NTA JEE Main 29th July 2022 Shift 1

NTA JEE Main 29th July 2022 Shift 1 - Question 1

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Given below are two statements: One is labelled as Assertion (A) and other is labelled as Reason (R)
Assertion (A): Time period of oscillation of a liquid drop depends on surface tension (S), if density of the liquid is $$\rho$$ and radius of the drop is $$r$$, then $$T = K\sqrt{\frac{\rho r^3}{S^{3/2}}}$$ is dimensionally correct, where K is dimensionless.
Reason (R): Using dimensional analysis we get R.H.S. having different dimension than that of time period.
In the light of above statements, choose the correct answer from the options given below.

A

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

B

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

C

(A) is true but (R) is false

D

(A) is false but (R) is true

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NTA JEE Main 29th July 2022 Shift 1 - Question 2

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A ball is thrown up vertically with a certain velocity so that it reaches a maximum height h. Find the ratio of the times in which it is at height $$\frac{h}{3}$$ while going up and coming down respectively.

A

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

B

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

C

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

D

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

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NTA JEE Main 29th July 2022 Shift 1 - Question 3

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If $$t = \sqrt{x} + 4$$, then $$\left(\frac{dx}{dt}\right)_{t=4}$$ is:

A

4

B

Zero

C

8

D

16

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NTA JEE Main 29th July 2022 Shift 1 - Question 4

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A ball is projected with kinetic energy E, at an angle of $$60^\circ$$ to the horizontal. The kinetic energy of this ball at the highest point of its flight will become:

A

Zero

B

$$\frac{E}{2}$$

C

$$\frac{E}{4}$$

D

E

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NTA JEE Main 29th July 2022 Shift 1 - Question 5

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A smooth circular groove has a smooth vertical wall as shown in figure. A block of mass m moves against the wall with a speed v. Which of the following curve represents the correct relation between the normal reaction on the block by the wall (N) and speed of the block (v)?

A

B

C

D

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NTA JEE Main 29th July 2022 Shift 1 - Question 6

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Two bodies of mass 1 kg and 3 kg have position vectors $$\hat{i} + 2\hat{j} + \hat{k}$$ and $$-3\hat{i} - 2\hat{j} + \hat{k}$$ respectively. The magnitude of position vector of centre of mass of this system will be similar to the magnitude of vector:

A

$$\hat{i} - 2\hat{j} + \hat{k}$$

B

$$-3\hat{i} - 2\hat{j} + \hat{k}$$

C

$$-2\hat{i} + 2\hat{k}$$

D

$$-2\hat{i} - \hat{j} + 2\hat{k}$$

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NTA JEE Main 29th July 2022 Shift 1 - Question 7

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If the length of a wire is made double and radius is halved of its respective values. Then, the Young's modulus of the material of the wire will:

A

Remain same

B

Become 8 times its initial value

C

Become $$\frac{1^{th}}{4}$$ of its initial value

D

Become 4 times its initial value

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NTA JEE Main 29th July 2022 Shift 1 - Question 8

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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): Clothes containing oil or grease stains cannot be cleaned by water wash.
Reason (R): Because the angle of contact between the oil/grease and water is obtuse.
In the light of the above statements, choose the correct answer from the option given below.

A

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

B

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

C

(A) is true but (R) is false

D

(A) is true but (R) is true

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NTA JEE Main 29th July 2022 Shift 1 - Question 9

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Two metallic wires of identical dimensions are connected in series. If $$\sigma_1$$ and $$\sigma_2$$ are the conductivities of these wires respectively, the effective conductivity of the combination is:

A

$$\frac{\sigma_1\sigma_2}{\sigma_1+\sigma_2}$$

B

$$\frac{2\sigma_1\sigma_2}{\sigma_1+\sigma_2}$$

C

$$\frac{\sigma_1+\sigma_2}{2\sigma_1\sigma_2}$$

D

$$\frac{\sigma_1+\sigma_2}{\sigma_1\sigma_2}$$

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NTA JEE Main 29th July 2022 Shift 1 - Question 10

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The time period of oscillation of a simple pendulum of length L suspended from the roof of a vehicle, which moves without friction down an inclined plane of inclination $$\alpha$$, is given by:

A

$$2\pi\sqrt{\frac{L}{g\cos\alpha}}$$

B

$$2\pi\sqrt{\frac{L}{g\sin\alpha}}$$

C

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

D

$$2\pi\sqrt{\frac{L}{g\tan\alpha}}$$

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NTA JEE Main 29th July 2022 Shift 1 - Question 11

