NTA JEE Main 6th September 2020 Shift 1

NTA JEE Main 6th September 2020 Shift 1 - Question 1

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A clock has a continuously moving second's hand of $$0.1\,\text{m}$$ length. The average acceleration of the tip of the hand (in units of $$\text{ms}^{-2}$$) is of the order of:

A

$$10^{-3}$$

B

$$10^{-4}$$

C

$$10^{-2}$$

D

$$10^{-1}$$

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NTA JEE Main 6th September 2020 Shift 1 - Question 2

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An insect is at the bottom of a hemispherical ditch of radius $$1\,\text{m}$$. It crawls up the ditch but starts slipping after it is at height $$h$$ from the bottom. If the coefficient of friction between the ground and the insect is $$0.75$$, then $$h$$ is: $$(g = 10\,\text{m s}^{-2})$$

A

0.20 m

B

0.45 m

C

0.60 m

D

0.80 m

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NTA JEE Main 6th September 2020 Shift 1 - Question 3

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If the potential energy between two molecules is given by $$U = \frac{A}{r^6} + \frac{B}{r^{12}}$$, then at equilibrium, separation between molecules, and the potential energy are:

A

$$\left(\frac{B}{2A}\right)^{1/6}$$, $$-\frac{A^2}{2B}$$

B

$$\left(\frac{B}{A}\right)^{1/6}$$, 0

C

$$\left(\frac{2B}{A}\right)^{1/6}$$, $$-\frac{A^2}{4B}$$

D

$$\left(\frac{B}{2A}\right)^{\frac{1}{6}}$$, $$\frac{A^2}{2B}$$

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NTA JEE Main 6th September 2020 Shift 1 - Question 4

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Shown in the figure is a hollow ice-cream cone (it is open at top). If its mass is M, radius of its top is R and height, H, then its moment of inertia about its axis is:

A

$$\frac{MR^2}{2}$$

B

$$\frac{M(R^2+H^2)}{4}$$

C

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

D

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

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NTA JEE Main 6th September 2020 Shift 1 - Question 5

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Four point masses, each of mass $$m$$, are fixed at the corners of a square of side $$l$$. The square is rotating with angular frequency $$\omega$$, about an axis passing through one of the corners of the square and parallel to its diagonal, as shown in the figure. The angular momentum of the square about the axis is:

A

$$ml^2\omega$$

B

$$4ml^2\omega$$

C

$$3ml^2\omega$$

D

$$2ml^2\omega$$

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NTA JEE Main 6th September 2020 Shift 1 - Question 6

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A satellite is in an elliptical orbit around a planet $$P$$. It is observed that the velocity of the satellite when it is farthest from the planet is 6 times less than that when it is closest to the planet. The ratio of distances between the satellite and the planet at closest and farthest points is:

A

1 : 6

B

1 : 3

C

1 : 2

D

3 : 4

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

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Molecules of an ideal gas are known to have three translational degrees of freedom. The gas is maintained at a temperature of T. The total internal energy, U of a mole of this gas, and the value of $$\gamma = \left(\frac{C_p}{C_v}\right)$$ are given, respectively, by:

A

$$U = \frac{5}{2}RT$$ and $$\gamma = \frac{6}{5}$$

B

$$U = 5RT$$ and $$\gamma = \frac{7}{5}$$

C

$$U = \frac{5}{2}RT$$ and $$\gamma = \frac{7}{5}$$

D

$$U = 5RT$$ and $$\gamma = \frac{6}{5}$$

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NTA JEE Main 6th September 2020 Shift 1 - Question 8

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An object of mass $$m$$ is suspended at the end of a massless wire of length $$L$$ and area of cross-section, A. Young modulus of the material of the wire is $$Y$$. If the mass is pulled down slightly its frequency of oscillation along the vertical direction is:

A

$$f = \frac{1}{2\pi}\sqrt{\frac{mL}{YA}}$$

B

$$f = \frac{1}{2\pi}\sqrt{\frac{YA}{mL}}$$

C

$$f = \frac{1}{2\pi}\sqrt{\frac{mA}{YL}}$$

D

$$f = \frac{1}{2\pi}\sqrt{\frac{YL}{mA}}$$

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NTA JEE Main 6th September 2020 Shift 1 - Question 9

