NTA JEE Main 7th January 2020 Shift 1

NTA JEE Main 7th January 2020 Shift 1 - Question 1

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A 60HP electric motor lifts an elevator having a maximum total load capacity of 2000 kg. If the frictional force on the elevator is 4000 N, the speed of the elevator at full load is close to: (1 HP = 746 W, $$g = 10$$ m s$$^{-2}$$)

A

1.7 m s$$^{-1}$$

B

1.9 m s$$^{-1}$$

C

1.5 m s$$^{-1}$$

D

2.0 m s$$^{-1}$$

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NTA JEE Main 7th January 2020 Shift 1 - Question 2

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Three point particles of masses 1.0 kg, 1.5 kg and 2.5 kg are placed at three corners of a right angle triangle of sides 4.0 cm, 3.0 cm and 5.0 cm as shown in the figure. The centre of mass of the system is at a point:

A

0.6 cm right and 2.0 cm above 1 kg mass.

B

1.5 cm right and 1.2 cm above 1 kg mass.

C

2.0 cm right and 0.9 cm above 1 kg mass.

D

0.9 cm right and 2.0 cm above 1 kg mass.

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NTA JEE Main 7th January 2020 Shift 1 - Question 3

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As shown in the figure, a bob of mass m is tied to a massless string whose other end portion is wound on a fly wheel (disc) of radius r and mass m. When released from rest the bob starts falling vertically. When it has covered a distance of h, the angular speed of the wheel will be:

A

$$\frac{1}{r}\sqrt{\frac{4gh}{3}}$$

B

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

C

$$\frac{1}{r}\sqrt{\frac{2gh}{3}}$$

D

$$r\sqrt{\frac{3}{4gh}}$$

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NTA JEE Main 7th January 2020 Shift 1 - Question 4

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The radius of gyration of a uniform rod of length $$l$$, about an axis passing through a point $$\frac{l}{4}$$ away from the centre of the rod, and perpendicular to it, is:

A

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

B

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

C

$$\sqrt{\frac{7}{48}}\,l$$

D

$$\sqrt{\frac{3}{8}}\,l$$

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NTA JEE Main 7th January 2020 Shift 1 - Question 5

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A satellite of mass $$M$$ is launched vertically upwards with an initial speed $$u$$ from the surface of the earth. After it reaches height $$R$$ ($$R$$ = radius of the earth), it ejects a rocket of mass $$\frac{M}{10}$$ so that subsequently the satellite moves in a circular orbit. The kinetic energy of the rocket is ($$G$$ is the gravitational constant; $$M_e$$ is the mass of the earth):

A

$$\frac{M}{20}\left(u^2 + \frac{113}{200}\frac{GM_e}{R}\right)$$

B

$$5M\left(u^2 - \frac{119}{200}\frac{GM_e}{R}\right)$$

C

$$\frac{3M}{8}\left(u + \sqrt{\frac{5GM_e}{6R}}\right)^2$$

D

$$\frac{M}{20}\left(u - \sqrt{\frac{2GM_e}{3R}}\right)^2$$

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NTA JEE Main 7th January 2020 Shift 1 - Question 6

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Speed of a transverse wave on a straight wire (mass 6.0 g, length 60 cm and area of cross-section 1.0 mm$$^2$$) is 90 m s$$^{-1}$$. If the Young's modulus of wire is $$16 \times 10^{11}$$ N m$$^{-2}$$, the extension of wire over its natural length is:

A

0.03 mm

B

0.02 mm

C

0.04 mm

D

0.01 mm

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NTA JEE Main 7th January 2020 Shift 1 - Question 7

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A litre of dry air at STP expands adiabatically to a volume of 3 litres. If $$\gamma = 1.40$$, the work done by air is: ($$3^{1.4} = 4.6555$$) [Take air to be an ideal gas]

A

60.7 J

B

90.5 J

C

100.8 J

D

48 J

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NTA JEE Main 7th January 2020 Shift 1 - Question 8

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Two moles of an ideal gas, with $$\frac{C_p}{C_v} = \frac{5}{3}$$, are mixed with three moles of another ideal gas $$\frac{C_p}{C_v} = \frac{4}{3}$$. The value of $$\frac{C_p}{C_v}$$ for the mixture is

