NTA JEE Main 3rd September 2020 Shift 2

NTA JEE Main 3rd September 2020 Shift 2 - Question 1

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Amount of solar energy received on the earth's surface per unit area per unit time is defined a solar constant. Dimension of solar constant is:

A

$$ML^2T^{-2}$$

B

$$ML^0T^{-3}$$

C

$$M^2L^0T^{-1}$$

D

$$MLT^{-2}$$

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NTA JEE Main 3rd September 2020 Shift 2 - Question 2

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A particle is moving unidirectional on a horizontal plane under the action of a constant power supplying energy source. The displacement (s) - time (t) graph that describes the motion of the particle is (graphs are drawn schematically and are not to scale):

A

B

C

D

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NTA JEE Main 3rd September 2020 Shift 2 - Question 3

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Hydrogen ion and singly ionized helium atom are accelerated, from rest, through the same potential difference. The ratio of final speeds of hydrogen and helium ions is close to:

A

$$1 : 2$$

B

$$10 : 7$$

C

$$2 : 1$$

D

$$5 : 7$$

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NTA JEE Main 3rd September 2020 Shift 2 - Question 4

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A block of mass 1.9 kg is at rest at the edge of a table, of height 1 m. A bullet of mass 0.1 kg collides with the block and sticks to it. If the velocity of the bullet is 20 m s$$^{-1}$$ in the horizontal direction just before the collision then the kinetic energy just before the combined system strikes the floor, is [Take g = 10 m s$$^{-2}$$. Assume there is no rotational motion and loss of energy after the collision is negligible.]

A

21 J

B

20 J

C

19 J

D

23 J

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NTA JEE Main 3rd September 2020 Shift 2 - Question 5

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A uniform rod of length '$$\ell$$' is pivoted at one of its ends on a vertical shaft of negligible radius. When the shaft rotates at angular speed $$\omega$$ the rod makes an angle $$\theta$$ with it (see figure). To find $$\theta$$ equate the rate of change of angular momentum (direction going into the paper) $$\frac{m\ell^2}{12}\omega^2 \sin\theta$$ about the centre of mass (CM) to the torque provided by the horizontal and vertical forces $$F_H$$ and $$F_v$$ about the CM. The value of $$\theta$$ is then such that:

A

$$\cos\theta = \frac{2g}{3\ell\omega^2}$$

B

$$\cos\theta = \frac{g}{2\ell\omega^2}$$

C

$$\cos\theta = \frac{g}{\ell\omega^2}$$

D

$$\cos\theta = \frac{3g}{2\ell\omega^2}$$

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NTA JEE Main 3rd September 2020 Shift 2 - Question 6

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The mass density of a planet of radius R varies with the distance r from its centre as $$\rho(r) = \rho_0\left(1 - \frac{r^2}{R^2}\right)$$. Then the gravitational field is maximum at:

A

$$r = \sqrt{\frac{3}{4}}R$$

B

$$r = R$$

C

$$r = \frac{1}{\sqrt{3}}R$$

D

$$r = \sqrt{\frac{5}{9}}R$$

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NTA JEE Main 3rd September 2020 Shift 2 - Question 7

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A metallic sphere cools from 50°C to 40°C in 300 s. If atmospheric temperature around is 20°C, then the sphere's temperature after the next 5 minutes will be close to:

A

31°C

B

33°C

C

28°C

D

35°C

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NTA JEE Main 3rd September 2020 Shift 2 - Question 8

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A calorimeter of water equivalent 20 g contains 180 g of water at 25°C. 'm' grams of steam at 100°C is mixed in it till the temperature of the mixture is 31°C. The value of 'm' is close to (Latent heat of water = 540 cal g$$^{-1}$$, specific heat of water = 1 cal g$$^{-1}$$°C$$^{-1}$$)

A

2

B

4

C

3.2

D

2.6

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NTA JEE Main 3rd September 2020 Shift 2 - Question 9

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To raise the temperature of a certain mass of gas by 50°C at a constant pressure, 160 calories of heat is required. When the same mass of gas is cooled by 100°C at constant volume, 240 calories of heat is released. How many degrees of freedom does each molecule of this gas have (assume gas to be ideal)?

