NTA JEE Main 27th July 2021 Shift 1

NTA JEE Main 27th July 2021 Shift 1 - Question 1

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Assertion $$A$$ : If $$A, B, C, D$$ are four points on a semi-circular arc with a centre at $$O$$ such that $$\left|\overrightarrow{AB}\right| = \left|\overrightarrow{BC}\right| = \left|\overrightarrow{CD}\right|$$. Then, $$\overrightarrow{AB} + \overrightarrow{AC} + \overrightarrow{AD} = 4\overrightarrow{AO} + \overrightarrow{OB} + \overrightarrow{OC}$$
Reason $$R$$ : Polygon law of vector addition yields $$\overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{CD} = \overrightarrow{AD} = 2\overrightarrow{AO}$$

In the light of the above statements, choose the most appropriate answer from the options given below.

A

$$A$$ is correct but $$R$$ is not correct.

B

$$A$$ is not correct but $$R$$ is correct.

C

Both $$A$$ and $$R$$ are correct and $$R$$ is the correct explanation of $$A$$.

D

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

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NTA JEE Main 27th July 2021 Shift 1 - Question 2

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A ball is thrown up with a certain velocity so that it reaches a height $$h$$. Find the ratio of the two different times of the ball reaching $$\frac{h}{3}$$ in both the directions.

A

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

B

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

C

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

D

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

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NTA JEE Main 27th July 2021 Shift 1 - Question 3

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Three objects $$A$$, $$B$$ and $$C$$ are kept in a straight line on a frictionless horizontal surface. The masses of $$A$$, $$B$$ and $$C$$ are $$m$$, $$2m$$ and $$2m$$ respectively. $$A$$ moves towards $$B$$ with a speed of 9 m s$$^{-1}$$ and makes an elastic collision with it. Thereafter $$B$$ makes a completely inelastic collision with $$C$$. All motions occur along the same straight line. The final speed of $$C$$ is:

A

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

B

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

C

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

D

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

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NTA JEE Main 27th July 2021 Shift 1 - Question 4

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List-I                                                                                                                                                                     List-II
(a) MI of the rod (length $$L$$, Mass $$M$$, about an axis $$\perp$$ to the rod passing through the midpoint)          (i) $$\frac{8ML^2}{3}$$
(b) MI of the rod (length $$L$$, Mass 2M, about an axis $$\perp$$ to the rod passing through one of its end)        (ii) $$\frac{ML^2}{3}$$
(c) MI of the rod (length 2L, Mass $$M$$, about an axis $$\perp$$ to the rod passing through its midpoint)          (iii) $$\frac{ML^2}{12}$$
(d) MI of the rod (Length 2L, Mass 2M, about an axis $$\perp$$ to the rod passing through one of its end)        (iv) $$\frac{2ML^2}{3}$$

Choose the correct answer from the options given below:

A

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

B

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

C

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

D

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

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NTA JEE Main 27th July 2021 Shift 1 - Question 5

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The figure shows two solid discs with radius $$R$$ and $$r$$ respectively. If mass per unit area is the same for both, what is the ratio of MI of bigger disc around axis $$AB$$ (Which is $$\perp$$ to the plane of the disc and passing through its centre) of MI of smaller disc around one of its diameters lying on its plane? Given $$M$$ is the mass of the larger disc.

A

$$R^2 : r^2$$

B

$$2r^4 : R^4$$

C

$$2R^2 : r^2$$

D

$$2R^4 : r^4$$

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NTA JEE Main 27th July 2021 Shift 1 - Question 6

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A light cylindrical vessel is kept on a horizontal surface. Area of the base is $$A$$. A hole of cross-sectional area $$a$$ is made just at its bottom side. The minimum coefficient of friction necessary to prevent sliding the vessel due to the impact force of the emerging liquid is

A

$$\frac{A}{2a}$$

B

None of these

C

$$\frac{2a}{A}$$

D

$$\frac{a}{A}$$

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NTA JEE Main 27th July 2021 Shift 1 - Question 7

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A body takes 4 min to cool from 61°C to 59°C. If the temperature of the surroundings is 30°C, the time taken by the body to cool from 51°C to 49°C is:

A

4 min.

B

3 min.

C

8 min.

D

6 min.

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NTA JEE Main 27th July 2021 Shift 1 - Question 8

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In the reported figure, there is a cyclic process $$ABCDA$$ on a sample of 1 mol of a diatomic gas. The temperature of the gas during the process $$A \rightarrow B$$ and $$C \rightarrow D$$ are $$T_1$$ and $$T_2$$ ($$T_1 > T_2$$) respectively.

Choose the correct option out of the following for work done if processes $$BC$$ and $$DA$$ are adiabatic.