A

1.45

B

1.50

C

1.47

D

1.42

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NTA JEE Main 7th January 2020 Shift 1 - Question 9

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Two infinite planes each with uniform surface charge density $$+\sigma$$ are kept in such a way that the angle between them is 30$$^\circ$$. The electric field in the region shown between them is given by:

A

$$\frac{\sigma}{2\varepsilon_0}\left[\left(1+\sqrt{3}\right)\hat{y} - \frac{\hat{x}}{2}\right]$$

B

$$\frac{\sigma}{\varepsilon_0}\left[\left(1+\frac{\sqrt{3}}{2}\right)\hat{y} + \frac{\hat{x}}{2}\right]$$

C

$$\frac{\sigma}{2\varepsilon_0}\left[\left(1+\sqrt{3}\right)\hat{y} + \frac{\hat{x}}{2}\right]$$

D

$$\frac{\sigma}{2\varepsilon_0}\left[\left(1-\frac{\sqrt{3}}{2}\right)\hat{y} - \frac{\hat{x}}{2}\right]$$

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NTA JEE Main 7th January 2020 Shift 1 - Question 10

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A parallel plate capacitor has plates of area A separated by distance d between them. It is filled with a dielectric which has a dielectric constant that varies as $$K(x) = K_0(1 + \alpha x)$$ where $$x$$ is the distance measured from one of the plates. If $$(\alpha d) \ll 1$$, the total capacitance of the system is best given by the expression:

A

$$\frac{AK_0\varepsilon_0}{d}\left(1 + \frac{\alpha d}{2}\right)$$

B

$$\frac{AK_0\varepsilon_0}{d}\left[1 + \left(\frac{\alpha d}{2}\right)^2\right]$$

C

$$\frac{AK_0\varepsilon_0}{d}\left(1 + \frac{\alpha^2 d^2}{2}\right)$$

D

$$\frac{AK_0\varepsilon_0}{d}(1 + \alpha d)$$

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NTA JEE Main 7th January 2020 Shift 1 - Question 11

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The current $$I_1$$ (in A) flowing through 1$$\Omega$$ resistor in the following circuit is:

A

0.4

B

0.5

C

0.2

D

0.25

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NTA JEE Main 7th January 2020 Shift 1 - Question 12

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A long solenoid of radius R carries a time $$(t)$$ dependent current $$I(t) = I_0 t(1 - t)$$. A ring of radius 2R is placed coaxially near its middle. During the time interval $$0 \le t \le 1$$, the induced current $$(I_R)$$ and the induced EMF $$(V_R)$$ in the ring change as:

A

Direction of $$I_R$$ remains unchanged and $$V_R$$ is maximum at $$t = 0.5$$

B

At $$t = 0.25$$ direction of $$I_R$$ reverses and $$V_R$$ is maximum

C

Direction of $$I_R$$ remains unchanged and $$V_R$$ is zero at $$t = 0.25$$

D

At $$t = 0.5$$ direction of $$I_R$$ reverses and $$V_R$$ is zero

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NTA JEE Main 7th January 2020 Shift 1 - Question 13

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Consider a circular coil of wire carrying constant current I, forming a magnetic dipole. The magnetic flux through an infinite plane that contains the circular coil and excluding the circular coil area is given by $$\phi_i$$. The magnetic flux through the area of the circular coil area is given by $$\phi_0$$. Which of the following option is correct?

A

$$\phi_i = \phi_0$$

B

$$\phi_i > \phi_0$$

C

$$\phi_i < \phi_0$$

D

$$\phi_i = -\phi_0$$

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NTA JEE Main 7th January 2020 Shift 1 - Question 14

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A LCR circuit behaves like a clamped harmonic oscillator. Comparing it with a physical spring-mass damped oscillator having damping constant 'b', the correct equivalence would be:

A

$$L \leftrightarrow m, C \leftrightarrow k, R \leftrightarrow b$$

B

$$L \leftrightarrow \frac{1}{b}, C \leftrightarrow \frac{1}{m}, R \leftrightarrow \frac{1}{k}$$

C

$$L \leftrightarrow k, C \leftrightarrow b, R \leftrightarrow m$$

D

$$L \leftrightarrow m, C \leftrightarrow \frac{1}{k}, R \leftrightarrow b$$

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NTA JEE Main 7th January 2020 Shift 1 - Question 15

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If the magnetic field in a plane electromagnetic wave is given by $$\vec{B} = 3 \times 10^{-8} \sin(1.6 \times 10^3 x + 48 \times 10^{10} t)\hat{j}\,T$$, then what will be the expression for electric field?

