NTA JEE Main 31st August 2021 Shift 2

NTA JEE Main 31st August 2021 Shift 2 - Question 1

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Statement I: If three forces $$\vec{F}_1$$, $$\vec{F}_2$$ and $$\vec{F}_3$$ are represented by three sides of a triangle and $$\vec{F}_1 + \vec{F}_2 = -\vec{F}_3$$, then these three forces are concurrent forces and satisfy the condition for equilibrium.
Statement II: A triangle made up of three forces $$\vec{F}_1$$, $$\vec{F}_2$$ and $$\vec{F}_3$$ as its sides were taken in the same order, satisfies the condition for translatory equilibrium.
In the light of the above statements, choose the most appropriate answer from the options given below:

A

Both Statement I and Statement II are true.

B

Statement I is true but Statement II is false.

C

Both Statement I and Statement II are false.

D

Statement I is false but Statement II is true.

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NTA JEE Main 31st August 2021 Shift 2 - Question 2

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Statement-I: Two forces $$\vec{P} + \vec{Q}$$ and $$\vec{P} - \vec{Q}$$ where $$\vec{P} \perp \vec{Q}$$, when act at an angle $$\theta_1$$ each other, the magnitude of their resultant is $$\sqrt{3P^2 + Q^2}$$, when they act at an angle $$\theta_2$$, the magnitude of their resultant becomes $$\sqrt{2P^2 + Q^2}$$. This is possible only when $$\theta_1 < \theta_2$$.
Statement-II: In the situation given above. $$\theta_1 = 60°$$ and $$\theta_2 = 90°$$
In the light of the above statement, choose the most appropriate answer from the options given below :

A

Statement I is false but Statement II is true.

B

Both Statement I and Statement II are true.

C

Both Statement I and Statement II are false.

D

Statement I is true but Statement II is false.

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NTA JEE Main 31st August 2021 Shift 2 - Question 3

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If velocity $$V$$, time $$T$$ and force $$F$$ are chosen as the base quantities, the dimensions of the mass will be:

A

$$FVT^{-1}$$

B

$$FT^{-1}V^{-1}$$

C

$$FT^2 V$$

D

$$FTV^{-1}$$

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NTA JEE Main 31st August 2021 Shift 2 - Question 4

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A block moving horizontally on a smooth surface with a speed of 40 m s$$^{-1}$$ splits into two parts with masses in the ratio of 1 : 2. If the smaller part moves at 60 m s$$^{-1}$$ in the same direction, then the fractional change in kinetic energy is:

A

$$\frac{1}{3}$$

B

$$\frac{2}{3}$$

C

$$\frac{1}{4}$$

D

$$\frac{1}{8}$$

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NTA JEE Main 31st August 2021 Shift 2 - Question 5

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A system consists of two identical spheres each of mass 1.5 kg and radius 50 cm at the ends of a light rod. The distance between the centres of the two spheres is 5 m. What will be the moment of inertia of the system about an axis perpendicular to the rod passing through its midpoint?

A

$$1.905 \times 10^5$$ kg m$$^2$$

B

18.75 kg m$$^2$$

C

19.05 kg m$$^2$$

D

$$1.875 \times 10^5$$ kg m$$^2$$

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NTA JEE Main 31st August 2021 Shift 2 - Question 6

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If $$R_E$$ be the radius of Earth, then the ratio between the acceleration due to gravity at a depth $$r$$ below and a height $$r$$ above the earth surface is: (Given: $$r \lt R_E$$)

A

$$1 + \frac{r}{R_E} - \frac{r^2}{R_E^2} - \frac{r^3}{R_E^3}$$

B

$$1 + \frac{r}{R_E} + \frac{r^2}{R_E^2} + \frac{r^3}{R_E^3}$$

C

$$1 - \frac{r}{R_E} - \frac{r^2}{R_E^2} - \frac{r^3}{R_E^3}$$

D

$$1 + \frac{r}{R_E} - \frac{r^2}{R_E^2} + \frac{r^3}{R_E^3}$$

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NTA JEE Main 31st August 2021 Shift 2 - Question 7

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Four identical hollow cylindrical columns of mild steel support a big structure of mass $$50 \times 10^3$$ kg. The inner and outer radii of each column are 50 cm and 100 cm respectively. Assuming uniform local distribution, calculate the compression strain of each column.
[use $$Y = 2.0 \times 10^{11}$$ Pa, $$g = 9.8$$ m s$$^{-2}$$]

A

$$1.87 \times 10^{-3}$$

B

$$2.60 \times 10^{-7}$$

C

$$3.60 \times 10^{-8}$$

D

$$7.07 \times 10^{-4}$$

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NTA JEE Main 31st August 2021 Shift 2 - Question 8

