NTA JEE Main 25th June 2022 Shift 1

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

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If $$Z = \frac{A^2 B^3}{C^4}$$, then the relative error in $$Z$$ will be

A

$$\frac{2\Delta A}{A} + \frac{3\Delta B}{B} + \frac{4\Delta C}{C}$$

B

$$\frac{\Delta A}{A} + \frac{\Delta B}{B} + \frac{\Delta C}{C}$$

C

$$\frac{2\Delta A}{A} + \frac{3\Delta B}{B} - \frac{4\Delta C}{C}$$

D

$$\frac{\Delta A}{A} + \frac{\Delta B}{B} - \frac{\Delta C}{C}$$

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NTA JEE Main 25th June 2022 Shift 1 - Question 2

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$$\vec{A}$$ is a vector quantity such that $$|\vec{A}|$$ = non-zero constant. Which of the following expression is true for $$\vec{A}$$?

A

$$\vec{A} \cdot \vec{A} = 0$$

B

$$\vec{A} \times \vec{A} < 0$$

C

$$\vec{A} \times \vec{A} = 0$$

D

$$\vec{A} \times \vec{A} > 0$$

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NTA JEE Main 25th June 2022 Shift 1 - Question 3

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Which of the following relations is true for two unit vectors $$\hat{A}$$ and $$\hat{B}$$ making an angle $$\theta$$ to each other?

A

$$|\hat{A} + \hat{B}| = |\hat{A} - \hat{B}|\frac{\tan\theta}{2}$$

B

$$|\hat{A} - \hat{B}| = |\hat{A} + \hat{B}|\frac{\tan\theta}{2}$$

C

$$|\hat{A} + \hat{B}| = |\hat{A} - \hat{B}|\frac{\cos\theta}{2}$$

D

$$|\hat{A} - \hat{B}| = |\hat{A} + \hat{B}|\frac{\cos\theta}{2}$$

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NTA JEE Main 25th June 2022 Shift 1 - Question 4

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If force $$\vec{F} = 3\hat{i} + 4\hat{j} - 2\hat{k}$$ acts on a particle having position vector $$2\hat{i} + \hat{j} + 2\hat{k}$$ then, the torque about the origin will be:

A

$$-10\hat{i} + 10\hat{j} + 5\hat{k}$$

B

$$3\hat{i} + 4\hat{j} - 2\hat{k}$$

C

$$10\hat{i} + 5\hat{j} - 10\hat{k}$$

D

$$10\hat{i} + \hat{j} - 5\hat{k}$$

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NTA JEE Main 25th June 2022 Shift 1 - Question 5

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The height of any point $$P$$ above the surface of earth is equal to diameter of earth. The value of acceleration due to gravity at point $$P$$ will be : (Given $$g$$ = acceleration due to gravity at the surface of earth).

A

$$\frac{g}{2}$$

B

$$\frac{g}{4}$$

C

$$\frac{g}{3}$$

D

$$\frac{g}{9}$$

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NTA JEE Main 25th June 2022 Shift 1 - Question 6

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The terminal velocity $$v_t$$ of the spherical rain drop depends on the radius $$r$$ of the spherical rain drop as

A

$$r$$

B

$$r^2$$

C

$$\frac{1}{r}$$

D

$$\frac{1}{r^2}$$

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NTA JEE Main 25th June 2022 Shift 1 - Question 7

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The relation between root mean square speed $$v_{rms}$$ and most probable speed $$v_p$$ for the molar mass $$M$$ of oxygen gas molecule at the temperature of $$300$$ K will be

A

$$v_{rms} = \sqrt{\frac{2}{3}}v_p$$

B

$$v_{rms} = \sqrt{\frac{1}{3}}v_p$$

C

$$v_{rms} = \sqrt{\frac{3}{2}}v_p$$

D

$$v_{rms} = v_p$$

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NTA JEE Main 25th June 2022 Shift 1 - Question 8

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In the figure, a very large plane sheet of positive charge is shown. $$P_1$$ and $$P_2$$ are two points at distance $$l$$ and $$2l$$ from the charge distribution. If $$\sigma$$ is the surface charge density, then the magnitude of electric fields $$E_1$$ and $$E_2$$ at $$P_1$$ and $$P_2$$ respectively are

A

$$E_1 = \frac{\sigma}{\varepsilon_0}, E_2 = \frac{\sigma}{2\varepsilon_0}$$

B

$$E_1 = \frac{2\sigma}{\varepsilon_0}, E_2 = \frac{\sigma}{\varepsilon_0}$$

C

$$E_1 = E_2 = \frac{\sigma}{2\varepsilon_0}$$

D

$$E_1 = E_2 = \frac{\sigma}{\varepsilon_0}$$

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NTA JEE Main 25th June 2022 Shift 1 - Question 9

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A teacher in his physics laboratory allotted an experiment to determine the resistance $$G$$ a galvanometer. Students took the observations for $$\frac{1}{3}$$ deflection in the galvanometer. Which of the below is true for measuring value of $$G$$?

