NTA JEE Main 16 th April 2018 Online

NTA JEE Main 16 th April 2018 Online - Question 1

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Let $$\vec{A} = (\hat{i} + \hat{j})$$ and $$\vec{B} = (2\hat{i} - \hat{j})$$. The magnitude of a coplanar vector $$\vec{C}$$ such that $$\vec{A} \cdot \vec{C} = \vec{B} \cdot \vec{C} = \vec{A} \cdot \vec{B}$$ is given by:

A

$$\sqrt{\frac{9}{12}}$$

B

$$\sqrt{\frac{20}{9}}$$

C

$$\sqrt{\frac{5}{9}}$$

D

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

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NTA JEE Main 16 th April 2018 Online - Question 2

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The percentage errors in quantities P, Q, R and S are 0.5%, 1%, 3% and 1.5% respectively in the measurement of a physical quantity $$A = \frac{P^3 Q^2}{\sqrt{RS}}$$. The maximum percentage error in the value of A will be:

A

6.5%

B

7.5%

C

6.0%

D

8.5%

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NTA JEE Main 16 th April 2018 Online - Question 3

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A body of mass m starts moving from rest along x-axis so that its velocity varies as $$v = a\sqrt{s}$$ where a is a constant and s is the distance covered by the body. The total work done by all the forces acting on the body in the first t second after the start of the motion is:

A

$$8ma^4t^2$$

B

$$\frac{1}{4}ma^4t^2$$

C

$$4ma^4t^2$$

D

$$\frac{1}{8}ma^4t^2$$

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NTA JEE Main 16 th April 2018 Online - Question 4

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Two particles of the same mass m are moving in circular orbits because of force, given by $$F(r) = -\frac{16}{r} - r^3$$. The first particle is at a distance r = 1, and the second, at r = 4. The best estimate for the ratio of kinetic energies of the first and the second particle is closest to:

A

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

B

$$6 \times 10^2$$

C

$$6 \times 10^{-2}$$

D

$$10^{-2}$$

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NTA JEE Main 16 th April 2018 Online - Question 5

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A thin circular disk is in the xy plane as shown in the figure. The ratio of its moment of inertia about z and z' axes will be:

A

1 : 4

B

1 : 5

C

1 : 3

D

1 : 2

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NTA JEE Main 16 th April 2018 Online - Question 6

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The relative uncertainty in the period of a satellite orbiting around the earth is $$10^{-2}$$. If the relative uncertainty in the radius of the orbit is negligible, the relative uncertainty in the mass of the earth is:

A

$$2 \times 10^{-2}$$

B

$$6 \times 10^{-2}$$

C

$$3 \times 10^{-2}$$

D

$$10^{-2}$$

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NTA JEE Main 16 th April 2018 Online - Question 7

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Suppose that the angular velocity of rotation of the Earth is increased. Then, as a consequence:

A

weight of the object, everywhere on the earth, will decrease.

B

weight of the object, everywhere on the earth, will increase.

C

except at poles, weight of the object on the earth, will decrease.

D

there will be no change in weight anywhere on the earth.

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NTA JEE Main 16 th April 2018 Online - Question 8

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A small soap bubble of radius 4 cm is trapped inside another bubble of radius 6 cm without any contact. Let P$$_2$$ be the pressure inside the inner bubble and P$$_0$$, the pressure outside the outer bubble. Radius of another bubble with pressure difference P$$_2$$ - P$$_0$$ between its inside and outside would be:

A

2.4 cm

B

12 cm

C

4.8 cm

D

6 cm

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NTA JEE Main 16 th April 2018 Online - Question 9

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One mole of an ideal monatomic gas is taken along the path ABCA as shown in the PV diagram. The maximum temperature attained by the gas along the path BC is given by:

A

$$\frac{25}{4} \frac{P_0 V_0}{R}$$

B

$$\frac{5}{8} \frac{P_0 V_0}{R}$$

C

$$\frac{25}{8} \frac{P_0 V_0}{R}$$

D

$$\frac{25}{16} \frac{P_0 V_0}{R}$$

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NTA JEE Main 16 th April 2018 Online - Question 10

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Two moles of helium are mixed with n moles of hydrogen. If $$\frac{C_p}{C_v} = \frac{3}{2}$$ for the mixture then the value of n is:

A

1

B

3

C

2

D

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

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NTA JEE Main 16 th April 2018 Online - Question 11

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An oscillator of mass M is at rest in its equilibrium position in a potential, $$V = \frac{1}{2}k(x - X)^2$$. A particle of mass m comes from the right with speed u and collides completely inelastic with M and sticks to it. This process repeats every time the oscillator crosses its equilibrium position. The amplitude of oscillations after 13 collisions is: (M = 10, m = 5, u = 1, k = 1)