A

5

B

6

C

3

D

7

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NTA JEE Main 3rd September 2020 Shift 2 - Question 10

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A block of mass m attached to a massless spring is performing oscillatory motion of amplitude 'A' on a frictionless horizontal plane. If half of the mass of the block breaks off when it is passing through its equilibrium point, the amplitude of oscillation for the remaining system become $$fA$$. The value of $$f$$ is:

A

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

B

$$1$$

C

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

D

$$\sqrt{2}$$

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NTA JEE Main 3rd September 2020 Shift 2 - Question 11

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Concentric metallic hollow spheres of radii R and 4R hold charges $$Q_1$$ and $$Q_2$$ respectively. Given that surface charge densities of the concentric spheres are equal, the potential difference V(R) - V(4R) is:

A

$$\frac{Q_2}{4\pi\varepsilon_0 R}$$

B

$$\frac{3Q_2}{4\pi\varepsilon_0 R}$$

C

$$\frac{3Q_1}{16\pi\varepsilon_0 R}$$

D

$$\frac{3Q_1}{4\pi\varepsilon_0 R}$$

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NTA JEE Main 3rd September 2020 Shift 2 - Question 12

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Two resistors 400 $$\Omega$$ and 800 $$\Omega$$ are connected in series across a 6V battery. The potential difference measured by a voltmeter of 10 k$$\Omega$$ across 400 $$\Omega$$ resistor is close to:

A

2 V

B

1.8 V

C

2.05 V

D

1.95 V

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NTA JEE Main 3rd September 2020 Shift 2 - Question 13

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Which of the following will NOT be observed when a multimeter (operating in resistance measuring mode) probes connected across a component, are just reversed?

A

Multimeter shows an equal deflection in both cases i.e., before and after reversing the probes if the chosen component is resistor

B

Multimeter shows NO deflection in both cases i.e. before and after reversing the probes if the chosen component is capacitor

C

Multimeter shows a deflection, accompanied by a splash of light out of connected component in one direction and NO deflection on reversing the probes if the chosen component is LED

D

Multimeter shows NO deflection in both cases i.e. before and after reversing the probes is the chosen component is metal wire.

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NTA JEE Main 3rd September 2020 Shift 2 - Question 14

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A perfectly diamagnetic sphere has a small spherical cavity at its centre, which is filled with a paramagnetic substance. The whole system is placed in a uniform magnetic field $$\vec{B}$$. Then the field inside the paramagnetic substance is:

A

$$\vec{B}$$

B

Zero

C

much larger than $$|\vec{B}|$$ and parallel to $$\vec{B}$$

D

much larger than $$|\vec{B}|$$ but opposite to $$\vec{B}$$

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NTA JEE Main 3rd September 2020 Shift 2 - Question 15

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A uniform magnetic field B exists in a direction perpendicular to the plane of a square loop made of a metal wire. The wire has a diameter of 4 mm and a total length of 30 cm. The magnetic field changes with time at a steady rate dB/dt = 0.032 Ts$$^{-1}$$. The induced current in the loop is close to (Resistivity of the metal wire is 1.23 $$\times$$ 10$$^{-8}$$ $$\Omega$$m)

A

0.43 A

B

0.61 A

C

0.34 A

D

0.53 A

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NTA JEE Main 3rd September 2020 Shift 2 - Question 16

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The electric field of a plane electromagnetic wave propagating along the x direction in vacuum is $$\vec{E} = E_0 \hat{j}\cos(\omega t - kx)$$. The magnetic field $$\vec{B}$$, at the moment t = 0 is:

A

$$\vec{B} = \frac{E_0}{\sqrt{\mu_0\varepsilon_0}}\cos(kx)\hat{k}$$

B

$$\vec{B} = E_0\sqrt{\mu_0\varepsilon_0}\cos(kx)\hat{j}$$

C

$$\vec{B} = E_0\sqrt{\mu_0\varepsilon_0}\cos(kx)\hat{k}$$

D

$$\vec{B} = \frac{E_0}{\sqrt{\mu_0\varepsilon_0}}\cos(kx)\hat{j}$$

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NTA JEE Main 3rd September 2020 Shift 2 - Question 17

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Two sources of light emit X-rays of wavelength 1 nm and visible light of wavelength 500 nm, respectively. Both the sources emit light of the same power 200 W. The ratio of the number density of photons of X-rays to the number density of photons of the visible light of the given wavelengths is:

A

$$\frac{1}{500}$$

B

250

C

$$\frac{1}{250}$$

D

500

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NTA JEE Main 3rd September 2020 Shift 2 - Question 18

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Two light waves having the same wavelength $$\lambda$$ in vacuum are in phase initially. Then the first wave travels a path $$L_1$$ through a medium of refractive index $$n_1$$ while the second wave travels a path of length $$L_2$$ through a medium of refractive index $$n_2$$. After this the phase difference between the two waves is:

A

$$\frac{2\pi}{\lambda}\left(\frac{L_2}{n_1} - \frac{L_1}{n_2}\right)$$

B

$$\frac{2\pi}{\lambda}\left(\frac{L_1}{n_1} - \frac{L_2}{n_2}\right)$$

C

$$\frac{2\pi}{\lambda}(n_1L_1 - n_2L_2)$$

D

$$\frac{2\pi}{\lambda}(n_2L_1 - n_1L_2)$$

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NTA JEE Main 3rd September 2020 Shift 2 - Question 19

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The radius R of a nucleus of mass number A can be estimated by the formula $$R = (1.3 \times 10^{-15})A^{1/3}$$ m. It follows that the mass density of n nucleus is of the order of: $$(M_{prot} \cong M_{neut} \simeq 1.67 \times 10^{-27}$$ kg)

A

$$10^3$$ kg m$$^{-3}$$

B

$$10^{10}$$ kg m$$^{-3}$$

C

$$10^{24}$$ kg m$$^{-3}$$

D

$$10^{17}$$ kg m$$^{-3}$$

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NTA JEE Main 3rd September 2020 Shift 2 - Question 20

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If a semiconductor photo diode can detect a photon with a maximum wavelength of 400 nm, then its band gap energy is: Planck's constant h = 6.63 $$\times$$ 10$$^{-34}$$ J.s, Speed of light c = 3 $$\times$$ 10$$^8$$ m s$$^{-1}$$

A

1.1 eV

B

2.0 eV

C

1.5 eV

D

3.1 eV

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NTA JEE Main 3rd September 2020 Shift 2 - Question 21

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A block starts moving up an inclined plane of inclination 30° with an initial velocity of $$v_0$$. It comes back to its initial position with velocity $$\frac{v_0}{2}$$. The value of the coefficient of kinetic friction between the block and the inclined plane is close to $$\frac{I}{1000}$$. The nearest integer to I is:

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NTA JEE Main 3rd September 2020 Shift 2 - Question 22

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An massless equilateral triangle EFG of side 'a' (As shown in figure) has three particles of mass m situated at its vertices. The moment of inertia of the system about the line EX perpendicular to EG in the plane of EFG is $$\frac{N}{20}ma^2$$ where N is an integer. The value of N is __________.

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NTA JEE Main 3rd September 2020 Shift 2 - Question 23

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If minimum possible work is done by a refrigerator in converting 100 grams of water at 0°C to ice, how much heat (in calories) is released to the surroundings at temperature 27°C (Latent heat of ice = 80 Cal/gram) to the nearest integer?

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NTA JEE Main 3rd September 2020 Shift 2 - Question 24

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A galvanometer coil has 500 turns and each turn has an average area of $$3 \times 10^{-4}$$ m$$^2$$. If a torque of 1.5 Nm is required to keep this coil parallel to a magnetic field when a current of 0.5A is flowing through it, the strength of the field (in T) is __________.

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NTA JEE Main 3rd September 2020 Shift 2 - Question 25

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When an object is kept at a distance of 30 cm from a concave mirror, the image is formed at a distance of 10 cm from the mirror. If the object is moved with a speed of 9 cm s$$^{-1}$$, the speed (in cm s$$^{-1}$$) with which image moves at that instant is

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NTA JEE Main 3rd September 2020 Shift 2 - Question 26

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The strengths of 5.6 volume hydrogen peroxide (of density 1 g/mL) in terms of mass percentage and molarity(M) respectively, are: (Take molar mass of hydrogen peroxide as 34 g/mol)

A

1.7 and 0.5

B

0.85 and 0.25

C

1.7 and 0.25

D

0.85 and 0.5

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NTA JEE Main 3rd September 2020 Shift 2 - Question 27

