NTA JEE Main 9th April 2019 Shift 1

NTA JEE Main 9th April 2019 Shift 1 - Question 1

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In the density measurement of a cube, the mass and edge length are measured as $$(10.00 \pm 0.10)$$ kg and $$(0.10 \pm 0.01)$$ m, respectively. The error in the measurement of density is:

A

0.31

B

0.10

C

0.07

D

0.01

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NTA JEE Main 9th April 2019 Shift 1 - Question 2

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A ball is thrown vertically up (taken as +z-axis) from the ground. The correct momentum-height (p-h) diagram is:

A

B

C

D

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NTA JEE Main 9th April 2019 Shift 1 - Question 3

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The stream of a river is flowing with a speed of 2 km h$$^{-1}$$. A swimmer can swim at a speed of 4 km h$$^{-1}$$. The direction of the swimmer with respect to the flow of the river, to cross the river straight, is:

A

150°

B

90°

C

120°

D

60°

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NTA JEE Main 9th April 2019 Shift 1 - Question 4

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A uniform cable of mass $$M$$ and length $$L$$ is placed on a horizontal surface such that its $$\left(\frac{1}{n}\right)^{th}$$ part is hanging below the edge of the surface. To lift the hanging part of the cable upto the surface, the work done should be:

A

$$\frac{MgL}{2n^2}$$

B

$$\frac{MgL}{n^2}$$

C

$$nMgL$$

D

$$\frac{2MgL}{n^2}$$

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NTA JEE Main 9th April 2019 Shift 1 - Question 5

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A body of mass 2 kg makes an elastic collision with a second body at rest and continues to move in the original direction but with one fourth of its original speed. What is the mass of the second body?

A

1.8 kg

B

1.2 kg

C

1.0 kg

D

1.5 kg

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NTA JEE Main 9th April 2019 Shift 1 - Question 6

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A stationary horizontal disc is free to rotate about its axis. When a torque is applied on it, its kinetic energy as a function of $$\theta$$, where $$\theta$$ is the angle by which it has rotated, is given as $$k\theta^2$$. If its moment of inertia is I then the angular acceleration of the disc is:

A

$$\frac{2k}{I}\theta$$

B

$$\frac{k}{2I}\theta$$

C

$$\frac{k}{4I}\theta$$

D

$$\frac{k}{I}\theta$$

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NTA JEE Main 9th April 2019 Shift 1 - Question 7

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The following bodies are made to roll up (without slipping) the same inclined plane from a horizontal plane: (i) a ring of radius R, (ii) a solid cylinder of radius $$\frac{R}{2}$$ and (iii) a solid sphere of radius $$\frac{R}{4}$$. If, in each case, the speed of the center of mass at the bottom of the incline is same, the ratio of the maximum heights they climb is:

A

2 : 3 : 4

B

20 : 15 : 14

C

4 : 3 : 2

D

14 : 15 : 20

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NTA JEE Main 9th April 2019 Shift 1 - Question 8

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A solid sphere of mass M and radius a is surrounded by a uniform concentric spherical shell of thickness 2a and mass 2M. The gravitational field at distance 3a from the centre will be:

A

$$\frac{GM}{9a^2}$$

B

$$\frac{GM}{3a^2}$$

C

$$\frac{2GM}{3a^2}$$

D

$$\frac{2GM}{9a^2}$$

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NTA JEE Main 9th April 2019 Shift 1 - Question 9

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If 'M' is the mass of water that rises in a capillary tube of radius 'r', then mass of water which will rise in a capillary tube of radius '2r' is:

A

4 M

B

2 M

C

$$\frac{M}{2}$$

D

M

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NTA JEE Main 9th April 2019 Shift 1 - Question 10

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Following figure shows two processes A and B for a gas. If $$\Delta Q_A$$ and $$\Delta Q_B$$ are the amount of heat absorbed by the system in two cases, and $$\Delta U_A$$ and $$\Delta U_B$$ are changes in internal energies, respectively, then:

A

$$\Delta Q_A < \Delta Q_B; \; \Delta U_A < \Delta U_B$$

B

$$\Delta Q_A = \Delta Q_B; \; \Delta U_A = \Delta U_B$$

C

$$\Delta Q_A > \Delta Q_B; \; \Delta U_A = \Delta U_B$$

D

$$\Delta Q_A > \Delta Q_B; \; \Delta U_A > \Delta U_B$$

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NTA JEE Main 9th April 2019 Shift 1 - Question 11

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For given gas at 1 atm pressure, rms speed of the molecules is 200 m/s at 127°C. At 2 atm pressure and at 227°C, the rms speed of the molecules will be:

A

100 m/s

B

80 m/s

C

$$80\sqrt{5}$$ m/s

D

$$100\sqrt{5}$$ m/s

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NTA JEE Main 9th April 2019 Shift 1 - Question 12

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An HCl molecule has rotational, translational and vibrational motions. If the rms velocity of HCl molecules in its gaseous phase is $$\bar{v}$$, m is its mass and k$$_B$$ is Boltzmann's constant, then its temperature will be:

A

$$\frac{m\bar{v}^2}{5k_B}$$

B

$$\frac{m\bar{v}^2}{6k_B}$$

C

$$\frac{m\bar{v}^2}{3k_B}$$

D

$$\frac{m\bar{v}^2}{7k_B}$$

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NTA JEE Main 9th April 2019 Shift 1 - Question 13

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A simple pendulum oscillating in air has period T. The bob of the pendulum is completely immersed in a non-viscous liquid. The density of the liquid is $$\frac{1}{16}$$th of the material of the bob. If the bob is inside liquid all the time, its period of oscillation in this liquid is:

A

$$2T\sqrt{\frac{1}{10}}$$

B

$$2T\sqrt{\frac{1}{14}}$$

C

$$4T\sqrt{\frac{1}{15}}$$

D

$$4T\sqrt{\frac{1}{14}}$$

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NTA JEE Main 9th April 2019 Shift 1 - Question 14

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A string is clamped at both the ends and it is vibrating in its 4th harmonic. The equation of the stationary wave is $$y = 0.3 \sin(0.157x) \cos(200\pi t)$$. The length of the string is: (All quantities are in SI units.)

A

20 m

B

60 m

C

40 m

D

80 m

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NTA JEE Main 9th April 2019 Shift 1 - Question 15

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The pressure wave, $$P = 0.01 \sin[1000t - 3x]$$ N m$$^{-2}$$, corresponds to the sound produced by a vibrating blade on a day when atmospheric temperature is 0°C. On some other day when temperature is T, the speed of sound produced by the same blade and at the same frequency is found to be 336 m s$$^{-1}$$. Approximate value of T is:

A

12°C

B

15°C

C

4°C

D

11°C

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NTA JEE Main 9th April 2019 Shift 1 - Question 16

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A system of three charges are placed as shown in the figure:

If $$D >> d$$, the potential energy of the system is best given by:

A

$$\frac{1}{4\pi\varepsilon_0}\left[-\frac{q^2}{d} - \frac{qQd}{D^2}\right]$$

B

$$\frac{1}{4\pi\varepsilon_0}\left[+\frac{q^2}{d} + \frac{qQd}{D^2}\right]$$

C

$$\frac{1}{4\pi\varepsilon_0}\left[-\frac{q^2}{d} - \frac{qQd}{2D^2}\right]$$

D

$$\frac{1}{4\pi\varepsilon_0}\left[-\frac{q^2}{d} + \frac{2qQd}{D^2}\right]$$

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NTA JEE Main 9th April 2019 Shift 1 - Question 17

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A capacitor with capacitance 5 $$\mu$$F is charged to 5 $$\mu$$C. If the plates are pulled apart to reduce the capacitance to 2 $$\mu$$F, how much work is done?

A

$$6.25 \times 10^{-6}$$ J

B

$$2.55 \times 10^{-6}$$ J

C

$$2.16 \times 10^{-6}$$ J

D

$$3.75 \times 10^{-6}$$ J

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NTA JEE Main 9th April 2019 Shift 1 - Question 18

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A wire of resistance R is bent to form a square ABCD as shown in the figure. The effective resistance between E and C is: (E is mid-point of arm CD)

A

R

B

$$\frac{7}{64}R$$

C

$$\frac{1}{16}R$$

D

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

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NTA JEE Main 9th April 2019 Shift 1 - Question 19

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Determine the charge on the capacitor in the following circuit:

A

2 $$\mu$$C

B

60 $$\mu$$C

C

10 $$\mu$$C

D

200 $$\mu$$C

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NTA JEE Main 9th April 2019 Shift 1 - Question 20

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A moving coil galvanometer has resistance 50 $$\Omega$$ and it indicates full deflection at 4 mA current. A voltmeter is made using this galvanometer and a 5 k$$\Omega$$ resistance. The maximum voltage, that can be measured using this voltmeter, will be close to:

A

40 V

B

15 V

C

20 V

D

10 V

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NTA JEE Main 9th April 2019 Shift 1 - Question 21

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A rigid square loop of side 'a' and carrying current $$I_2$$ is lying on a horizontal surface near a long current $$I_1$$ carrying wire in the same plane as shown in figure. The net force on the loop due to the wire will be:

A

Attractive and equal to $$\frac{\mu_0 I_1 I_2}{3\pi}$$

B

Repulsive and equal to $$\frac{\mu_0 I_1 I_2}{4\pi}$$

C

Repulsive and equal to $$\frac{\mu_0 I_1 I_2}{2\pi}$$

D

Zero

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NTA JEE Main 9th April 2019 Shift 1 - Question 22

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A rectangular coil (Dimension 5 cm $$\times$$ 2.5 cm) with 100 turns, carrying a current of 3 A in the clock-wise direction, is kept centered at the origin and in the X-Z plane. A magnetic field of 1 T is applied along X-axis. If the coil is tilted through 45° about Z-axis, then the torque on the coil is:

A

0.42 N m

B

0.55 N m

C

0.38 N m

D

0.27 N m

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NTA JEE Main 9th April 2019 Shift 1 - Question 23

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The total number of turns and cross-section area in a solenoid is fixed. However, its length L is varied by adjusting the separation between windings. The inductance of solenoid will be proportional to:

A

L

B

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

C

$$\frac{1}{L}$$

D

$$L^2$$

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NTA JEE Main 9th April 2019 Shift 1 - Question 24

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The magnetic field of a plane electromagnetic wave is given by $$\vec{B} = B_0[\hat{i}\cos(kz - \omega t)] + B_1[\hat{j}\cos(kz + \omega t)]$$, where $$B_0 = 3 \times 10^{-5}$$ T and $$B_1 = 2 \times 10^{-6}$$ T. The RMS value of the force experienced by a stationary charge $$Q = 10^{-4}$$ C at z = 0 is closest to:

A

0.1 N

B

0.9 N

C

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

D

0.6 N

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NTA JEE Main 9th April 2019 Shift 1 - Question 25

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A concave mirror for face viewing has a focal length of 0.4 m. The distance at which you hold the mirror from your face in order to see your image upright with a magnification of 5 is:

A

1.60 m

B

0.16 m

C

0.32 m

D

0.24 m

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NTA JEE Main 9th April 2019 Shift 1 - Question 26

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The figure shows a Young's double slit experimental setup. It is observed that when a thin transparent sheet of thickness t and refractive index $$\mu$$ is put in front of one of the slits, the central maximum gets shifted by a distance equal to n fringe width. If the wavelength of light used is $$\lambda$$ then t will be:

A

$$\frac{D\lambda}{a(\mu - 1)}$$

B

$$\frac{n\lambda}{(\mu - 1)}$$

C

$$\frac{2n\lambda}{(\mu - 1)}$$

D

$$\frac{2D\lambda}{a(\mu - 1)}$$

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NTA JEE Main 9th April 2019 Shift 1 - Question 27

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The electric field of light wave is given as $$\vec{E} = 10^{-3} \cos\left(\frac{2\pi x}{5 \times 10^{-7}} - 2\pi \times 6 \times 10^{14}t\right) \hat{x} \frac{N}{C}$$. This light falls on a metal plate of work function 2 eV. The stopping potential of the photo-electrons is: Given, E (in eV) = $$\frac{12375}{\lambda(\text{in } \mathring{A})}$$

A

0.72 V

B

2.0 V

C

2.48 V

D

0.48 V

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NTA JEE Main 9th April 2019 Shift 1 - Question 28

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Taking the wavelength of first Balmer line in hydrogen spectrum (n = 3 to n = 2) as 660 nm, the wavelength of the 2nd Balmer line (n = 4 to n = 2) will be:

A

889.2 nm

B

488.9 nm

C

388.9 nm

D

642.7 nm

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NTA JEE Main 9th April 2019 Shift 1 - Question 29

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An NPN transistor is used in common emitter configuration as an amplifier with 1 k$$\Omega$$ load resistance. Signal voltage of 10 mV is applied across the base-emitter. This produces a 3 mA change in the collector current and 15 $$\mu$$A change in the base current of the amplifier. The input resistance and voltage gain are:

A

0.33 k$$\Omega$$, 1.5

B

0.67 k$$\Omega$$, 300

C

0.67 k$$\Omega$$, 200

D

0.33 k$$\Omega$$, 300

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NTA JEE Main 9th April 2019 Shift 1 - Question 30

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A signal A cos$$\omega$$t is transmitted using $$v_0 \sin\omega_0 t$$ as carrier wave. The correct amplitude modulated (AM) signal is:

A

$$v_0 \sin[\omega_0(1 + 0.01A\sin\omega t)t]$$

B

$$(v_0 + A)\cos\omega t \sin\omega_0 t$$

C

$$v_0 \sin\omega_0 t + A\cos\omega t$$

D

$$v_0 \sin\omega_0 t + \frac{A}{2}\sin(\omega_0 - \omega)t + \frac{A}{2}\sin(\omega_0 + \omega)t$$

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NTA JEE Main 9th April 2019 Shift 1 - Question 31

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For a reaction, N$$_2$$(g) + 3H$$_2$$(g) $$\rightarrow$$ 2NH$$_3$$(g), identify di-hydrogen (H$$_2$$) as a limiting reagent in the following reaction mixtures.