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Two thin metallic spherical shells of radii $$r_1$$ and $$r_2$$ ($$r_1 < r_2$$) are placed with their centres coinciding. A material of thermal conductivity $$K$$ is filled in the space between the shells. The inner shell is maintained at temperature $$\theta_1$$ and the outer shell at temperature $$\theta_2$$ ($$\theta_1 < \theta_2$$). The rate at which heat flows radially through the material is:

A

$$\frac{K\theta_2 - \theta_1}{r_2 - r_1}$$

B

$$\frac{K(\theta_2 - \theta_1)r_2 r_1}{4\pi r_1 r_2}$$

C

$$\frac{\pi K r_1 r_2(\theta_2 - \theta_1)}{r_2 - r_1}$$

D

$$\frac{4\pi K r_1 r_2(\theta_2 - \theta_1)}{r_2 - r_1}$$

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NTA JEE Main 31st August 2021 Shift 2 - Question 9

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A mixture of hydrogen and oxygen has volume 500 cm$$^3$$, temperature 300 K, pressure 400 kPa and mass 0.76 g. The ratio of masses of oxygen to hydrogen will be:

A

16 : 3

B

3 : 8

C

8 : 3

D

3 : 16

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NTA JEE Main 31st August 2021 Shift 2 - Question 10

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A bob of mass $$m$$ suspended by a thread of length $$\ell$$ undergoes simple harmonic oscillations with time period $$T$$. If the bob is immersed in a liquid that has density $$\frac{1}{4}$$ times that of the bob and the length of the thread is increased by $$\frac{1}{3}$$rd of the original length, then the time period of the simple harmonic oscillations will be:

A

$$\frac{3}{4}T$$

B

$$\frac{4}{3}T$$

C

$$T$$

D

$$\frac{3}{2}T$$

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NTA JEE Main 31st August 2021 Shift 2 - Question 11

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For a body executing S.H.M.:
(a) Potential energy is always equal to its K.E.
(b) Average potential and kinetic energy over any given time interval are always equal.
(c) Sum of the kinetic and potential energy at any point of time is constant.
(d) Average K.E. in one time period is equal to average potential energy in one time period.
Choose the most appropriate option from the options given below:

A

(c) and (d)

B

only (c)

C

only (b)

D

(b) and (c)

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NTA JEE Main 31st August 2021 Shift 2 - Question 12

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Choose the incorrect statement:
(a) The electric lines of force entering into a Gaussian surface provide negative flux.
(b) A charge $$q$$ is placed at the centre of a cube. The flux through all the faces will be the same.
(c) In a uniform electric field net flux through a closed Gaussian surface containing no net charge, is zero.
(d) When an electric field is parallel to a Gaussian surface, it provides a finite non-zero flux.
Choose the most appropriate answer from the options given below:

A

(b) and (d) Only

B

(c) and (d) Only

C

(a) and (c) Only

D

(d) Only

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NTA JEE Main 31st August 2021 Shift 2 - Question 13

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The equivalent resistance of the given circuit between the terminals $$A$$ and $$B$$ is:

A

0 $$\Omega$$

B

$$\frac{9}{2}$$ $$\Omega$$

C

3 $$\Omega$$

D

1 $$\Omega$$

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NTA JEE Main 31st August 2021 Shift 2 - Question 14

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A current of 1.5 A is flowing through a triangle, of side 9 cm each. The magnetic field at the centroid of the triangle is:
(Assume that the current is flowing in the clockwise direction.)

A

$$2\sqrt{3} \times 10^{-5}$$ T, inside the plane of triangle

B

$$2\sqrt{3} \times 10^{-7}$$ T, outside the plane of triangle

C

$$3 \times 10^{-7}$$ T, outside the plane of triangle

D

$$3 \times 10^{-5}$$ T, inside the plane of triangle

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NTA JEE Main 31st August 2021 Shift 2 - Question 15

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A coil is placed in a magnetic field $$\vec{B}$$ as shown below:

A current is induced in the coil because $$\vec{B}$$ is:

A

parallel to the plane of coil and increasing with time

B

outward and increasing with time

C

outward and decreasing with time

D

parallel to the plane of coil and decreasing with time

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NTA JEE Main 31st August 2021 Shift 2 - Question 16

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The magnetic field vector of an electromagnetic wave is given by $$B = B_0\frac{\hat{i}+\hat{j}}{\sqrt{2}}\cos kz - \omega t$$ where $$\hat{i}$$, $$\hat{j}$$ represents unit vector along x and y-axis respectively. At $$t = 0$$ s, two electric charges $$q_1$$ of $$4\pi$$ coulomb and $$q_2$$ of $$2\pi$$ coulomb located at $$\left(0, 0, \frac{\pi}{k}\right)$$ and $$\left(0, 0, \frac{3\pi}{k}\right)$$, respectively, have the same velocity of $$0.5c\hat{i}$$, (where $$c$$ is the velocity of light). The ratio of the force acting on charge $$q_1$$ to $$q_2$$ is:

A

$$1 : \sqrt{2}$$

B

$$2\sqrt{2} : 1$$

C

$$\sqrt{2} : 1$$

D

$$2 : 1$$

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NTA JEE Main 31st August 2021 Shift 2 - Question 17

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Consider two separate ideal gases of electrons and protons having same number of particles. The temperature of both the gases are same. The ratio of the uncertainty in determining the position of an electron to that of a proton is proportional to:

A

$$\sqrt{\frac{m_p}{m_e}}$$

B

$$\frac{m_p}{m_e}$$

C

$$\frac{m_p^{3/2}}{m_e}$$

D

$$\sqrt{\frac{m_e}{m_p}}$$

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NTA JEE Main 31st August 2021 Shift 2 - Question 18

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A free electron of 2.6 eV energy collides with a H$$^+$$ ion. This results in the formation of a hydrogen atom in the first excited state and a photon is released. Find the frequency of the emitted photon. (h = $$6.6 \times 10^{-34}$$ J s)

A

$$9.0 \times 10^{27}$$ MHz

B

$$1.45 \times 10^9$$ MHz

C

$$0.19 \times 10^{15}$$ MHz

D

$$1.45 \times 10^{16}$$ MHz

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NTA JEE Main 31st August 2021 Shift 2 - Question 19

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If $$V_A$$ and $$V_B$$ are the input voltages (either 5 V or 0 V) and $$V_0$$ is the output voltage then the two gates represented in the following circuits A and B are:

A

OR and NOT Gate

B

AND and NOT Gate

C

AND and OR Gate

D

NAND and NOR Gate

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NTA JEE Main 31st August 2021 Shift 2 - Question 20

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Statement I: To get a steady DC output from the pulsating voltage received from a full wave rectifier we can connect a capacitor across the output parallel to the load $$R_L$$.
Statement II: To get a steady DC output from the pulsating voltage received from a full wave rectifier we can connect an inductor in series with $$R_L$$.
In the light of the above statements, choose the most appropriate answer from the options given below:

A

Both Statement I and Statement II are false

B

Both Statement I and Statement II are true

C

Statement I is false but Statement II is true

D

Statement I is true but Statement II is false

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NTA JEE Main 31st August 2021 Shift 2 - Question 21

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A particle is moving with constant acceleration $$a$$. Following graph shows $$v^2$$ versus $$x$$ (displacement) plot. The acceleration of the particle is _________ m s$$^{-2}$$.

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NTA JEE Main 31st August 2021 Shift 2 - Question 22

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A sample of gas with $$\gamma = 1.5$$ is taken through an adiabatic process in which the volume is compressed from 1200 cm$$^3$$ to 300 cm$$^3$$. If the initial pressure is 200 kPa. The absolute value of the workdone by the gas in the process = _________ J.

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NTA JEE Main 31st August 2021 Shift 2 - Question 23

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A parallel plate capacitor of capacitance 200$$\mu$$F is connected to a battery of 200 V. A dielectric slab of dielectric constant 2 is now inserted into the space between plates of capacitor while the battery remain connected. The change in the electrostatic energy in the capacitor will be _________ J.

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NTA JEE Main 31st August 2021 Shift 2 - Question 24

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A resistor dissipates 192 J of energy in 1 s when a current of 4 A is passed through it. Now, when the current is doubled, the amount of thermal energy dissipated in 5 s is _________ J.

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NTA JEE Main 31st August 2021 Shift 2 - Question 25

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A long solenoid with 1000 turns m$$^{-1}$$ has a core material with relative permeability 500 and volume $$10^3$$ cm$$^3$$. If the core material is replaced by another material having relative permeability of 750 with same volume maintaining same current of 0.75 A in the solenoid, the fractional change in the magnetic moment of the core would be approximately $$\frac{x}{499}$$. Find the value of $$x$$.

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NTA JEE Main 31st August 2021 Shift 2 - Question 26

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At very high frequencies, the effective impedance of the given circuit will be _________ $$\Omega$$.

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NTA JEE Main 31st August 2021 Shift 2 - Question 27

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Cross-section view of a prism is the equilateral triangle $$ABC$$ shown in the figure. The minimum deviation is observed using this prism when the angle of incidence is equal to the prism angle. The time taken by light to travel from P (midpoint of BC) to A is _________ $$\times 10^{-10}$$ s. (Given, speed of light in vacuum = $$3 \times 10^8$$ m s$$^{-1}$$ and $$\cos 30° = \frac{\sqrt{3}}{2}$$)

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NTA JEE Main 31st August 2021 Shift 2 - Question 28

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In a Young's double slit experiment, the slits are separated by 0.3 mm and the screen is 1.5 m away from the plane of slits. Distance between fourth bright fringes on both sides of central bright fringe is 2.4 cm. The frequency of light used is $$x \times 10^{14}$$ Hz.