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A sound source S is moving along a straight track with speed $$v$$, and is emitting sound of frequency $$v_0$$. An observer is standing at a finite distance, at the point O, from the track. The time variation of frequency heard by the observer is best represented by: ($$t_0$$ represents the instant when the distance between the source and observer is minimum)

A

B

C

D

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NTA JEE Main 6th September 2020 Shift 1 - Question 10

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Charges $$Q_1$$ and $$Q_2$$ are at points A and B of a right-angled triangle OAB. The resultant electric field at point O is perpendicular to the hypotenuse, then $$Q_1/Q_2$$ is proportional to:

A

$$\frac{x_1^3}{x_2^3}$$

B

$$\frac{x_2}{x_1}$$

C

$$\frac{x_1}{x_2}$$

D

$$\frac{x_2^2}{x_1^2}$$

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NTA JEE Main 6th September 2020 Shift 1 - Question 11

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For the given input voltage waveform $$V_{in}(t)$$, the output voltage waveform $$V_0(t)$$, across the capacitor is correctly depicted by:

A

B

C

D

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NTA JEE Main 6th September 2020 Shift 1 - Question 12

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A particle of charge $$q$$ and mass $$m$$ is moving with a velocity $$-v\hat{i}$$ $$(v \neq 0)$$ towards a large screen placed in the $$Y-Z$$ plane at distance $$d$$. If there is a magnetic field $$\vec{B} = B_0\hat{k}$$, the minimum value of $$v$$ for which the particle will not hit the screen is:

A

$$\frac{qdB_0}{3m}$$

B

$$\frac{2qdB_0}{m}$$

C

$$\frac{qdB_0}{m}$$

D

$$\frac{qdB_0}{2m}$$

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NTA JEE Main 6th September 2020 Shift 1 - Question 13

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An electron is moving along $$+x$$ direction with a velocity of $$6 \times 10^6\,\text{ms}^{-1}$$. It enters a region of uniform electric field of $$300\,V/cm$$ pointing along $$+y$$ direction. The magnitude and direction of the magnetic field set up in this region such that the electron keeps moving along the $$x$$ direction will be:

A

$$3 \times 10^{-4}\,\text{T}$$, along $$+z$$ direction

B

$$5 \times 10^{-3}\,\text{T}$$, along $$-z$$ direction

C

$$5 \times 10^{-3}\,\text{T}$$, along $$+z$$ direction

D

$$3 \times 10^{-4}\,\text{T}$$, along $$-z$$ direction

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NTA JEE Main 6th September 2020 Shift 1 - Question 14

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An AC circuit has $$R = 100\,\Omega$$, $$C = 2\,\mu F$$ and $$L = 80\,\text{mH}$$, connected in series. The quality factor of the circuit is:

A

2

B

0.5

C

20

D

400

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NTA JEE Main 6th September 2020 Shift 1 - Question 15

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A point like object is placed at distance of $$1\,\text{m}$$ in front of a convex lens of focal length $$0.5\,\text{m}$$. A plane mirror is placed at a distance of $$2\,\text{m}$$ behind the lens. The position and nature of the image formed by the system is:

A

$$2.6\,\text{m}$$ from the mirror, real

B

$$1\,\text{m}$$ from the mirror, virtual

C

$$1\,\text{m}$$ from the mirror, real

D

$$2.6\,\text{m}$$ from the mirror, virtual

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NTA JEE Main 6th September 2020 Shift 1 - Question 16

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In the figure below, $$P$$ and $$Q$$ are two equally intense coherent sources emitting radiation of wavelength $$20\,\text{m}$$. The separation between P and Q is $$5\,\text{m}$$ and the phase of P is ahead of that of Q by $$90^\circ$$. A, B and C are three distinct point of observation, each equidistant from the midpoint of PQ. The intensities of radiation at A, B, C will be in the ratio:

A

0 : 1 : 4

B

2 : 1 : 0

C

0 : 1 : 2

D

4 : 1 : 0

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NTA JEE Main 6th September 2020 Shift 1 - Question 17