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A spherically symmetric charge distribution is considered with charge density varying as $$\rho(r) = \begin{cases} \rho_0\left(\frac{3}{4} - \frac{r}{R}\right) & \text{for } r \leq R \\ 0 & \text{for } r > R \end{cases}$$
Where, $$r(r < R)$$ is the distance from the centre O (as shown in figure). The electric field at point P will be:

A

$$\frac{\rho_0 r}{4\varepsilon_0}\left(\frac{3}{4} - \frac{r}{R}\right)$$

B

$$\frac{\rho_0 r}{3\varepsilon_0}\left(\frac{3}{4} - \frac{r}{R}\right)$$

C

$$\frac{\rho_0 r}{4\varepsilon_0}\left(1 - \frac{r}{R}\right)$$

D

$$\frac{\rho_0 r}{5\varepsilon_0}\left(1 - \frac{r}{R}\right)$$

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NTA JEE Main 29th July 2022 Shift 1 - Question 12

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Given below are two statements.
Statement I: Electric potential is constant within and at the surface of each conductor.
Statement II: Electric field just outside a charged conductor is perpendicular to the surface of the conductor at every point.
In the light of the above statements, choose the most appropriate answer from the options given below.

A

Both statement I and statement II are correct

B

Both statement I and statement II are incorrect

C

Statement I is correct but statement II is incorrect

D

Statement I is incorrect but statement II is correct

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NTA JEE Main 29th July 2022 Shift 1 - Question 13

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An alternating emf $$E = 440\sin 100\pi t$$ is applied to a circuit containing an inductance of $$\frac{\sqrt{2}}{\pi}$$ H. If an a.c. ammeter is connected in the circuit, its reading will be:

A

4.4 A

B

1.55 A

C

2.2 A

D

3.11 A

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NTA JEE Main 29th July 2022 Shift 1 - Question 14

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A coil of inductance 1 H and resistance 100 $$\Omega$$ is connected to a battery of 6 V. Determine approximately:
(a) The time elapsed before the current acquires half of its steady-state value
(b) The energy stored in the magnetic field associated with the coil at an instant 15 ms after the circuit is switched on.
(Given $$\ln 2 = 0.693$$, $$e^{-3/2} = 0.25$$)

A

$$t = 10$$ ms; $$U = 2$$ mJ

B

$$t = 10$$ ms; $$U = 1$$ mJ

C

$$t = 7$$ ms; $$U = 1$$ mJ

D

$$t = 7$$ ms; $$U = 2$$ mJ

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NTA JEE Main 29th July 2022 Shift 1 - Question 15

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Match List-I with List-II

List-I	List-II
(a) UV rays	(i) Diagnostic tool in medicine
(b) X-rays	(ii) Water purification
(c) Microwave	(iii) Communication, Radar
(d) Infrared wave	(iv) Improving visibility in foggy days

Choose the correct answer from the options given below :

A

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

B

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

C

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

D

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

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NTA JEE Main 29th July 2022 Shift 1 - Question 16

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The kinetic energy of emitted electron is E when the light incident on the metal has wavelength $$\lambda$$. To double the kinetic energy, the incident light must have wavelength:

A

$$\frac{hc}{E\lambda - hc}$$

B

$$\frac{hc\lambda}{E\lambda + hc}$$

C

$$\frac{h\lambda}{E\lambda + hc}$$

D

$$\frac{hc\lambda}{E\lambda - hc}$$

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NTA JEE Main 29th July 2022 Shift 1 - Question 17

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Find the ratio of energies of photons produced due to transition of an electron of hydrogen atom from its (i) second permitted energy level to the first level, and (ii) the highest permitted energy level to the first permitted level.

A

3 : 4

B

4 : 3

C

1 : 4

D

4 : 1

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NTA JEE Main 29th July 2022 Shift 1 - Question 18

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Find the modulation index of an AM wave having 8 V variation where maximum amplitude of the AM wave is 9 V.

A

0.8

B

0.5

C

0.2

D

0.1

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NTA JEE Main 29th July 2022 Shift 1 - Question 19

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A travelling microscope has 20 divisions per cm on the main scale while its Vernier scale has total 50 divisions and 25 Vernier scale divisions are equal to 24 main scale divisions, what is the least count of the travelling microscope?