A

$$\frac{1}{3}$$ deflection method cannot be used for determining the resistance of the galvanometer.

B

$$\frac{1}{3}$$ deflection method can be used and in this case the $$G$$ equals to twice the value of shunt resistance(s).

C

$$\frac{1}{3}$$ deflection method can be used and in this case, the $$G$$ equals to three times the value of shunt resistance(s).

D

$$\frac{1}{3}$$ deflection method can be used and in this case the $$G$$ value equals to the shunt resistance(s).

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NTA JEE Main 25th June 2022 Shift 1 - Question 10

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A long straight wire with a circular cross-section having radius $$R$$, is carrying a steady current $$I$$. The current $$I$$ is uniformly distributed across this cross-section. Then the variation of magnetic field due to current $$I$$ with distance $$r$$ ($$r < R$$) from its centre will be

A

$$B \propto r$$

B

$$B \propto \frac{1}{r}$$

C

$$B \propto r^2$$

D

$$B \propto \frac{1}{r^2}$$

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NTA JEE Main 25th June 2022 Shift 1 - Question 11

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Match List - I with List - II.

List-I	List-II
(A) AC generator	(I) Detects the presence of current in the circuit
(B) Galvanometer	(II) Converts mechanical energy into electrical energy
(C) Transformer	(III) Works on the principle of resonance in AC circuit
(D) Metal detector	(IV) Changes an alternating voltage for smaller or greater value

A

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

B

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

C

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

D

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

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NTA JEE Main 25th June 2022 Shift 1 - Question 12

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If wattless current flows in the AC circuit, then the circuit is :

A

Purely Resistive circuit

B

Purely Inductive circuit

C

LCR series circuit

D

RC series circuit only

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NTA JEE Main 25th June 2022 Shift 1 - Question 13

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The electric field in an electromagnetic wave is given by $$E = 56.5 \sin\omega\left(\frac{t - x}{c}\right)$$ NC$$^{-1}$$. Find the intensity of the wave if it is propagating along $$x$$-axis in the free space. (Given $$\varepsilon_0 = 8.85 \times 10^{-12}$$ C$$^2$$ N$$^{-1}$$ m$$^{-2}$$)

A

$$5.65$$ Wm$$^{-2}$$

B

$$1.9 \times 10^{-7}$$ Wm$$^{-2}$$

C

$$4.24$$ Wm$$^{-2}$$

D

$$56.5$$ Wm$$^{-2}$$

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NTA JEE Main 25th June 2022 Shift 1 - Question 14

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A light wave travelling linearly in a medium of dielectric constant $$4$$, incidents on the horizontal interface separating medium with air. The angle of incidence for which the total intensity of incident wave will be reflected back into the same medium will be :
(Given : relative permeability of medium $$\mu_r = 1$$)

A

$$10°$$

B

$$20°$$

C

$$30°$$

D

$$60°$$

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NTA JEE Main 25th June 2022 Shift 1 - Question 15

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The difference of speed of light in the two media $$A$$ and $$B$$ ($$v_A - v_B$$) is $$2.6 \times 10^7$$ m s$$^{-1}$$. If the refractive index of medium $$B$$ is $$1.47$$, then the ratio of refractive index of medium $$B$$ to medium $$A$$ is: (Given : speed of light in vacuum $$c = 3 \times 10^8$$ m s$$^{-1}$$)

A

1.303

B

1.318

C

1.13

D

0.12

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NTA JEE Main 25th June 2022 Shift 1 - Question 16

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The two light beams having intensities $$I$$ and $$9I$$ interfere to produce a fringe pattern on a screen. The phase difference between the beams is $$\frac{\pi}{2}$$ at point $$P$$ and $$\pi$$ at point $$Q$$. Then the difference between the resultant intensities at $$P$$ and $$Q$$ will be :

A

$$2I$$

B

$$6I$$

C

$$5I$$

D

$$7I$$

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NTA JEE Main 25th June 2022 Shift 1 - Question 17

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Given below are two statements :
Statement I : Davisson-Germer experiment establishes the wave nature of electrons.
Statement II : If electrons have wave nature, they can interfere and show diffraction.
In the light of the above statements choose the correct answer from the option given below :

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 25th June 2022 Shift 1 - Question 18

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The ratio for the speed of the electron in the $$3^{rd}$$ orbit of $$He^+$$ to the speed of the electron in the $$3^{rd}$$ orbit of hydrogen atom will be :

A

$$1 : 1$$

B

$$1 : 2$$

C

$$4 : 1$$

D

$$2 : 1$$

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NTA JEE Main 25th June 2022 Shift 1 - Question 19

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The photodiode is used to detect the optical signals. These diodes are preferably operated in reverse biased mode because

A

fractional change in majority carriers produce higher forward bias current

B

fractional change in majority carriers produce higher reverse bias current

C

fractional change in minority carriers produce higher forward bias current

D

fractional change in minority carriers produce higher reverse bias current

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NTA JEE Main 25th June 2022 Shift 1 - Question 20

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A signal of $$100$$ THz frequency can be transmitted with maximum efficiency by

A

Coaxial cable

B

Optical fibre

C

Twisted pair of copper wires

D

Water

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NTA JEE Main 25th June 2022 Shift 1 - Question 21

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A uniform chain of $$6$$ m length is placed on a table such that a part of its length is hanging over the edge of the table. The system is at rest. The co-efficient of static friction between the chain and the surface of the table is $$0.5$$, the maximum length of the chain hanging from the table is ______ m.