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NTA JEE Main 31st August 2021 Shift 2 - Question 29

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A bandwidth of 6 MHz is available for A.M. transmission. If the maximum audio signal frequency used for modulating the carrier wave is not to exceed 6 kHz. The number of stations that can be broadcasted within this band simultaneously without interfering with each other will be _________.

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NTA JEE Main 31st August 2021 Shift 2 - Question 30

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The diameter of a spherical bob is measured using a vernier callipers. 9 divisions of the main scale, in the vernier callipers, are equal to 10 divisions of vernier scale. One main scale division is 1 mm. The main scale reading is 10 mm and 8th division of vernier scale was found to coincide exactly with one of the main scale division. If the given vernier callipers has positive zero error of 0.04 cm, then the radius of the bob is _________ $$\times 10^{-2}$$ cm.

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NTA JEE Main 31st August 2021 Shift 2 - Question 31

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The incorrect expression among the following is:

A

For isothermal process $$w_{reversible} = -nRT \ln\frac{V_f}{V_i}$$

B

$$\ln K = \frac{\Delta H° - T\Delta S°}{RT}$$

C

$$\frac{\Delta G_{System}}{\Delta S_{Total}} = -T$$ (at constant P)

D

$$K = e^{-\Delta G°/RT}$$

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NTA JEE Main 31st August 2021 Shift 2 - Question 32

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In which one of the following sets all species show disproportionation reaction?

A

ClO$$_4^-$$, MnO$$_4^-$$, ClO$$_2^-$$ and F$$_2$$

B

MnO$$_4^{2-}$$, ClO$$_2$$, Cl$$_2$$ and Mn$$^{3+}$$

C

Cr$$_2$$O$$_7^{2-}$$, MnO$$_4^-$$, ClO$$_2^-$$ and Cl$$_2$$

D

ClO$$_2^-$$, F$$_2$$, MnO$$_4^-$$ and Cr$$_2$$O$$_7^{2-}$$

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NTA JEE Main 31st August 2021 Shift 2 - Question 33

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Which one of the following statements is incorrect?

A

Atomic hydrogen is produced when H$$_2$$ molecules at a high temperature are irradiated with UV radiation.

B

At around 2000 K, the dissociation of dihydrogen into its atoms is nearly 8.1%

C

Bond dissociation enthalpy of H$$_2$$ is highest among diatomic gaseous molecules which contain a single bond.

D

Dihydrogen is produced on reacting zinc with HCl as well as NaOH$$(aq)$$.

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NTA JEE Main 31st August 2021 Shift 2 - Question 34

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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): Lithium salts are hydrated.
Reason (R): Lithium has higher polarising power than other alkali metal group members.
In the light of the above statements, choose the most appropriate answer from the options given below:

A

(A) is not correct but (R) is correct.

B

(A) is correct but (R) is not correct.

C

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

D

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

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NTA JEE Main 31st August 2021 Shift 2 - Question 35

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Match List - I with List - II:
List - I (Metal ion)     List - II (Group in Qualitative analysis)
(a) Mn$$^{2+}$$     (i) Group - III
(b) As$$^{3+}$$     (ii) Group - IIA
(c) Cu$$^{2+}$$     (iii) Group - IV
(d) Al$$^{3+}$$     (iv) Group - IIB
Choose the most appropriate answer from the options given below:

A

(a)-(iii), (b)-(iv), (c)-(ii), (d)-(i)

B

(a)-(iv), (b)-(ii), (c)-(iii), (d)-(i)

C

(a)-(i), (b)-(iv), (c)-(ii), (d)-(iii)

D

(a)-(i), (b)-(ii), (c)-(iii), (d)-(iv)

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NTA JEE Main 31st August 2021 Shift 2 - Question 36

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Arrange the following conformational isomers of n-butane in order of their increasing potential energy:

A

I < III < IV < II

B

I < IV < III < II

C

II < IV < III < I

D

II < III < IV < I

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NTA JEE Main 31st August 2021 Shift 2 - Question 37

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Identify correct A, B and C in the reaction sequence given below:

A

B

C

D

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NTA JEE Main 31st August 2021 Shift 2 - Question 38

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For the following sequence of reactions, the correct products are:

A

B

C

D

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NTA JEE Main 31st August 2021 Shift 2 - Question 39

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The deposition of X and Y on ground surfaces is referred as wet and dry depositions, respectively. X and Y are:

A

X = CO$$_2$$, Y = SO$$_2$$

B

X = Ammonium salts, Y = SO$$_2$$

C

X = Ammonium salts, Y = CO$$_2$$

D

X = SO$$_2$$, Y = Ammonium salts

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NTA JEE Main 31st August 2021 Shift 2 - Question 40

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Match List - I with List - II:
List - I (Parameter)                                         List - II (Unit)
(a) Cell constant                                            (i) S cm$$^2$$ mol$$^{-1}$$
(b) Molar conductivity                                   (ii) Dimensionless
(c) Conductivity                                            (iii) m$$^{-1}$$
(d) Degree of dissociation of electrolyte     (iv) $$\Omega^{-1}$$ m$$^{-1}$$
Choose the most appropriate answer:

A

(a)-(iii), (b)-(i), (c)-(iv), (d)-(ii)

B

(a)-(iii), (b)-(i), (c)-(ii), (d)-(iv)

C

(a)-(iii), (b)-(iv), (c)-(i), (d)-(ii)

D

(a)-(ii), (b)-(i), (c)-(iii), (d)-(iv)

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NTA JEE Main 31st August 2021 Shift 2 - Question 41

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The number of S = O bonds present in sulphurous acid, peroxodisulphuric acid and pyrosulphuric acid, respectively are:

A

1, 4 and 3

B

1, 4 and 4

C

2, 3 and 4

D

2, 4 and 3

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NTA JEE Main 31st August 2021 Shift 2 - Question 42

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Which one of the following correctly represents the order of stability of oxides, X$$_2$$O; (X = halogen)?

A

Cl > I > Br

B

Br > Cl > I

C

I > Cl > Br

D

Br > I > Cl

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NTA JEE Main 31st August 2021 Shift 2 - Question 43

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The Eu$$^{2+}$$ ion is a strong reducing agent in spite of its ground state electronic configuration (outermost):
[Atomic number of Eu = 63]

A

$$4f^6 6s^2$$

B

$$4f^7$$

C

$$4f^7 6s^2$$

D

$$4f^6$$

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NTA JEE Main 31st August 2021 Shift 2 - Question 44

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Spin only magnetic moment in BM of FeCO$$_4$$C$$_2$$O$$_4^+$$ is:

A

5.92

B

1

C

0

D

1.73

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NTA JEE Main 31st August 2021 Shift 2 - Question 45

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The major product of the following reaction is:

A

B

C

D

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NTA JEE Main 31st August 2021 Shift 2 - Question 46

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For the reaction given below:

The compound which is not formed as a product in the reaction is a:

A

diol

B

monocarboxylic acid

C

compound with both alcohol and acid functional groups

D

dicarboxylic acid

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NTA JEE Main 31st August 2021 Shift 2 - Question 47

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The structures of A and B formed in the following reaction are: Ph = -C$$_6$$H$$_5$$

A

B

C

D

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NTA JEE Main 31st August 2021 Shift 2 - Question 48

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The major products A and B formed in the following reaction sequence are:

A

B

C

D

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NTA JEE Main 31st August 2021 Shift 2 - Question 49

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Which among the following is not a polyester?

A

Glyptal

B

PHBV

C

Novolac

D

Dacron

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NTA JEE Main 31st August 2021 Shift 2 - Question 50

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Which of the following is NOT an example of fibrous protein?

A

Keratin

B

Albumin

C

Myosin

D

Collagen

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NTA JEE Main 31st August 2021 Shift 2 - Question 51

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The value of magnetic quantum number of the outermost electron of Zn$$^+$$ ion is _________. (Integer answer)

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NTA JEE Main 31st August 2021 Shift 2 - Question 52

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According to molecular orbital theory, the number of unpaired electron(s) in O$$_2^{2-}$$ is _________.

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NTA JEE Main 31st August 2021 Shift 2 - Question 53

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The pH of a solution obtained by mixing 50 mL of 1 M HCl and 30 mL of 1 M NaOH is $$x \times 10^{-4}$$. The value of x is _________. (Nearest integer)
log 2.5 = 0.3979

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NTA JEE Main 31st August 2021 Shift 2 - Question 54

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The transformation occurring in Duma's method is given below
$$C_2H_7N + 2x + \frac{y}{2}CuO \rightarrow xCO_2 + \frac{y}{2}H_2O + \frac{z}{2}N_2 + 2x + \frac{y}{2}Cu$$
The value of y is _________. (Integer answer)

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NTA JEE Main 31st August 2021 Shift 2 - Question 55

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The empirical formula for a compound with a cubic close packed arrangement of anions and with cations occupying all the octahedral sites in A$$_x$$B. The value of x is _________. (Integer answer)

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NTA JEE Main 31st August 2021 Shift 2 - Question 56

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Sodium oxide reacts with water to produce sodium hydroxide. 20.0 g of sodium oxide is dissolved in 500 mL of water. Neglecting the change in volume, the concentration of the resulting NaOH solution is _________ $$\times 10^{-1}$$ M. (Nearest integer)
[Atomic mass : Na = 23.0, O = 16.0, H = 1.0]

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NTA JEE Main 31st August 2021 Shift 2 - Question 57

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1.22 g of an organic acid is separately dissolved in 100 g of benzene (K$$_b$$ = 2.6 K kg mol$$^{-1}$$) and 100 g of acetone (K$$_b$$ = 1.7 K kg mol$$^{-1}$$). The acid is known to dimerize in benzene but remain as a monomer in acetone. The boiling point of the solution in acetone increases by 0.17°C. The increase in boiling point of solution in benzene in °C is $$x \times 10^{-2}$$. The value of x is _________. (Nearest integer)
[Atomic mass : C = 12.0, H = 1.0, O = 16.0]

