NTA JEE Main 24th February 2021 Shift 2

NTA JEE Main 24th February 2021 Shift 2 - Question 1

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The period of oscillation of a simple pendulum is $$T = 2\pi\sqrt{\frac{L}{g}}$$. Measured value of $$L$$ is 1.0 m from meter scale having a minimum division of 1 mm and time of one complete oscillation is 1.95 s measured from stopwatch of 0.01 s resolution. The percentage error in the determination of $$g$$ will be:

A

1.03%

B

1.33%

C

1.30%

D

1.13%

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NTA JEE Main 24th February 2021 Shift 2 - Question 2

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A particle is projected with velocity $$v_0$$ along $$x$$-axis. A damping force is acting on the particle which is proportional to the square of the distance from the origin i.e. $$ma = -\alpha x^2$$. The distance at which the particle stops:

A

$$\left(\frac{2v_0}{3\alpha}\right)^{\frac{1}{3}}$$

B

$$\left(\frac{3mv_0^2}{2\alpha}\right)^{\frac{1}{3}}$$

C

$$\left(\frac{3v_0^2}{2\alpha}\right)^{\frac{1}{2}}$$

D

$$\left(\frac{2v_0^2}{3\alpha}\right)^{\frac{1}{2}}$$

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NTA JEE Main 24th February 2021 Shift 2 - Question 3

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A circular hole of radius $$\frac{a}{2}$$ is cut out of a circular disc of radius $$a$$ as shown in figure. The centroid of the remaining circular portion with respect to point O will be:

A

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

B

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

C

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

D

$$\frac{10}{11}a$$

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NTA JEE Main 24th February 2021 Shift 2 - Question 4

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A body weighs 49 N on a spring balance at the north pole. What will be its weight recorded on the same weighing machine, if it is shifted to the equator?
[Use $$g = \frac{GM}{R^2} = 9.8$$ m s$$^{-2}$$ and radius of earth, $$R = 6400$$ km.]

A

49.17 N

B

48.83 N

C

49.83 N

D

49 N

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NTA JEE Main 24th February 2021 Shift 2 - Question 5

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If one mole of an ideal gas at $$(P_1, V_1)$$ is allowed to expand reversibly and isothermally (A to B) its pressure is reduced to one-half of the original pressure (see figure). This is followed by a constant volume cooling till its pressure is reduced to one-fourth of the initial value $$(B \to C)$$. Then it is restored to its initial state by a reversible adiabatic compression (C to A). The net work done by the gas is equal to:

A

0

B

$$RT\ln(2)$$

C

$$-\frac{RT}{2(\gamma - 1)}$$

D

$$RT\left[\ln(2) - \frac{1}{2(\gamma - 1)}\right]$$

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NTA JEE Main 24th February 2021 Shift 2 - Question 6

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On the basis of kinetic theory of gases, the gas exerts pressure because its molecules:

A

continuously stick to the walls of container.

B

suffer change in momentum when impinge on the walls of container.

C

continuously lose their energy till it reaches wall.

D

are attracted by the walls of container.

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NTA JEE Main 24th February 2021 Shift 2 - Question 7

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In the given figure, a body of mass $$M$$ is held between two massless springs, on a smooth inclined plane. The free ends of the springs are attached to firm supports. If each spring has spring constant $$k$$, the frequency of oscillation of given body is:

A

$$\frac{1}{2\pi}\sqrt{\frac{2k}{Mg\sin\alpha}}$$

B

$$\frac{1}{2\pi}\sqrt{\frac{k}{Mg}\sin\alpha}$$

C

$$\frac{1}{2\pi}\sqrt{\frac{k}{2M}}$$

D

$$\frac{1}{2\pi}\sqrt{\frac{2k}{M}}$$

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NTA JEE Main 24th February 2021 Shift 2 - Question 8

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When a particle executes SHM, the nature of graphical representation of velocity as a function of displacement is:

A

straight line

B

elliptical

C

circular

D

parabolic

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NTA JEE Main 24th February 2021 Shift 2 - Question 9

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Which of the following equations represents a travelling wave?