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NTA JEE Main 25th June 2022 Shift 1 - Question 22

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A force on an object of mass $$100$$ g is $$(10\hat{i} + 5\hat{j})$$ N. The position of that object at $$t = 2$$ s is $$a\hat{i} + b\hat{j}$$ m after starting from rest. The value of $$\frac{a}{b}$$ will be ______.

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NTA JEE Main 25th June 2022 Shift 1 - Question 23

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A $$0.5$$ kg block moving at a speed of $$12$$ ms$$^{-1}$$ compresses a spring through a distance $$30$$ cm when its speed is halved. The spring constant of the spring will be ______ Nm$$^{-1}$$

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NTA JEE Main 25th June 2022 Shift 1 - Question 24

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The velocity of upper layer of water in a river is $$36$$ km h$$^{-1}$$. Shearing stress between horizontal layers of water is $$10^{-3}$$ N m$$^{-2}$$. Depth of the river is ______ m. (Co-efficient of viscosity of water is $$10^{-2}$$ Pa s)

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

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A steam engine intakes $$50$$ g of steam at $$100°$$C per minute and cools it down to $$20°$$C. If latent heat of vaporization of steam is $$540$$ cal g$$^{-1}$$, then the heat rejected by the steam engine per minute is ______ $$\times 10^3$$ cal
(Given : specific heat capacity of water : $$1$$ cal g$$^{-1}$$ °C$$^{-1}$$)

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NTA JEE Main 25th June 2022 Shift 1 - Question 26

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The first overtone frequency of an open organ pipe is equal to the fundamental frequency of a closed organ pipe. If the length of the closed organ pipe is $$20$$ cm. The length of the open organ pipe is ______ cm

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NTA JEE Main 25th June 2022 Shift 1 - Question 27

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The equivalent capacitance between points $$A$$ and $$B$$ in below shown figure will be ______ $$\mu$$F.

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

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A resistor develops $$300$$ J of thermal energy in $$15$$ s, when a current of $$2$$ A is passed through it. If the current increases to $$3$$ A, the energy developed in $$10$$ s is ______ J.

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NTA JEE Main 25th June 2022 Shift 1 - Question 29

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The total current supplied to the circuit as shown in the figure by the $$5$$ V battery is ______ A.

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NTA JEE Main 25th June 2022 Shift 1 - Question 30

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The current in a coil of self inductance $$L = 2.0$$ H is increasing according to the law $$i = 2\sin t^2$$. Find the amount of energy spent (in J) during the period when the current changes from $$0$$ to $$2$$ A.

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NTA JEE Main 25th June 2022 Shift 1 - Question 31

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The pair, in which ions are isoelectronic with $$Al^{3+}$$ is

A

$$Br^-$$ and $$Be^{2+}$$

B

$$Cl^-$$ and $$Li^+$$

C

$$S^{2-}$$ and $$K^+$$

D

$$O^{2-}$$ and $$Mg^{2+}$$

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NTA JEE Main 25th June 2022 Shift 1 - Question 32

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Bonding in which of the following diatomic molecule(s) become(s) stronger, on the basis of MO Theory, by removal of an electron?
(A) NO
(B) $$N_2$$
(C) $$O_2$$
(D) $$C_2$$
(E) $$B_2$$
Choose the most appropriate answer from the options given below :

A

A, B, C only

B

B, C, E only

C

A, C only

D

D only

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NTA JEE Main 25th June 2022 Shift 1 - Question 33

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Number of electron deficient molecules among the following $$PH_3, B_2H_6, CCl_4, NH_3, LiH$$ and $$BCl_3$$ is

A

0

B

1

C

2

D

3

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NTA JEE Main 25th June 2022 Shift 1 - Question 34

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Which one of the following alkaline earth metal ions has the highest ionic mobility in its aqueous solution?

A

$$Sr^{2+}$$

B

$$Mg^{2+}$$

C

$$Ca^{2+}$$

D

$$Be^{2+}$$

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NTA JEE Main 25th June 2022 Shift 1 - Question 35

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Phenol on reaction with dilute nitric acid, gives two products. Which method will be most efficient for large scale separation?

A

Chromatographic separation

B

Steam distillation

C

Fractional Crystallisation

D

Sublimation

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NTA JEE Main 25th June 2022 Shift 1 - Question 36

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In the following structures, which one is having staggered conformation with maximum dihedral angle?