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NTA JEE Main 31st August 2021 Shift 2 - Question 58

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For the reaction A $$\rightarrow$$ B, the rate constant k (in s$$^{-1}$$) is given by
$$\log_{10} k = 20.35 - \frac{2.47 \times 10^3}{T}$$
The energy of activation in kJ mol$$^{-1}$$ is _________. (Nearest integer)
[Given : R = 8.314 J K$$^{-1}$$ mol$$^{-1}$$]

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NTA JEE Main 31st August 2021 Shift 2 - Question 59

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CH$$_4$$ is adsorbed on 1 g charcoal at 0°C following the Freundlich adsorption isotherm. 10.0 mL of CH$$_4$$ is adsorbed at 100 mm of Hg, whereas 15.0 mL is adsorbed at 200 mm of Hg. The volume of CH$$_4$$ adsorbed at 300 mm of Hg is $$10^x$$ mL. The value of x is _________ $$\times 10^{-2}$$. (Nearest integer)
[Use $$\log_{10} 2 = 0.3010$$, $$\log_{10} 3 = 0.4771$$]

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NTA JEE Main 31st August 2021 Shift 2 - Question 60

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In the electrolytic refining of blister copper, the total number of main impurities, from the following, removed as anode mud is _________.
Pb, Sb, Se, Te, Ru, Ag, Au and Pt

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NTA JEE Main 31st August 2021 Shift 2 - Question 61

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The sum of the roots of the equation, $$x+1-2\log_2\left(3+2^x\right)+2\log_4\left(10-2^{-x}\right)=0$$, is:

A

$$\log_2 14$$

B

$$\log_2 12$$

C

$$\log_2 13$$

D

$$\log_2 11$$

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NTA JEE Main 31st August 2021 Shift 2 - Question 62

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The number of solutions of the equation $$32^{\tan^2 x} + 32^{\sec^2 x} = 81$$, $$0 \leq x \leq \frac{\pi}{4}$$ is:

A

0

B

2

C

1

D

3

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NTA JEE Main 31st August 2021 Shift 2 - Question 63

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If $$z$$ is a complex number such that $$\frac{z-i}{z-1}$$ is purely imaginary, then the minimum value of $$|z - (3 + 3i)|$$ is:

A

$$3\sqrt{2}$$

B

$$2\sqrt{2}$$

C

$$2\sqrt{2} - 1$$

D

$$6\sqrt{2}$$

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NTA JEE Main 31st August 2021 Shift 2 - Question 64

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Let $$a_1, a_2, a_3, \ldots$$ be an A.P. If $$\frac{a_1 + a_2 + \ldots + a_{10}}{a_1 + a_2 + \ldots + a_p} = \frac{100}{p^2}$$, $$p \neq 10$$, then $$\frac{a_{11}}{a_{10}}$$ is equal to:

A

$$\frac{19}{21}$$

B

$$\frac{100}{121}$$

C

$$\frac{21}{19}$$

D

$$\frac{121}{100}$$

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NTA JEE Main 31st August 2021 Shift 2 - Question 65

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Let $$A$$ be the set of all points $$\alpha, \beta$$ such that the area of triangle formed by the points $$(5, 6)$$, $$(3, 2)$$ and $$(\alpha, \beta)$$ is 12 square units. Then the least possible length of a line segment joining the origin to a point in $$A$$, is:

A

$$\frac{8}{\sqrt{5}}$$

B

$$\frac{12}{\sqrt{5}}$$

C

$$\frac{16}{\sqrt{5}}$$

D

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

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NTA JEE Main 31st August 2021 Shift 2 - Question 66

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The locus of mid-points of the line segments joining -3, -5 and the points on the ellipse $$\frac{x^2}{4} + \frac{y^2}{9} = 1$$ is:

A

$$36x^2 + 16y^2 + 90x + 56y + 145 = 0$$

B

$$36x^2 + 16y^2 + 108x + 80y + 145 = 0$$

C

$$9x^2 + 4y^2 + 18x + 8y + 145 = 0$$

D

$$36x^2 + 16y^2 + 72x + 32y + 145 = 0$$

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NTA JEE Main 31st August 2021 Shift 2 - Question 67

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If $$\alpha = \lim_{x \to \pi/4} \frac{\tan^3 x - \tan x}{\cos x + \frac{\pi}{4}}$$ and $$\beta = \lim_{x \to 0} \cos x^{\cot x}$$ are the roots of the equation, $$ax^2 + bx - 4 = 0$$, then the ordered pair $$a, b$$ is:

A

-1, 3

B

1, -3

C

1, 3

D

-1, -3

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NTA JEE Main 31st August 2021 Shift 2 - Question 68