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An electron, a doubly ionized helium ion $$(He^{++})$$ and proton are having the same kinetic energy. The relation between their respective de-Broglie wavelength $$\lambda_e$$, $$\lambda_{He^{++}}$$ and $$\lambda_p$$ is:

A

$$\lambda_e > \lambda_{He^{++}} > \lambda_p$$

B

$$\lambda_e < \lambda_{He^{++}} = \lambda_p$$

C

$$\lambda_e > \lambda_p > \lambda_{He^{++}}$$

D

$$\lambda_e < \lambda_p < \lambda_{He^{++}}$$

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NTA JEE Main 6th September 2020 Shift 1 - Question 18

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You are given that $$^7_3\text{Li} = 7.0160\,\text{u}$$, Mass of $$^4_2\text{He} = 4.0026\,\text{u}$$ and Mass of $$^1_1\text{He} = 1.0079\,\text{H}$$. When $$20\,\text{g}$$ of $$^7_3\text{Li}$$ is converted into $$^4_2\text{He}$$ by proton capture, the energy liberated, (in kWh), is: [Mass of nucleon $$= 1\,\text{GeV}/c^2$$]

A

$$4.5 \times 10^5$$

B

$$8 \times 10^6$$

C

$$6.82 \times 10^5$$

D

$$1.33 \times 10^6$$

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NTA JEE Main 6th September 2020 Shift 1 - Question 19

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Identify the correct output signal $$Y$$ in the given combination of gates (as shown $$n$$) for the given inputs $$A$$ and $$B$$:

A

B

C

D

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NTA JEE Main 6th September 2020 Shift 1 - Question 20

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A screw gauge has 50 divisions on its circular scale. The circular scale is 4 units ahead of the pitch scale marking, prior to use. Upon one complete rotation of the circular scale, a displacement of $$0.5\,\text{mm}$$ is noticed on the pitch scale. The nature of zero error involved and the least count of the screw gauge, are respectively:

A

Negative, $$2\mu m$$

B

Positive $$10\mu m$$

C

Positive $$0.1\,mm$$

D

Positive, $$0.1\mu m$$

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NTA JEE Main 6th September 2020 Shift 1 - Question 21

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The density of a solid metal sphere is diameter. The maximum error in the density of the sphere is $$\left(\frac{x}{100}\right)\%$$. If the relative errors in measuring the mass and the diameter are $$6.0\%$$ and $$1.5\%$$ respectively, the value of $$x$$ is ___

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NTA JEE Main 6th September 2020 Shift 1 - Question 22

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Two bodies of the same mass are moving with the same speed, but in different directions in a plane. They have a completely inelastic collision and move together thereafter with a final speed which is half of their initial velocities of the two bodies (in degree) is ___

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NTA JEE Main 6th September 2020 Shift 1 - Question 23

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Initially a gas of diatomic molecules is contained in a cylinder of volume $$V_1$$ at a pressure $$P_1$$ and temperature $$250\,\text{K}$$. Assuming that $$25\%$$ of the molecules get dissociated causing a change in number of moles. The pressure of the resulting gas at temperature $$2000\,\text{K}$$, when contained in a volume $$2V_1$$ is given by $$P_2$$. The ratio $$P_2/P_1$$ is ___

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NTA JEE Main 6th September 2020 Shift 1 - Question 24

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Suppose that intensity of a laser is $$\left(\frac{315}{\pi}\right)\,\text{W m}^{-2}$$. The rms electric field, in units of $$\text{V m}^{-1}$$ associated with this source is close to the nearest integer is ___ ($$\varepsilon_0 = 8.86 \times 10^{-12}\,\text{C}^2\,\text{N m}^{-2}$$; $$c = 3 \times 10^8\,\text{m s}^{-1}$$)

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NTA JEE Main 6th September 2020 Shift 1 - Question 25

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A part of a complete circuit is shown in the figure. At some instant, the value of current I is $$1\,\text{A}$$ and it is decreasing at a rate of $$10^2\,\text{As}^{-1}$$. The value of the potential difference $$V_P - V_Q$$, (in volts) at that instant is___

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NTA JEE Main 6th September 2020 Shift 1 - Question 26