A

$$\vec{E} = \left(60 \sin(1.6 \times 10^3 x + 48 \times 10^{10} t)\hat{k}\; \frac{V}{m}\right)$$

B

$$\vec{E} = \left(9 \sin(1.6 \times 10^3 x + 48 \times 10^{10} t)\hat{k}\; \frac{V}{m}\right)$$

C

$$\vec{E} = \left(3 \times 10^{-8} \sin(1.6 \times 10^3 x + 48 \times 10^{10} t)\hat{j}\; \frac{V}{m}\right)$$

D

$$\vec{E} = \left(3 \times 10^{-8} \sin(1.6 \times 10^3 x + 48 \times 10^{10} t)\hat{i}\; \frac{V}{m}\right)$$

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NTA JEE Main 7th January 2020 Shift 1 - Question 16

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If we need a magnification of 375 from a compound microscope of tube length 150mm and an objective of focal length 5mm, the focal length of the eye-piece, should be close to:

A

22 mm

B

2 mm

C

4 mm

D

33 mm

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NTA JEE Main 7th January 2020 Shift 1 - Question 17

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A polarizer - analyser set is adjusted such that the intensity of light coming out of the analyser is just 36% of the original intensity. Assuming that the polarizer - analyser set does not absorb any light, the angle by which the analyser needs to be rotated further, to reduce the output intensity to zero, is ($$\sin^{-1}\left(\frac{3}{5}\right) = 37^\circ$$)

A

53$$^\circ$$

B

37$$^\circ$$

C

90$$^\circ$$

D

45$$^\circ$$

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NTA JEE Main 7th January 2020 Shift 1 - Question 18

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Visible light of wavelength $$6000 \times 10^{-8}$$ cm falls normally on a single slit and produces a diffraction pattern. It is found that the second diffraction minimum is at 60$$^\circ$$ from the central maximum. If the first minimum is produced at $$\theta_1$$, then $$\theta_1$$ is close to

A

20$$^\circ$$

B

30$$^\circ$$

C

25$$^\circ$$

D

45$$^\circ$$

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NTA JEE Main 7th January 2020 Shift 1 - Question 19

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The time period of revolution of electron in its ground state orbit in a hydrogen atom is $$1.6 \times 10^{-16}$$ s. The frequency of revolution of the electron in its first excited state (in s$$^{-1}$$) is:

A

$$1.6 \times 10^{14}$$

B

$$7.8 \times 10^{14}$$

C

$$6.2 \times 10^{15}$$

D

$$5.6 \times 10^{12}$$

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NTA JEE Main 7th January 2020 Shift 1 - Question 20

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Which of the following gives a reversible operation?

A

B

C

D

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NTA JEE Main 7th January 2020 Shift 1 - Question 21

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A particle (m = 1 kg) slides down a frictionless track (AOC) starting from rest at a point A (height 2m). After reaching C, the particle continues to move freely in air as a projectile. When it reaches its highest point P (height 1m), the kinetic energy of the particle (in J) is: (Figure drawn is schematic and not to scale; take $$g = 10$$ ms$$^{-2}$$)

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NTA JEE Main 7th January 2020 Shift 1 - Question 22

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A non-isotropic solid metal cube has coefficients of linear expansion as: $$5 \times 10^{-5}$$/$$^\circ$$C along the x-axis and $$5 \times 10^{-6}$$/$$^\circ$$C along the y and the z-axis. If the coefficient of volume expansion of the solid is $$C \times 10^{-6}$$/$$^\circ$$C then the value of C is

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NTA JEE Main 7th January 2020 Shift 1 - Question 23