A

$$W_{AB} = W_{DC}$$

B

$$W_{AD} = W_{BC}$$

C

$$W_{BC} + W_{DA} > 0$$

D

$$W_{AB} < W_{CD}$$

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NTA JEE Main 27th July 2021 Shift 1 - Question 9

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The number of molecules in one litre of an ideal gas at 300 K and 2 atmospheric pressure with mean kinetic energy $$2 \times 10^{-9}$$ J per molecule is:

A

$$0.75 \times 10^{11}$$

B

$$3 \times 10^{11}$$

C

$$1.5 \times 10^{11}$$

D

$$6 \times 10^{11}$$

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NTA JEE Main 27th July 2021 Shift 1 - Question 10

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A particle starts executing simple harmonic motion (SHM) of amplitude $$a$$ and total energy $$E$$. At any instant, its kinetic energy is $$\frac{3E}{4}$$, then its displacement $$y$$ is given by:

A

$$y = a$$

B

$$y = \frac{a}{\sqrt{2}}$$

C

$$y = \frac{a\sqrt{3}}{2}$$

D

$$y = \frac{a}{2}$$

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NTA JEE Main 27th July 2021 Shift 1 - Question 11

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Two identical tennis balls each having mass $$m$$ and charge $$q$$ are suspended from a fixed point by threads of length $$l$$. What is the equilibrium separation when each thread makes a small angle $$\theta$$ with the vertical?

A

$$x = \left(\frac{q^2 l}{2\pi\varepsilon_0 mg}\right)^{\frac{1}{2}}$$

B

$$x = \left(\frac{q^2 l}{2\pi\varepsilon_0 mg}\right)^{\frac{1}{3}}$$

C

$$x = \left(\frac{q^2 l^2}{2\pi\varepsilon_0 m^2 g}\right)^{\frac{1}{3}}$$

D

$$x = \left(\frac{q^2 l^2}{2\pi\varepsilon_0 m^2 g^2}\right)^{\frac{1}{3}}$$

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NTA JEE Main 27th July 2021 Shift 1 - Question 12

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The relative permittivity of distilled water is 81. The velocity of light in it will be: (Given $$\mu_r = 1$$)

A

$$4.33 \times 10^7$$ m s$$^{-1}$$

B

$$2.33 \times 10^7$$ m s$$^{-1}$$

C

$$3.33 \times 10^7$$ m s$$^{-1}$$

D

$$5.33 \times 10^7$$ m s$$^{-1}$$

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NTA JEE Main 27th July 2021 Shift 1 - Question 13

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A capacitor of capacitance $$C = 1 \, \mu$$F is suddenly connected to a battery of 100 V through a resistance $$R = 100 \, \Omega$$. The time taken for the capacitor to be charged to get 50 V is:
(Take ln 2 = 0.69)

A

$$1.44 \times 10^{-4}$$ s

B

$$3.33 \times 10^{-4}$$ s

C

$$0.69 \times 10^{-4}$$ s

D

$$0.30 \times 10^{-4}$$ s

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NTA JEE Main 27th July 2021 Shift 1 - Question 14

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In the reported figure, a capacitor is formed by placing a compound dielectric between the plates of parallel plate capacitor. The expression for the capacity of the said capacitor will be: (Given the area of the plate = $$A$$)

A

$$\frac{15}{34} \frac{K\varepsilon_0 A}{d}$$

B

$$\frac{15}{6} \frac{K\varepsilon_0 A}{d}$$

C

$$\frac{25}{6} \frac{K\varepsilon_0 A}{d}$$

D

$$\frac{9}{6} \frac{K\varepsilon_0 A}{d}$$

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NTA JEE Main 27th July 2021 Shift 1 - Question 15

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Two capacitors of capacities $$2C$$ and $$C$$ are joined in parallel and charged up to potential $$V$$. The battery is removed and the capacitor of capacity $$C$$ is filled completely with a medium of dielectric constant $$K$$. The potential difference across the capacitors will now be:

A

$$\frac{V}{K+2}$$

B

$$\frac{V}{K}$$

C

$$\frac{3V}{K+2}$$

D

$$\frac{3V}{K}$$

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NTA JEE Main 27th July 2021 Shift 1 - Question 16

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In the given figure, a battery of emf $$E$$ is connected across a conductor $$PQ$$ of length $$l$$ and different area of cross-sections having radii $$r_1$$ and $$r_2$$ ($$r_2 < r_1$$).

Choose the correct option as one moves from $$P$$ to $$Q$$.

A

Drift velocity of electron increases.

B

Electric field decreases.

C

Electron current decreases.

D

All of these

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NTA JEE Main 27th July 2021 Shift 1 - Question 17

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A 0.07 H inductor and a 12 $$\Omega$$ resistor are connected in series to a 220 V, 50 Hz AC source. The approximate current in the circuit and the phase angle between current and source voltage are respectively. [Take $$\pi$$ as $$\frac{22}{7}$$]

A

8.8 A and $$\tan^{-1}\left(\frac{11}{6}\right)$$

B

88 A and $$\tan^{-1}\left(\frac{11}{6}\right)$$

C

0.88 A and $$\tan^{-1}\left(\frac{11}{6}\right)$$

D

8.8 A and $$\tan^{-1}\left(\frac{6}{11}\right)$$

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NTA JEE Main 27th July 2021 Shift 1 - Question 18

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In Young's double slit experiment, if the source of light changes from orange to blue then:

A

the central bright fringe will become a dark fringe.

B

the distance between consecutive fringes will decrease.

C

the distance between consecutive fringes will increase.

D

the intensity of the minima will increase.