A

0.001 cm

B

0.002 mm

C

0.002 cm

D

0.005 cm

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NTA JEE Main 29th July 2022 Shift 1 - Question 20

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In an experiment to find out the diameter of wire using screw gauge, the following observation were noted:
(a) Screw moves 0.5 mm on main scale in one complete rotation
(b) Total divisions on circular scale = 50
(c) Main scale reading is 2.5 mm
(d) 45th division of circular scale is in the pitch line
(e) Instrument has 0.03 mm negative error
Then the diameter of wire is:

A

2.92 mm

B

2.54 mm

C

2.98 mm

D

3.45 mm

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NTA JEE Main 29th July 2022 Shift 1 - Question 21

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An object is projected in the air with initial velocity u at an angle $$\theta$$. The projectile motion is such that the horizontal range R, is maximum. Another object is projected in the air with a horizontal range half of the range of first object. The initial velocity remains same in both the case. The value of the angle of projection, at which the second object is projected, will be _____ degree.

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NTA JEE Main 29th July 2022 Shift 1 - Question 22

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If the acceleration due to gravity experienced by a point mass at a height h above the surface of earth is same as that of the acceleration due to gravity at a depth $$\alpha h$$ ($$ h \ll R_E$$) from the earth surface. The value of $$\alpha$$ will be _____. (use $$R_E = 6400$$ km)

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NTA JEE Main 29th July 2022 Shift 1 - Question 23

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The pressure $$P_1$$ and density $$d_1$$ of diatomic gas $$(\gamma = \frac{7}{5})$$ changes suddenly to $$P_2(> P_1)$$ and $$d_2$$ respectively during an adiabatic process. The temperature of the gas increases and becomes _____ times of its initial temperature. (Given $$\frac{d_2}{d_1} = 32$$)

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NTA JEE Main 29th July 2022 Shift 1 - Question 24

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One mole of a monoatomic gas is mixed with three moles of a diatomic gas. The molecular specific heat of mixture at constant volume is $$\frac{\alpha^2}{4}R$$ J mol$$^{-1}$$ K$$^{-1}$$; then the value of $$\alpha$$ will be _____. (Assume that the given diatomic gas has no vibrational mode.)

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NTA JEE Main 29th July 2022 Shift 1 - Question 25

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Two light beams of intensities 4I and 9I interfere on a screen. The phase difference between these beams on the screen at point A is zero and at point B is $$\pi$$. The difference of resultant intensities, at the point A and B, will be _____ I.

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NTA JEE Main 29th July 2022 Shift 1 - Question 26

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The current I flowing through the given circuit will be _____ A.

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NTA JEE Main 29th July 2022 Shift 1 - Question 27

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A closely wound circular coil of radius 5 cm produces a magnetic field of $$37.68 \times 10^{-4}$$ T at its center. The current through the coil is _____ A. [Given, number of turns in the coil is 100 and $$\pi = 3.14$$]

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NTA JEE Main 29th July 2022 Shift 1 - Question 28

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A wire of length 314 cm carrying current of 14 A is bent to form a circle. The magnetic moment of the coil is _____ A-m$$^2$$. [Given $$\pi = 3.14$$]

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NTA JEE Main 29th July 2022 Shift 1 - Question 29

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The X-Y plane be taken as the boundary between two transparent media $$M_1$$ and $$M_2$$. $$M_1$$ in $$Z \geq 0$$ has a refractive index of $$\sqrt{2}$$ and $$M_2$$ with $$Z < 0$$ has a refractive index of $$\sqrt{3}$$. A ray of light travelling in $$M_1$$ along the direction given by the vector $$\vec{A} = 4\sqrt{3}\hat{i} - 3\sqrt{3}\hat{j} - 5\hat{k}$$, is incident on the plane of separation. The value of difference between the angle of incident in $$M_1$$ and the angle of refraction in $$M_2$$ will be _____ degree.