A

$$y = A\sin(15x - 2t)$$

B

$$y = Ae^x\cos(\omega t - \theta)$$

C

$$y = Ae^{-x^2}(vt + \theta)$$

D

$$y = A\sin x \cos\omega t$$

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NTA JEE Main 24th February 2021 Shift 2 - Question 10

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Two electrons each are fixed at a distance 2d. A third charge proton placed at the midpoint is displaced slightly by a distance $$x (x \ll d)$$ perpendicular to the line joining the two fixed charges. Proton will execute simple harmonic motion having angular frequency:
($$m$$ = mass of charged particle)

A

$$\left(\frac{\pi\varepsilon_0 md^3}{2q^2}\right)^{\frac{1}{2}}$$

B

$$\left(\frac{2\pi\varepsilon_0 md^3}{q^2}\right)^{\frac{1}{2}}$$

C

$$\left(\frac{q^2}{2\pi\varepsilon_0 md^3}\right)^{\frac{1}{2}}$$

D

$$\left(\frac{2q^2}{\pi\varepsilon_0 md^3}\right)^{\frac{1}{2}}$$

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NTA JEE Main 24th February 2021 Shift 2 - Question 11

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A soft ferromagnetic material is placed in an external magnetic field. The magnetic domains:

A

have no relation with external magnetic field.

B

decrease in size and changes orientation.

C

increase in size but no change in orientation.

D

may increase or decrease in size and change its orientation.

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NTA JEE Main 24th February 2021 Shift 2 - Question 12

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The figure shows a circuit that contains four identical resistors with resistance $$R = 2.0$$ $$\Omega$$, two identical inductors with inductance $$L = 2.0$$ mH and an ideal battery with E.M.F. $$E = 9$$ V. The current $$i$$ just after the switch $$S$$ is closed will be:

A

9 A

B

3.37 A

C

2.25 A

D

3.0 A

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NTA JEE Main 24th February 2021 Shift 2 - Question 13

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An X-ray tube is operated at 1.24 million volt. The shortest wavelength of the produced photon will be:

A

$$10^{-3}$$ nm

B

$$10^{-2}$$ nm

C

$$10^{-4}$$ nm

D

$$10^{-1}$$ nm

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NTA JEE Main 24th February 2021 Shift 2 - Question 14

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Match List - I with List - II.

List-I	List-II
(a) Source of microwave frequency	(i) Radioactive decay of nucleus
(b) Source of infrared frequency	(ii) Magnetron
(c) Source of Gamma Rays	(iii) Inner shell electrons
(d) Source of X-rays	(iv) Vibration of atoms and molecules
	(v) LASER
	(vi) RC circuit

Choose the correct answer from the options given below:

A

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

B

(a)-(vi), (b)-(iv), (c)-(i), (d)-(v)

C

(a)-(vi), (b)-(v), (c)-(i), (d)-(iv)

D

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

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NTA JEE Main 24th February 2021 Shift 2 - Question 15

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If the source of light used in a Young's double slit experiment is changed from red to violet:

A

consecutive fringe lines will come closer.

B

the fringes will become brighter.

C

the central bright fringe will become a dark fringe.

D

the intensity of minima will increase.

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NTA JEE Main 24th February 2021 Shift 2 - Question 16

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The de Broglie wavelength of a proton and $$\alpha$$-particle are equal. The ratio of their velocities is

A

4 : 1

B

4 : 3

C

1 : 4

D

4 : 2

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NTA JEE Main 24th February 2021 Shift 2 - Question 17

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According to Bohr atom model, in which of the following transitions will the frequency be maximum?

A

$$n = 3$$ to $$n = 2$$

B

$$n = 5$$ to $$n = 4$$

C

$$n = 4$$ to $$n = 3$$

D

$$n = 2$$ to $$n = 1$$

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NTA JEE Main 24th February 2021 Shift 2 - Question 18

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The logic circuit shown above is equivalent to:

A

B

C

D

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NTA JEE Main 24th February 2021 Shift 2 - Question 19

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Zener breakdown occurs in a $$p - n$$ junction having $$p$$ and $$n$$ both:

A

lightly doped and have narrow depletion layer.

B

heavily doped and have wide depletion layer.

C

lightly doped and have wide depletion layer.

D

heavily doped and have narrow depletion layer.

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NTA JEE Main 24th February 2021 Shift 2 - Question 20

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Given below are two statements:
Statement I: $$p - n$$ junction diodes can be used to function as a transistor, simply by connecting two diodes, back to back, which acts as the base terminal.
Statement II: In the study of transistors, the amplification factor $$\beta$$ indicates ratio of the collector current to the base current.
In the light of the above statements, choose the correct answer from the options given below.

A

Both Statement I and Statement II are false

B

Statement I is true but Statement II is false

C

Statement I is false but Statement II is true

D

Both Statement I and Statement II are true

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NTA JEE Main 24th February 2021 Shift 2 - Question 21

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Two solids $$A$$ and $$B$$ of mass 1 kg and 2 kg respectively are moving with equal linear momentum. The ratio of their kinetic energies $$(K.E.)_A : (K.E.)_B$$ will be $$\frac{A}{1}$$, so the value of $$A$$ will be ______.