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A solution of two components containing $$n_1$$ moles of the 1st component and $$n_2$$ moles of the 2nd component is prepared. $$M_1$$ and $$M_2$$ are the molecular weights of component 1 and 2 respectively. If $$d$$ is the density of the solution in $$\text{g mL}^{-1}$$, $$C_2$$ is the molarity and $$x_2$$ is the mole fraction of the 2nd component, then $$C_2$$ can be expressed as:

A

$$C_2 = \frac{1000\,x_2}{M_1 + x_2(M_2 - M_1)}$$

B

$$C_2 = \frac{dx_2}{M_1 + x_2(M_2 - M_1)}$$

C

$$C_2 = \frac{1000\,dx_2}{M_1 + x_2(M_2 - M_1)}$$

D

$$C_2 = \frac{dx_1}{M_2 + x_2(M_2 - M_1)}$$

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NTA JEE Main 6th September 2020 Shift 1 - Question 27

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The variation of equilibrium constant with temperature is given below:
Temperature: ,        Equilibrium Constant: 
$$T_1 = 25^\circ\text{C}$$              $$K_1 = 10$$
$$T_2 = 100^\circ\text{C}$$,           $$K_2 = 100$$
The values of $$\Delta H^\circ$$, $$\Delta G^\circ$$ at $$T_1$$ and $$\Delta G^\circ$$ at $$T_2$$ (in $$\text{kJ mol}^{-1}$$) respectively, are close to [use $$R = 8.314\,\text{JK}^{-1}\text{mol}^{-1}$$]

A

28.4, $$-7.14$$ and $$-5.71$$

B

0.64, $$-7.14$$ and $$-5.71$$

C

28.4, $$-5.71$$ and $$-14.29$$

D

0.64, $$-5.71$$ and $$-14.29$$

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NTA JEE Main 6th September 2020 Shift 1 - Question 28

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For the reaction
$$\text{Fe}_2\text{N}(s) + \frac{3}{2}\text{H}_2(g) \rightleftharpoons 2\text{Fe}(s) + \text{NH}_3(g)$$

A

$$K_c = K_p(RT)$$

B

$$K_c = K_p(RT)^{-1/2}$$

C

$$K_c = K_p(RT)^{1/2}$$

D

$$K_c = K_p(RT)^{3/2}$$

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NTA JEE Main 6th September 2020 Shift 1 - Question 29

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Arrange the following solutions in the decreasing order of pOH:
(A) 0.01 M HCl
(B) 0.01 M NaOH
(C) 0.01 M $$\text{CH}_3\text{COONa}$$
(D) 0.01 M NaCl

A

$$(A) \gt (C) \gt (D) \gt (B)$$

B

$$(A) \gt (D) \gt (C) \gt (B)$$

C

$$(B) \gt (C) \gt (D) \gt (A)$$

D

$$(B) \gt (D) \gt (C) \gt (A)$$

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NTA JEE Main 6th September 2020 Shift 1 - Question 30

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Among the sulphates of alkaline earth metals, the solubilities of $$\text{BeSO}_4$$ and $$\text{MgSO}_4$$ in water, respectively, are:

A

poor and poor

B

high and poor

C

high and high

D

poor and high

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NTA JEE Main 6th September 2020 Shift 1 - Question 31

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Which of the following compounds shows geometrical isomerism?

A

2-methylpent-2-ene

B

4-methylpent-2-ene

C

4-methylpent-1-ene

D

2-methylpent-1-ene

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NTA JEE Main 6th September 2020 Shift 1 - Question 32

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Consider the following reactions:

'A' is:

A

B

C

D

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NTA JEE Main 6th September 2020 Shift 1 - Question 33

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The major product obtained from the following reaction is:

A

B

C

D

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NTA JEE Main 6th September 2020 Shift 1 - Question 34

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The major products of the following reaction are:

A

B

C

D

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NTA JEE Main 6th September 2020 Shift 1 - Question 35

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The major product of the following reaction is:

A

B

C

D

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NTA JEE Main 6th September 2020 Shift 1 - Question 36

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Consider the following reactions:
$$A \to P1;\; B \to P2;\; C \to P3;\; D \to P4$$
The order of the above reactions are a, b, c and d, respectively. The following graph is obtained when log[rate] vs log[conc.] are plotted:

Among the following, the correct sequence for the order of the reactions is:

A

$$d > a > b > c$$

B

$$a > b > c > d$$

C

$$c > a > b > d$$

D

$$d > b > a > c$$

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NTA JEE Main 6th September 2020 Shift 1 - Question 37

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Kraft temperature is the temperature:

A

below which the aqueous solution of detergents starts freezing

B

below which the formation of micelles takes place

C

above which the aqueous solution of detergents starts boiling

D

above which the formation of micelles takes place

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NTA JEE Main 6th September 2020 Shift 1 - Question 38

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The correct statement with respect to dinitrogen is:

A

$$N_2$$ is paramagnetic in nature

B

it can combine with dioxygen at $$25^\circ\text{C}$$

C

liquid dinitrogen is not used in cryosurgery

D

it can be used as an inert diluent for reactive chemicals

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NTA JEE Main 6th September 2020 Shift 1 - Question 39

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The presence of soluble fluoride ion upto 1 ppm concentration in drinking water, is:

A

harmful for teeth

B

harmful to skin

C

harmful to bones

D

safe for teeth

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NTA JEE Main 6th September 2020 Shift 1 - Question 40

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The INCORRECT statement is:

A

bronze is an alloy of copper and tin

B

cast iron is used to manufacture wrought iron

C

german silver is an alloy of zinc, copper and nickel

D

brass is an alloy of copper and nickel

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NTA JEE Main 6th September 2020 Shift 1 - Question 41

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The set that contains atomic numbers of only transition elements, is:

A

37, 42, 50, 64

B

21, 25, 42, 72

C

9, 17, 34, 38

D

21, 32, 53, 64

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NTA JEE Main 6th September 2020 Shift 1 - Question 42

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The lanthanoid that does NOT show +4 oxidation state is:

A

Dy

B

Ce

C

Eu

D

Tb

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NTA JEE Main 6th September 2020 Shift 1 - Question 43

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The species that has a spin-only magnetic moment of 5.9 BM, is: ($$T_d$$ = tetrahedral)

A

$$[\text{Ni(CN)}_4]^{2-}$$ (square planar)

B

$$[\text{NiCl}_4]^{2-}$$ ($$T_d$$)

C

$$\text{Ni(CO)}_4$$ ($$T_d$$)

D

$$[\text{MnBr}_4]^{2-}$$ ($$T_d$$)

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NTA JEE Main 6th September 2020 Shift 1 - Question 44

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The increasing order of $$pK_b$$ values of the following compounds is:

A

$$II < IV < III < I$$

B

$$I < II < IV < III$$

C

$$II < I < III < IV$$

D

$$I < II < III < IV$$

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NTA JEE Main 6th September 2020 Shift 1 - Question 45

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Consider the Assertion and Reason given below.
Assertion (A): Ethene polymerized in the presence of Ziegler Natta Catalyst at high temperature and pressure is used to make buckets and dustbins.
Reason (R): High density polymers are closely packed and are chemically inert. Choose the correct answer from the following:

A

(A) is correct but (R) is wrong

B

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

C

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

D

(A) and (R) both are wrong

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NTA JEE Main 6th September 2020 Shift 1 - Question 46

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A spherical balloon of radius $$3\,\text{cm}$$ containing helium gas has a pressure of $$48 \times 10^{-3}\,\text{bar}$$. At the same temperature, the pressure, of a spherical balloon of radius $$12\,\text{cm}$$ containing the same amount of gas will be............ $$\times 10^{-6}\,\text{bar}$$.

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NTA JEE Main 6th September 2020 Shift 1 - Question 47

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In an estimation of bromine by Carius method, $$1.6\,\text{g}$$ of an organic compound gave $$1.88\,\text{g}$$ of AgBr. The mass percentage of bromine in the compound is........ (Atomic mass, Ag = 108, Br = 80 $$\text{g mol}^{-1}$$)

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NTA JEE Main 6th September 2020 Shift 1 - Question 48

