^k}\right)^{12}, x \neq 0$$ is $$2^8 \cdot l$$, where $$l$$ is an odd integer, is ______

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NTA JEE Main 28th June 2022 Shift 1 - Question 84

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A ray of light passing through the point $$P(2, 3)$$ reflects on the $$X$$-axis at point $$A$$ and the reflected ray passes through the point $$Q(5, 4)$$. Let $$R$$ be the point that divides the line segment $$AQ$$ internally into the ratio $$2 : 1$$. Let the co-ordinates of the foot of the perpendicular $$M$$ from $$R$$ on the bisector of the angle $$PAQ$$ be $$(\alpha, \beta)$$. Then, the value of $$7\alpha + 3\beta$$ is equal to ______

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NTA JEE Main 28th June 2022 Shift 1 - Question 85

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Let the lines $$y + 2x = \sqrt{11} + 7\sqrt{7}$$ and $$2y + x = 2\sqrt{11} + 6\sqrt{7}$$ be normal to a circle $$C : (x-h)^2 + (y-k)^2 = r^2$$. If the line $$\sqrt{11}y - 3x = \frac{5\sqrt{77}}{3} + 11$$ is tangent to the circle $$C$$, then the value of $$(5h - 8k)^2 + 5r^2$$ is equal to ______

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NTA JEE Main 28th June 2022 Shift 1 - Question 86

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The mean and standard deviation of 15 observations are found to be $$8$$ and $$3$$ respectively. On rechecking it was found that, in the observations, $$20$$ was misread as $$5$$. Then, the correct variance is equal to ______

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NTA JEE Main 28th June 2022 Shift 1 - Question 87

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Let $$R_1$$ and $$R_2$$ be relations on the set $$\{1, 2, \ldots, 50\}$$ such that $$R_1 = \{(p, p^n) : p$$ is a prime and $$n \geq 0$$ is an integer$$\}$$ and $$R_2 = \{(p, p^n) : p$$ is a prime and $$n = 0$$ or $$1\}$$. Then, the number of elements in $$R_1 - R_2$$ is ______

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NTA JEE Main 28th June 2022 Shift 1 - Question 88

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Let $$A = \{1, a_1, a_2, \ldots a_{18}, 77\}$$ be a set of integers with $$1 < a_1 < a_2 < \ldots < a_{18} < 77$$. Let the set $$A + A = \{x + y : x, y \in A\}$$ contain exactly $$39$$ elements. Then, the value of $$a_1 + a_2 + \ldots + a_{18}$$ is equal to ______

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NTA JEE Main 28th June 2022 Shift 1 - Question 89

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Let $$l$$ be a line which is normal to the curve $$y = 2x^2 + x + 2$$ at a point $$P$$ on the curve. If the point $$Q(6, 4)$$ lies on the line $$l$$ and $$O$$ is origin, then the area of the triangle $$OPQ$$ is equal to ______

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NTA JEE Main 28th June 2022 Shift 1 - Question 90

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If $$\vec{a} = 2\hat{i} + \hat{j} + 3\hat{k}, \vec{b} = 3\hat{i} + 3\hat{j} + \hat{k}$$ and $$\vec{c} = c_1\hat{i} + c_2\hat{j} + c_3\hat{k}$$ are coplanar vectors and $$\vec{a} \cdot \vec{c} = 5, \vec{b} \perp \vec{c}$$, then $$122(c_1 + c_2 + c_3)$$ is equal to ______

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