A

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

B

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

C

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

D

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

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NTA JEE Main 16 th April 2018 Online - Question 12

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A particle executes simple harmonic motion and it is located at x = a, b and c at time t$$_0$$, 2t$$_0$$ and 3t$$_0$$ respectively. The frequency of the oscillation is:

A

$$\frac{1}{2\pi t_0} \cos^{-1}\left(\frac{a+c}{2b}\right)$$

B

$$\frac{1}{2\pi t_0} \cos^{-1}\left(\frac{a+2b}{3c}\right)$$

C

$$\frac{1}{2\pi t_0} \cos^{-1}\left(\frac{a+b}{2c}\right)$$

D

$$\frac{1}{2\pi t_0} \cos^{-1}\left(\frac{2a+3c}{b}\right)$$

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NTA JEE Main 16 th April 2018 Online - Question 13

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The end correction of a resonance column is 1 cm. If the shortest length resonating with the tuning fork is 10 cm, the next resonating length should be:

A

32 cm

B

40 cm

C

28 cm

D

36 cm

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NTA JEE Main 16 th April 2018 Online - Question 14

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Two sitar strings, A and B playing the note 'Dha' are slightly out of tune and produce beats of frequency 5 Hz. The tension of the string B is slightly increased and the beat frequency is found to decrease by 3 Hz. If the frequency of A is 425 Hz. The original frequency of B is:

A

428 Hz

B

430 Hz

C

422 Hz

D

420 Hz

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NTA JEE Main 16 th April 2018 Online - Question 15

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Two identical conducting spheres A and B carry an equal charge. They are separated by a distance much larger than their diameters, and the force between them is F. A third identical conducting sphere, C, is uncharged. Sphere C is first touched to A, then to B, and then removed. As a result, force between A and B would be equal to:

A

$$\frac{3F}{4}$$

B

$$\frac{F}{2}$$

C

$$\frac{3F}{8}$$

D

F

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NTA JEE Main 16 th April 2018 Online - Question 16

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In the following circuit the switch S is closed at t = 0. The charge on the capacitor C$$_1$$ as a function of time will be given by $$\left(C_{eq} = \frac{C_1 C_2}{C_1 + C_2}\right)$$:

A

$$C_1 E\left[1 - \exp\left(-\frac{tR}{C_1}\right)\right]$$

B

$$C_{eq} E \exp\left[-\frac{t}{RC_{eq}}\right]$$

C

$$C_{eq} E\left[1 - \exp\left(-\frac{t}{RC_{eq}}\right)\right]$$

D

$$C_2 E\left[1 - \exp\left(-\frac{t}{RC_2}\right)\right]$$

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NTA JEE Main 16 th April 2018 Online - Question 17

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A heating element has a resistance of 100 $$\Omega$$ at room temperature. When it is connected to a supply of 220 V, a steady current of 2 A passes in it and temperature is 500$$^\circ$$C more than the room temperature. The temperature coefficient of resistance of the heating element is:

A

$$5 \times 10^{-4}$$ $$^\circ$$C$$^{-1}$$

B

$$2 \times 10^{-4}$$ $$^\circ$$C$$^{-1}$$

C

$$1 \times 10^{-4}$$ $$^\circ$$C$$^{-1}$$

D

$$0.5 \times 10^{-4}$$ $$^\circ$$C$$^{-1}$$

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NTA JEE Main 16 th April 2018 Online - Question 18

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A galvanometer with its coil resistance 25 $$\Omega$$ requires a current of 1 mA for its full deflection. In order to construct an ammeter to read up to a current of 2 A the approximate value of the shunt resistance should be:

A

$$1.25 \times 10^{-2}$$ $$\Omega$$

B

$$2.5 \times 10^{-3}$$ $$\Omega$$

C

$$2.5 \times 10^{-2}$$ $$\Omega$$

D

$$1.25 \times 10^{-3}$$ $$\Omega$$

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NTA JEE Main 16 th April 2018 Online - Question 19

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In a circuit for finding the resistance of a galvanometer by half deflection method, a 6 V battery and a high resistance of 11 k$$\Omega$$ are used. The figure of merit of the galvanometer is 60 $$\mu$$A division$$^{-1}$$. In the absence of shunt resistance, the galvanometer produces a deflection of $$\theta$$ = 9 divisions when current flows in the circuit. The value of the shunt resistance that can cause the deflection of $$\frac{\theta}{2}$$, is closest to:

A

550 $$\Omega$$

B

220 $$\Omega$$

C

55 $$\Omega$$

D

110 $$\Omega$$

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NTA JEE Main 16 th April 2018 Online - Question 20