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NTA JEE Main 27th July 2021 Shift 1 - Question 19

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If $$f$$ denotes the ratio of the number of nuclei decayed $$(N_d)$$ to the number of nuclei at $$t = 0$$, $$(N_0)$$ then for a collection of radioactive nuclei, the rate of change of $$f$$ with respect to time is given as: [$$\lambda$$ is the radioactive decay constant]

A

$$-\lambda(1 - e^{-\lambda t})$$

B

$$\lambda(1 - e^{-\lambda t})$$

C

$$\lambda e^{-\lambda t}$$

D

$$-\lambda e^{-\lambda t}$$

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NTA JEE Main 27th July 2021 Shift 1 - Question 20

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Assertion $$A$$ : If in five complete rotations of the circular scale, the distance travelled on the main scale of the screw gauge is 5 mm and there are 50 total divisions on a circular scale, then the least count is 0.001 cm.
Reason $$R$$ : Least Count = $$\frac{\text{Pitch}}{\text{Total divisions on circular scale}}$$
In the light of the above statements, choose the most appropriate answer from the options given below.

A

$$A$$ is not correct but $$R$$ is correct.

B

Both $$A$$ and $$R$$ are correct and $$R$$ is the correct explanation of $$A$$.

C

$$A$$ is correct but $$R$$ is not correct.

D

Both $$A$$ and $$R$$ are correct and $$R$$ is NOT the correct explanation of $$A$$.

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NTA JEE Main 27th July 2021 Shift 1 - Question 21

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Suppose two planets (spherical in shape) of radii $$R$$ and $$2R$$, but mass $$M$$ and $$9M$$ respectively have a centre to centre separation $$8R$$ as shown in the figure. A satellite of mass $$m$$ is projected from the surface of the planet of mass $$M$$ directly towards the centre of the second planet. The minimum speed $$v$$ required for the satellite to reach the surface of the second planet is $$\sqrt{\frac{a}{7} \frac{GM}{R}}$$, then the value of $$a$$ is
[Given: The two planets are fixed in their position]

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NTA JEE Main 27th July 2021 Shift 1 - Question 22

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A stone of mass 20 g is projected from a rubber catapult of length 0.1 m and area of cross section $$10^{-6}$$ m$$^2$$ stretched by an amount 0.04 m. The velocity of the projected stone is _________ m s$$^{-1}$$. (Young's modulus of rubber = $$0.5 \times 10^9$$ N m$$^{-2}$$)

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NTA JEE Main 27th July 2021 Shift 1 - Question 23

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In a uniform magnetic field, the magnetic needle has a magnetic moment $$9.85 \times 10^{-2}$$ A m$$^{-2}$$ and moment of inertia $$5 \times 10^{-6}$$ kg m$$^2$$. If it performs 10 complete oscillations in 5 seconds then the magnitude of the magnetic field is _________ mT [Take $$\pi^2$$ as 9.85]

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NTA JEE Main 27th July 2021 Shift 1 - Question 24

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Consider an electrical circuit containing a two way switch $$S$$. Initially $$S$$ is open and then $$T_1$$ is connected to $$T_2$$. As the current in $$R = 6 \, \Omega$$ attains a maximum value of steady-state level, $$T_1$$ is disconnected from $$T_2$$ and immediately connected to $$T_3$$. Potential drop across $$r = 3 \, \Omega$$ resistor immediately after $$T_1$$ is connected to $$T_3$$ is _________ V. (Round off to the Nearest Integer)

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NTA JEE Main 27th July 2021 Shift 1 - Question 25

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A prism of refractive index $$n_1$$ and another prism of refractive index $$n_2$$ are stuck together (as shown in the figure). $$n_1$$ and $$n_2$$ depend on $$\lambda$$, the wavelength of light, according to the relation $$n_1 = 1.2 + \frac{10.8 \times 10^{-14}}{\lambda^2}$$ and $$n_2 = 1.45 + \frac{1.8 \times 10^{-14}}{\lambda^2}$$
The wavelength for which rays incident at any angle on the interface $$BC$$ pass through without bending at that interface will be _________ nm.

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NTA JEE Main 27th July 2021 Shift 1 - Question 26

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A particle of mass $$9.1 \times 10^{-31}$$ kg travels in a medium with a speed of $$10^6$$ m s$$^{-1}$$ and a photon of radiation of linear momentum $$10^{-27}$$ kg m s$$^{-1}$$ travels in a vacuum. The wavelength of the photon is _________ times the wavelength of the particle.

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NTA JEE Main 27th July 2021 Shift 1 - Question 27

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In Bohr's atomic model, the electron is assumed to revolve in a circular orbit of radius 0.5 $$\mathring{A}$$. If the speed of electron is $$2.2 \times 10^6$$ m s$$^{-1}$$. Then the current associated with the electron will be _________ $$\times 10^{-2}$$ mA. [Take $$\pi$$ as $$\frac{22}{7}$$]

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NTA JEE Main 27th July 2021 Shift 1 - Question 28

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A radioactive sample has an average life of 30 ms and is decaying. A capacitor of capacitance 200 $$\mu$$F is first charged and later connected with resistor $$R$$. If the ratio of the charge on the capacitor to the activity of the radioactive sample is fixed with respect to time then the value of $$R$$ should be _________ $$\Omega$$.