A

B

C

D

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NTA JEE Main 25th June 2022 Shift 1 - Question 37

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The IUPAC name of ethylidene chloride is

A

1-chloroethene

B

1, 2-dichloroethane

C

1, 1-dichloroethane

D

1-chloroethyne

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NTA JEE Main 25th June 2022 Shift 1 - Question 38

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The product formed in the following reaction.

is:

A

B

C

D

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NTA JEE Main 25th June 2022 Shift 1 - Question 39

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The eutrophication of water body results in

A

Increase in biodiversity

B

Loss in biodiversity

C

Break down of organic matter

D

decrease in BOD.

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NTA JEE Main 25th June 2022 Shift 1 - Question 40

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Incorrect statement for Tyndall effect is

A

The refractive indices of the dispersed phase and the dispersion medium differ greatly in magnitude.

B

The diameter of the dispersed particles is much smaller than the wavelength of the light used.

C

It is used to distinguish a true solution from a colloidal solution.

D

During projection of movies in the cinemas hall, Tyndall effect is noticed.

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NTA JEE Main 25th June 2022 Shift 1 - Question 41

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Leaching of gold with dilute aqueous solution of NaCN in presence of oxygen gives complex A, which on reaction with zinc forms the elemental gold and another complex B. A and B, respectively are

A

$$AuCN_2^-$$ and $$ZnCN_4^{2-}$$

B

$$AuCN_4^-$$ and $$ZnCN_2OH_2^{2+}$$

C

$$AuCN_2^-$$ and $$ZnOH_4^{2-}$$

D

$$AuCN_4^{2-}$$ and $$ZnCN_6^{4-}$$

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NTA JEE Main 25th June 2022 Shift 1 - Question 42

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Cerium IV has a noble gas configuration. Which of the following is the correct statement about it?

A

It will prefer to gain electron and act as an oxidizing agent

B

It will prefer to give away an electron and behave as reducing agent

C

It will not prefer to undergo redox reactions.

D

It acts as both, oxidizing and reducing agent.

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NTA JEE Main 25th June 2022 Shift 1 - Question 43

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Among the following, which is the strongest oxidizing agent?

A

$$Mn^{3+}$$

B

$$Ti^{3+}$$

C

$$Fe^{3+}$$

D

$$Cr^{3+}$$

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NTA JEE Main 25th June 2022 Shift 1 - Question 44

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White precipitate of AgCl dissolves in aqueous ammonia solution due to formation of

A

$$[Ag(NH_3)_4]Cl_2$$

B

$$AgCl_2(NH_3)_2$$

C

$$[Ag(NH_3)_2]Cl$$

D

$$[Ag(NH_3)]ClCl$$

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NTA JEE Main 25th June 2022 Shift 1 - Question 45

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The major product in the reaction

A

$$t$$-Butyl ethyl ether

B

2-Methyl pent-1-ene

C

2, 2-Dimethyl butane

D

2-Methyl prop-1-ene

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NTA JEE Main 25th June 2022 Shift 1 - Question 46

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The intermediate X, in the reaction

A

B

C

D

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NTA JEE Main 25th June 2022 Shift 1 - Question 47

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In the following reaction :

The compounds A and B respectively are

A

B

C

D

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NTA JEE Main 25th June 2022 Shift 1 - Question 48

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The reaction of 

with bromine and KOH gives $$RNH_2$$ as the end product. Which one of the following is the intermediate product formed in this reaction?

A

B

$$R-NH-Br$$

C

$$R-N=C=O$$

D

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NTA JEE Main 25th June 2022 Shift 1 - Question 49

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Using very little soap while washing clothes, does not serve the purpose of cleaning of clothes, because

A

soap particles remain floating in water as ions.

B

colloidal structure of soap in water is completely disturbed.

C

the hydrophobic part of soap is not able to take away grease.

D

the micelles are not formed due to concentration of soap, below its CMC value.

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NTA JEE Main 25th June 2022 Shift 1 - Question 50

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Which one of the following is an example of artificial sweetner?

A

Bithional

B

Alitame

C

Lactose

D

Salvarsan

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NTA JEE Main 25th June 2022 Shift 1 - Question 51

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The number of N atoms in $$681$$ g of $$C_7H_5N_3O_6$$ is $$x \times 10^{21}$$. The value of $$x$$ is ______ (Nearest Integer)
$$N_A = 6.02 \times 10^{23}$$ mol$$^{-1}$$

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NTA JEE Main 25th June 2022 Shift 1 - Question 52

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$$1$$ L aqueous solution of $$H_2SO_4$$ contains $$0.02$$ m mol $$H_2SO_4$$. $$50\%$$ of this solution is diluted with deionized water to give $$1$$ L solution A. In solution A, $$0.01$$ m mol of $$H_2SO_4$$ are added. Total m mols of $$H_2SO_4$$ in the final solution is ______ $$\times 10^{-3}$$ m moles.