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A charge q is spread uniformly over an insulated loop of radius r. If it is rotated with an angular velocity $$\omega$$ with respect to normal axis then magnetic moment of the loop is:

A

$$\frac{3}{2}q\omega r^2$$

B

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

C

$$q\omega r^2$$

D

$$\frac{4}{3}q\omega r^2$$

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NTA JEE Main 16 th April 2018 Online - Question 21

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A coil of cross-sectional area A having n turns is placed in a uniform magnetic field B. When it is rotated with an angular velocity $$\omega$$, the maximum e.m.f. induced in the coil will be:

A

$$\frac{3}{2}nBA\omega$$

B

$$3nBA\omega$$

C

$$nBA\omega$$

D

$$\frac{1}{2}nBA\omega$$

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NTA JEE Main 16 th April 2018 Online - Question 22

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A power transmission line feeds input power at 2300 V to a step-down transformer with its primary windings having 4000 turns giving the output power at 230 V. If the current in the primary coil of the transformer is 5 A and its efficiency is 90%, the output current would be:

A

45 A

B

50 A

C

20 A

D

25 A

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NTA JEE Main 16 th April 2018 Online - Question 23

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A plane electromagnetic wave of wavelength $$\lambda$$ has an intensity I. It is propagating along the positive Y-direction. The allowed expressions for the electric and magnetic fields are given by:

A

$$\vec{E} = \sqrt{\frac{2I}{\varepsilon_0 c}} \cos\left[\frac{2\pi}{\lambda}(y-ct)\right]\hat{k}$$; $$\vec{B} = +\frac{1}{c}E\hat{i}$$

B

$$\vec{E} = \sqrt{\frac{I}{\varepsilon_0 c}} \cos\left[\frac{2\pi}{\lambda}(y-ct)\right]\hat{k}$$; $$\vec{B} = +\frac{1}{c}E\hat{i}$$

C

$$\vec{E} = \sqrt{\frac{2I}{\varepsilon_0 c}} \cos\left[\frac{2\pi}{\lambda}(y+ct)\right]\hat{k}$$; $$\vec{B} = +\frac{1}{c}E\hat{i}$$

D

$$\vec{E} = \sqrt{\frac{I}{\varepsilon_0 c}} \cos\left[\frac{2\pi}{\lambda}(y-ct)\right]\hat{i}$$; $$\vec{B} = +\frac{1}{c}E\hat{k}$$

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NTA JEE Main 16 th April 2018 Online - Question 24

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A ray of light is incident at an angle of 60$$^\circ$$ on one face of a prism of angle 30$$^\circ$$. The emergent ray of light makes an angle of 30$$^\circ$$ with incident ray. The angle made by the emergent ray with second face of prism will be:

A

0$$^\circ$$

B

90$$^\circ$$

C

30$$^\circ$$

D

45$$^\circ$$

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NTA JEE Main 16 th April 2018 Online - Question 25

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Unpolarized light of intensity I is incident on a system of two polarizers, A followed by B. The intensity of emergent light is $$\frac{I}{2}$$. If a third polarizer C is placed between A and B, the intensity of emergent light is reduced to $$\frac{I}{3}$$. The angle between the polarizers A and C is $$\theta$$, then:

A

$$\cos\theta = \left(\frac{2}{3}\right)^{\frac{1}{4}}$$

B

$$\cos\theta = \left(\frac{1}{3}\right)^{\frac{1}{4}}$$

C

$$\cos\theta = \left(\frac{1}{3}\right)^{\frac{1}{2}}$$

D

$$\cos\theta = \left(\frac{2}{3}\right)^{\frac{1}{2}}$$

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NTA JEE Main 16 th April 2018 Online - Question 26

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The de-Broglie wavelength ($$\lambda_B$$) associated with the electron orbiting in the second excited state of hydrogen atom is related to that in the ground state ($$\lambda_G$$) by:

A

$$\lambda_B = 3\lambda_G$$

B

$$\lambda_B = 2\lambda_G$$

C

$$\lambda_B = \lambda_{G/3}$$

D

$$\lambda_B = \lambda_{G/2}$$

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NTA JEE Main 16 th April 2018 Online - Question 27

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Both the nucleus and the atom of some element are in their respective first excited states. They get de-excited by emitting photons of wavelengths $$\lambda_N$$, $$\lambda_A$$ respectively. The ratio $$\frac{\lambda_N}{\lambda_A}$$ is closest to:

A

$$10^{-1}$$

B

$$10^{-6}$$

C

10

D

$$10^{-10}$$

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NTA JEE Main 16 th April 2018 Online - Question 28