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A Carnot engine operates between two reservoirs of temperatures 900K and 300K. The engine performs 1200J of work per cycle. The heat energy (in J) delivered by the engine to the low temperature reservoir, in a cycle, is

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NTA JEE Main 7th January 2020 Shift 1 - Question 24

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A loop ABCDEFA of straight edges has six corner points A(0, 0, 0), B(5, 0, 0), C(5, 5, 0), D(0, 5, 0), E(0, 5, 5) and F(0, 0, 5). The magnetic field in this region is $$\vec{B} = (3\hat{i} + 4\hat{k})$$ T. The quantity of flux through the loop ABCDEFA (in Wb) is

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NTA JEE Main 7th January 2020 Shift 1 - Question 25

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A beam of electromagnetic radiation of intensity $$6.4 \times 10^{-5}$$ W/cm$$^2$$ is comprised of wavelength, $$\lambda = 310$$ nm. It falls normally on a metal (work function $$\varphi = 2$$ eV) of surface area of 1 cm$$^2$$. If one in $$10^3$$ photons ejects an electron, total number of electrons ejected in 1s is $$10^x$$. ($$hc = 1240$$ eVnm, $$1$$ eV $$= 1.6 \times 10^{-19}$$ J), then $$x$$ is

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NTA JEE Main 7th January 2020 Shift 1 - Question 26

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Amongst the following statements, that which was not proposed by Dalton was:

A

chemical reactions involve reorganization of atoms. These are neither created nor destroyed in a chemical reaction.

B

all the atoms of a given element have identical properties including identical mass. Atoms of different elements differ in mass.

C

when gases combine or reproduced in a chemical reaction they do so in a simple ratio by volume provided all gases are at the same T & P.

D

matter consists of indivisible atoms.

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NTA JEE Main 7th January 2020 Shift 1 - Question 27

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The number of orbitals associated with quantum numbers $$n = 5$$, $$m_s = +\frac{1}{2}$$ is:

A

11

B

25

C

50

D

15

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NTA JEE Main 7th January 2020 Shift 1 - Question 28

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The electron gain enthalpy (in kJ/mol) of fluorine, chlorine, bromine and iodine, respectively, are

A

$$-296, -325, -333$$ and $$-349$$

B

$$-349, -333, -325$$ and $$-296$$

C

$$-333, -349, -325$$ and $$-296$$

D

$$-333, -325, -349$$ and $$-296$$

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NTA JEE Main 7th January 2020 Shift 1 - Question 29

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The dipole moments of CCl$$_4$$, CHCl$$_3$$ and CH$$_4$$ are in the order

A

CHCl$$_3$$ < CH$$_4$$ = CCl$$_4$$

B

CCl$$_4$$ < CH$$_4$$ < CHCl$$_3$$

C

CH$$_4$$ < CCl$$_4$$ < CHCl$$_3$$

D

CH$$_4$$ = CCl$$_4$$ < CHCl$$_3$$

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NTA JEE Main 7th January 2020 Shift 1 - Question 30

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The relative strength of the interionic/ intermolecular forces in a decreasing order is:

A

dipole-dipole > ion-dipole > ion-ion

B

ion-dipole > ion-ion > dipole-dipole

C

ion-dipole > dipole-dipole > ion-ion

D

ion-ion > ion-dipole > dipole-dipole

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NTA JEE Main 7th January 2020 Shift 1 - Question 31

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Oxidation number of potassium in K$$_2$$O, K$$_2$$O$$_2$$ and KO$$_2$$, respectively, is:

A

$$+2, +1$$ and $$+\frac{1}{2}$$

B

$$+1, +1$$ and $$+1$$

C

$$+1, +4$$ and $$+2$$

D

$$+1, +2$$ and $$+4$$

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NTA JEE Main 7th January 2020 Shift 1 - Question 32

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In comparison to the zeolite process for the removal of permanent hardness, the synthetic resin method is

A

less efficient as it exchanges only anions.

B

more efficient as it can exchange both cations as well as anions.

C

less efficient as the resins cannot be regenerated.

D

more efficient as it can exchange only cations.