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NTA JEE Main 29th July 2022 Shift 1 - Question 30

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If the potential barrier across a p-n junction is 0.6 V. Then the electric field intensity, in the depletion region having the width of $$6 \times 10^{-6}$$ m, will be _____ $$\times 10^5$$ N C$$^{-1}$$

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NTA JEE Main 29th July 2022 Shift 1 - Question 31

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$$N_2(g) + 3H_2(g) \rightleftharpoons 2NH_3(g)$$
Consider the reaction: . If 20 g of dinitrogen reacts with 5 g of dihydrogen, then the limiting reagent of the reaction and number of moles of $$NH_3$$ formed respectively are

A

$$H_2$$, 1.42 moles

B

$$H_2$$, 0.71 moles

C

$$N_2$$, 1.42 moles

D

$$N_2$$, 0.71 moles

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NTA JEE Main 29th July 2022 Shift 1 - Question 32

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The first ionization enthalpy of Na and Mg, respectively, are: 496 & 737  kJ mol$$^{-1}$$. The first ionization enthalpy (kJ mol$$^{-1}$$) of Al is

A

487

B

768

C

577

D

856

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NTA JEE Main 29th July 2022 Shift 1 - Question 33

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Which of the following pair of molecules contain odd electron molecule and an expanded octet molecule?

A

$$BCl_3$$ and $$SF_6$$

B

$$NO$$ and $$H_2SO_4$$

C

$$SF_6$$ and $$H_2SO_4$$

D

$$BCl_3$$ and $$NO$$

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NTA JEE Main 29th July 2022 Shift 1 - Question 34

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Number of lone pairs of electrons in the central atom of $$SCl_2$$, $$O_3$$, $$ClF_3$$ and $$SF_6$$, respectively, are

A

0, 1, 2 and 2

B

2, 1, 2 and 0

C

1, 2, 2 and 0

D

None of these

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NTA JEE Main 29th July 2022 Shift 1 - Question 35

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The reaction of zinc with excess of aqueous alkali, evolves hydrogen gas and gives

A

$$Zn(OH)_2$$

B

$$ZnO$$

C

$$[Zn(OH)_4]^{2-}$$

D

$$[ZnO_2]^{2-}$$

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NTA JEE Main 29th July 2022 Shift 1 - Question 36

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Lithium nitrate and sodium nitrate, when heated separately, respectively, give

A

$$LiNO_2$$ and $$NaNO_2$$

B

$$Li_2O$$ and $$Na_2O$$

C

$$Li_2O$$ and $$NaNO_2$$

D

$$LiNO_2$$ and $$Na_2O$$

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NTA JEE Main 29th July 2022 Shift 1 - Question 37

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Which among the following pairs of structures will give different products on ozonolysis? (Consider the double bonds in the structures are rigid and not delocalized.)

A

B

C

D

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NTA JEE Main 29th July 2022 Shift 1 - Question 38

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Which among the following pairs has only herbicides?

A

Aldrin and Dieldrin

B

Sodium chlorate and Aldrin

C

Sodium arsinate and Dieldrin

D

Sodium chlorate and sodium arsenite

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NTA JEE Main 29th July 2022 Shift 1 - Question 39

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100 mL of 5% (w/v) solution of NaCl in water was prepared in 250 mL beaker. Albumin from the egg was poured into NaCl solution and stirred well. This resulted in a/an

A

Lyophilic sol

B

Lyophobic sol

C

Emulsion

D

Precipitate

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NTA JEE Main 29th July 2022 Shift 1 - Question 40

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In metallurgy the term "gangue" is used for

A

Contamination of undesired earthy materials.

B

Contamination of metals, other than desired metal

C

Minerals which are naturally occurring in pure form

D

Magnetic impurities in an ore.

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NTA JEE Main 29th July 2022 Shift 1 - Question 41

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In following pairs, the one in which both transition metal ions are colourless is

A

$$Sc^{3+}, Zn^{2+}$$

B

$$Ti^{4+}, Cu^{2+}$$

C

$$V^{2+}, Ti^{3+}$$

D

$$Zn^{2+}, Mn^{2+}$$

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NTA JEE Main 29th July 2022 Shift 1 - Question 42

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In neutral or faintly alkaline medium, $$KMnO_4$$ being a powerful oxidant can oxidise, thiosulphate almost quantitatively, to sulphate. In this reaction overall change in oxidation state of manganese will be

A

5

B

1

C

0

D

3

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NTA JEE Main 29th July 2022 Shift 1 - Question 43

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Considering the above reactions, the compound 'A' and compound 'B' respectively are

A

B

C

D

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NTA JEE Main 29th July 2022 Shift 1 - Question 44

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Consider the above reaction sequence, the Product 'C' is

A

B

C

D

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NTA JEE Main 29th July 2022 Shift 1 - Question 45

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A compound 'X' is acidic and it is soluble in NaOH solution, but insoluble in $$NaHCO_3$$ solution. Compound 'X' also gives violet colour with neutral $$FeCl_3$$ solution. The compound 'X' is

A

B

C

D

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NTA JEE Main 29th July 2022 Shift 1 - Question 46

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Consider the following reaction sequence. The product 'B' is

A

B

C

D

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NTA JEE Main 29th July 2022 Shift 1 - Question 47

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Which among the following is the strongest Bronsted base?