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At some instant, a radioactive sample S$$_1$$ having an activity 5$$\mu$$Ci has twice the number of nuclei as another sample S$$_2$$ which has an activity of 10$$\mu$$Ci. The half lives of S$$_1$$ and S$$_2$$ are:

A

4 : 1

B

1 : 4

C

1 : 2

D

2 : 1

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NTA JEE Main 16 th April 2018 Online - Question 29

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In the given circuit the current through zener diode is:

A

3.3 mA

B

2.5 mA

C

5.5 mA

D

6.7 mA

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NTA JEE Main 16 th April 2018 Online - Question 30

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A carrier wave of peak voltage 14 V is used for transmitting a message signal. The peak voltage of the modulating signal given to achieve a modulation index of 80% will be:

A

22.4 V

B

7 V

C

11.2 V

D

28 V

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NTA JEE Main 16 th April 2018 Online - Question 31

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An unknown chlorohydrocarbon has 3.55% of chlorine. If each molecule of the hydrocarbon has one chlorine atom only; chlorine atoms present in 1 g of chlorohydrocarbon are: (Atomic wt. of Cl = 35.5 u; Avogadro constant = 6.023 $$\times$$ 10$$^{23}$$ mol$$^{-1}$$)

A

$$6.023 \times 10^{9}$$

B

$$6.023 \times 10^{23}$$

C

$$6.023 \times 10^{21}$$

D

$$6.023 \times 10^{20}$$

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NTA JEE Main 16 th April 2018 Online - Question 32

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Which of the following statements is false?

A

Splitting of spectral lines in electrical field is called Stark effect

B

Frequency of emitted radiation from a black body goes from a higher wavelength to lower wavelength as the temperature increases

C

Photon has momentum as well as wavelength

D

Rydberg constant has unit of energy

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NTA JEE Main 16 th April 2018 Online - Question 33

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Which of the following conversions involves change in both shape and hybridisation?

A

$$H_2O \rightarrow H_3O^+$$

B

$$BF_3 \rightarrow BF_4^-$$

C

$$CH_4 \rightarrow C_2H_6$$

D

$$NH_3 \rightarrow NH_4^+$$

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NTA JEE Main 16 th April 2018 Online - Question 34

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The most polar compound among the following is:

A

B

C

D

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NTA JEE Main 16 th April 2018 Online - Question 35

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The incorrect geometry is represented by:

A

NF$$_3$$ - trigonal planar

B

BF$$_3$$ - trigonal planar

C

AsF$$_5$$ - trigonal bipyramidal

D

H$$_2$$O - bent

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NTA JEE Main 16 th April 2018 Online - Question 36

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Assuming ideal gas behavior, the ratio of density of ammonia to that of hydrogen chloride at same temperature and pressure is: (Atomic weight of Cl is 35.5 u)

A

1.46

B

1.64

C

0.46

D

0.64

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NTA JEE Main 16 th April 2018 Online - Question 37

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At 320 K, a gas A$$_2$$ is 20% dissociated to A(g). The standard Gibbs free energy change at 320 K and 1 atm in J mol$$^{-1}$$ is approximately: (R = 8.314 J K$$^{-1}$$ mol$$^{-1}$$; ln 2 = 0.693; ln 3 = 1.098)

A

1844

B

2068

C

4281

D

4763

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NTA JEE Main 16 th April 2018 Online - Question 38

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For which of the following processes, $$\Delta S$$ is negative?

A

C (diamond) $$\rightarrow$$ C (graphite)

B

N$$_2$$(g, 1 atm) $$\rightarrow$$ N$$_2$$(g, 5 atm)

C

N$$_2$$(g, 273K) $$\rightarrow$$ N$$_2$$(g, 300K)

D

H$$_2$$(g) $$\rightarrow$$ 2H(g)

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NTA JEE Main 16 th April 2018 Online - Question 39

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For standardizing NaOH solution, which of the following is used as a primary standard?

A

Sodium tetra borate

B

Ferrous Ammonium sulphate

C

Oxalic acid

D

Dil. HCl

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NTA JEE Main 16 th April 2018 Online - Question 40

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The gas phase reaction 2NO$$_2$$(g) $$\rightarrow$$ N$$_2$$O$$_4$$(g) is an exothermic reaction. The decomposition of N$$_2$$O$$_4$$, in equilibrium mixture of NO$$_2$$(g) and N$$_2$$O$$_4$$(g) can be increased by:

A

Addition of an inert gas at constant pressure

B

Lowering the temperature

C

Increasing the pressure

D

Addition of an inert gas at constant volume

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NTA JEE Main 16 th April 2018 Online - Question 41