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NTA JEE Main 7th January 2020 Shift 1 - Question 33

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A solution of m-chloroaniline, m-chlorophenol and m-chlorobenzoic acid in ethyl acetate was extracted initially with a saturated solution of NaHCO$$_3$$ to give fraction A. The left over organic phase was extracted with dilute NaOH solution to give fraction B. The final organic layer was labelled as fraction C. Fractions A, B and C, contain respectively:

A

m-chlorobenzoic acid, m-chloroaniline and m-chlorophenol

B

m-chlorobenzoic acid, m-chlorophenol and m-chloroaniline

C

m-chlorophenol, m-chlorobenzoic acid and m-chloroaniline

D

m-chloroaniline, m-chlorobenzoic acid and m-chlorophenol

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NTA JEE Main 7th January 2020 Shift 1 - Question 34

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The increasing order of pK$$_b$$ for the following compounds will be:

A

(B) < (C) < (A)

B

(A) < (B) < (C)

C

(C) < (A) < (B)

D

(B) < (A) < (C)

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NTA JEE Main 7th January 2020 Shift 1 - Question 35

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Consider the following reactions:
(a) $$(CH_3)_3CCH(OH)CH_3 \xrightarrow{\text{conc.} H_2SO_4}$$
(b) $$(CH_3)_2CHCH(Br)CH_3 \xrightarrow{\text{alc. KOH}}$$
(c) $$(CH_3)_2CHCH(Br)CH_3 \xrightarrow{(CH_3)_3O^\ominus K^\oplus}$$
(d)

Which of the reaction(s) will not produce Saytzeff product?

A

(a), (c) and (d)

B

(d) only

C

(c) only

D

(b) and (d)

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NTA JEE Main 7th January 2020 Shift 1 - Question 36

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At 35 $$^\circ$$C, the vapour pressure of CS$$_2$$ is 512 mmHg and that of acetone is 144 mmHg. A solution of CS$$_2$$ in acetone has a total vapour pressure of 600 mmHg. The false statement amongst the following is:

A

Raoult's law is not obeyed by this system

B

a mixture of 100 mL CS$$_2$$ and 100 mL acetone has a volume < 200 mL

C

CS$$_2$$ and acetone are less attracted to each other than to themselves

D

heat must be absorbed in order to produce the solution at 35$$^\circ$$C

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NTA JEE Main 7th January 2020 Shift 1 - Question 37

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Given that the standard potentials (E$$^\circ$$) of Cu$$^{2+}$$/Cu and Cu$$^+$$/Cu are 0.34V and 0.522V respectively, the E$$^\circ$$ of Cu$$^{2+}$$/Cu$$^+$$ is:

A

0.182V

B

+0.158V

C

-0.182V

D

-0.158V

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NTA JEE Main 7th January 2020 Shift 1 - Question 38

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The purest form of commercial iron is:

A

pig iron

B

wrought iron

C

cast iron

D

scrap iron and pig iron

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NTA JEE Main 7th January 2020 Shift 1 - Question 39

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The atomic radius of Ag is closest to

A

Au

B

Ni

C

Cu

D

Hg

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NTA JEE Main 7th January 2020 Shift 1 - Question 40

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The IUPAC name of the complex [Pt(NH$$_3$$)$$_2$$Cl(NH$$_2$$CH$$_3$$)]Cl is

A

Diamminechlorido (methanamine) platinum (II) chloride

B

Diammine(methanamine)chlorido platinum (II) chloride

C

Diamminechlorido (aminomethane) platinum (II) chloride

D

Bisammine (methanamine) chlorido platinum (II) chloride

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NTA JEE Main 7th January 2020 Shift 1 - Question 41

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The theory that can completely/properly explain the nature of bonding in [Ni(CO)$$_4$$] is:

A

Werner's theory

B

Molecular orbital theory

C

Crystal field theory

D

Valence bond theory

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NTA JEE Main 7th January 2020 Shift 1 - Question 42

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1-methyl ethylene oxide when treated with an excess of HBr produces

A

B

C

D

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NTA JEE Main 7th January 2020 Shift 1 - Question 43

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What is the product of the following reaction?