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Consider the hypothetical situation where the azimuthal quantum number, $$\ell$$, takes values 0, 1, 2, ......... n + 1. Where n is the principal quantum number. Then, the element with atomic number:

A

9 is the first alkali metal

B

13 has a half-filled valence subshell

C

8 is the first noble gas

D

6 has a 2p-valence subshell

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NTA JEE Main 3rd September 2020 Shift 2 - Question 28

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The five successive ionization enthalpies of an element are 800, 2427, 3658, 35024, 32824 kJ mol$$^{-1}$$. The number of valence electrons in the element is:

A

5

B

4

C

3

D

2

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NTA JEE Main 3rd September 2020 Shift 2 - Question 29

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Consider the following molecules and statements related to them:

(a) (B) is more likely to be crystalline than (A)
(b) (B) has higher boiling point than (A)
(c) (B) dissolves more readily than (A) in water
Identify the correct option from below:

A

(a) and (b) are true

B

(a) and (c) are true

C

only (a) is true

D

(b) and (c) are true

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NTA JEE Main 3rd September 2020 Shift 2 - Question 30

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A mixture of one mole each of $$H_2$$, He and $$O_2$$ each are enclosed in a cylinder of volume V at temperature T. If the partial pressure of $$H_2$$ is 2 atm, the total pressure of the gases in the cylinder is:

A

6 atm

B

38 atm

C

14 atm

D

22 atm

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NTA JEE Main 3rd September 2020 Shift 2 - Question 31

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100 mL of 0.1 M HCl is taken in a beaker and to it 100 mL 0.1 M NaOH of is added in steps of 2 mL and the pH is continuously measured. Which of the following graphs correctly depicts the change in pH?

A

B

C

D

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NTA JEE Main 3rd September 2020 Shift 2 - Question 32

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Among the statements (I - IV), the correct ones are:
(I) Be has smaller atomic radius compared to Mg.
(II) Be has higher ionization enthalpy than Al.
(III) Charge/radius ratio of Be is greater than that of Al.
(IV) Both Be and Al form mainly covalent compounds

A

(I), (II) and (IV)

B

(I), (III) and (IV)

C

(II), (III) and (IV)

D

(I), (II) and (III)

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NTA JEE Main 3rd September 2020 Shift 2 - Question 33

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The incorrect statement(s) among (a) - (d) regarding acid rain is (are):
(a) It can corrode water pipes
(b) It can damage structures made up of stone.
(c) It cannot cause respiratory ailments in animals
(d) It is not harmful for trees

A

(a), (c) and (d)

B

(c) only

C

(a), (b) and (d)

D

(c) and (d)

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NTA JEE Main 3rd September 2020 Shift 2 - Question 34

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For the reaction $$2A + 3B + \frac{3}{2}C \to 3P$$, which statement is correct?

A

$$\frac{dn_A}{dt} = \frac{3}{2}\frac{dn_B}{dt} = \frac{3}{4}\frac{dn_C}{dt}$$

B

$$\frac{dn_A}{dt} = \frac{dn_B}{dt} = \frac{dn_C}{dt}$$

C

$$\frac{dn_A}{dt} = \frac{2}{3}\frac{dn_B}{dt} = \frac{4}{3}\frac{dn_C}{dt}$$

D

$$\frac{dn_A}{dt} = \frac{2}{3}\frac{dn_B}{dt} = \frac{3}{4}\frac{dn_C}{dt}$$

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NTA JEE Main 3rd September 2020 Shift 2 - Question 35

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The incorrect statement is:

A

Manganate and permanganate ions are tetrahedral

B

In manganate and permanganate ions, the $$\pi$$-bonding takes place by overlap of p-orbitals of oxygen and d-orbitals of manganese

C

Manganate and permanganate ions are paramagnetic

D

Manganate ion is green in colour and permanganate ion is purple in colour

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NTA JEE Main 3rd September 2020 Shift 2 - Question 36

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Complex A has a composition of $$H_{12}O_6Cl_3Cr$$. If the complex on treatment with conc. $$H_2SO_4$$ loses 13.5% of its original mass, the correct molecular formula of A is:
[Given: atomic mass of Cr = 52 amu and Cl = 35 amu]