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NTA JEE Main 24th February 2021 Shift 2 - Question 22

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A uniform thin bar of mass 6 kg and length 2.4 meter is bent to make an equilateral hexagon. The moment of inertia about an axis passing through the centre of mass and perpendicular to the plane of hexagon is ______ $$\times 10^{-1}$$ kg m$$^2$$.

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NTA JEE Main 24th February 2021 Shift 2 - Question 23

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A uniform metallic wire is elongated by 0.04 m when subjected to a linear force $$F$$. The elongation, if its length and diameter is doubled and subjected to the same force will be ______ cm.

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NTA JEE Main 24th February 2021 Shift 2 - Question 24

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The root-mean-square speed of molecules of a given mass of a gas at 27°C and 1 atmosphere pressure is 200 m s$$^{-1}$$. The root-mean-square speed of molecules of the gas at 127°C and 2 atmosphere pressure is $$\frac{x}{\sqrt{3}}$$ m s$$^{-1}$$. The value of $$x$$ will be ______.

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NTA JEE Main 24th February 2021 Shift 2 - Question 25

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Two cars are approaching each other at an equal speed of 7.2 km hr$$^{-1}$$. When they see each other, both blow horns having a frequency of 676 Hz. The beat frequency heard by each driver will be ______ Hz. [Velocity of sound in air is 340 m s$$^{-1}$$.]

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NTA JEE Main 24th February 2021 Shift 2 - Question 26

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A point charge of $$+12$$ $$\mu C$$ is at a distance 6 cm vertically above the centre of a square of side 12 cm as shown in figure. The magnitude of the electric flux through the square will be ______ $$\times 10^3$$ N m$$^2$$ C$$^{-1}$$.

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NTA JEE Main 24th February 2021 Shift 2 - Question 27

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A cylindrical wire of radius 0.5 mm and conductivity $$5 \times 10^7$$ S m$$^{-1}$$ is subjected to an electric field of 10 mV m$$^{-1}$$. The expected value of current in the wire will be $$x^3\pi$$ mA. The value of $$x$$ is ______.

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NTA JEE Main 24th February 2021 Shift 2 - Question 28

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A series LCR circuit is designed to resonate at an angular frequency $$\omega_0 = 10^5$$ rad s$$^{-1}$$. The circuit draws 16 W power from 120 V source at resonance. The value of resistance $$R$$ in the circuit is ______ $$\Omega$$.

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NTA JEE Main 24th February 2021 Shift 2 - Question 29

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An electromagnetic wave of frequency 3 GHz enters a dielectric medium of relative electric permittivity 2.25 from vacuum. The wavelength of this wave in that medium will be ______ $$\times 10^{-2}$$ cm.

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NTA JEE Main 24th February 2021 Shift 2 - Question 30

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A signal of 0.1 kW is transmitted in a cable. The attenuation of cable is $$-5$$ dB per km and cable length is 20 km. The power received at the receiver is $$10^{-x}$$ W. The value of $$x$$ is ______. [Gain in dB = $$10 \log_{10}\left(\frac{P_O}{P_i}\right)$$]

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NTA JEE Main 24th February 2021 Shift 2 - Question 31

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According to Bohr's atomic theory:
(A) Kinetic energy of electron is $$\propto \frac{Z^2}{n^2}$$.
(B) The product of velocity (v) of electron and principal quantum number (n), $$'vn' \propto Z^2$$.
(C) Frequency of revolution of electron in an orbit is $$\propto \frac{Z^3}{n^3}$$.
(D) Coulombic force of attraction on the electron is $$\propto \frac{Z^3}{n^4}$$.
Choose the most appropriate answer from the options given below:

A

(A), (C) and (D) only

B

(A) and (D) only

C

(C) only

D

(A) only

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NTA JEE Main 24th February 2021 Shift 2 - Question 32

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The correct set from the following in which both pairs are in correct order of melting point is:

A

LiF > LiCl; NaCl > MgO

B

LiCl > LiF; MgO > NaCl

C

LiCl > LiF; NaCl > MgO

D

LiF > LiCl; MgO > NaCl

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NTA JEE Main 24th February 2021 Shift 2 - Question 33

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The correct shape and I - I - I bond angles respectively in $$I_3^-$$ ion are:

A

T-shaped; 180° and 90°

B

Distorted trigonal planar; 135° and 90°

C

Trigonal planar; 120°

D

Linear; 180°

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NTA JEE Main 24th February 2021 Shift 2 - Question 34

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Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A: Hydrogen is the most abundant element in the Universe, but it is not the most abundant gas in the troposphere.
Reason R: Hydrogen is the lightest element.
In the light of the above statements, choose the correct answer from the options given below:

A

Both A and R are true but R is NOT the correct explanation of A

B

Both A and R are true and R is the correct explanation of A

C

A is true but R is false

D

A is false but R true

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NTA JEE Main 24th February 2021 Shift 2 - Question 35

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Match List - I with List - II.