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NTA JEE Main 27th July 2021 Shift 1 - Question 29

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A transistor is connected in common emitter circuit configuration, the collector supply voltage is 10 V and the voltage drop across a resistor of 1000 $$\Omega$$ in the collector circuit is 0.6 V. If the current gain factor $$(\beta)$$ is 24, then the base current is _________ $$\mu$$A. (Round off to the Nearest Integer)

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NTA JEE Main 27th July 2021 Shift 1 - Question 30

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The amplitude of upper and lower side bands of AM wave where a carrier signal with frequency 11.21 MHz, peak voltage 15 V is amplitude modulated by a 7.7 kHz sine wave of 5 V amplitude are $$\frac{a}{10}$$ V and $$\frac{b}{10}$$ V respectively. Then the value of $$\frac{a}{b}$$ is _________.

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NTA JEE Main 27th July 2021 Shift 1 - Question 31

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Given below are two statements:
Statement I : Rutherford's gold foil experiment cannot explain the line spectrum of hydrogen atom.
Statement II : Bohr's model of hydrogen atom contradicts Heisenberg's uncertainty principle.
In the light of the above statements, choose the most appropriate answer from the options given below:

A

Statement I is false but statement II is true.

B

Statement I is true but statement II is false.

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 27th July 2021 Shift 1 - Question 32

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Match List - I with List - II:
List-I                         List-II
(a) NaOH       (i) Acidic
(b) Be(OH)$$_2$$   (ii) Basic
(c) Ca(OH)$$_2$$   (iii) Amphoteric
(d) B(OH)$$_3$$  
(e) Al(OH)$$_3$$
Choose the most appropriate answer from the options given below:

A

(a)-(ii), (b)-(ii), (c)-(iii), (d)-(ii), (e)-(ii)

B

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

C

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

D

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

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NTA JEE Main 27th July 2021 Shift 1 - Question 33

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Given below are two statements: One is labelled as Assertion A and the other labelled as Reason R.
Assertion A : Lithium halides are somewhat covalent in nature.
Reason R : Lithium possess high polarisation capability.
In the light of the above statements, choose the most appropriate answer from the options given below:

A

A is true but R is false

B

A is false but R is true

C

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

D

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

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NTA JEE Main 27th July 2021 Shift 1 - Question 34

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The oxidation states of $$P$$ in H$$_4$$P$$_2$$O$$_7$$, H$$_4$$P$$_2$$O$$_5$$ and H$$_4$$P$$_2$$O$$_6$$, respectively, are:

A

7, 5 and 6

B

5, 4 and 3

C

5, 3 and 4

D

6, 4 and 5

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NTA JEE Main 27th July 2021 Shift 1 - Question 35

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Given below are two statements:
Statement I : Aniline is less basic than acetamide.
Statement II : In aniline, the lone pair of electrons on nitrogen atom is delocalised over benzene ring due to resonance and hence less available to a proton.
Choose the most appropriate option:

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 true.

D

Both statement I and statement II are false.

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NTA JEE Main 27th July 2021 Shift 1 - Question 36

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Staggered and eclipsed conformers of ethane are:

A

Polymers

B

Rotamers

C

Enantiomers

D

Mirror images

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NTA JEE Main 27th July 2021 Shift 1 - Question 37

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The correct order of stability of given carbocation is:

A

A > C > B > D

B

D > B > C > A

C

D > B > A > C

D

C > A > D > B

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NTA JEE Main 27th July 2021 Shift 1 - Question 38

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Presence of which reagent will affect the reversibility of the following reaction, and change it to a irreversible reaction:
$$CH_4 + I_2 \underset{\text{Reversible}}{\overset{h\nu}{\rightleftharpoons}} CH_3 - I + HI$$

A

HOCl

B

dilute HNO$$_2$$

C

Liquid NH$$_3$$

D

Concentrated HIO$$_3$$

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NTA JEE Main 27th July 2021 Shift 1 - Question 39

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Which one of the following statements is NOT correct?

A

Eutrophication indicates that water body is polluted

B

The dissolved oxygen concentration below 6 ppm inhibits fish growth

C

Eutrophication leads to increase in the oxygen level in water

D

Eutrophication leads to anaerobic conditions

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NTA JEE Main 27th July 2021 Shift 1 - Question 40

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The parameters of the unit cell of a substance are a = 2.5, b = 3.0, c = 4.0, $$\alpha$$ = 90°, $$\beta$$ = 120°, $$\gamma$$ = 90°. The crystal system of the substance is:

A

Hexagonal

B

Orthorhombic

C

Monoclinic

D

Triclinic

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NTA JEE Main 27th July 2021 Shift 1 - Question 41

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For a reaction of order n, the unit of the rate constant is:

A

$$mol^{1-n} \, L^{1-n} \, s$$

B

$$mol^{1-n} \, L^{2n} \, s^{-1}$$

C

$$mol^{1-n} \, L^{n-1} \, s^{-1}$$

D

$$mol^{1-n} \, L^{1-n} \, s^{-1}$$

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NTA JEE Main 27th July 2021 Shift 1 - Question 42

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The statement that is INCORRECT about Ellingham diagram is

A

provides idea about the reaction rate.