A

28 g of N$$_2$$ + 6 g of H$$_2$$

B

35 g of N$$_2$$ + 8 g of H$$_2$$

C

56 g of N$$_2$$ + 10 g of H$$_2$$

D

14 g of N$$_2$$ + 4 g of H$$_2$$

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NTA JEE Main 9th April 2019 Shift 1 - Question 32

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For any given series of spectral lines of atomic hydrogen, let $$\Delta\bar{v} = \bar{v}_{max} - \bar{v}_{min}$$ be the difference in maximum and minimum wave number in cm$$^{-1}$$. The ratio $$\Delta\bar{v}_{Lyman}/\Delta\bar{v}_{Balmer}$$ is:

A

5 : 4

B

27 : 5

C

4 : 1

D

9 : 4

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NTA JEE Main 9th April 2019 Shift 1 - Question 33

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The element having the greatest difference between its first and second ionization energies, is:

A

Ba

B

K

C

Ca

D

Sc

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NTA JEE Main 9th April 2019 Shift 1 - Question 34

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Among the following, the molecule expected to be stabilized by anion formation is: C$$_2$$, O$$_2$$, NO, F$$_2$$

A

F$$_2$$

B

C$$_2$$

C

O$$_2$$

D

NO

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NTA JEE Main 9th April 2019 Shift 1 - Question 35

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Consider the van der Waal's constants, a and b, for the following gases.

Which gas is expected to have the highest critical temperature?

A

Kr

B

Ar

C

Xe

D

Ne

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NTA JEE Main 9th April 2019 Shift 1 - Question 36

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Among the following, the set of parameters that represents path functions, is:
i) q + w
ii) q
iii) w
iv) H - TS

A

(ii) and (iii)

B

(i), (ii), and (iii)

C

(ii), (iii) and (iv)

D

(i) and (iv)

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NTA JEE Main 9th April 2019 Shift 1 - Question 37

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Magnesium powder burns in air to give:

A

MgO and Mg$$_3$$N$$_2$$

B

MgO only

C

Mg(NO$$_3$$)$$_2$$ and Mg$$_3$$N$$_2$$

D

MgO and Mg(NO$$_3$$)$$_2$$

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NTA JEE Main 9th April 2019 Shift 1 - Question 38

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C$$_{60}$$, an allotrope of carbon contains:

A

12 hexagons and 20 pentagons.

B

16 hexagons and 16 pentagons.

C

18 hexagons and 14 pentagons.

D

20 hexagons and 12 pentagons.

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NTA JEE Main 9th April 2019 Shift 1 - Question 39

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The correct IUPAC name of the following compound is:

A

5-chloro-4-methyl-1-nitrobenzene.

B

2-chloro-1-methyl-4-nitrobenzene.

C

2-methyl-5-nitro-1-chlorobenzene.

D

3-chloro-4-methyl-1-nitrobenzene.

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NTA JEE Main 9th April 2019 Shift 1 - Question 40

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The increasing order of reactivity of the following compounds towards aromatic electrophilic substitution reaction is:

A

A < B < C < D

B

B < C < A < D

C

D < B < A < C

D

D < A < C < B

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NTA JEE Main 9th April 2019 Shift 1 - Question 41

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The major product of the following reaction is:

A

B

C

D

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NTA JEE Main 9th April 2019 Shift 1 - Question 42

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The major product of the following reaction is:

A

CH$$_3$$CD(I)CHD(Cl)

B

CH$$_3$$CD(Cl)CHD(I)

C

CH$$_3$$CD$$_2$$CH(Cl)(I)

D

CH$$_3$$C(I)(Cl)CHD$$_2$$

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NTA JEE Main 9th April 2019 Shift 1 - Question 43

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Excessive release of CO$$_2$$ into the atmosphere results in:

A

global warming.

B

polar vortex.

C

depletion of ozone.

D

formation of smog.