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Negation of the statement $$(p \vee r) \Rightarrow (q \vee r)$$ is:

A

$$\sim p \wedge q \wedge \sim r$$

B

$$\sim p \wedge q \wedge r$$

C

$$p \wedge \sim q \wedge \sim r$$

D

$$p \wedge q \wedge r$$

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NTA JEE Main 31st August 2021 Shift 2 - Question 69

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The mean and variance of 7 observations are 8 and 16 respectively. If two observations are 6 and 8, then the variance of the remaining 5 observations is:

A

$$\frac{92}{5}$$

B

$$\frac{134}{5}$$

C

$$\frac{112}{5}$$

D

$$\frac{536}{25}$$

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NTA JEE Main 31st August 2021 Shift 2 - Question 70

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If $$\alpha + \beta + \gamma = 2\pi$$, then the system of equations
$$x + \cos\gamma y + \cos\beta z = 0$$
$$\cos\gamma x + y + \cos\alpha z = 0$$
$$\cos\beta x + \cos\alpha y + z = 0$$
has:

A

infinitely many solutions

B

a unique solution

C

no solution

D

exactly two solutions

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NTA JEE Main 31st August 2021 Shift 2 - Question 71

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Let $$f : N \rightarrow N$$ be a function such that $$f(m+n) = f(m) + f(n)$$ for every $$m, n \in N$$. If $$f(6) = 18$$ then $$f(2) \cdot f(3)$$ is equal to:

A

54

B

6

C

36

D

18

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NTA JEE Main 31st August 2021 Shift 2 - Question 72

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The domain of the function, $$f(x) = \sin^{-1}\frac{3x^2+x-1}{(x-1)^2} + \cos^{-1}\frac{x-1}{x+1}$$ is:

A

$$\left(0, \frac{1}{2}\right)$$

B

$$\left(0, \frac{1}{4}\right)$$

C

$$\left(\frac{1}{4}, \frac{1}{2}\right) \cup \{0\}$$

D

$$\left(-2, 0\right) \cup \left(\frac{1}{4}, \frac{1}{2}\right)$$

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NTA JEE Main 31st August 2021 Shift 2 - Question 73

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An angle of intersection of the curves, $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ and $$x^2 + y^2 = ab$$, $$a \gt b$$, is:

A

$$\tan^{-1} 2\sqrt{ab}$$

B

$$\tan^{-1}\frac{a+b}{\sqrt{ab}}$$

C

$$\tan^{-1}\frac{a-b}{\sqrt{ab}}$$

D

$$\tan^{-1}\frac{a-b}{2\sqrt{ab}}$$

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NTA JEE Main 31st August 2021 Shift 2 - Question 74

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Let $$f$$ be any continuous function on $$[0, 2]$$ and twice differentiable on $$(0, 2)$$. If $$f(0) = 0$$, $$f(1) = 1$$ and $$f(2) = 2$$, then:

A

$$f''(x) > 0$$ for all $$x \in (0, 2)$$

B

$$f'(x) = 0$$ for some $$x \in (0, 2)$$

C

$$f''(x) = 0$$ for some $$x \in (0, 2)$$

D

$$f'(x) = 0$$ for all $$x \in (0, 2)$$

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NTA JEE Main 31st August 2021 Shift 2 - Question 75

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If $$x$$ is the greatest integer $$\leq x$$, then $$\pi^2 \int_0^2 \sin\frac{\pi x}{2} x - x^{[x]} dx$$ is equal to:

A

$$2(\pi + 1)$$

B

$$4(\pi - 1)$$

C

$$2(\pi - 1)$$

D

$$4(\pi + 1)$$

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NTA JEE Main 31st August 2021 Shift 2 - Question 76

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If $$y\frac{dy}{dx} = x\frac{y^2}{x^2} + \frac{\phi\frac{y^2}{x^2}}{\phi'\frac{y^2}{x^2}}$$, $$x > 0$$, $$\phi > 0$$, and $$y(1) = -1$$, then $$\phi\frac{y^2}{4}$$ is equal to:

A

$$2\phi(1)$$

B

$$\phi(1)$$

C

$$4\phi(2)$$

D

$$4\phi(1)$$

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NTA JEE Main 31st August 2021 Shift 2 - Question 77

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If $$\frac{dy}{dx} = \frac{2^x y + 2^y \cdot 2^x}{2^x + 2^x + y\log_e 2}$$, $$y(0) = 0$$, then for $$y = 1$$, the value of $$x$$ lies in the interval:

A

(1, 2)

B

$$\left(\frac{1}{2}, 1\right)$$

C

(2, 3)

D

$$\left(0, \frac{1}{2}\right)$$

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NTA JEE Main 31st August 2021 Shift 2 - Question 78