List-I (Salt)	List-II (Flame colour wavelength)
(a) LiCl	(i) 455.5 nm
(b) NaCl	(ii) 670.8 nm
(c) RbCl	(iii) 780.0 nm
(d) CsCl	(iv) 589.2 nm

Choose the correct answer from the options given below:

A

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

B

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

C

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

D

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

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NTA JEE Main 24th February 2021 Shift 2 - Question 36

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Which one of the following compounds is non-aromatic?

A

B

C

D

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NTA JEE Main 24th February 2021 Shift 2 - Question 37

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What is the correct sequence of reagents used for converting nitrobenzene into m-dibromobenzene?

A

$$\xrightarrow{Sn/HCl} \xrightarrow{KBr} \xrightarrow{Br_2} \xrightarrow{H^+}$$

B

$$\xrightarrow{Br_2/Fe} \xrightarrow{Sn/HCl} \xrightarrow{NaNO_2/HCl} \xrightarrow{CuBr/HBr}$$

C

$$\xrightarrow{NaNO_2} \xrightarrow{HCl} \xrightarrow{KBr} \xrightarrow{H^+}$$

D

$$\xrightarrow{Sn/HCl} \xrightarrow{Br_2} \xrightarrow{NaNO_2} \xrightarrow{NaBr}$$

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NTA JEE Main 24th February 2021 Shift 2 - Question 38

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The correct order of the following compounds showing increasing tendency towards nucleophilic substitution reaction is:

A

(iv) < (i) < (ii) < (iii)

B

(i) < (ii) < (iii) < (iv)

C

(iv) < (i) < (iii) < (ii)

D

(iv) < (iii) < (ii) < (i)

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NTA JEE Main 24th February 2021 Shift 2 - Question 39

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Given below are two statements:
Statement I: The value of the parameter "Biochemical Oxygen Demand (BOD)" is important for survival of aquatic life.
Statement II: The optimum value of BOD is 6.5 ppm.
In the light of the above statements, choose the most appropriate answer from the options given below:

A

Statement I is true but Statement II is false

B

Statement I is false but Statement II is true

C

Both Statement I and Statement II are false

D

Both Statement I and Statement II are true

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NTA JEE Main 24th February 2021 Shift 2 - Question 40

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Most suitable salt which can be used for efficient clotting of blood will be?

A

$$FeCl_3$$

B

$$Mg(HCO_3)_2$$

C

$$FeSO_4$$

D

$$NaHCO_3$$

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NTA JEE Main 24th February 2021 Shift 2 - Question 41

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Match List - I with List - II.

List-I (Metal)	List-II (Ores)
(a) Aluminium	(i) Siderite
(b) Iron	(ii) Calamine
(c) Copper	(iii) Kaolinite
(d) Zinc	(iv) Malachite

Choose the correct answer from the options given below:

A

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

B

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

C

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

D

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

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NTA JEE Main 24th February 2021 Shift 2 - Question 42

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The incorrect statement among the following is:

A

$$Cr_2O_3$$ is an amphoteric oxide

B

$$VOSO_4$$ is a reducing agent

C

$$RuO_4$$ is an oxidizing agent

D

Red colour of ruby is due to the presence of $$Co^{3+}$$

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NTA JEE Main 24th February 2021 Shift 2 - Question 43

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What is the correct order of the following elements with respect to their density?

A

Zn < Cu < Co < Fe < Cr

B

Cr < Zn < Co < Cu < Fe

C

Zn < Cr < Fe < Co < Cu

D

Cr < Fe < Co < Cu < Zn

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NTA JEE Main 24th February 2021 Shift 2 - Question 44

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The calculated magnetic moments (spin only value) for species $$[FeCl_4]^{2-}$$, $$[Co(C_2O_4)_3]^{3-}$$ and $$MnO_4^{2-}$$ respectively are:

A

4.90, 0 and 1.73 BM

B

4.90, 0 and 2.83 BM

C

5.82, 0 and 0 BM

D

5.92, 4.90 and 0 BM

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NTA JEE Main 24th February 2021 Shift 2 - Question 45

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Which of the following reagent is suitable for the preparation of the product in the above reaction?

A

Red P + $$Cl_2$$

B

$$NaBH_4$$

C

$$NH_2 - NH_2 / C_2H_5ONa$$

D

Ni / $$H_2$$

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NTA JEE Main 24th February 2021 Shift 2 - Question 46

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Which one of the following carbonyl compounds cannot be prepared by addition of water on an alkyne in the presence of $$HgSO_4$$ and $$H_2SO_4$$?