A

$$[Cr(H_2O_6)]Cl_3$$

B

$$[Cr(H_2O)_3Cl_3] \cdot 3H_2O$$

C

$$[Cr(H_2O)_5Cl]Cl_2 \cdot H_2O$$

D

$$[Cr(H_2O)_4Cl_2]Cl \cdot 2H_2O$$

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NTA JEE Main 3rd September 2020 Shift 2 - Question 37

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The d-electron configuration of $$[Ru(en)_3]Cl_2$$, and $$[Fe(H_2O)_6]Cl_2$$ respectively are:

A

$$t_{2g}^6 e_g^0$$ and $$t_{2g}^6 e_g^0$$

B

$$t_{2g}^4 e_g^2$$ and $$t_{2g}^6 e_g^0$$

C

$$t_{2g}^6 e_g^0$$ and $$t_{2g}^4 e_g^2$$

D

$$t_{2g}^4 e_g^2$$ and $$t_{2g}^4 e_g^2$$

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NTA JEE Main 3rd September 2020 Shift 2 - Question 38

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The decreasing order of reactivity of the following compounds towards nucleophilic substitution ($$S_N2$$) is:

A

(II) > (III) > (I) > (IV)

B

(II) > (III) > (IV) > (I)

C

(III) > (II) > (IV) > (I)

D

(IV) > (II) > (III) > (I)

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NTA JEE Main 3rd September 2020 Shift 2 - Question 39

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The major product in the following reaction is:

A

B

C

D

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NTA JEE Main 3rd September 2020 Shift 2 - Question 40

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Consider the following reaction:

The product 'P' gives positive ceric ammonium nitrate test. This is because of the presence of which of these -OH group(s)?

A

(b) only

B

(c) and (d)

C

(d) only

D

(b) and (d)

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NTA JEE Main 3rd September 2020 Shift 2 - Question 41

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The increasing order of the reactivity of the following compounds in nucleophile addition reaction is:
Propanal, Benzaldehyde, Propanone, Butanone

A

Benzaldehyde > Butanone < Propanone < Propanal

B

Butanone < Propanone < Benzaldehyde < Propanal

C

Propanal < Propanone < Butanone < Benzaldehyde

D

Benzaldehyde < Propanal < Propanone < Butanone

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NTA JEE Main 3rd September 2020 Shift 2 - Question 42

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The compound A in the following reactions is:
$$A \xrightarrow[(ii)Conc. H_{2}SO_{4}/ \Delta]{(i)CH_{3}MgBr/H_{2}O} B \xrightarrow[(ii) Zn/H_{2}O]{(i) O_{3}} C + D$$

A

B

C

D

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NTA JEE Main 3rd September 2020 Shift 2 - Question 43

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Three isomers A, B and C (mol. formula $$C_8H_{11}N$$) give the following results:
A and C $$\xrightarrow{Diazotization}$$ P + Q $$\xrightarrow[(ii) acidation(KMnO_4+H^+)]{(i) Hydrolysis}$$ R (product of A) + S (product of C)
R has lower boiling point than S
B $$\xrightarrow{C_6H_5SO_2Cl}$$ alkali-insoluble product
A, B and C, respectively are:

A

B

C

D

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NTA JEE Main 3rd September 2020 Shift 2 - Question 44

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An ionic micelle is formed on the addition of:

A

liquid diethyl ether to aqueous NaCl solution

B

C

D

Sodium stearate to pure toluene

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NTA JEE Main 3rd September 2020 Shift 2 - Question 45

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Match the following drugs with their therapeutic actions:
(i) Ranitidine                                        (a) Antidepressant
(ii) Nardil (Phenelzine)                        (b) Antibiotic
(iii) Chloramphenicol                           (c) Antihistamine
(iv) Dimetane (Brompheniramine)   (d) Antacid
                                                                (e) Analgesic

A

(i)-(a); (ii)-(c); (iii)-(b); (iv)-(e)

B

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

C

(i)-(e); (ii)-(a); (iii)-(c); (iv)-(d)

D

(i)-(d); (ii)-(c); (iii)-(a); (iv)-(e)

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NTA JEE Main 3rd September 2020 Shift 2 - Question 46

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$$0.023 \times 10^{22}$$ molecules are present in 10g of a substance 'x'. The molarity of a solution containing 5g of substance 'x' in 2 L solution is _________ $$\times 10^{-3}$$

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NTA JEE Main 3rd September 2020 Shift 2 - Question 47

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If 250 cm$$^3$$ of an aqueous solution containing 0.73g of a protein A is isotonic with one litre of another aqueous solution containing 1.65g of a protein B, at 298K, the ratio of the molecular masses of A and B is _________ $$\times 10^{-2}$$ (to the nearest integer).