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NTA JEE Main 9th April 2019 Shift 1 - Question 44

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The osmotic pressure of a dilute solution of an ionic compound XY in water is four times that of a solution of 0.01 M BaCl$$_2$$ in water. Assuming complete dissociation of the given ionic compounds in water, the concentration of XY (in mol L$$^{-1}$$) in solution is:

A

$$4 \times 10^{-2}$$

B

$$16 \times 10^{-4}$$

C

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

D

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

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NTA JEE Main 9th April 2019 Shift 1 - Question 45

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Liquid M and liquid N form an ideal solution. The vapour pressures of pure liquids M and N are 450 and 700 mmHg, respectively, at the same temperature. Then correct statements is:
(x$$_M$$ = Mole fraction of 'M' in solution; x$$_N$$ = Mole fraction of 'N' in solution; y$$_M$$ = Mole fraction of 'M' in vapour phase; y$$_N$$ = Mole fraction of 'N' in vapour phase)

A

$$\frac{x_M}{x_N} > \frac{y_M}{y_N}$$

B

$$\frac{x_M}{x_N} = \frac{y_M}{y_N}$$

C

$$\frac{x_M}{x_N} < \frac{y_M}{y_N}$$

D

$$(x_M - y_M) < (x_N - y_N)$$

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NTA JEE Main 9th April 2019 Shift 1 - Question 46

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The standard Gibbs energy for the given cell reaction in kJ mol$$^{-1}$$ at 298 K is:
Zn(s) + Cu$$^{2+}$$(aq) $$\rightarrow$$ Zn$$^{2+}$$(aq) + Cu(s),
E$$^0$$ = 2 V at 298 K
(Faraday's constant, F = 96000 C mol$$^{-1}$$)

A

-192

B

192

C

384

D

-384

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NTA JEE Main 9th April 2019 Shift 1 - Question 47

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The given plots represent the variation of the concentration of a reactant R with time for two different reactions (i) and (ii). The respective orders of the reaction are:

A

1, 1

B

0, 1

C

1, 0

D

0, 2

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NTA JEE Main 9th April 2019 Shift 1 - Question 48

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The aerosol is a kind of colloid in which:

A

solid is dispersed in gas.

B

liquid is dispersed in water.

C

gas is dispersed in solid.

D

gas is dispersed in liquid.

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NTA JEE Main 9th April 2019 Shift 1 - Question 49

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The ore that contains the metal in the form of fluoride is known as which of the following?

A

cryolite

B

magnetite

C

malachite

D

sphalerite

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NTA JEE Main 9th April 2019 Shift 1 - Question 50

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Match the catalysts (Column I) with products (Column II)

A

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

B

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

C

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

D

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

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NTA JEE Main 9th April 2019 Shift 1 - Question 51

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The correct order of the oxidation states of nitrogen in NO, N$$_2$$O, NO$$_2$$ and N$$_2$$O$$_3$$ is:

A

N$$_2$$O < NO < N$$_2$$O$$_3$$ < NO$$_2$$

B

NO$$_2$$ < N$$_2$$O$$_3$$ < NO < N$$_2$$O

C

NO$$_2$$ < NO < N$$_2$$O$$_3$$ < N$$_2$$O

D

N$$_2$$O < N$$_2$$O$$_3$$ < NO < NO$$_2$$

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NTA JEE Main 9th April 2019 Shift 1 - Question 52

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The number of water molecules not coordinated to copper ion directly in CuSO$$_4$$ . 5H$$_2$$O, is:

A

4

B

2

C

1

D

3

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NTA JEE Main 9th April 2019 Shift 1 - Question 53

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The degenerate orbitals of [Cr(H$$_2$$O)$$_6$$]$$^{3+}$$ are:

A

$$d_{x^2-y^2}$$ and $$d_{xy}$$

B

$$d_{z^2}$$ and $$d_{xz}$$

C

$$d_{yz}$$ and $$d_{z^2}$$

D

$$d_{xz}$$ and $$d_{yz}$$

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NTA JEE Main 9th April 2019 Shift 1 - Question 54

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The one that will show optical activity is (en = ethane-1, 2-diamine):

A

B

C

D

Option (4)

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NTA JEE Main 9th April 2019 Shift 1 - Question 55

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The major product of the following reaction is:

1. PBr$$_3$$
2. KOH (alc.)

A

B

C

D

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NTA JEE Main 9th April 2019 Shift 1 - Question 56

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The organic compound that gives following qualitative analysis is:

A

B

C

D

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NTA JEE Main 9th April 2019 Shift 1 - Question 57

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The major product of the following reaction is:

A

B

C

D

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NTA JEE Main 9th April 2019 Shift 1 - Question 58

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The major product of the following reaction is:
CH$$_3$$CH = CHCO$$_2$$CH$$_3$$ $$\xrightarrow{\text{LiAlH}_4}$$

A

CH$$_3$$CH$$_2$$CH$$_2$$CO$$_2$$CH$$_3$$

B

CH$$_3$$CH = CHCH$$_2$$OH

C

CH$$_3$$CH$$_2$$CH$$_2$$CH$$_2$$OH

D

CH$$_3$$CH$$_2$$CH$$_2$$CHO

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NTA JEE Main 9th April 2019 Shift 1 - Question 59

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Aniline dissolved in dilute HCl is reacted with sodium nitrite at 0°C. This solution was added dropwise to a solution containing an equimolar mixture of aniline and phenol in dilute HCl. The structure of the major product is:

A

B

C

D

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NTA JEE Main 9th April 2019 Shift 1 - Question 60

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Which of the following statements is not true about sucrose?