A

B

C

D

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NTA JEE Main 7th January 2020 Shift 1 - Question 44

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Consider the following reaction:

The product 'X' is used:

A

in protein estimation as an alternative to ninhydrin

B

in acid base titration as an indicator

C

as food grade colourant

D

in laboratory test for phenols

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NTA JEE Main 7th January 2020 Shift 1 - Question 45

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Match the following:
(i) Riboflavin            (a) Beriberi
(ii) Thiamine            (b) Scurvy
(iii) Pyridoxine        (c) Cheilosis
(iv) Ascorbic acid    (d) Convulsions

A

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

B

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

C

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

D

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

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NTA JEE Main 7th January 2020 Shift 1 - Question 46

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For the reaction:
A(l) $$\rightarrow$$ 2B(g)
$$\Delta U = 2.1$$ kcal, $$\Delta S = 20$$ cal K$$^{-1}$$ at 300K.
Hence $$\Delta G$$ in kcal is

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NTA JEE Main 7th January 2020 Shift 1 - Question 47

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Two solutions, A and B, each of 100L was made by dissolving 4g of NaOH and 9.8g of H$$_2$$SO$$_4$$ in water, respectively. The pH of the resultant solution obtained from mixing 40L of solution A and 10L of solution B is (log 2 = 0.3)

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NTA JEE Main 7th January 2020 Shift 1 - Question 48

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During the nuclear explosion, one of the products is $$^{90}$$Sr with half life of 6.93 years. If 1$$\mu$$g of $$^{90}$$Sr was absorbed in the bones of a newly born baby in place of Ca, how much time, in years, is required to reduce it by 90% if it is not lost metabolically

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NTA JEE Main 7th January 2020 Shift 1 - Question 49

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Chlorine reacts with hot and concentrated NaOH and produces compounds (X) and (Y). Compound (X) gives white precipitate with silver nitrate solution. The average bond order between Cl and O atoms in (Y) is

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NTA JEE Main 7th January 2020 Shift 1 - Question 50

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The number of chiral carbons in chloramphenicol is

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NTA JEE Main 7th January 2020 Shift 1 - Question 51

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Let $$\alpha$$ and $$\beta$$ be two real roots of the equation $$(k + 1)\tan^2 x - \sqrt{2} \cdot \lambda \tan x = (1 - k)$$, where $$k(\neq -1)$$ and $$\lambda$$ are real numbers. If $$\tan^2(\alpha + \beta) = 50$$, then a value of $$\lambda$$ is

A

$$10\sqrt{2}$$

B

10

C

5

D

$$5\sqrt{2}$$

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NTA JEE Main 7th January 2020 Shift 1 - Question 52

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If $$\text{Re}\left(\frac{z-1}{2z+i}\right) = 1$$, where $$z = x + iy$$, then the point $$(x, y)$$ lies on a

A

circle whose centre is at $$\left(-\frac{1}{2}, -\frac{3}{2}\right)$$

B

straight line whose slope is $$-\frac{2}{3}$$

C

straight line whose slope is $$\frac{3}{2}$$

D

circle whose diameter is $$\frac{\sqrt{5}}{2}$$

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NTA JEE Main 7th January 2020 Shift 1 - Question 53

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Total number of 6-digit numbers in which only and all the five digits 1, 3, 5, 7 and 9 appears, is

A

$$\frac{1}{2}(6!)$$

B

6!

C

$$5^6$$

D

$$\frac{5}{2}(6!)$$

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NTA JEE Main 7th January 2020 Shift 1 - Question 54

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Five numbers are in A.P., whose sum is 25 and product is 2520. If one of these five numbers is $$-\frac{1}{2}$$, then the greatest number amongst them is

A

27

B

7

C

$$\frac{21}{2}$$

D

16

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NTA JEE Main 7th January 2020 Shift 1 - Question 55

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The greatest positive integer $$k$$, for which $$49^k + 1$$ is a factor of the sum $$49^{125} + 49^{124} + \ldots + 49^2 + 49 + 1$$, is

A

32

B

63

C

60

D

35

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NTA JEE Main 7th January 2020 Shift 1 - Question 56