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NTA JEE Main 25th June 2022 Shift 1 - Question 53

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Number of grams of bromine that will completely react with $$5.0$$ g of pent-1-ene is ______ $$\times 10^{-2}$$ g. (Atomic mass of Br $$= 80$$ g/mol) [Nearest integer]

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NTA JEE Main 25th June 2022 Shift 1 - Question 54

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The longest wavelength of light that can be used for the ionisation of lithium ion $$Li^{2+}$$ is $$x \times 10^{-8}$$ m. The value of $$x$$ is ______ (Nearest Integer)
(Given : Energy of the electron in the first shell of the hydrogen atom is $$-2.2 \times 10^{-18}$$ J; $$h = 6.63 \times 10^{-34}$$ Js and $$c = 3 \times 10^{8}$$ ms$$^{-1}$$)

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NTA JEE Main 25th June 2022 Shift 1 - Question 55

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The standard entropy change for the reaction
$$4Fe(s) + 3O_2(g) \to 2Fe_2O_3(s)$$ is $$-550$$ J K$$^{-1}$$ at $$298$$ K
[Given : The standard enthalpy change for the reaction is $$-165$$ kJ mol$$^{-1}$$]. The temperature in K at which the reaction attains equilibrium is (Nearest Integer) ______

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NTA JEE Main 25th June 2022 Shift 1 - Question 56

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The standard free energy change $$\Delta G°$$ for $$50\%$$ dissociation of $$N_2O_4$$ into $$NO_2$$ at $$27°$$C and $$1$$ atm pressure is $$-x$$ J mol$$^{-1}$$. The value of $$x$$ is ______ J. (Nearest Integer)
[Given : $$R = 8.31$$ J K$$^{-1}$$ mol$$^{-1}$$, $$\log 1.33 = 0.1239$$, $$\ln 10 = 2.3$$]

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NTA JEE Main 25th June 2022 Shift 1 - Question 57

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The distance between $$Na^+$$ and $$Cl^-$$ ions in solid NaCl of density $$43.1$$ g cm$$^{-3}$$ is ______ $$\times 10^{-10}$$ m. (Nearest Integer)
(Given : $$N_A = 6.02 \times 10^{23}$$ mol$$^{-1}$$)

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NTA JEE Main 25th June 2022 Shift 1 - Question 58

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In a cell, the following reactions take place
$$Fe^{2+} \to Fe^{3+} + e^-$$ $$E^\circ_{Fe^{3+}/Fe^{2+}} = 0.77$$ V
$$2I^- \to I_2 + 2e^-$$ $$E^\circ_{I_2/I^-} = 0.54$$ V
The standard electrode potential for the spontaneous reaction in the cell is $$x \times 10^{-2}$$ V at $$298$$ K. The value of $$x$$ is ______ (Nearest Integer)

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NTA JEE Main 25th June 2022 Shift 1 - Question 59

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For a given chemical reaction $$\gamma_1 A + \gamma_2 B \to \gamma_3 C + \gamma_4 D$$. Concentration of C changes from $$10$$ mmol dm$$^{-3}$$ to $$20$$ mmol dm$$^{-3}$$ in $$10$$ s. Rate of appearance of D is $$1.5$$ times the rate of disappearance of B which is twice the rate of disappearance of A. The rate of appearance of D has been experimentally determined to be $$9$$ mmol dm$$^{-3}$$ s$$^{-1}$$. Therefore the rate of reaction is ______ mmol dm$$^{-3}$$ s$$^{-1}$$. (Nearest Integer)

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NTA JEE Main 25th June 2022 Shift 1 - Question 60

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If $$[CuH_2O_4]^{2+}$$ absorbs a light of wavelength $$600$$ nm for d-d transition, then the value of octahedral crystal field splitting energy for $$[CuH_2O_6]^{2+}$$ will be ______ $$\times 10^{-21}$$ J [Nearest integer]
(Given : $$h = 6.63 \times 10^{-34}$$ Js and $$c = 3.08 \times 10^{8}$$ ms$$^{-1}$$)

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NTA JEE Main 25th June 2022 Shift 1 - Question 61

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Let a circle $$C$$ in complex plane pass through the points $$z_1 = 3 + 4i, z_2 = 4 + 3i$$ and $$z_3 = 5i$$. If $$z \neq z_1$$ is a point on $$C$$ such that the line through $$z$$ and $$z_1$$ is perpendicular to the line through $$z_2$$ and $$z_3$$, then $$\arg z$$ is equal to

A

$$\tan^{-1}\frac{24}{7} - \pi$$

B

$$\tan^{-1}\frac{2}{\sqrt{5}} - \pi$$

C

$$\tan^{-1}3 - \pi$$

D

$$\tan^{-1}\frac{3}{4} - \pi$$

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NTA JEE Main 25th June 2022 Shift 1 - Question 62

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If $$\frac{1}{2 \cdot 3^{10}} + \frac{1}{2^2 \cdot 3^9} + \cdots + \frac{1}{2^{10} \cdot 3} = \frac{K}{2^{10} \cdot 3^{10}}$$, then the remainder when $$K$$ is divided by $$6$$ is