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The elevation of boiling point of $$0.10\,\text{m}$$ aqueous $$\text{CrCl}_3 \cdot x\text{NH}_3$$ solution is two times that of $$0.05\,\text{m}$$ aqueous $$\text{CaCl}_2$$ solution. The value of $$x$$ is............ [Assume 100% ionisation of the complex and $$\text{CaCl}_2$$, coordination number of Cr as 6, and that all $$\text{NH}_3$$ molecules are present inside the coordination sphere]

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NTA JEE Main 6th September 2020 Shift 1 - Question 49

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Potassium chlorate is prepared by the electrolysis of KCl in basic solution
$$6\text{OH}^- + \text{Cl}^- \to \text{ClO}_3^- + 3\text{H}_2\text{O} + 6e^-$$. If only 60% of the current is utilized in the reaction, the time (rounded to the nearest hour) required to produce $$10\,\text{g}$$ of $$\text{KClO}_3$$ using a current of $$2\,\text{A}$$ is........ (Given: F = 96,500 C/mol; molar mass of $$\text{KClO}_3$$ = 122 g mol$$^{-1}$$)

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NTA JEE Main 6th September 2020 Shift 1 - Question 50

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The number of $$\text{Cl} = \text{O}$$ bonds in perchloric acid is, $$n$$.........

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NTA JEE Main 6th September 2020 Shift 1 - Question 51

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If $$\alpha$$ and $$\beta$$ be two roots of the equation $$x^2 - 64x + 256 = 0$$. Then the value of $$\left(\frac{\alpha^3}{\beta^5}\right)^{1/8} + \left(\frac{\beta^3}{\alpha^5}\right)^{1/8}$$ is:

A

2

B

3

C

1

D

4

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NTA JEE Main 6th September 2020 Shift 1 - Question 52

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The region represented by $$\{z = x + iy \in \mathbb{C} : |z| - \text{Re}(z) \leq 1\}$$ is also given by the inequality:

A

$$y^2 \geq 2(x+1)$$

B

$$y^2 \leq 2\left(x + \frac{1}{2}\right)$$

C

$$y^2 \leq \left(x + \frac{1}{2}\right)$$

D

$$y^2 \geq x + 1$$

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NTA JEE Main 6th September 2020 Shift 1 - Question 53

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Two families with three members each and one family with four members are to be seated in a row. In how many ways can they be seated so that the same family members are not separated?

A

$$2!\,3!\,4!$$

B

$$(3!)^3 \cdot (4!)$$

C

$$(3!)2 \cdot (4!)$$

D

$$3!\,(4!)^3$$

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NTA JEE Main 6th September 2020 Shift 1 - Question 54

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Let $$a, b, c, d$$ and $$p$$ be non-zero distinct real numbers such that $$(a^2 + b^2 + c^2)p^2 - 2(ab + bc + cd)p + (b^2 + c^2 + d^2) = 0$$. Then:

A

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

B

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

C

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

D

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

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NTA JEE Main 6th September 2020 Shift 1 - Question 55

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If $$\{p\}$$ denotes the fractional part of the number $$p$$, then $$\left\{\frac{3^{200}}{8}\right\}$$ is equal to:

A

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

B

$$\frac{7}{8}$$

C

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

D

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

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NTA JEE Main 6th September 2020 Shift 1 - Question 56

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A ray of light coming from the point $$(2, 2\sqrt{3})$$ is incident at an angle $$30^\circ$$ on the line $$x = 1$$ at the point A. The ray gets reflected on the line $$x = 1$$ and meets $$x$$-axis at the point B. Then, the line AB passes through the point:

A

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

B

$$\left(4, -\frac{\sqrt{3}}{2}\right)$$

C

$$\left(3, -\sqrt{3}\right)$$

D

$$\left(4, -\sqrt{3}\right)$$

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NTA JEE Main 6th September 2020 Shift 1 - Question 57

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Let $$L_1$$ be a tangent to the parabola $$y^2 = 4(x+1)$$ and $$L_2$$ be a tangent to the parabola $$y^2 = 8(x+2)$$ such that $$L_1$$ and $$L_2$$ intersect at right angles. Then $$L_1$$ and $$L_2$$ meet on the straight line:

A

$$x + 3 = 0$$

B

$$2x + 1 = 0$$

C

$$x + 2 = 0$$

D

$$x + 2y = 0$$

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NTA JEE Main 6th September 2020 Shift 1 - Question 58

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Which of the following points lies on the locus of the foot of perpendicular drawn upon any tangent to the ellipse, $$\frac{x^2}{4} + \frac{y^2}{2} = 1$$ from any of its foci?