A

It is also named as invert sugar.

B

It is a non-reducing sugar.

C

The glycosidic linkage is present between C$$_1$$ of $$\alpha$$-glucose and C$$_1$$ of $$\beta$$-fructose.

D

On hydrolysis, it produces glucose and fructose.

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NTA JEE Main 9th April 2019 Shift 1 - Question 61

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Let $$p, q \in Q$$. If $$2 - \sqrt{3}$$ is a root of the quadratic equation $$x^2 + px + q = 0$$, then:

A

$$p^2 - 4q + 12 = 0$$

B

$$q^2 + 4p + 14 = 0$$

C

$$p^2 - 4q - 12 = 0$$

D

$$q^2 - 4p - 16 = 0$$

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NTA JEE Main 9th April 2019 Shift 1 - Question 62

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All the points in the set $$S = \left\{\frac{\alpha + i}{\alpha - i}, \alpha \in R\right\}$$, $$i = \sqrt{-1}$$ lie on a:

A

straight line whose slope is -1

B

circle whose radius is $$\sqrt{2}$$

C

circle whose radius is 1

D

straight line whose slope is 1

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NTA JEE Main 9th April 2019 Shift 1 - Question 63

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A committee of 11 members is to be formed from 8 males and 5 females. If $$m$$ is the number of ways the committee is formed with at least 6 males and $$n$$ is the number of ways the committee is formed with at least 3 females, then:

A

$$m = n = 68$$

B

$$n = m - 8$$

C

$$m = n = 78$$

D

$$m + n = 68$$

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NTA JEE Main 9th April 2019 Shift 1 - Question 64

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Let the sum of the first $$n$$ terms of a non-constant A.P., $$a_1, a_2, a_3, \ldots, a_n$$ be $$50n + \frac{n(n-7)}{2}A$$, where A is a constant. If $$d$$ is the common difference of this A.P., then the ordered pair $$(d, a_{50})$$ is equal to:

A

$$(50, 50 + 46A)$$

B

$$(A, 50 + 45A)$$

C

$$(50, 50 + 45A)$$

D

$$(A, 50 + 46A)$$

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NTA JEE Main 9th April 2019 Shift 1 - Question 65

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If the fourth term in the Binomial expansion of $$\left(\frac{2}{x} + x^{\log_8 x}\right)^6$$, $$(x > 0)$$ is $$20 \times 8^7$$, then a value of $$x$$ is:

A

$$8^{-2}$$

B

8

C

$$8^3$$

D

$$8^2$$

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NTA JEE Main 9th April 2019 Shift 1 - Question 66

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The value of $$\cos^2 10° - \cos 10° \cos 50° + \cos^2 50°$$ is:

A

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

B

$$\frac{3}{4} + \cos 20°$$

C

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

D

$$\frac{3}{2}(1 + \cos 20°)$$

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NTA JEE Main 9th April 2019 Shift 1 - Question 67

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Let $$S = \{\theta \in [-2\pi, 2\pi] : 2\cos^2\theta + 3\sin\theta = 0\}$$. Then the sum of the elements of S is:

A

$$\pi$$

B

$$\frac{13\pi}{6}$$

C

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

D

$$2\pi$$

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NTA JEE Main 9th April 2019 Shift 1 - Question 68

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Slope of a line passing through $$P(2, 3)$$ and intersecting the line $$x + y = 7$$ at a distance of 4 units from $$P$$, is:

A

$$\frac{\sqrt{7} - 1}{\sqrt{7} + 1}$$

B

$$\frac{1 - \sqrt{7}}{1 + \sqrt{7}}$$

C

$$\frac{\sqrt{5} - 1}{\sqrt{5} + 1}$$

D

$$\frac{1 - \sqrt{5}}{1 + \sqrt{5}}$$

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NTA JEE Main 9th April 2019 Shift 1 - Question 69

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If a tangent to the circle $$x^2 + y^2 = 1$$ intersects the coordinate axes at distinct points P and Q, then the locus of the mid-point of PQ is:

A

$$x^2 + y^2 - 16x^2y^2 = 0$$

B

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

C

$$x^2 + y^2 - 2xy = 0$$

D

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

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NTA JEE Main 9th April 2019 Shift 1 - Question 70

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If one end of a focal chord of the parabola, $$y^2 = 16x$$ is at $$(1, 4)$$, then the length of this focal chord is:

A

24

B

25

C

22

D

20

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NTA JEE Main 9th April 2019 Shift 1 - Question 71

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If the line $$y = mx + 7\sqrt{3}$$ is normal to the hyperbola $$\frac{x^2}{24} - \frac{y^2}{18} = 1$$, then a value of $$m$$ is:

A

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

B

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

C

$$\frac{\sqrt{15}}{2}$$

D

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

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NTA JEE Main 9th April 2019 Shift 1 - Question 72

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For any two statement $$p$$ and $$q$$, the negative of the expression $$p \lor (\sim p \land q)$$ is:

A

$$\sim p \lor \sim q$$

B

$$p \land q$$

C

$$\sim p \land \sim q$$

D

$$p \leftrightarrow q$$

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NTA JEE Main 9th April 2019 Shift 1 - Question 73

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If the standard deviation of the numbers $$-1, 0, 1, k$$ is $$\sqrt{5}$$ where $$k \gt 0$$, then $$k$$ is equal to:

A

$$\sqrt{6}$$

B

$$4\sqrt{\frac{5}{3}}$$

C

$$2\sqrt{\frac{10}{3}}$$

D

$$2\sqrt{6}$$

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NTA JEE Main 9th April 2019 Shift 1 - Question 74

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If $$\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix} \cdots \begin{bmatrix} 1 & n-1 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 78 \\ 0 & 1 \end{bmatrix}$$, then the inverse of $$\begin{bmatrix} 1 & n \\ 0 & 1 \end{bmatrix}$$ is:

A

$$\begin{bmatrix} 1 & -12 \\ 0 & 1 \end{bmatrix}$$

B

$$\begin{bmatrix} 1 & 0 \\ 12 & 1 \end{bmatrix}$$

C

$$\begin{bmatrix} 1 & 0 \\ 13 & 1 \end{bmatrix}$$

D

$$\begin{bmatrix} 1 & -13 \\ 0 & 1 \end{bmatrix}$$

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NTA JEE Main 9th April 2019 Shift 1 - Question 75

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Let $$\alpha$$ and $$\beta$$ be the roots of the equation $$x^2 + x + 1 = 0$$. Then for $$y \neq 0$$ in R, $$\begin{vmatrix} y+1 & \alpha & \beta \\ \alpha & y+\beta & 1 \\ \beta & 1 & y+\alpha \end{vmatrix}$$ is equal to:

A

$$y^3$$

B

$$y(y^2 - 1)$$

C

$$y^3 - 1$$

D

$$y(y^2 - 3)$$

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NTA JEE Main 9th April 2019 Shift 1 - Question 76

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If the function $$f: R - \{1, -1\} \rightarrow A$$ defined by $$f(x) = \frac{x^2}{1 - x^2}$$, is surjective, then $$A$$ is equal to:

A

$$[0, \infty)$$

B

$$R - \{-1\}$$

C

$$R - [-1, 0)$$

D

$$R - (-1, 0)$$

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NTA JEE Main 9th April 2019 Shift 1 - Question 77

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Let $$f(x) = 15 - |x - 10|$$; $$x \in R$$. Then the set of all values of $$x$$, at which the function $$g(x) = f(f(x))$$ is not differentiable, is:

A

{5, 10, 15}

B

{10}

C

{10, 15}

D

{5, 10, 15, 20}

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NTA JEE Main 9th April 2019 Shift 1 - Question 78

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If the function $$f$$ defined on $$\left(\frac{\pi}{6}, \frac{\pi}{3}\right)$$ by $$f(x) = \begin{cases} \frac{\sqrt{2}\cos x - 1}{\cot x - 1}, & x \neq \frac{\pi}{4} \\ k, & x = \frac{\pi}{4} \end{cases}$$ is continuous, then $$k$$ is equal to:

A

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

B

1

C

2

D

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

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NTA JEE Main 9th April 2019 Shift 1 - Question 79

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Let $$\sum_{k=1}^{10} f(a + k) = 16(2^{10} - 1)$$, where the function $$f$$ satisfies $$f(x + y) = f(x)f(y)$$ for all natural numbers $$x$$, $$y$$ and $$f(1) = 2$$. Then the natural number 'a' is:

A

3

B

16

C

4

D

2

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NTA JEE Main 9th April 2019 Shift 1 - Question 80

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If $$f(x)$$ is a non-zero polynomial of degree four, having local extreme points at $$x = -1, 0, 1$$; then the set $$S = \{x \in R : f(x) = f(0)\}$$ contains exactly:

A

Two irrational and two rational numbers

B

Four rational numbers

C

Two irrational and one rational number

D

Four irrational numbers

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NTA JEE Main 9th April 2019 Shift 1 - Question 81

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If the tangent to the curve, $$y = x^3 + ax - b$$ at the point $$(1, -5)$$ is perpendicular to the line, $$-x + y + 4 = 0$$, then which one of the following points lies on the curve?