A

2

B

3

C

4

D

5

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NTA JEE Main 25th June 2022 Shift 1 - Question 63

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Let a circle $$C$$ touch the lines $$L_1: 4x - 3y + K_1 = 0$$ and $$L_2: 4x - 3y + K_2 = 0$$, $$K_1, K_2 \in R$$. If a line passing through the centre of the circle $$C$$ intersects $$L_1$$ at $$(-1, 2)$$ and $$L_2$$ at $$(3, -6)$$, then the equation of the circle $$C$$ is

A

$$(x-1)^2 + (y-2)^2 = 4$$

B

$$(x-1)^2 + (y+2)^2 = 16$$

C

$$(x+1)^2 + (y-2)^2 = 4$$

D

$$(x-1)^2 + (y-2)^2 = 16$$

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NTA JEE Main 25th June 2022 Shift 1 - Question 64

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If $$y = m_1 x + c_1$$ and $$y = m_2 x + c_2$$, $$m_1 \neq m_2$$ are two common tangents of circle $$x^2 + y^2 = 2$$ and parabola $$y^2 = x$$, then the value of $$8|m_1 m_2|$$ is equal to

A

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

B

$$6\sqrt{2} - 4$$

C

$$-5 + 6\sqrt{2}$$

D

$$3 + 4\sqrt{2}$$

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NTA JEE Main 25th June 2022 Shift 1 - Question 65

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Let $$x = 2t, y = \frac{t^2}{3}$$ be a conic. Let $$S$$ be the focus and $$B$$ be the point on the axis of the conic such that $$SA \perp BA$$, where $$A$$ is any point on the conic. If $$k$$ is the ordinate of the centroid of the $$\triangle SAB$$, then $$\lim_{t \to 1} k$$ is equal to

A

$$\frac{17}{18}$$

B

$$\frac{19}{18}$$

C

$$\frac{11}{18}$$

D

$$\frac{13}{18}$$

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NTA JEE Main 25th June 2022 Shift 1 - Question 66

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Let $$f(x)$$ be a polynomial function such that $$f(x) + f'(x) + f''(x) = x^5 + 64$$. Then, the value of $$\lim_{x \to 1} \frac{f(x)}{x - 1}$$ is equal to

A

$$-15$$

B

$$15$$

C

$$-60$$

D

$$60$$

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NTA JEE Main 25th June 2022 Shift 1 - Question 67

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Consider the following two propositions :
$$P_1:\sim(p\to\sim q)$$

$$P_2:(p\land\sim q)\land(\sim p\lor q)$$
If the proposition $$p\to(\sim p\lor q) $$ is evaluated as FALSE, then

A

$$P_1$$ is TRUE and $$P_2$$ is FALSE

B

$$P_1$$ is FALSE and $$P_2$$ is TRUE

C

Both $$P_1$$ and $$P_2$$ are FALSE

D

Both $$P_1$$ and $$P_2$$ are TRUE

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NTA JEE Main 25th June 2022 Shift 1 - Question 68

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Let $$a, b$$ and $$c$$ be the length of sides of a triangle $$ABC$$ such that $$\frac{a+b}{7} = \frac{b+c}{8} = \frac{c+a}{9}$$. If $$r$$ and $$R$$ are the radius of incircle and radius of circumcircle of the triangle $$ABC$$, respectively, then the value of $$\frac{R}{r}$$ is equal to

A

$$2$$

B

$$\frac{3}{5}$$

C

$$\frac{5}{2}$$

D

$$1$$

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NTA JEE Main 25th June 2022 Shift 1 - Question 69

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Let $$A = \begin{pmatrix} 0 & -2 \\ 2 & 0 \end{pmatrix}$$. If $$M$$ and $$N$$ are two matrices given by $$M = \sum_{k=1}^{10} A^{2k}$$ and $$N = \sum_{k=1}^{10} A^{2k-1}$$ then $$MN^2$$ is

A

a non-identity symmetric matrix

B

a skew-symmetric matrix

C

neither symmetric nor skew-symmetric matrix

D

an identity matrix

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NTA JEE Main 25th June 2022 Shift 1 - Question 70

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Let $$A$$ be a $$3\times 3$$ real matrix such that

$$A\begin{pmatrix}1\\1\\0\end{pmatrix}=\begin{pmatrix}1\\1\\0\end{pmatrix},\qquad A\begin{pmatrix}1\\0\\1\end{pmatrix}=\begin{pmatrix}-1\\0\\1\end{pmatrix},\qquad A\begin{pmatrix}0\\0\\1\end{pmatrix}=\begin{pmatrix}1\\1\\2\end{pmatrix}.$$

If $$X=(x_1,x_2,x_3)^T$$ and $$I$$ is an identity matrix of order $$3$$, then the system

$$(A-2I)X=\begin{pmatrix}4\\1\\1\end{pmatrix}$$

has ____________.