B

provides idea about free energy change.

C

provides idea about changes in the phases during the reaction.

D

provides idea about reduction of metal oxide.

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NTA JEE Main 27th July 2021 Shift 1 - Question 43

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The product obtained from the electrolytic oxidation of acidified sulphate solutions, is:

A

$$HSO_4^-$$

B

$$HO_3SOOSO_3H$$

C

$$HO_2SOSO_2H$$

D

$$HO_3SOSO_3H$$

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NTA JEE Main 27th July 2021 Shift 1 - Question 44

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The type of hybridisation and magnetic property of the complex $$[MnCl_6]^{3-}$$, respectively, are:

A

$$sp^3d^2$$ and diamagnetic

B

$$d^2sp^3$$ and diamagnetic

C

$$d^2sp^3$$ and paramagnetic

D

$$sp^3d^2$$ and paramagnetic

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NTA JEE Main 27th July 2021 Shift 1 - Question 45

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The number of geometrical isomers found in the metal complexes $$[PtCl_2(NH_3)_2]$$, $$[Ni(CO)_4]$$, $$[Ru(H_2O)_3Cl_3]$$ and $$[CoCl_2(NH_3)_4]^+$$ respectively, are:

A

1, 1, 1, 1

B

2, 0, 2, 2

C

2, 1, 2, 2

D

2, 1, 2, 1

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NTA JEE Main 27th July 2021 Shift 1 - Question 46

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Which one of the following compounds will give orange precipitate when treated with 2,4-dinitrophenyl hydrazine?

A

B

C

D

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NTA JEE Main 27th July 2021 Shift 1 - Question 47

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Consider the above reaction and identify the Product P:

A

B

C

D

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NTA JEE Main 27th July 2021 Shift 1 - Question 48

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Match List-I (Drug) with List-II (Class of Drug):
(a) Furacin              (i) Antibiotic
(b) Arsphenamine   (ii) Tranquilizers
(c) Dimetone          (iii) Antiseptic
(d) Valium              (iv) Synthetic antihistamines
Choose the most appropriate match:

A

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

B

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

C

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

D

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

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NTA JEE Main 27th July 2021 Shift 1 - Question 49

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Which one among the following chemical tests is used to distinguish monosaccharide from disaccharide?

A

Seliwanoff's test

B

Iodine test

C

Barfoed test

D

Tollen's test

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NTA JEE Main 27th July 2021 Shift 1 - Question 50

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The compound 'A' is a complementary base of _________ in DNA strands.

A

Uracil

B

Guanine

C

Adenine

D

Cytosine

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NTA JEE Main 27th July 2021 Shift 1 - Question 51

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The density of NaOH solution is 1.2 g cm$$^{-3}$$. The molality of this solution is _________ m (Round off to the Nearest Integer):
[Use: Atomic masses: Na : 23.0u, O : 16.0u, H : 1.0u
Density of H$$_2$$O : 1.0 g cm$$^{-3}$$]

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NTA JEE Main 27th July 2021 Shift 1 - Question 52

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In gaseous triethyl amine the C-N-C bond angle is _________ degree.

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NTA JEE Main 27th July 2021 Shift 1 - Question 53

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The difference between bond orders of CO and NO$$^{\oplus}$$ is $$\frac{x}{2}$$ where x = _________ (Round off to the Nearest Integer)

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NTA JEE Main 27th July 2021 Shift 1 - Question 54

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For water at 100°C and 1 bar,
$$\Delta_{vap}H - \Delta_{vap}U = \_ \times 10^2$$ J mol$$^{-1}$$
(Round off to the Nearest Integer)
[Use: R = 8.31 J mol$$^{-1}$$ K$$^{-1}$$]
[Assume volume of H$$_2$$O(l) is much smaller than volume of H$$_2$$O(g). Assume H$$_2$$O(g) treated as an ideal gas]

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NTA JEE Main 27th July 2021 Shift 1 - Question 55

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$$PCl_5 \rightleftharpoons PCl_3 + Cl_2 \quad K_c = 1.844$$
3.0 moles of PCl$$_5$$ is introduced in a 1L closed reaction vessel at 380 K. The number of moles of PCl$$_5$$ at equilibrium is _________ $$\times 10^{-3}$$ (Round off to the Nearest Integer)

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NTA JEE Main 27th July 2021 Shift 1 - Question 56

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An organic compound is subjected to chlorination to get compound A using 5.0 g of chlorine. When 0.5 g of compound A is reacted with AgNO$$_3$$ [Carius Method], the percentage of chlorine in compound A is when it forms 0.3849 g of AgCl. (Round off to the Nearest Integer)
(Atomic masses of Ag and Cl are 107.87 and 35.5 respectively)

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NTA JEE Main 27th July 2021 Shift 1 - Question 57

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1.46 g of a biopolymer dissolved in a 100 mL water at 300 K exerted an osmotic pressure of $$2.42 \times 10^{-3}$$ bar.
The molar mass of the biopolymer is _________ $$\times 10^4$$ g mol$$^{-1}$$. (Round off to the Nearest Integer)
[Use: R = 0.083 L bar mol$$^{-1}$$ K$$^{-1}$$]