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NTA JEE Main 3rd September 2020 Shift 2 - Question 48

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An acidic solution of dichromate is electrolyzed for 8 minutes using 2 A current. As per the following equation $$Cr_2O_7^{2-} + 14H^+ + 6e^- \to 2Cr^{3+} + 7H_2O$$
The amount of $$Cr^{3+}$$ obtained was 0.104g. The efficiency of the process (in %) is (Take: F = 960000C, At. mass of chromium = 52)

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NTA JEE Main 3rd September 2020 Shift 2 - Question 49

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The volume (in mL) of 0.1 N NaOH required to neutralise 10 mL of 0.1 N phosphinic acid is __________.

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NTA JEE Main 3rd September 2020 Shift 2 - Question 50

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The number of 

groups present in a tripeptide Asp - Glu - Lys is ____

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NTA JEE Main 3rd September 2020 Shift 2 - Question 51

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The set of all real values of $$\lambda$$ for which the quadratic equation $$(\lambda^2 + 1)x^2 - 4\lambda x + 2 = 0$$ always have exactly one root in the interval (0, 1) is:

A

$$(-3, -1)$$

B

$$(0, 2)$$

C

$$(1, 3]$$

D

$$(2, 4]$$

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NTA JEE Main 3rd September 2020 Shift 2 - Question 52

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If $$z_1, z_2$$ are complex numbers such that $$Re(z_1) = |z_1 - 1|$$ and $$Re(z_2) = |z_2 - 1|$$ and $$\arg(z_1 - z_2) = \frac{\pi}{6}$$, then $$Im(z_1 + z_2)$$ is equal to:

A

$$2\sqrt{3}$$

B

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

C

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

D

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

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NTA JEE Main 3rd September 2020 Shift 2 - Question 53

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If the sum of the series $$20 + 19\frac{3}{5} + 19\frac{1}{5} + 18\frac{4}{5} + \ldots$$ up to $$n^{th}$$ term is 488 and the $$n^{th}$$ term is negative, then:

A

$$n^{th}$$ term is $$-4\frac{2}{5}$$

B

$$n = 41$$

C

$$n^{th}$$ term is $$-4$$

D

$$n = 60$$

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NTA JEE Main 3rd September 2020 Shift 2 - Question 54

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If the term independent of $$x$$ in the expansion of $$\left(\frac{3}{2}x^2 - \frac{1}{3x}\right)^9$$ is $$k$$, then $$18k$$ is equal to:

A

11

B

5

C

9

D

7

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NTA JEE Main 3rd September 2020 Shift 2 - Question 55

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If a $$\triangle ABC$$ has vertices $$A(-1, 7)$$, $$B(-7, 1)$$ and $$C(5, -5)$$, then its orthocentre has coordinates:

A

$$(-3, 3)$$

B

$$(3, -3)$$

C

$$\left(-\frac{3}{5}, \frac{3}{5}\right)$$

D

$$\left(\frac{3}{5}, -\frac{3}{5}\right)$$

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NTA JEE Main 3rd September 2020 Shift 2 - Question 56

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Let the latus rectum of the parabola $$y^2 = 4x$$ be the common chord to the circles $$C_1$$ and $$C_2$$ each of them having radius $$2\sqrt{5}$$. Then, the distance between the centres of the circles $$C_1$$ and $$C_2$$ is:

A

12

B

8

C

$$8\sqrt{5}$$

D

$$4\sqrt{5}$$

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NTA JEE Main 3rd September 2020 Shift 2 - Question 57

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Let $$e_1$$ and $$e_2$$ be the eccentricities of the ellipse $$\frac{x^2}{25} + \frac{y^2}{b^2} = 1$$ $$(b < 5)$$ and the hyperbola $$\frac{x^2}{16} - \frac{y^2}{b^2} = 1$$ respectively satisfying $$e_1 e_2 = 1$$. If $$\alpha$$ and $$\beta$$ are the distances between the foci of the ellipse and the foci of the hyperbola respectively, then the ordered pair $$(\alpha, \beta)$$ is equal to:

A

$$(8, 10)$$

B

$$\left(\frac{20}{3}, 12\right)$$

C

$$(8, 12)$$

D

$$\left(\frac{24}{5}, 10\right)$$

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NTA JEE Main 3rd September 2020 Shift 2 - Question 58

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$$\lim_{x \to a}\frac{(a+2x)^{\frac{1}{3}} - (3x)^{\frac{1}{3}}}{(3a+x)^{\frac{1}{3}} - (4x)^{\frac{1}{3}}}$$ $$(a \neq 0)$$ is equal to:

A

$$\left(\frac{2}{9}\right)\left(\frac{2}{3}\right)^{\frac{1}{3}}$$

B

$$\left(\frac{2}{3}\right)^{\frac{4}{3}}$$

C

$$\left(\frac{2}{9}\right)^{\frac{4}{3}}$$

D

$$\left(\frac{2}{3}\right)\left(\frac{2}{9}\right)^{\frac{1}{3}}$$

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NTA JEE Main 3rd September 2020 Shift 2 - Question 59

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Let $$p$$, $$q$$, $$r$$ be three statements such that the truth value of $$(p \wedge q) \to (\sim q \vee r)$$ is $$F$$. Then the truth values of $$p$$, $$q$$, $$r$$ are respectively:

A

T, T, F

B

T, T, T

C

T, F, T

D

F, T, F

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NTA JEE Main 3rd September 2020 Shift 2 - Question 60

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Let $$x_i (1 \leq i \leq 10)$$ be ten observations of a random variable X. If $$\sum_{i=1}^{10}(x_i - p) = 3$$ and $$\sum_{i=1}^{10}(x_i - p)^2 = 9$$ where $$0 \neq p \in R$$, then the standard deviation of these observations is:

A

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

B

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

C

$$\frac{9}{10}$$

D

$$\frac{7}{10}$$

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NTA JEE Main 3rd September 2020 Shift 2 - Question 61

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Let $$R_1$$ and $$R_2$$ be two relations defined as follows:
$$R_1 = \{(a, b) \in R^2 : a^2 + b^2 \in Q\}$$ and $$R_2 = \{(a, b) \in R^2 : a^2 + b^2 \notin Q\}$$, where Q is the set of all rational numbers, then

A

$$R_1$$ is transitive but $$R_2$$ is not transitive.

B

$$R_2$$ is transitive but $$R_1$$ is not transitive.

C

Neither $$R_1$$ nor $$R_2$$ is transitive.

D

$$R_1$$ and $$R_2$$ are both transitive.

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NTA JEE Main 3rd September 2020 Shift 2 - Question 62

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Let A be a $$3 \times 3$$ matrix such that adj $$A = \begin{bmatrix} 2 & -1 & 1 \\ -1 & 0 & 2 \\ 1 & -2 & -1 \end{bmatrix}$$ and $$B = $$ adj(adjA). If $$|A| = \lambda$$ and $$\left|(B^{-1})^T\right| = \mu$$, then the ordered pair $$(|\lambda|, \mu)$$ is equal to

A

$$\left(3, \frac{1}{81}\right)$$

B

$$\left(9, \frac{1}{9}\right)$$

C

$$(3, 81)$$

D

$$\left(9, \frac{1}{81}\right)$$

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NTA JEE Main 3rd September 2020 Shift 2 - Question 63

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Suppose $$f(x)$$ is a polynomial of degree four having critical points at -1, 0, 1. If $$T = \{x \in R | f(x) = f(0)\}$$, then the sum of squares of all the elements of $$T$$ is:

A

4

B

6

C

2

D

8

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NTA JEE Main 3rd September 2020 Shift 2 - Question 64

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If the surface area of a cube is increasing at a rate of 3.6 cm$$^2$$/sec, retaining its shape; then the rate of change of its volume (in cm$$^3$$/sec), when the length of a side of the cube is 10 cm, is:

A

20

B

10

C

18

D

9

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NTA JEE Main 3rd September 2020 Shift 2 - Question 65