A

no solution

B

infinitely many solutions

C

unique solution

D

exactly two solutions

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NTA JEE Main 25th June 2022 Shift 1 - Question 71

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Let $$f : N \to R$$ be a function such that $$f(x + y) = 2f(x)f(y)$$ for natural numbers $$x$$ and $$y$$. If $$f(1) = 2$$, then the value of $$\alpha$$ for which $$\sum_{k=1}^{10} f(\alpha + k) = \frac{512}{3}2^{20} - 1$$ holds, is

A

$$3$$

B

$$4$$

C

$$5$$

D

$$6$$

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NTA JEE Main 25th June 2022 Shift 1 - Question 72

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Let $$f : R \to R$$ be defined as $$f(x) = x^3 + x - 5$$. If $$g(x)$$ is a function such that $$f(g(x)) = x, \forall x \in R$$, then $$g'(63)$$ is equal to

A

$$49$$

B

$$\frac{1}{49}$$

C

$$\frac{43}{49}$$

D

$$\frac{3}{49}$$

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NTA JEE Main 25th June 2022 Shift 1 - Question 73

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Let $$f : R \to R$$ and $$g : R \to R$$ be two functions defined by $$f(x) = \log_e(x^2 + 1) - e^{-x} + 1$$ and $$g(x) = \frac{1 - 2e^{2x}}{e^x}$$. Then, for which of the following range of $$\alpha$$, the inequality $$f\left(g\left(\frac{\left(\alpha-1\right)^2}{3}\right)\right)>f\left(g\left(\alpha-\frac{5}{3}\right)\right)$$ holds?

A

$$(-2, -1)$$

B

$$(2, 3)$$

C

$$(1, 2)$$

D

$$(-1, 1)$$

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NTA JEE Main 25th June 2022 Shift 1 - Question 74

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Let $$g : (0, \infty) \to R$$ be a differentiable function such that $$\int (\frac{x(\cos x - \sin x)}{e^x + 1} + \frac{g(x)(e^x + 1 - xe^x)}{(e^x + 1)^2} )dx = \frac{xg(x)}{e^x + 1} + C$$, for all $$x > 0$$, where $$C$$ is an arbitrary constant. Then

A

$$g$$ is decreasing in $$\left(0, \frac{\pi}{4}\right)$$

B

$$g - g'$$ is increasing in $$\left(0, \frac{\pi}{2}\right)$$

C

$$g'$$ is increasing in $$\left(0, \frac{\pi}{4}\right)$$

D

$$g + g'$$ is increasing in $$\left(0, \frac{\pi}{2}\right)$$

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NTA JEE Main 25th June 2022 Shift 1 - Question 75

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Evaluate

$$I=\int_0^{\pi}\frac{e^{\cos x}\sin x}{(1+\cos^2x)e^{\cos x}+e^{-\cos x}}\,dx.$$

A

$$\frac{\pi^2}{4}$$

B

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

C

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

D

$$\frac{\pi^2}{6}$$

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NTA JEE Main 25th June 2022 Shift 1 - Question 76

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Let $$y = y(x)$$ be the solution of the differential equation $$(x + 1)y' - y = e^{3x}(x + 1)^2$$, with $$y(0) = \frac{1}{3}$$. Then, the point $$x = -\frac{4}{3}$$ for the curve $$y = y(x)$$ is

A

not a critical point

B

a point of local minima

C

a point of local maxima

D

a point of inflection

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NTA JEE Main 25th June 2022 Shift 1 - Question 77

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If the solution curve $$y = y(x)$$ of the differential equation $$y^2 dx + (x^2 - xy + y^2)dy = 0$$, which passes through the point $$(1, 1)$$ and intersects the line $$y = \sqrt{3}x$$ at the point $$(\alpha, \sqrt{3}\alpha)$$, then value of $$\log_e \sqrt{3}\alpha$$ is equal to

A

$$\frac{\pi}{2}$$

B

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

C

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

D

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

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NTA JEE Main 25th June 2022 Shift 1 - Question 78

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Let $$\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}, a_i > 0, i = 1, 2, 3$$ be a vector which makes equal angles with the coordinate axes $$OX, OY$$ and $$OZ$$. Also, let the projection of $$\vec{a}$$ on the vector $$3\hat{i} + 4\hat{j}$$ be $$7$$. Let $$\vec{b}$$ be a vector obtained by rotating $$\vec{a}$$ with $$90°$$. If $$\vec{a}, \vec{b}$$ and x-axis are coplanar, then projection of a vector $$\vec{b}$$ on $$3\hat{i} + 4\hat{j}$$ is equal to

A

$$\sqrt{7}$$

B

$$\sqrt{2}$$

C

$$2$$

D

$$7$$

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NTA JEE Main 25th June 2022 Shift 1 - Question 79

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Let $$Q$$ be the mirror image of the point $$P(1, 0, 1)$$ with respect to the plane $$S : x + y + z = 5$$. If a line $$L$$ passing through $$(1, -1, -1)$$, parallel to the line $$PQ$$ meets the plane $$S$$ at $$R$$, then $$QR^2$$ is equal to