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NTA JEE Main 27th July 2021 Shift 1 - Question 58

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The conductivity of a weak acid HA of concentration 0.001 mol L$$^{-1}$$ is $$2.0 \times 10^{-5}$$ S cm$$^{-1}$$. If $$\Lambda_m^0(HA) = 190$$ S cm$$^2$$ mol$$^{-1}$$, the ionization constant (K$$_a$$) of HA is equal to _________ $$\times 10^{-6}$$ (Round off to the Nearest Integer)

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NTA JEE Main 27th July 2021 Shift 1 - Question 59

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CO$$_2$$ gas adsorbs on charcoal following Freundlich adsorption isotherm. For a given amount of charcoal, the mass of CO$$_2$$ adsorbed becomes 64 times when the pressure of CO$$_2$$ is doubled.
The value of n in the Freundlich isotherm equation is _________ $$\times 10^2$$. (Round off to the Nearest Integer)

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NTA JEE Main 27th July 2021 Shift 1 - Question 60

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The number of geometrical isomers possible in triamminetrinitrocobalt (III) is X and in trioxalatochromate (III) is Y. Then the value of X + Y is _________.

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NTA JEE Main 27th July 2021 Shift 1 - Question 61

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Let $$\alpha, \beta$$ be two roots of the equation $$x^2 + (20)^{1/4}x + (5)^{1/2} = 0$$. Then $$\alpha^8 + \beta^8$$ is equal to:

A

10

B

100

C

50

D

160

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NTA JEE Main 27th July 2021 Shift 1 - Question 62

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Let $$C$$ be the set of all complex numbers. Let
$$S_1 = \{z \in C \mid |z - 3 - 2i|^2 = 8\}$$,
$$S_2 = \{z \in C \mid \text{Re}(z) \geq 5\}$$ and
$$S_3 = \{z \in C \mid |z - \bar{z}| \geq 8\}$$.
Then the number of elements in $$S_1 \cap S_2 \cap S_3$$ is equal to

A

1

B

0

C

2

D

Infinite

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NTA JEE Main 27th July 2021 Shift 1 - Question 63

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If the coefficients of $$x^7$$ in $$\left(x^2 + \frac{1}{bx}\right)^{11}$$ and $$x^{-7}$$ in $$\left(x - \frac{1}{bx^2}\right)^{11}$$, $$b \neq 0$$, are equal, then the value of $$b$$ is equal to:

A

2

B

$$-1$$

C

1

D

$$-2$$

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NTA JEE Main 27th July 2021 Shift 1 - Question 64

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If $$\sin \theta + \cos \theta = \frac{1}{2}$$, then $$16(\sin(2\theta) + \cos(4\theta) + \sin(6\theta))$$ is equal to:

A

23

B

$$-27$$

C

$$-23$$

D

27

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NTA JEE Main 27th July 2021 Shift 1 - Question 65

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Two tangents are drawn from the point $$P(-1, 1)$$ to the circle $$x^2 + y^2 - 2x - 6y + 6 = 0$$. If these tangents touch the circle at points $$A$$ and $$B$$, and if $$D$$ is a point on the circle such that length of the segments $$AB$$ and $$AD$$ are equal, then the area of the triangle $$ABD$$ is equal to:

A

2

B

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

C

4

D

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

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NTA JEE Main 27th July 2021 Shift 1 - Question 66

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Let $$P$$ and $$Q$$ be two distinct points on a circle which has center at $$C(2, 3)$$ and which passes through origin $$O$$. If $$OC$$ is perpendicular to both the line segments $$CP$$ and $$CQ$$, then the set $$\{P, Q\}$$ is equal to

A

$$\{(4, 0), (0, 6)\}$$

B

$$\{(2 + 2\sqrt{2}, 3 - \sqrt{5}), (2 - 2\sqrt{2}, 3 + \sqrt{5})\}$$

C

$$\{(2 + 2\sqrt{2}, 3 + \sqrt{5}), (2 - 2\sqrt{2}, 3 - \sqrt{5})\}$$

D

$$\{(-1, 5), (5, 1)\}$$

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NTA JEE Main 27th July 2021 Shift 1 - Question 67

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Let
$$A = \{(x, y) \in R \times R \mid 2x^2 + 2y^2 - 2x - 2y = 1\}$$
$$B = \{(x, y) \in R \times R \mid 4x^2 + 4y^2 - 16y + 7 = 0\}$$ and
$$C = \{(x, y) \in R \times R \mid x^2 + y^2 - 4x - 2y + 5 \leq r^2\}$$.
Then the minimum value of $$|r|$$ such that $$A \cup B \subseteq C$$ is equal to

A

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

B

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

C

$$\frac{3 + 2\sqrt{5}}{2}$$

D

$$1 + \sqrt{5}$$

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NTA JEE Main 27th July 2021 Shift 1 - Question 68

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A ray of light through $$(2, 1)$$ is reflected at a point $$P$$ on the $$y$$-axis and then passes through the point $$(5, 3)$$. If this reflected ray is the directrix of an ellipse with eccentricity $$\frac{1}{3}$$ and the distance of the nearer focus from this directrix is $$\frac{8}{\sqrt{53}}$$, then the equation of the other directrix can be:

A

$$11x + 7y + 8 = 0$$ or $$11x + 7y - 15 = 0$$

B

$$11x - 7y - 8 = 0$$ or $$11x + 7y + 15 = 0$$

C

$$2x - 7y + 29 = 0$$ or $$2x - 7y - 7 = 0$$

D

$$2x - 7y - 39 = 0$$ or $$2x - 7y - 7 = 0$$

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NTA JEE Main 27th July 2021 Shift 1 - Question 69

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Let $$f : R \rightarrow R$$ be a function such that $$f(2) = 4$$ and $$f'(2) = 1$$. Then, the value of $$\lim_{x \to 2} \frac{x^2 f(2) - 4f(x)}{x - 2}$$ is equal to:

A

4

B

8

C

16

D

12

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NTA JEE Main 27th July 2021 Shift 1 - Question 70

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The compound statement $$(P \vee Q) \wedge (\sim P) \Rightarrow Q$$ equivalent to:

A

$$P \vee Q$$

B

$$P \wedge \sim Q$$

C

$$\sim(P \Rightarrow Q)$$

D

$$\sim(P \Rightarrow Q) \Leftrightarrow P \wedge \sim Q$$

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NTA JEE Main 27th July 2021 Shift 1 - Question 71

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If the mean and variance of the following data: 6, 10, 7, 13, $$a$$, 12, $$b$$, 12 are 9 and $$\frac{37}{4}$$ respectively, then $$(a - b)^2$$ is equal to:

A

24

B

12

C

32

D

16

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NTA JEE Main 27th July 2021 Shift 1 - Question 72

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Let $$A = \begin{bmatrix} 1 & 2 \\ -1 & 4 \end{bmatrix}$$. If $$A^{-1} = \alpha I + \beta A$$, $$\alpha, \beta \in R$$, $$I$$ is a $$2 \times 2$$ identity matrix, then $$4(\alpha - \beta)$$ is equal to:

A

5

B

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

C

2

D

4

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NTA JEE Main 27th July 2021 Shift 1 - Question 73

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Let $$f : \left(-\frac{\pi}{4}, \frac{\pi}{4}\right) \rightarrow R$$ be defined as,
$$f(x) = \begin{cases} (1 + |\sin x|)^{\frac{3a}{|\sin x|}}, & -\frac{\pi}{4} < x < 0 \\ b, & x = 0 \\ e^{\cot 4x / \cot 2x}, & 0 < x < \frac{\pi}{4} \end{cases}$$
If $$f$$ is continuous at $$x = 0$$ then the value of $$6a + b^2$$ is equal to:

A

$$1 - e$$

B

$$e - 1$$

C

$$1 + e$$

D

$$e$$

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NTA JEE Main 27th July 2021 Shift 1 - Question 74

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The value of $$\lim_{n \to \infty} \frac{1}{n} \sum_{j=1}^{n} \frac{(2j-1) + 8n}{(2j-1) + 4n}$$ is equal to:

A

$$5 + \log_e\left(\frac{3}{2}\right)$$

B

$$2 - \log_e\left(\frac{2}{3}\right)$$

C

$$3 + 2\log_e\left(\frac{2}{3}\right)$$

D

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

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NTA JEE Main 27th July 2021 Shift 1 - Question 75

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The value of the definite integral $$\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{dx}{(1 + e^{x\cos x})(\sin^4 x + \cos^4 x)}$$ is equal to:

A

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

B

$$\frac{\pi}{2\sqrt{2}}$$

C

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

D

$$\frac{\pi}{\sqrt{2}}$$

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NTA JEE Main 27th July 2021 Shift 1 - Question 76

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If the area of the bounded region $$R = \{(x, y) : \max\{0, \log_e x\} \leq y \leq 2^x, \frac{1}{2} \leq x \leq 2\}$$ is, $$\alpha(\log_e 2)^{-1} + \beta(\log_e 2) + \gamma$$ then the value of $$(\alpha + \beta - 2\gamma)^2$$ is equal to:

A

8

B

2

C

4

D

1

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NTA JEE Main 27th July 2021 Shift 1 - Question 77

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Let $$y = y(x)$$ be solution of the differential equation $$\log_e\left(\frac{dy}{dx}\right) = 3x + 4y$$, with $$y(0) = 0$$. If $$y\left(-\frac{2}{3}\log_e 2\right) = \alpha \log_e 2$$, then the value of $$\alpha$$ is equal to:

A

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

B

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

C

2

D

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

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NTA JEE Main 27th July 2021 Shift 1 - Question 78

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Let $$\vec{a} = \hat{i} + \hat{j} + 2\hat{k}$$ and $$\vec{b} = -\hat{i} + 2\hat{j} + 3\hat{k}$$. Then the vector product $$\left(\vec{a} + \vec{b}\right) \times \left(\left(\vec{a} \times \left(\left(\vec{a} - \vec{b}\right) \times \vec{b}\right)\right) \times \vec{b}\right)$$ is equal to:

A

$$5(34\hat{i} - 5\hat{j} + 3\hat{k})$$

B

$$7(34\hat{i} - 5\hat{j} + 3\hat{k})$$

C

$$7(30\hat{i} - 5\hat{j} + 7\hat{k})$$

D

$$5(30\hat{i} - 5\hat{j} + 7\hat{k})$$

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NTA JEE Main 27th July 2021 Shift 1 - Question 79

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Let the plane passing through the point $$(-1, 0, -2)$$ and perpendicular to each of the planes $$2x + y - z = 2$$ and $$x - y - z = 3$$ be $$ax + by + cz + 8 = 0$$. Then the value of $$a + b + c$$ is equal to:

A

3

B

8

C

5

D

4

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NTA JEE Main 27th July 2021 Shift 1 - Question 80

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The probability that a randomly selected 2-digit number belongs to the set $$\{n \in N : (2^n - 2)$$ is a multiple of 3$$\}$$ is equal to

A

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

B

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

C

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

D

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

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NTA JEE Main 27th July 2021 Shift 1 - Question 81

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If $$\log_3 2, \log_3(2^x - 5), \log_3\left(2^x - \frac{7}{2}\right)$$ are in an arithmetic progression, then the value of $$x$$ is equal to _________.

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NTA JEE Main 27th July 2021 Shift 1 - Question 82

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For real numbers $$\alpha$$ and $$\beta$$, consider the following system of linear equations: $$x + y - z = 2$$, $$x + 2y + \alpha z = 1$$ and $$2x - y + z = \beta$$. If the system has infinite solutions, then $$\alpha + \beta$$ is equal to _________.

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NTA JEE Main 27th July 2021 Shift 1 - Question 83

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Let $$f(x) = \begin{vmatrix} \sin^2 x & -2 + \cos^2 x & \cos 2x \\ 2 + \sin^2 x & \cos^2 x & \cos 2x \\ \sin^2 x & \cos^2 x & 1 + \cos 2x \end{vmatrix}$$, $$x \in [0, \pi]$$. Then the maximum value of $$f(x)$$ is equal to _________.

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NTA JEE Main 27th July 2021 Shift 1 - Question 84

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Let the domain of the function $$f(x) = \log_4(\log_5(\log_3(18x - x^2 - 77)))$$ be $$(a, b)$$. Then the value of the integral $$\int_a^b \frac{\sin^3 x}{\sin^3 x + \sin^3(a + b - x)}$$ is equal to _________.

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NTA JEE Main 27th July 2021 Shift 1 - Question 85

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Let $$S = \{1, 2, 3, 4, 5, 6, 7\}$$. Then the number of possible functions $$f : S \rightarrow S$$ such that $$f(m \cdot n) = f(m) \cdot f(n)$$ for every $$m, n \in S$$ and $$m \cdot n \in S$$, is equal to _________.

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NTA JEE Main 27th July 2021 Shift 1 - Question 86

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Let $$f : [0, 3] \rightarrow R$$ be defined by $$f(x) = \min\{x - [x], 1 + [x] - x\}$$ where $$[x]$$ is the greatest integer less than or equal to $$x$$. Let $$P$$ denote the set containing all $$x \in [0, 3]$$ where $$f$$ is discontinuous, and $$Q$$ denote the set containing all $$x \in (0, 3)$$ where $$f$$ is not differentiable. Then the sum of number of elements in $$P$$ and $$Q$$ is equal to _________.

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NTA JEE Main 27th July 2021 Shift 1 - Question 87

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Let $$F : [3, 5] \rightarrow R$$ be a twice differentiable function on $$(3, 5)$$ such that $$F(x) = e^{-x} \int_3^x (3t^2 + 2t + 4F'(t)) \, dt$$. If $$F'(4) = \frac{\alpha e^\beta - 224}{(e^\beta - 4)^2}$$, then $$\alpha + \beta$$ is equal to _________.

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NTA JEE Main 27th July 2021 Shift 1 - Question 88

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If $$y = y(x)$$, $$y \in \left[0, \frac{\pi}{2}\right)$$ is the solution of the differential equation $$\sec y \frac{dy}{dx} - \sin(x + y) - \sin(x - y) = 0$$, with $$y(0) = 0$$, then $$5y'\left(\frac{\pi}{2}\right)$$ is equal to _________.

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NTA JEE Main 27th July 2021 Shift 1 - Question 89

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Let $$\vec{a} = \hat{i} + \hat{j} + \hat{k}$$, $$\vec{b}$$ and $$\vec{c} = \hat{j} - \hat{k}$$ be three vectors such that $$\vec{a} \times \vec{b} = \vec{c}$$ and $$\vec{a} \cdot \vec{b} = 1$$. If the length of projection vector of the vector $$\vec{b}$$ on the vector $$\vec{a} \times \vec{c}$$ is $$l$$, then the value of $$3l^2$$ is equal to _________.

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NTA JEE Main 27th July 2021 Shift 1 - Question 90

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Let a plane $$P$$ pass through the point $$(3, 7, -7)$$ and contain the line, $$\frac{x - 2}{-3} = \frac{y - 3}{2} = \frac{z + 2}{1}$$. If distance of the plane $$P$$ from the origin is $$d$$, then $$d^2$$ is equal to _________.

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