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Let $$\vec{a}$$, $$\vec{b}$$, $$\vec{c}$$ be three vectors mutually perpendicular to each other and have same magnitude. If a vector $$\vec{r}$$ satisfies $$\vec{a} \times \{\vec{r} - \vec{b} \times \vec{a}\} + \vec{b} \times \{\vec{r} - \vec{c} \times \vec{b}\} + \vec{c} \times \{\vec{r} - \vec{a} \times \vec{c}\} = \vec{0}$$, then $$\vec{r}$$ is equal to:

A

$$\frac{1}{3}(\vec{a} + \vec{b} + \vec{c})$$

B

$$\frac{1}{3}(2\vec{a} + \vec{b} - \vec{c})$$

C

$$\frac{1}{2}(\vec{a} + \vec{b} + \vec{c})$$

D

$$\frac{1}{2}(\vec{a} + \vec{b} + 2\vec{c})$$

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NTA JEE Main 31st August 2021 Shift 2 - Question 79

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The distance of the point $$(-1, 2, -2)$$ from the line of intersection of the planes $$2x + 3y + 2z = 0$$ and $$x - 2y + z = 0$$ is:

A

$$\frac{1}{\sqrt{2}}$$

B

$$\frac{\sqrt{42}}{2}$$

C

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

D

$$\frac{\sqrt{34}}{2}$$

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NTA JEE Main 31st August 2021 Shift 2 - Question 80

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Let $$S = \{1, 2, 3, 4, 5, 6\}$$. Then the probability that a randomly chosen onto function $$g$$ from $$S$$ to $$S$$ satisfies $$g(3) = 2g(1)$$ is:

A

$$\frac{1}{15}$$

B

$$\frac{1}{5}$$

C

$$\frac{1}{30}$$

D

$$\frac{1}{10}$$

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NTA JEE Main 31st August 2021 Shift 2 - Question 81

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The number of 4-digit numbers which are neither multiple of 7 nor multiple of 3 is _________.

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NTA JEE Main 31st August 2021 Shift 2 - Question 82

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If $$S = \frac{7}{5} + \frac{9}{5^2} + \frac{13}{5^3} + \frac{19}{5^4} + \ldots$$, then $$160 S$$ is equal to _________.

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NTA JEE Main 31st August 2021 Shift 2 - Question 83

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If the coefficient of $$a^7 b^8$$ in the expansion of $$(a + 2b + 4ab)^{10}$$ is $$K \cdot 2^{16}$$, then $$K$$ is equal to _________.

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NTA JEE Main 31st August 2021 Shift 2 - Question 84

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Let $$B$$ be the centre of the circle $$x^2 + y^2 - 2x + 4y + 1 = 0$$. Let the tangents at two points $$P$$ and $$Q$$ on the circle intersect at the point $$A(3, 1)$$. Then $$8 \cdot \frac{\text{area } \triangle APQ}{\text{area } \triangle BPQ}$$ is equal to _________.

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NTA JEE Main 31st August 2021 Shift 2 - Question 85

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A tangent line $$L$$ is drawn at the point $$(2, -4)$$ on the parabola $$y^2 = 8x$$. If the line $$L$$ is also tangent to the circle $$x^2 + y^2 = a$$, then $$a$$ is equal to _________.

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NTA JEE Main 31st August 2021 Shift 2 - Question 86

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The number of elements in the set $$\{A = \begin{pmatrix} a & b \\ 0 & d \end{pmatrix} : a, b, d \in \{-1, 0, 1\}$$ and $$(I - A)^3 = I - A^3\}$$, where $$I$$ is $$2 \times 2$$ identity matrix, is _________.

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NTA JEE Main 31st August 2021 Shift 2 - Question 87

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Let $$f(x)$$ be a cubic polynomial with $$f(1) = -10$$, $$f(-1) = 6$$, and has a local minima at $$x = 1$$, and $$f'(x)$$ has a local minima at $$x = -1$$. Then $$f(3)$$ is equal to _________.

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NTA JEE Main 31st August 2021 Shift 2 - Question 88

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If $$\int \frac{\sin x}{\sin^3 x + \cos^3 x} dx = \alpha\log_e |1 + \tan x| + \beta\log_e|1 - \tan x + \tan^2 x| + \gamma\tan^{-1}\frac{2\tan x - 1}{\sqrt{3}} + C$$, when $$C$$ is constant of integration, then the value of $$18\left(\alpha+\beta+\gamma^2\right)$$ is _________.

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NTA JEE Main 31st August 2021 Shift 2 - Question 89

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If the line $$y = mx$$ bisects the area enclosed by the lines $$x = 0$$, $$y = 0$$, $$x = \frac{3}{2}$$ and the curve $$y = 1 + 4x - x^2$$, then $$12m$$ is equal to _________.

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NTA JEE Main 31st August 2021 Shift 2 - Question 90

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Suppose the line $$\frac{x-2}{\alpha} = \frac{y-2}{-5} = \frac{z+2}{2}$$ lies on the plane $$x + 3y - 2z + \beta = 0$$. Then $$(\alpha + \beta)$$ is equal to _________.

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