A

$$2$$

B

$$5$$

C

$$7$$

D

$$11$$

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NTA JEE Main 25th June 2022 Shift 1 - Question 80

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Let $$E_1$$ and $$E_2$$ be two events such that the conditional probabilities $$P(E_1 | E_2) = \frac{1}{2}$$, $$P(E_2 | E_1) = \frac{3}{4}$$ and $$P(E_1 \cap E_2) = \frac{1}{8}$$. Then

A

$$P(E_1 \cap E_2) = P(E_1) \cdot P(E_2)$$

B

$$P(E_1' \cap E_2') = P(E_1') \cdot P(E_2)$$

C

$$P(E_1 \cap E_2') = P(E_1) \cdot P(E_2)$$

D

$$P(E_1 \cup E_2) = P(E_1) P(E_2)$$

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NTA JEE Main 25th June 2022 Shift 1 - Question 81

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For a natural number $$n$$, let $$\alpha_n = 19^n - 12^n$$. Then, the value of $$\frac{31\alpha_9 - \alpha_{10}}{57\alpha_8}$$ is ______.

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NTA JEE Main 25th June 2022 Shift 1 - Question 82

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The number of 3-digit odd numbers, whose sum of digits is a multiple of $$7$$, is ______.

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NTA JEE Main 25th June 2022 Shift 1 - Question 83

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The greatest integer less than or equal to the sum of first $$100$$ terms of the sequence $$\frac{1}{3}, \frac{5}{9}, \frac{19}{27}, \frac{65}{81}, \ldots$$ is equal to ______.

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NTA JEE Main 25th June 2022 Shift 1 - Question 84

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Let $$C_r$$ denote the binomial coefficient of $$x^r$$ in the expansion of $$(1 + x)^{10}$$. If for $$\alpha, \beta \in R$$,
$$C_{1} + 3 \cdot 2C_{2} + 5 \cdot 3C_{3} + \ldots $$ upto 10 terms $$= \frac{\alpha \times 2^{11}}{2^{\beta} - 1}(C_{0} + \frac{C_{1}}{2} + \frac{C_{2}}{3} + \ldots $$ upto 10 terms ) then the value of $$\alpha + \beta $$ is equal to ______.

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NTA JEE Main 25th June 2022 Shift 1 - Question 85

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The number of values of $$x$$ in the interval $$\left[\frac{\pi}{4}, \frac{7\pi}{4}\right]$$ for which $$14\cosec^2 x - 2\sin^2 x = 21 - 4\cos^2 x$$ holds, is ______.

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NTA JEE Main 25th June 2022 Shift 1 - Question 86

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Let the abscissae of the two points $$P$$ and $$Q$$ be the roots of $$2x^2 - rx + p = 0$$ and the ordinates of $$P$$ and $$Q$$ be the roots of $$x^2 - sx - q = 0$$. If the equation of the circle described on $$PQ$$ as diameter is $$2(x^2 + y^2) - 11x - 14y - 22 = 0$$, then $$2r + s - 2q + p$$ is equal to ______.

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NTA JEE Main 25th June 2022 Shift 1 - Question 87

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Let $$A$$ be a $$3 \times 3$$ matrix having entries from the set $$\{-1, 0, 1\}$$. The number of all such matrices $$A$$ having sum of all the entries equal to $$5$$, is ______.

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NTA JEE Main 25th June 2022 Shift 1 - Question 88

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Let $$f:\mathbb{R}\to\mathbb{R}$$ be a function defined by $$f(x)=\left(2\left(1-\frac{x^{25}}{2}\right)(2+x^{25})\right)^{\frac{1}{50}}.$$ If the function $$g(x)=f(f(f(x)))+f(f(x)),$$ then the greatest integer less than or equal to $$g(1)$$ is ____________.

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NTA JEE Main 25th June 2022 Shift 1 - Question 89

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Let $$\theta$$ be the angle between the vectors $$\vec{a}$$ and $$\vec{b}$$, where $$|\vec{a}| = 4, |\vec{b}| = 3$$ and $$\theta \in \left[\frac{\pi}{4}, \frac{\pi}{3}\right]$$. Then $$|\vec{a} - \vec{b} \times \vec{a} + \vec{b}|^2 + 4|\vec{a} \cdot \vec{b}|^2$$ is equal to ______.

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NTA JEE Main 25th June 2022 Shift 1 - Question 90

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Let the lines $$L_1 : \vec{r} = \lambda(\hat{i} + 2\hat{j} + 3\hat{k}), \lambda \in R$$ and $$L_2 : \vec{r} = \hat{i} + 3\hat{j} + \hat{k} + \mu(\hat{i} + \hat{j} + 5\hat{k}); \mu \in R$$, intersect at the point $$S$$. If a plane $$ax + by - z + d = 0$$ passes through $$S$$ and is parallel to the lines $$L_1$$ and $$L_2$$, then the value of $$a + b + d$$ is equal to ______.

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