A

$$(2, -2)$$

B

$$(2, -1)$$

C

$$(-2, 1)$$

D

$$(-2, 2)$$

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NTA JEE Main 9th April 2019 Shift 1 - Question 82

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Let $$S$$ be the set of all values of $$x$$ for which the tangent to the curve $$y = f(x) = x^3 - x^2 - 2x$$ at $$(x, y)$$ is parallel to the line segment joining the points $$(1, f(1))$$ and $$(-1, f(-1))$$, then $$S$$ is equal to:

A

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

B

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

C

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

D

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

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NTA JEE Main 9th April 2019 Shift 1 - Question 83

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$$\int \sec^2 x \cdot \cot^{4/3} x \, dx$$ is equal to:

A

$$3\tan^{-1/3}x + C$$

B

$$-\frac{3}{4}\tan^{-4/3}x + C$$

C

$$-3\tan^{-1/3}x + C$$

D

$$-3\cot^{-1/3}x + C$$

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NTA JEE Main 9th April 2019 Shift 1 - Question 84

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The value of $$\int_0^{\pi/2} \frac{\sin^3 x}{\sin x + \cos x} dx$$ is:

A

$$\frac{\pi - 1}{2}$$

B

$$\frac{\pi - 2}{8}$$

C

$$\frac{\pi - 1}{4}$$

D

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

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NTA JEE Main 9th April 2019 Shift 1 - Question 85

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The area (in sq. units) of the region $$A = \{(x, y) : x^2 \le y \le x + 2\}$$ is:

A

$$\frac{13}{6}$$

B

$$\frac{31}{6}$$

C

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

D

$$\frac{10}{3}$$

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NTA JEE Main 9th April 2019 Shift 1 - Question 86

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The solution of the differential equation $$x\frac{dy}{dx} + 2y = x^2$$, $$(x \neq 0)$$ with $$y(1) = 1$$, is:

A

$$y = \frac{x^3}{5} + \frac{1}{5x^2}$$

B

$$y = \frac{3}{4}x^2 + \frac{1}{4x^2}$$

C

$$y = \frac{x^2}{4} + \frac{3}{4x^2}$$

D

$$y = \frac{4}{5}x^3 + \frac{1}{5x^2}$$

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NTA JEE Main 9th April 2019 Shift 1 - Question 87

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Let $$\vec{\alpha} = 3\hat{i} + \hat{j}$$ and $$\vec{\beta} = 2\hat{i} - \hat{j} + 3\hat{k}$$. If $$\vec{\beta} = \vec{\beta_1} - \vec{\beta_2}$$, where $$\vec{\beta_1}$$ is parallel to $$\vec{\alpha}$$ and $$\vec{\beta_2}$$ is perpendicular to $$\vec{\alpha}$$, then $$\vec{\beta_1} \times \vec{\beta_2}$$ is equal to:

A

$$\frac{1}{2}(-3\hat{i} + 9\hat{j} + 5\hat{k})$$

B

$$3\hat{i} - 9\hat{j} - 5\hat{k}$$

C

$$-3\hat{i} + 9\hat{j} + 5\hat{k}$$

D

$$\frac{1}{2}(3\hat{i} - 9\hat{j} + 5\hat{k})$$

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NTA JEE Main 9th April 2019 Shift 1 - Question 88

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A plane passing through the points $$(0, -1, 0)$$ and $$(0, 0, 1)$$ and making an angle $$\frac{\pi}{4}$$ with the plane $$y - z + 5 = 0$$, also passes through the point:

A

$$(\sqrt{2}, -1, 4)$$

B

$$(\sqrt{2}, 1, 4)$$

C

$$(-\sqrt{2}, -1, -4)$$

D

$$(-\sqrt{2}, 1, -4)$$

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NTA JEE Main 9th April 2019 Shift 1 - Question 89

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If the line, $$\frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 2}{4}$$ meets the plane, $$x + 2y + 3z = 15$$ at a point P, then the distance of P from the origin is:

A

$$2\sqrt{5}$$

B

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

C

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

D

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

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NTA JEE Main 9th April 2019 Shift 1 - Question 90

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Four persons can hit a target correctly with probabilities $$\frac{1}{2}$$, $$\frac{1}{3}$$, $$\frac{1}{4}$$ and $$\frac{1}{8}$$ respectively. If all hit at the target independently, then the probability that the target would be hit, is:

A

$$\frac{25}{192}$$

B

$$\frac{7}{32}$$

C

$$\frac{1}{192}$$

D

$$\frac{25}{32}$$

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