A

B

C

D

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NTA JEE Main 24th February 2021 Shift 2 - Question 47

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The diazonium salt of which of the following compounds will form a coloured dye on reaction with $$\beta$$-Naphthol in NaOH?

A

B

C

D

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NTA JEE Main 24th February 2021 Shift 2 - Question 48

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Match List - I with List - II.

Choose the correct answer from the options given below:

A

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

B

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

C

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

D

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

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NTA JEE Main 24th February 2021 Shift 2 - Question 49

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In polymer Buna-S, 'S' stands for:

A

Strength

B

Sulphur

C

Sulphonation

D

Styrene

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NTA JEE Main 24th February 2021 Shift 2 - Question 50

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Match List - I with List - II.

List-I	List-II
(a) Valium	(i) Antifertility drug
(b) Morphine	(ii) Pernicious anaemia
(c) Norethindrone	(iii) Analgesic
(d) Vitamin B$$_{12}$$	(iv) Tranquilizer

Choose the correct answer from the options given below:

A

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

B

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

C

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

D

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

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NTA JEE Main 24th February 2021 Shift 2 - Question 51

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The formula of a gaseous hydrocarbon which requires 6 times of its own volume of $$O_2$$ for complete oxidation and produces 4 times its own volume of $$CO_2$$ is $$C_xH_y$$. The value of y is ______.

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NTA JEE Main 24th February 2021 Shift 2 - Question 52

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1.86 g of aniline completely reacts to form acetanilide. 10% of the product is lost during purification. Amount of acetanilide obtained after purification (in g) is ______ $$\times 10^{-2}$$.

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NTA JEE Main 24th February 2021 Shift 2 - Question 53

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The volume occupied by 4.75 g of acetylene gas at 50°C and 740 mm Hg pressure is ______ L.
(Rounded off to the nearest integer)
[Given R = 0.0826 L atm K$$^{-1}$$ mol$$^{-1}$$]

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NTA JEE Main 24th February 2021 Shift 2 - Question 54

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Assuming ideal behaviour, the magnitude of log K for the following reaction at 25°C is $$x \times 10^{-1}$$. The value of x is ______. (Integer answer)
$$3HC \equiv CH_{(g)} \rightleftharpoons C_6H_{6(l)}$$
[Given: $$\Delta_f G^{\circ}(HC \equiv CH) = -2.04 \times 10^5$$ J mol$$^{-1}$$; $$\Delta_f G^{\circ}(C_6H_6) = -1.24 \times 10^5$$ J mol$$^{-1}$$; R = 8.314 J K$$^{-1}$$ mol$$^{-1}$$]

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NTA JEE Main 24th February 2021 Shift 2 - Question 55

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The solubility product of $$PbI_2$$ is $$8.0 \times 10^{-9}$$. The solubility of lead iodide in 0.1 molar solution of lead nitrate is $$x \times 10^{-6}$$ mol/L. The value of x is ______ (Rounded off to the nearest integer)
[Given $$\sqrt{2} = 1.41$$]

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NTA JEE Main 24th February 2021 Shift 2 - Question 56

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$$C_6H_6$$ freezes at 5.5°C. The temperature at which a solution of 10 g of $$C_4H_{10}$$ in 200 g of $$C_6H_6$$ freeze is ______ °C. (nearest integer value), (The molal freezing point depression constant of $$C_6H_6$$ is 5.12°C/m.)

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NTA JEE Main 24th February 2021 Shift 2 - Question 57

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The magnitude of the change in oxidising power of the $$MnO_4^- / Mn^{2+}$$ couple is $$x \times 10^{-4}$$ V if the $$H^+$$ concentration is decreased from 1 M to $$10^{-4}$$ M at 25°C. (Assume concentration of $$MnO_4^-$$ and $$Mn^{2+}$$ to be same on change in $$H^+$$ concentration). The value of x is ______. (Rounded off to the nearest integer)
[Given: $$\frac{2.303RT}{F} = 0.059$$]

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NTA JEE Main 24th February 2021 Shift 2 - Question 58

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Sucrose hydrolyses in acid solution into glucose and fructose following first order rate law with a half-life of 3.33 h at 25°C. After 9 h, the fraction of sucrose remaining is f. The value of $$\log_{10}\left(\frac{1}{f}\right)$$ is ______ $$\times 10^{-2}$$.
(Rounded off to the nearest integer)
[Assume: ln10 = 2.303, ln 2 = 0.693]

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NTA JEE Main 24th February 2021 Shift 2 - Question 59

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Among the following allotropic forms of sulphur, the number of allotropic forms, which will show paramagnetism is ______.
(A) $$\alpha$$-sulphur
(B) $$\beta$$-sulphur
(C) $$S_2$$-form

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NTA JEE Main 24th February 2021 Shift 2 - Question 60

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The total number of amines among the following which can be synthesized by Gabriel synthesis is ______.