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A group 13 element 'X' reacts with chlorine gas to produce a compound XCl$$_3$$. XCl$$_3$$ is electron deficient and easily reacts with NH$$_3$$ to form Cl$$_3$$X $$\leftarrow$$ NH$$_3$$ adduct; however, XCl$$_3$$ does not dimerize. X is:

A

B

B

Al

C

In

D

Ga

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NTA JEE Main 16 th April 2018 Online - Question 42

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The major product of the following reaction is:

A

B

C

D

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NTA JEE Main 16 th April 2018 Online - Question 43

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The mass of a non-volatile, non-electrolyte solute (molar mass = 50 g mol$$^{-1}$$) needed to be dissolved in 114 g octane to reduce its vapour pressure by 75%, is:

A

37.5 g

B

75 g

C

150 g

D

50 g

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NTA JEE Main 16 th April 2018 Online - Question 44

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When 9.65 ampere current was passed for 1.0 hour into nitrobenzene in acidic medium, the amount of p-aminophenol produced is:

A

109.0 g

B

98.1 g

C

9.81 g

D

10.9 g

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NTA JEE Main 16 th April 2018 Online - Question 45

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If 50% of a reaction occurs in 100 second and 75% of the reaction occurs in 200 second, the order of this reaction is:

A

2

B

3

C

Zero

D

1

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NTA JEE Main 16 th April 2018 Online - Question 46

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Which one of the following is not a property of physical adsorption?

A

Higher the pressure, more the adsorption

B

Greater the surface area, more the adsorption

C

Lower the temperature, more the adsorption

D

Unilayer adsorption occurs

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NTA JEE Main 16 th April 2018 Online - Question 47

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In the extraction of copper from its sulphide ore, metal is finally obtained by the oxidation of cuprous sulphide with:

A

SO$$_2$$

B

Fe$$_2$$O$$_3$$

C

Cu$$_2$$O

D

CO

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NTA JEE Main 16 th April 2018 Online - Question 48

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Among the oxides of nitrogen: N$$_2$$O$$_3$$, N$$_2$$O$$_4$$ and N$$_2$$O$$_5$$; the molecule(s) having nitrogen-nitrogen bond is/are:

A

N$$_2$$O$$_3$$ and N$$_2$$O$$_4$$

B

N$$_2$$O$$_4$$ and N$$_2$$O$$_5$$

C

N$$_2$$O$$_3$$ and N$$_2$$O$$_5$$

D

Only N$$_2$$O$$_5$$

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NTA JEE Main 16 th April 2018 Online - Question 49

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When XO$$_2$$ is fused with an alkali metal hydroxide in presence of an oxidizing agent such as KNO$$_3$$; a dark green product is formed which disproportionate in acidic solution to afford a dark purple solution. X is:

A

Mn

B

Cr

C

V

D

Ti

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NTA JEE Main 16 th April 2018 Online - Question 50

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The incorrect statement is:

A

Cu$$^{2+}$$ ion gives chocolate coloured precipitate with potassium ferrocyanide solution

B

Cu$$^{2+}$$ and Ni$$^{2+}$$ ions give black precipitate with H$$_2$$S in presence of HCl solution

C

Ferric ion gives blood red colour with potassium thiocyanate

D

Cu$$^{2+}$$ salts give red coloured borax bead test in reducing flame

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NTA JEE Main 16 th April 2018 Online - Question 51

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In a complexometric titration of metal ion with ligand M (Metal ion) + L(Ligand) $$\rightarrow$$ C (Complex). End point is estimated spectrophotometrically (through light absorption). If 'M' and 'C' do not absorb light and only 'L' absorbs then the titration plot between absorbed light (A) versus volume of ligand 'L' (V) would look like:

A

B

C

D

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NTA JEE Main 16 th April 2018 Online - Question 52

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In Wilkinson's catalyst, the hybridization of central metal ion and its shape are respectively:

A

$$sp^3d$$, trigonal bipyramidal

B

$$d^2sp^3$$, octahedral

C

$$dsp^2$$, square planar

D

$$sp^3$$, tetrahedral

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NTA JEE Main 16 th April 2018 Online - Question 53

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Which of the following complexes will show geometrical isomerism?

A

Potassium tris(oxalato)chromate(III)

B

Pentaaquachlorochromium(II)chloride

C

Aquachlorobis(ethylenediamine)cobalt(II)chloride

D

Potassiumamminetrichloroplatinate(II)

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NTA JEE Main 16 th April 2018 Online - Question 54

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The major product of the following reaction is:

A

B

C

D

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NTA JEE Main 16 th April 2018 Online - Question 55

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The major product of the following reaction is:

A

B

C

D

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NTA JEE Main 16 th April 2018 Online - Question 56

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Which of the following compounds will most readily be dehydrated to give alkene under acidic condition?