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If $$\int \sin^{-1}\left(\frac{\sqrt{x}}{1+x}\right)dx = A(x)\tan^{-1}(\sqrt{x}) + B(x) + C$$, where C is a constant of integration, then the ordered pair $$(A(x), B(x))$$ can be:

A

$$(x - 1, \sqrt{x})$$

B

$$(x - 1, -\sqrt{x})$$

C

$$(x + 1, \sqrt{x})$$

D

$$(x + 1, -\sqrt{x})$$

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NTA JEE Main 3rd September 2020 Shift 2 - Question 66

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If the value of the integral $$\int_0^{\frac{1}{2}}\frac{x^2}{(1-x^2)^{\frac{3}{2}}}dx$$ is $$\frac{k}{6}$$, then $$k$$ is equal to:

A

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

B

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

C

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

D

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

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NTA JEE Main 3rd September 2020 Shift 2 - Question 67

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If $$x^3 dy + xy \cdot dx = x^2 dy + 2y dx$$; $$y(2) = e$$ and $$x > 1$$, then $$y(4)$$ is equal to:

A

$$\frac{\sqrt{e}}{2}$$

B

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

C

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

D

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

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NTA JEE Main 3rd September 2020 Shift 2 - Question 68

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Let $$a, b, c \in R$$ be such that $$a^2 + b^2 + c^2 = 1$$. If $$a\cos\theta = b\cos\left(\theta + \frac{2\pi}{3}\right) = c\cos\left(\theta + \frac{4\pi}{3}\right)$$, where $$\theta = \frac{\pi}{9}$$, then the angle between the vectors $$a\hat{i} + b\hat{j} + c\hat{k}$$ and $$b\hat{i} + c\hat{j} + a\hat{k}$$ is:

A

0

B

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

C

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

D

$$\frac{\pi}{9}$$

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NTA JEE Main 3rd September 2020 Shift 2 - Question 69

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The plane which bisects the line joining the points (4, -2, 3) and (2, 4, -1) at right angles also passes through the point:

A

(0, -1, 1)

B

(4, 0, -1)

C

(4, 0, 1)

D

(0, 1, -1)

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NTA JEE Main 3rd September 2020 Shift 2 - Question 70

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The probability that a randomly chosen 5-digit number is made from exactly two digits is:

A

$$\frac{135}{10^4}$$

B

$$\frac{150}{10^4}$$

C

$$\frac{134}{10^4}$$

D

$$\frac{121}{10^4}$$

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NTA JEE Main 3rd September 2020 Shift 2 - Question 71

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The total number of 3-digit numbers whose sum of digits is 10, is ..........

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NTA JEE Main 3rd September 2020 Shift 2 - Question 72

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If m arithmetic means (A.Ms) and three geometric means (G.Ms) are inserted between 3 and 243 such that $$4^{th}$$ A.M. is equal to $$2^{nd}$$ G.M., then $$m$$ is equal to:

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NTA JEE Main 3rd September 2020 Shift 2 - Question 73

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Let $$S$$ be the set of all integer solutions $$(x, y, z)$$ of the system of equations
$$x - 2y + 5z = 0$$
$$-2x + 4y + z = 0$$
$$-7x + 14y + 9z = 0$$
such that $$15 \leq x^2 + y^2 + z^2 \leq 150$$. Then, the number of elements in the set $$S$$ is equal to ..........

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NTA JEE Main 3rd September 2020 Shift 2 - Question 74

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If the tangent to the curve $$y = e^x$$ at a point $$(c, e^c)$$ and the normal to the parabola $$y^2 = 4x$$ at the point (1, 2) intersect at the same point on the $$x$$-axis, then the value of $$c$$ is .....

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NTA JEE Main 3rd September 2020 Shift 2 - Question 75

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Let a plane $$P$$ contain two lines $$\vec{r} = \hat{i} + \lambda(\hat{i} + \hat{j})$$, $$\lambda \in R$$ and $$\vec{r} = -\hat{j} + \mu(\hat{j} - \hat{k})$$, $$\mu \in R$$. If $$Q(\alpha, \beta, \gamma)$$ is the foot of the perpendicular drawn from the point M(1, 0, 1) to P, then $$3(\alpha + \beta + \gamma)$$ equals .......

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