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NTA JEE Main 24th February 2021 Shift 2 - Question 61

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Let $$a, b, c$$ be in arithmetic progression. Let the centroid of the triangle with vertices $$(a, c)$$, $$(2, b)$$ and $$(a, b)$$ be $$\left(\frac{10}{3}, \frac{7}{3}\right)$$. If $$\alpha, \beta$$ are the roots of the equation $$ax^2 + bx + 1 = 0$$, then the value of $$\alpha^2 + \beta^2 - \alpha\beta$$ is:

A

$$-\frac{71}{256}$$

B

$$\frac{69}{256}$$

C

$$\frac{71}{256}$$

D

$$-\frac{69}{256}$$

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NTA JEE Main 24th February 2021 Shift 2 - Question 62

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If $$n \geq 2$$ is a positive integer, then the sum of the series $${}^{n+1}C_2 + 2({}^2C_2 + {}^3C_2 + {}^4C_2 + \ldots + {}^nC_2)$$ is

A

$$\frac{n(n-1)(2n+1)}{6}$$

B

$$\frac{n(n+1)(2n+1)}{6}$$

C

$$\frac{n(n+1)^2(n+2)}{12}$$

D

$$\frac{n(2n+1)(3n+1)}{6}$$

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NTA JEE Main 24th February 2021 Shift 2 - Question 63

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If $$P$$ is a point on the parabola $$y = x^2 + 4$$ which is closest to the straight line $$y = 4x - 1$$, then the co-ordinates of $$P$$ are:

A

$$(-2, 8)$$

B

$$(1, 5)$$

C

$$(2, 8)$$

D

$$(3, 13)$$

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NTA JEE Main 24th February 2021 Shift 2 - Question 64

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The negation of the statement $$\sim p \wedge (p \vee q)$$ is:

A

$$\sim p \vee q$$

B

$$\sim p \wedge q$$

C

$$p \vee \sim q$$

D

$$p \wedge \sim q$$

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NTA JEE Main 24th February 2021 Shift 2 - Question 65

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For the statements $$p$$ and $$q$$, consider the following compound statements:
$$(a)$$ $$(\sim q \wedge (p \to q)) \to \sim p$$
$$(b)$$ $$((p \vee q) \wedge \sim p) \to q$$
Then which of the following statements is correct?

A

$$(b)$$ is a tautology but not $$(a)$$.

B

$$(a)$$ and $$(b)$$ both are tautologies.

C

$$(a)$$ and $$(b)$$ both are not tautologies.

D

$$(a)$$ is a tautology but not $$(b)$$.

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NTA JEE Main 24th February 2021 Shift 2 - Question 66

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The angle of elevation of a jet plane from a point A on the ground is 60°. After a flight of 20 seconds at the speed of 432 km/hour, the angle of elevation changes to 30°. If the jet plane is flying at a constant height, then its height is:

A

$$1200\sqrt{3}$$ m

B

$$2400\sqrt{3}$$ m

C

$$1800\sqrt{3}$$ m

D

$$3600\sqrt{3}$$ m

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NTA JEE Main 24th February 2021 Shift 2 - Question 67

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For the system of linear equations:
$$x - 2y = 1$$, $$x - y + kz = -2$$, $$ky + 4z = 6$$, $$k \in R$$
Consider the following statements:
(A) The system has unique solution if $$k \neq 2, k \neq -2$$.
(B) The system has unique solution if $$k = -2$$.
(C) The system has unique solution if $$k = 2$$.
(D) The system has no-solution if $$k = 2$$.
(E) The system has infinite number of solutions if $$k \neq -2$$.
Which of the following statements are correct?

A

(A) and (E) only

B

(B) and (E) only

C

(A) and (D) only

D

(C) and (D) only

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NTA JEE Main 24th February 2021 Shift 2 - Question 68

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Let $$A$$ and $$B$$ be $$3 \times 3$$ real matrices such that $$A$$ is a symmetric matrix and $$B$$ is a skew-symmetric matrix. Then the system of linear equations $$(A^2B^2 - B^2A^2)X = O$$, where $$X$$ is a $$3 \times 1$$ column matrix of unknown variables and $$O$$ is a $$3 \times 1$$ null matrix, has:

A

exactly two solutions

B

infinitely many solutions

C

a unique solution

D

no solution

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NTA JEE Main 24th February 2021 Shift 2 - Question 69

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A possible value of $$\tan\left(\frac{1}{4}\sin^{-1}\frac{\sqrt{63}}{8}\right)$$ is:

A

$$2\sqrt{2} - 1$$

B

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

C

$$\sqrt{7} - 1$$

D

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

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NTA JEE Main 24th February 2021 Shift 2 - Question 70

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For which of the following curves, the line $$x + \sqrt{3}y = 2\sqrt{3}$$ is the tangent at the point $$\left(\frac{3\sqrt{3}}{2}, \frac{1}{2}\right)$$?