A

$$\left(-2, \sqrt{3}\right)$$

B

$$\left(-1, \sqrt{2}\right)$$

C

$$\left(-1, \sqrt{3}\right)$$

D

$$(1, 2)$$

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NTA JEE Main 6th September 2020 Shift 1 - Question 59

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The negation of the Boolean expression $$p \vee (\sim p \wedge q)$$ is equivalent to:

A

$$p \wedge \sim q$$

B

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

C

$$\sim p \vee \sim q$$

D

$$\sim p \vee q$$

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NTA JEE Main 6th September 2020 Shift 1 - Question 60

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If $$\sum_{i=1}^{n}(x_i - a) = n$$ and $$\sum_{i=1}^{n}(x_i - a)^2 = na$$, $$(n, a \gt 1)$$, then the standard deviation of $$n$$ observations $$x_1, x_2, \ldots, x_n$$ is:

A

$$a - 1$$

B

$$n\sqrt{(a-1)}$$

C

$$\sqrt{n(a-1)}$$

D

$$\sqrt{(a-1)}$$

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NTA JEE Main 6th September 2020 Shift 1 - Question 61

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Let m and M be respectively the minimum and maximum values of $$\begin{vmatrix} \cos^2 x & 1 + \sin^2 x & \sin 2x \\ 1 + \cos^2 x & \sin^2 x & \sin 2x \\ \cos^2 x & \sin^2 x & 1 + \sin 2x \end{vmatrix}$$. Then the ordered pair (m, M) is equal to:

A

$$(3, 3)$$

B

$$(-3, -1)$$

C

$$(4, 1)$$

D

$$(1, 3)$$

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NTA JEE Main 6th September 2020 Shift 1 - Question 62

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The values of $$\lambda$$ and $$\mu$$ for which the system of linear equations $$x + y + z = 2$$, $$x + 2y + 3z = 5$$, $$x + 3y + \lambda z = \mu$$ has infinitely many solutions, are respectively:

A

6 and 8

B

5 and 7

C

5 and 8

D

4 and 9

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NTA JEE Main 6th September 2020 Shift 1 - Question 63

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If $$f(x+y) = f(x)f(y)$$ and $$\sum_{x=1}^{\infty} f(x) = 2$$, $$x, y \in N$$, where $$N$$ is the set of all natural numbers, then the value of $$\frac{f(4)}{f(2)}$$ is:

A

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

B

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

C

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

D

$$\frac{4}{9}$$

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NTA JEE Main 6th September 2020 Shift 1 - Question 64

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The position of a moving car at time $$t$$ is given by $$f(t) = at^2 + bt + c$$, $$t > 0$$, where $$a$$, $$b$$ and $$c$$ are real numbers greater than 1. Then the average speed of the car over the time interval $$[t_1, t_2]$$ is attained at the point:

A

$$\frac{(t_2 - t_1)}{2}$$

B

$$a(t_2 - t_1) + b$$

C

$$\frac{(t_1 + t_2)}{2}$$

D

$$2a(t_1 + t_2) + b$$

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NTA JEE Main 6th September 2020 Shift 1 - Question 65

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If $$I_1 = \int_0^1 (1-x^{50})^{100}\,dx$$ and $$I_2 = \int_0^1 (1-x^{50})^{101}\,dx$$ such that $$I_2 = \alpha I_1$$ then $$\alpha$$ equals to:

A

$$\frac{5049}{5050}$$

B

$$\frac{5050}{5049}$$

C

$$\frac{5050}{5051}$$

D

$$\frac{5051}{5050}$$

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NTA JEE Main 6th September 2020 Shift 1 - Question 66

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$$\lim_{x \to 1}\left(\frac{\int_0^{(x-1)^2} t\cos t^2\,dt}{(x-1)\sin(x-1)}\right)$$

A

is equal to $$\frac{1}{2}$$

B

is equal to 1

C

is equal to $$-\frac{1}{2}$$

D

is equal to 0

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NTA JEE Main 6th September 2020 Shift 1 - Question 67