A

4-Hydroxypentan-2-one

B

3-Hydroxypentan-2-one

C

1-Pentanol

D

2-Hydroxycyclopentanone

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NTA JEE Main 16 th April 2018 Online - Question 57

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The major product B formed in the following reaction sequence is:

A

B

C

D

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NTA JEE Main 16 th April 2018 Online - Question 58

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Products A and B formed in the following reactions are respectively:

A

B

C

D

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NTA JEE Main 16 th April 2018 Online - Question 59

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The correct match between items of List-I and List-II is:

A

(A)-(S), (B)-(R), (C)-(Q), (D)-(P)

B

(A)-(R), (B)-(S), (C)-(P), (D)-(Q)

C

(A)-(R), (B)-(S), (C)-(Q), (D)-(P)

D

(A)-(S), (B)-(R), (C)-(P), (D)-(Q)

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NTA JEE Main 16 th April 2018 Online - Question 60

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Among the following, the incorrect statement is:

A

cellulose and amylose has 1,4-glycosidic linkage.

B

lactose contains $$\beta$$-D-galactose and $$\beta$$-D-glucose.

C

maltose and lactose has 1,4-glycosidic linkage.

D

sucrose and amylose has 1,2-glycosidic linkage.

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NTA JEE Main 16 th April 2018 Online - Question 61

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Let p, q and r be real numbers ($$p \neq q, r \neq 0$$), such that the roots of the equation $$\frac{1}{x+p} + \frac{1}{x+q} = \frac{1}{r}$$ are equal in magnitude but opposite in sign, then the sum of squares of these roots is equal to:

A

$$p^2 + q^2$$

B

$$\frac{p^2 + q^2}{2}$$

C

$$2(p^2 + q^2)$$

D

$$p^2 + q^2 + r^2$$

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NTA JEE Main 16 th April 2018 Online - Question 62

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If an angle A of a $$\triangle ABC$$ satisfies $$5\cos A + 3 = 0$$, then the roots of the quadratic equation $$9x^2 + 27x + 20 = 0$$ are:

A

$$\sec A$$, $$\cot A$$

B

$$\sec A$$, $$\tan A$$

C

$$\tan A$$, $$\cos A$$

D

$$\sin A$$, $$\sec A$$

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NTA JEE Main 16 th April 2018 Online - Question 63

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The least positive integer n for which $$\left(\frac{1 + i\sqrt{3}}{1 - i\sqrt{3}}\right)^n = 1$$ is:

A

2

B

5

C

6

D

3

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NTA JEE Main 16 th April 2018 Online - Question 64

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The number of numbers between 2,000 and 5,000 that can be formed with the digits 0, 1, 2, 3, 4 (repetition of digits is not allowed) and are multiple of 3 is:

A

36

B

30

C

24

D

48

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NTA JEE Main 16 th April 2018 Online - Question 65

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Let $$\frac{1}{x_1}, \frac{1}{x_2}, \ldots, \frac{1}{x_n}$$ ($$x_i \neq 0$$ for i = 1, 2, ..., n) be in A.P. such that $$x_1 = 4$$ and $$x_{21} = 20$$. If n is the least positive integer for which $$x_n \gt 50$$, then $$\sum_{i=1}^{n}\left(\frac{1}{x_i}\right)$$ is equal to:

A

3

B

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

C

$$\frac{13}{4}$$

D

$$\frac{13}{8}$$

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NTA JEE Main 16 th April 2018 Online - Question 66

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The sum of the first 20 terms of the series $$1 + \frac{3}{2} + \frac{7}{4} + \frac{15}{8} + \frac{31}{16} + \ldots$$ is:

A

$$39 + \frac{1}{2^{19}}$$

B

$$38 + \frac{1}{2^{20}}$$

C

$$38 + \frac{1}{2^{19}}$$

D

$$39 + \frac{1}{2^{20}}$$

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NTA JEE Main 16 th April 2018 Online - Question 67

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The coefficient of $$x^2$$ in the expansion of the product $$(2 - x^2)\left\{(1 + 2x + 3x^2)^6 + (1 - 4x^2)^6\right\}$$ is:

A

107

B

108

C

155

D

106

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NTA JEE Main 16 th April 2018 Online - Question 68

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The locus of the point of intersection of the lines $$\sqrt{2}x - y + 4\sqrt{2}k = 0$$ and $$\sqrt{2}kx + ky - 4\sqrt{2} = 0$$ (k is any non-zero real parameter) is:

A

an ellipse whose eccentricity is $$\frac{1}{\sqrt{3}}$$

B

a hyperbola whose eccentricity is $$\sqrt{3}$$

C

a hyperbola with length of its transverse axis $$8\sqrt{2}$$

D

an ellipse with length of its major axis $$8\sqrt{2}$$

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NTA JEE Main 16 th April 2018 Online - Question 69

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If a circle C, whose radius is 3, touches externally the circle $$x^2 + y^2 + 2x - 4y - 4 = 0$$ at the point (2, 2), then the length of the intercept cut by this circle C on the x-axis is equal to:

A

$$2\sqrt{3}$$

B

$$\sqrt{5}$$

C

$$3\sqrt{2}$$

D

$$2\sqrt{5}$$

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NTA JEE Main 16 th April 2018 Online - Question 70

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Let P be a point on the parabola $$x^2 = 4y$$. If the distance of P from the center of the circle $$x^2 + y^2 + 6x + 8 = 0$$ is minimum, then the equation of the tangent to the parabola at P is:

A

$$x + y + 1 = 0$$

B

$$x + 4y - 2 = 0$$

C

$$x + 2y = 0$$

D

$$x - y + 3 = 0$$

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NTA JEE Main 16 th April 2018 Online - Question 71

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If the length of the latus rectum of an ellipse is 4 units and the distance between a focus and its nearest vertex on the major axis is $$\frac{3}{2}$$ units, then its eccentricity is:

A

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

B

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

C

$$\frac{1}{9}$$

D

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

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NTA JEE Main 16 th April 2018 Online - Question 72

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$$\lim_{x \to 0} \frac{(27+x)^{\frac{1}{3}} - 3}{9 - (27+x)^{\frac{2}{3}}}$$ equals:

A

$$-\frac{1}{6}$$

B

$$\frac{1}{6}$$

C

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

D

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

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NTA JEE Main 16 th April 2018 Online - Question 73

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If $$p \to (\sim p \vee \sim q)$$ is false, then the truth values of p and q are, respectively:

A

F, F

B

T, T

C

F, T

D

T, F

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NTA JEE Main 16 th April 2018 Online - Question 74

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The mean and the standard deviation (S.D.) of five observations are 9 and 0, respectively. If one of the observation is increased such that the mean of the new set of five observations becomes 10, then their S.D. is:

A

0

B

2

C

4

D

1

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NTA JEE Main 16 th April 2018 Online - Question 75

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A man on the top of a vertical tower observes a car moving at a uniform speed towards the tower on a horizontal road. If it takes 18 min for the angle of depression of the car to change from 30$$^\circ$$ to 45$$^\circ$$, then the time taken (in min) by the car to reach the foot of the tower is:

A

$$\frac{9}{2}(\sqrt{3} - 1)$$

B

$$18(1 + \sqrt{3})$$

C

$$18(\sqrt{3} - 1)$$

D

$$9(1 + \sqrt{3})$$

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NTA JEE Main 16 th April 2018 Online - Question 76

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Let N denote the set of all natural numbers. Define two binary relations on N as $$R_1 = \{(x, y) \in N \times N : 2x + y = 10\}$$ and $$R_2 = \{(x, y) \in N \times N : x + 2y = 10\}$$. Then:

A

both $$R_1$$ and $$R_2$$ are transitive relations

B

range of $$R_2$$ is $$\{1, 2, 3, 4\}$$

C

range of $$R_1$$ is $$\{2, 4, 8\}$$

D

both $$R_1$$ and $$R_2$$ are symmetric relations

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NTA JEE Main 16 th April 2018 Online - Question 77

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Let $$A = \begin{bmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{bmatrix}$$ and $$B = A^{20}$$. Then the sum of the elements of the first column of B is:

A

210

B

211

C

251

D

231

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NTA JEE Main 16 th April 2018 Online - Question 78

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The number of values of k for which the system of linear equations $$(k+2)x + 10y = k$$ and $$kx + (k+3)y = k - 1$$ has no solution is:

A

1

B

2

C

3

D

4

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NTA JEE Main 16 th April 2018 Online - Question 79

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If the function f defined as $$f(x) = \frac{1}{x} - \frac{k-1}{e^{2x} - 1}$$, $$x \neq 0$$ is continuous at $$x = 0$$, then ordered pair (k, f(0)) is equal to: 

A

(2, 1)

B

(3, 1)

C

(3, 2)

D

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

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NTA JEE Main 16 th April 2018 Online - Question 80

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If $$x = \sqrt{2^{\text{cosec}^{-1}t}}$$ and $$y = \sqrt{2^{\text{sec}^{-1}t}}$$, ($$|t| \geq 1$$), then $$\frac{dy}{dx}$$ is equal to:

A

$$\frac{y}{x}$$

B

$$-\frac{y}{x}$$

C

$$-\frac{x}{y}$$

D

$$\frac{x}{y}$$

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NTA JEE Main 16 th April 2018 Online - Question 81

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Let M and m be respectively the absolute maximum and the absolute minimum values of the function, $$f(x) = 2x^3 - 9x^2 + 12x + 5$$ in the interval [0, 3]. Then M - m is equal to:

A

9

B

4

C

1

D

5

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NTA JEE Main 16 th April 2018 Online - Question 82

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If $$\int \frac{\tan x}{1 + \tan x + \tan^2 x} dx = x - \frac{K}{\sqrt{A}} \tan^{-1}\left(\frac{K\tan x + 1}{\sqrt{A}}\right) + C$$, (C is a constant of integration), then the ordered pair (K, A) is equal to:

A

(2, 1)

B

(2, 3)

C

(-2, 1)

D

(-2, 3)

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NTA JEE Main 16 th April 2018 Online - Question 83

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If $$f(x) = \int_0^x t(\sin x - \sin t)dt$$, then:

A

$$f'''(x) - f''(x) = \cos x - 2x\sin x$$

B

$$f'''(x) + f''(x) - f'(x) = \cos x$$

C

$$f'''(x) + f''(x) = \sin x$$

D

$$f'''(x) + f'(x) = \cos x - 2x\sin x$$

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NTA JEE Main 16 th April 2018 Online - Question 84

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If the area of the region bounded by the curves, $$y = x^2$$, $$y = \frac{1}{x}$$ and the lines $$y = 0$$ and $$x = t$$ (t > 1) is 1 sq. unit, then t is equal to:

A

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

B

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

C

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

D

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

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NTA JEE Main 16 th April 2018 Online - Question 85

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The differential equation representing the family of ellipses having foci either on the x-axis or on the y-axis, center at the origin and passing through the point (0, 3) is:

A

$$xyy' - y^2 + 9 = 0$$

B

$$xyy'' + x(y')^2 - yy' = 0$$

C

$$xyy' + y^2 - 9 = 0$$

D

$$x + yy'' = 0$$

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NTA JEE Main 16 th April 2018 Online - Question 86

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Let $$\vec{a} = \hat{i} + \hat{j} + \hat{k}$$, $$\vec{c} = \hat{j} - \hat{k}$$ and a vector $$\vec{b}$$ be such that $$\vec{a} \times \vec{b} = \vec{c}$$ and $$\vec{a} \cdot \vec{b} = 3$$. Then $$|\vec{b}|$$ equals:

A

$$\frac{11}{3}$$

B

$$\frac{11}{\sqrt{3}}$$

C

$$\sqrt{\frac{11}{3}}$$

D

$$\frac{\sqrt{11}}{3}$$

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NTA JEE Main 16 th April 2018 Online - Question 87

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The sum of the intercepts on the coordinate axes of the plane passing through the point (-2, -2, 2) and containing the line joining the points (1, -1, 2) and (1, 1, 1) is:

A

4

B

12

C

-8

D

-4

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NTA JEE Main 16 th April 2018 Online - Question 88

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If the angle between the lines $$\frac{x}{2} = \frac{y}{2} = \frac{z}{1}$$ and $$\frac{5-x}{-2} = \frac{7y-14}{P} = \frac{z-3}{4}$$ is $$\cos^{-1}\left(\frac{2}{3}\right)$$, then P is equal to:

A

$$\frac{2}{7}$$

B

$$\frac{7}{2}$$

C

$$-\frac{4}{7}$$

D

$$-\frac{7}{4}$$

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NTA JEE Main 16 th April 2018 Online - Question 89

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Two different families A and B are blessed with equal number of children. There are 3 tickets to be distributed amongst the children of these families so that no child gets more than one ticket. If the probability that all the tickets go to the children of the family B is $$\frac{1}{12}$$, then the number of children in each family is:

A

6

B

5

C

3

D

4

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NTA JEE Main 16 th April 2018 Online - Question 90

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Let A, B and C be three events, which are pair-wise independent and $$\bar{E}$$ denotes the complement of an event E. If $$P(A \cap B \cap C) = 0$$ and $$P(C) > 0$$, then $$P\left[(\bar{A} \cap \bar{B}) | C\right]$$ is equal to:

A

$$P(\bar{A}) - P(B)$$

B

$$P(\bar{A}) - P(\bar{B})$$

C

$$P(\bar{A}) + P(\bar{B})$$

D

$$P(A) + P(\bar{B})$$

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