A

$$2x^2 - 18y^2 = 9$$

B

$$y^2 = \frac{1}{6\sqrt{3}}x$$

C

$$x^2 + 9y^2 = 9$$

D

$$x^2 + y^2 = 7$$

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NTA JEE Main 24th February 2021 Shift 2 - Question 71

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Let $$f : R \to R$$ be defined as
$$f(x) = \begin{cases} -55x, & \text{if } x < -5 \\ 2x^3 - 3x^2 - 120x, & \text{if } -5 \leq x \leq 4 \\ 2x^3 - 3x^2 - 36x - 336, & \text{if } x > 4 \end{cases}$$
Let $$A = \{x \in R : f \text{ is increasing}\}$$. Then $$A$$ is equal to:

A

$$(-5, \infty)$$

B

$$(-5, -4) \cup (4, \infty)$$

C

$$(-\infty, -5) \cup (-4, \infty)$$

D

$$(-\infty, -5) \cup (4, \infty)$$

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NTA JEE Main 24th February 2021 Shift 2 - Question 72

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If the curve $$y = ax^2 + bx + c$$, $$x \in R$$, passes through the point (1, 2) and the tangent line to this curve at origin is $$y = x$$, then the possible values of $$a, b, c$$ are:

A

$$a = -1, b = 1, c = 1$$

B

$$a = 1, b = 0, c = 1$$

C

$$a = 1, b = 1, c = 0$$

D

$$a = \frac{1}{2}, b = \frac{1}{2}, c = 1$$

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NTA JEE Main 24th February 2021 Shift 2 - Question 73

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The value of the integral, $$\int_1^3 [x^2 - 2x - 2] dx$$, where $$[x]$$ denotes the greatest integer less than or equal to $$x$$, is

A

-4

B

-5

C

$$-\sqrt{2} - \sqrt{3} + 1$$

D

$$-\sqrt{2} - \sqrt{3} - 1$$

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NTA JEE Main 24th February 2021 Shift 2 - Question 74

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The area of the region: $$R = \{(x, y) : 5x^2 \leq y \leq 2x^2 + 9\}$$ is

A

$$9\sqrt{3}$$ square units

B

$$12\sqrt{3}$$ square units

C

$$11\sqrt{3}$$ square units

D

$$6\sqrt{3}$$ square units

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NTA JEE Main 24th February 2021 Shift 2 - Question 75

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Let $$f$$ be a twice differentiable function defined on $$R$$ such that $$f(0) = 1$$, $$f'(0) = 2$$ and $$f'(x) \neq 0$$ for all $$x \in R$$. If $$\begin{vmatrix} f(x) & f'(x) \\ f'(x) & f''(x) \end{vmatrix} = 0$$, for all $$x \in R$$, then the value of $$f(1)$$ lies in the interval

A

(9, 12)

B

(3, 6)

C

(0, 3)

D

(6, 9)

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NTA JEE Main 24th February 2021 Shift 2 - Question 76

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If a curve $$y = f(x)$$ passes through the point (1, 2) and satisfies $$x\frac{dy}{dx} + y = bx^4$$, then for what value of $$b$$, $$\int_1^2 f(x)dx = \frac{62}{5}$$?

A

$$\frac{31}{5}$$

B

10

C

5

D

$$\frac{62}{5}$$

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NTA JEE Main 24th February 2021 Shift 2 - Question 77

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Let $$f(x)$$ be a differentiable function defined on $$[0, 2]$$ such that $$f'(x) = f'(2 - x)$$ for all $$x \in (0, 2)$$, $$f(0) = 1$$ and $$f(2) = e^2$$. Then the value of $$\int_0^2 f(x)dx$$ is

A

$$2(1 + e^2)$$

B

$$1 + e^2$$

C

$$1 - e^2$$

D

$$2(1 - e^2)$$

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NTA JEE Main 24th February 2021 Shift 2 - Question 78

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The vector equation of the plane passing through the intersection of the planes $$\vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 1$$ and $$\vec{r} \cdot (\hat{i} - 2\hat{j}) = -2$$, and the point (1, 0, 2) is:

A

$$\vec{r} \cdot (\hat{i} + 7\hat{j} + 3\hat{k}) = 7$$

B

$$\vec{r} \cdot (\hat{i} - 7\hat{j} + 3\hat{k}) = \frac{7}{3}$$

C

$$\vec{r} \cdot (\hat{i} + 7\hat{j} + 3\hat{k}) = \frac{7}{3}$$

D

$$\vec{r} \cdot (3\hat{i} + 7\hat{j} + 3\hat{k}) = 7$$

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NTA JEE Main 24th February 2021 Shift 2 - Question 79

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Let $$a, b \in R$$. If the mirror image of the point $$P(a, 6, 9)$$ with respect to the line $$\frac{x - 3}{7} = \frac{y - 2}{5} = \frac{z - 1}{-9}$$ is $$(20, b, -a - 9)$$, then $$|a + b|$$ is equal to:

A

86

B

90

C

84

D

88

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NTA JEE Main 24th February 2021 Shift 2 - Question 80

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The probability that two randomly selected subsets of the set $$\{1, 2, 3, 4, 5\}$$ have exactly two elements in their intersection, is:

A

$$\frac{65}{2^8}$$

B

$$\frac{65}{2^7}$$

C

$$\frac{35}{2^7}$$

D

$$\frac{135}{2^9}$$

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NTA JEE Main 24th February 2021 Shift 2 - Question 81

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The number of the real roots of the equation $$(x + 1)^2 + |x - 5| = \frac{27}{4}$$ is ______.

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NTA JEE Main 24th February 2021 Shift 2 - Question 82

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Let $$i = \sqrt{-1}$$. If $$\frac{(-1 + i\sqrt{3})^{21}}{(1 - i)^{24}} + \frac{(1 + i\sqrt{3})^{21}}{(1 + i)^{24}} = k$$, and $$n = [|k|]$$ be the greatest integral part of $$|k|$$. Then $$\sum_{j=0}^{n+5} (j + 5)^2 - \sum_{j=0}^{n+5} (j + 5)$$ is equal to ______.

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NTA JEE Main 24th February 2021 Shift 2 - Question 83

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The students $$S_1, S_2, \ldots, S_{10}$$ are to be divided into 3 groups $$A$$, $$B$$ and $$C$$ such that each group has at least one student and the group $$C$$ has at most 3 students. Then the total number of possibilities of forming such groups is ______.

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NTA JEE Main 24th February 2021 Shift 2 - Question 84

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The sum of first four terms of a geometric progression (G.P.) is $$\frac{65}{12}$$ and the sum of their respective reciprocals is $$\frac{65}{18}$$. If the product of first three terms of the G.P. is 1, and the third term is $$\alpha$$, then $$2\alpha$$ is ______.

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NTA JEE Main 24th February 2021 Shift 2 - Question 85

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For integers $$n$$ and $$r$$, let $$\binom{n}{r} = \begin{cases} {}^nC_r, & \text{if } n \geq r \geq 0 \\ 0, & \text{otherwise} \end{cases}$$. The maximum value of $$k$$ for which the sum $$\sum_{i=0}^{k} \binom{10}{i}\binom{15}{k-i} + \sum_{i=0}^{k+1} \binom{12}{i}\binom{13}{k+1-i}$$ is maximum, is equal to ______.

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NTA JEE Main 24th February 2021 Shift 2 - Question 86

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Let a point $$P$$ be such that its distance from the point (5, 0) is thrice the distance of $$P$$ from the point (-5, 0). If the locus of the point $$P$$ is a circle of radius $$r$$, then $$4r^2$$ (in the nearest integer) is equal to ______.

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NTA JEE Main 24th February 2021 Shift 2 - Question 87

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If the variance of 10 natural numbers 1, 1, 1, ..., 1, $$k$$ is less than 10, then the maximum possible value of $$k$$ is ______.

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NTA JEE Main 24th February 2021 Shift 2 - Question 88

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If $$a + \alpha = 1, b + \beta = 2$$ and $$af(x) + \alpha f\left(\frac{1}{x}\right) = bx + \frac{\beta}{x}, x \neq 0$$, then the value of the expression $$\frac{f(x) + f\left(\frac{1}{x}\right)}{x + \frac{1}{x}}$$ is ______.

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NTA JEE Main 24th February 2021 Shift 2 - Question 89

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If the area of the triangle formed by the $$x$$-axis, the normal and the tangent to the circle $$(x - 2)^2 + (y - 3)^2 = 25$$ at the point (5, 7) is $$A$$, then $$24A$$ is equal to ______.

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NTA JEE Main 24th February 2021 Shift 2 - Question 90

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Let $$\lambda$$ be an integer. If the shortest distance between the lines $$x - \lambda = 2y - 1 = -2z$$ and $$x = y + 2\lambda = z - \lambda$$ is $$\frac{\sqrt{7}}{2\sqrt{2}}$$, then the value of $$|\lambda|$$ is ______.

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