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If $$y = mx + 4$$ is a tangent to both the parabolas, $$y^2 = 4x$$ and $$x^2 = 2by$$, then $$b$$ is equal to

A

$$-32$$

B

$$-64$$

C

$$-128$$

D

128

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NTA JEE Main 7th January 2020 Shift 1 - Question 57

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If the distance between the foci of an ellipse is 6 and the distance between its directrices is 12, then the length of its latus rectum is

A

$$\sqrt{3}$$

B

$$3\sqrt{2}$$

C

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

D

$$2\sqrt{3}$$

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NTA JEE Main 7th January 2020 Shift 1 - Question 58

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For two statements $$p$$ and $$q$$, the logical statement $$(p \rightarrow q) \wedge (q \rightarrow \sim p)$$ is equivalent to

A

$$p$$

B

$$q$$

C

$$\sim p$$

D

$$\sim q$$

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NTA JEE Main 7th January 2020 Shift 1 - Question 59

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Let $$\alpha$$ be a root of the equation $$x^2 + x + 1 = 0$$ and the matrix $$A = \frac{1}{\sqrt{3}}\begin{bmatrix} 1 & 1 & 1 \\ 1 & \alpha & \alpha^2 \\ 1 & \alpha^2 & \alpha^4 \end{bmatrix}$$, then the matrix $$A^{31}$$ is equal to

A

$$A^3$$

B

$$I_3$$

C

$$A^2$$

D

$$A$$

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NTA JEE Main 7th January 2020 Shift 1 - Question 60

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If the system of linear equations
$$2x + 2ay + az = 0$$
$$2x + 3by + bz = 0$$
$$2x + 4cy + cz = 0$$,
where $$a, b, c \in R$$ are non-zero and distinct; has a non-zero solution, then

A

$$\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$$ are in A.P.

B

$$a, b, c$$ are in G.P.

C

$$a + b + c = 0$$

D

$$a, b, c$$ are in A.P.

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NTA JEE Main 7th January 2020 Shift 1 - Question 61

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If $$g(x) = x^2 + x - 1$$ and $$(g \circ f)(x) = 4x^2 - 10x + 5$$, then $$f\left(\frac{5}{4}\right)$$ is equal to

A

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

B

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

C

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

D

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

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NTA JEE Main 7th January 2020 Shift 1 - Question 62

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If $$y(\alpha) = \sqrt{2\left(\frac{\tan\alpha + \cot\alpha}{1+\tan^2\alpha}\right) + \frac{1}{\sin^2\alpha}}$$, $$\alpha \in \left(\frac{3\pi}{4}, \pi\right)$$, then $$\frac{dy}{d\alpha}$$ at $$\alpha = \frac{5\pi}{6}$$ is

A

4

B

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

C

$$-4$$

D

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

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NTA JEE Main 7th January 2020 Shift 1 - Question 63

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Let $$x^k + y^k = a^k$$, $$(a, k > 0)$$ and $$\frac{dy}{dx} + \left(\frac{y}{x}\right)^{\frac{1}{3}} = 0$$, then $$k$$ is

A

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

B

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

C

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

D

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

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NTA JEE Main 7th January 2020 Shift 1 - Question 64

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Let the function $$f : [-7, 0] \rightarrow R$$ be continuous on $$[-7, 0]$$ and differentiable on $$(-7, 0)$$. If $$f(-7) = -3$$ and $$f'(x) \le 2$$ for all $$x \in (-7, 0)$$, then for all such functions $$f$$, $$f(-1) + f(0)$$ lies in the interval

A

$$(-\infty, 20]$$

B

$$[-3, 11]$$

C

$$(-\infty, 11]$$

D

$$[-6, 20]$$

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NTA JEE Main 7th January 2020 Shift 1 - Question 65

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If $$f(a + b + 1 - x) = f(x)$$, for all $$x$$, where $$a$$ and $$b$$ are fixed positive real numbers, then $$\frac{1}{a+b}\int_a^b x(f(x) + f(x + 1))dx$$ is equal to