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The area (in sq. units) of the region $$A = \{(x, y) : |x| + |y| \leq 1,\; 2y^2 \geq |x|\}$$

A

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

B

$$\frac{7}{6}$$

C

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

D

$$\frac{5}{6}$$

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NTA JEE Main 6th September 2020 Shift 1 - Question 68

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The general solution of the differential equation $$\sqrt{1 + x^2 + y^2 + x^2y^2} + xy\frac{dy}{dx} = 0$$ (where C is a constant of integration)

A

$$\sqrt{1+y^2} + \sqrt{1+x^2} = \frac{1}{2}\log_e\left(\frac{\sqrt{1+x^2}-1}{\sqrt{1+x^2}+1}\right) + C$$

B

$$\sqrt{1+y^2} - \sqrt{1+x^2} = \frac{1}{2}\log_e\left(\frac{\sqrt{1+x^2}-1}{\sqrt{1+x^2}+1}\right) + C$$

C

$$\sqrt{1+y^2} + \sqrt{1+x^2} = \frac{1}{2}\log_e\left(\frac{\sqrt{1+x^2}+1}{\sqrt{1+x^2}-1}\right) + C$$

D

$$\sqrt{1+y^2} - \sqrt{1+x^2} = \frac{1}{2}\log_e\left(\frac{\sqrt{1+x^2}+1}{\sqrt{1+x^2}-1}\right) + C$$

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NTA JEE Main 6th September 2020 Shift 1 - Question 69

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The shortest distance between the lines $$\frac{x-1}{0} = \frac{y+1}{-1} = \frac{z}{1}$$ and $$x + y + z + 1 = 0$$, $$2x - y + z + 3 = 0$$ is:

A

1

B

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

C

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

D

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

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NTA JEE Main 6th September 2020 Shift 1 - Question 70

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Out of 11 consecutive natural numbers if three numbers are selected at random (without repetition), then the probability that they are in A.P. with positive common difference is:

A

$$\frac{15}{101}$$

B

$$\frac{5}{101}$$

C

$$\frac{5}{33}$$

D

$$\frac{10}{99}$$

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NTA JEE Main 6th September 2020 Shift 1 - Question 71

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The angle of elevation of the top of a hill from a point on the horizontal plane passing through the foot of the hill is found to be $$45^\circ$$. After walking a distance of $$80$$ meters towards the top, up a slope inclined at angle of $$30^\circ$$ to the horizontal plane the angle of elevation of the top of the hill becomes $$75^\circ$$. Then the height of the hill (in meters) is_____.

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NTA JEE Main 6th September 2020 Shift 1 - Question 72

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Set $$A$$ has $$m$$ elements and set $$B$$ has $$n$$ elements. If the total number of subsets of $$A$$ is 112 more than the total number of subsets of $$B$$, then the value of $$m \cdot n$$ is___.

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NTA JEE Main 6th September 2020 Shift 1 - Question 73

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Let $$f : \mathbb{R} \to \mathbb{R}$$ be defined as $$f(x) = \begin{cases} x^5\sin\left(\frac{1}{x}\right) + 5x^2, & x < 0 \\ 0, & x = 0 \\ x^5\cos\left(\frac{1}{x}\right) + \lambda x^2, & x > 0 \end{cases}$$. The value of $$\lambda$$ for which $$f''(0)$$ exists, is___.

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NTA JEE Main 6th September 2020 Shift 1 - Question 74

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Let $$AD$$ and $$BC$$ be two vertical poles at $$A$$ and $$B$$ respectively on a horizontal ground. If $$AD = 8\,\text{m}$$, $$BC = 11\,\text{m}$$, $$AB = 10\,\text{m}$$; then the distance (in meters) of a point M lying in between AB from the point A such that $$MD^2 + MC^2$$ is minimum, is___.

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NTA JEE Main 6th September 2020 Shift 1 - Question 75

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If $$\vec{a}$$ and $$\vec{b}$$ are unit vectors, then the greatest value of $$\sqrt{3}|\vec{a} + \vec{b}| + |\vec{a} - \vec{b}|$$ is___.

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