A

$$\int_{a-1}^{b-1} f(x+1)dx$$

B

$$\int_{a-1}^{b-1} f(x)dx$$

C

$$\int_{a+1}^{b+1} f(x)dx$$

D

$$\int_{a+1}^{b+1} f(x+1)dx$$

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NTA JEE Main 7th January 2020 Shift 1 - Question 66

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The area of the region (in sq. units), enclosed by the circle $$x^2 + y^2 = 2$$ which is not common to the region bounded by the parabola $$y^2 = x$$ and the straight line $$y = x$$, is

A

$$\frac{1}{6}(24\pi - 1)$$

B

$$\frac{1}{3}(6\pi - 1)$$

C

$$\frac{1}{3}(12\pi - 1)$$

D

$$\frac{1}{6}(12\pi - 1)$$

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NTA JEE Main 7th January 2020 Shift 1 - Question 67

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If $$y = y(x)$$ is the solution of the differential equation, $$e^y\left(\frac{dy}{dx} - 1\right) = e^x$$ such that $$y(0) = 0$$, then $$y(1)$$ is equal to

A

$$1 + \log_e 2$$

B

$$2 + \log_e 2$$

C

2e

D

$$\log_e 2$$

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NTA JEE Main 7th January 2020 Shift 1 - Question 68

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A vector $$\vec{a} = \alpha\hat{i} + 2\hat{j} + \beta\hat{k}$$ $$(\alpha, \beta \in R)$$ lies in the plane of the vectors, $$\vec{b} = \hat{i} + \hat{j}$$ and $$\vec{c} = \hat{i} - \hat{j} + 4\hat{k}$$. If $$\vec{a}$$ bisects the angle between $$\vec{b}$$ and $$\vec{c}$$, then

A

$$\vec{a} \cdot \hat{i} + 3 = 0$$

B

$$\vec{a} \cdot \hat{i} + 1 = 0$$

C

$$\vec{a} \cdot \hat{k} + 2 = 0$$

D

$$\vec{a} \cdot \hat{k} + 4 = 0$$

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NTA JEE Main 7th January 2020 Shift 1 - Question 69

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Let P be a plane passing through the points (2, 1, 0), (4, 1, 1) and (5, 0, 1) and R be any point (2, 1, 6). Then the image of R in the plane P is

A

(6, 5, 2)

B

(6, 5, -2)

C

(4, 3, 2)

D

(3, 4, -2)

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NTA JEE Main 7th January 2020 Shift 1 - Question 70

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An unbiased coin is tossed 5 times. Suppose that a variable X is assigned the value k when k consecutive heads are obtained for $$k = 3, 4, 5$$, otherwise X takes the value $$-1$$. Then the expected value of X, is

A

$$\frac{3}{16}$$

B

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

C

$$-\frac{3}{16}$$

D

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

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NTA JEE Main 7th January 2020 Shift 1 - Question 71

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If the sum of the coefficients of all even powers of $$x$$ in the product $$(1 + x + x^2 + \ldots + x^{2n})(1 - x + x^2 - x^3 + \ldots + x^{2n})$$ is 61, then $$n$$ is equal to

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NTA JEE Main 7th January 2020 Shift 1 - Question 72

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Let $$A(1, 0)$$, $$B(6, 2)$$ and $$C\left(\frac{3}{2}, 6\right)$$ be the vertices of a triangle ABC. If P is a point inside the triangle ABC such that the triangles APC, APB and BPC have equal areas, then the length of the line segment PQ, where Q is the point $$\left(-\frac{7}{6}, -\frac{1}{3}\right)$$, is

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NTA JEE Main 7th January 2020 Shift 1 - Question 73

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$$\lim_{x \to 2} \frac{3^x + 3^{3-x} - 12}{3^{-x/2} - 3^{1-x}}$$ is equal to

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NTA JEE Main 7th January 2020 Shift 1 - Question 74

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If the variance of the first $$n$$ natural numbers is 10 and the variance of the first $$m$$ even natural numbers is 16, then the value of $$m + n$$ is equal to

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NTA JEE Main 7th January 2020 Shift 1 - Question 75

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Let S be the set of points where the function $$f(x) = |2 - |x - 3||$$, $$x \in R$$, is not differentiable. Then $$\sum_{x \in S} f(f(x))$$ is equal to

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