A

B

C

D

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NTA JEE Main 29th July 2022 Shift 1 - Question 48

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Consider the above reaction, the compound 'A' is

A

B

C

D

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NTA JEE Main 29th July 2022 Shift 1 - Question 49

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Which among the following represent reagent 'A'?

A

B

C

D

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NTA JEE Main 29th July 2022 Shift 1 - Question 50

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Which of the following compounds is an example of hypnotic drug?

A

Seldane

B

Amytal

C

Aspartame

D

Prontosil

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NTA JEE Main 29th July 2022 Shift 1 - Question 51

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The minimum uncertainty in the speed of an electron in one dimensional region of length $$2a_0$$ (Where $$a_0$$ = Bohr radius = 52.9 pm) is _____ km s$$^{-1}$$ (Nearest integer) (Given: Mass of electron $$= 9.1 \times 10^{-31}$$ kg, Planck's constant $$h = 6.63 \times 10^{-34}$$ Js)

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NTA JEE Main 29th July 2022 Shift 1 - Question 52

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When 600 mL of 0.2 M $$HNO_3$$ is mixed with 400 mL of 0.1 M NaOH solution in a flask, the rise in temperature of the flask is _____ $$\times 10^{-2}$$ °C (Enthalpy of neutralisation = 57 kJ mol$$^{-1}$$ and Specific heat of water = 4.2 J K$$^{-1}$$ g$$^{-1}$$) (Neglect heat capacity of flask)

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NTA JEE Main 29th July 2022 Shift 1 - Question 53

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If the solubility product of PbS is $$8 \times 10^{-28}$$, then the solubility of PbS in pure water at 298 K is $$x \times 10^{-16}$$ mol L$$^{-1}$$. The value of x is _____ (Nearest integer) [Given $$\sqrt{2} = 1.41$$]

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NTA JEE Main 29th July 2022 Shift 1 - Question 54

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In bromination of Propyne, with Bromine 1,1,2,2-tetrabromopropane is obtained in 27% yield. The amount of 1,1,2,2-tetrabromopropane obtained from 1 g of Bromine in this reaction is _____ $$\times 10^{-1}$$ g. (Molar Mass: Bromine = 80 g/mol)

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NTA JEE Main 29th July 2022 Shift 1 - Question 55

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Ionic radii of cation $$A^+$$ and anion $$B^-$$ are 102 and 181 pm respectively. These ions are allowed to crystallize into an ionic solid. This crystal has cubic close packing for $$B^-$$. $$A^+$$ is present in all octahedral voids. The edge length of the unit cell of the crystal AB is _____ pm.

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NTA JEE Main 29th July 2022 Shift 1 - Question 56

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If $$O_2$$ gas is bubbled through water at 303 K, the number of millimoles of $$O_2$$ gas that dissolve in 1 litre of water is _____ (Nearest integer) (Given: Henry's Law constant for $$O_2$$ at 303 K is 46.82k bar and partial pressure of $$O_2$$ = 0.920 bar) (Assume solubility of $$O_2$$ in water is too small, nearly negligible)

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NTA JEE Main 29th July 2022 Shift 1 - Question 57

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Resistance of a conductivity cell (cell constant 129 m$$^{-1}$$) filled with 74.5 ppm solution of KCl is 100 $$\Omega$$ (labelled as solution 1). When the same cell is filled with KCl 149 ppm solution of KCl, the resistance is 50 $$\Omega$$ (labelled as solution 2). The ratio of molar conductivity of solution 1 and solution 2 is i.e. $$\frac{\Lambda_1}{\Lambda_2} = x \times 10^{-3}$$. The value of x is _____ (Given, molar mass of KCl is 74.5 g mol$$^{-1}$$)

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NTA JEE Main 29th July 2022 Shift 1 - Question 58

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The reaction between X and Y is first order with respect to X and zero order with respect to Y.

Experiment	[X] mol L$$^{-1}$$	[Y] mol L$$^{-1}$$	Initial rate mol L$$^{-1}$$ min$$^{-1}$$
I	0.1	0.1	$$2 \times 10^{-3}$$
II	L	0.2	$$4 \times 10^{-3}$$
III	0.4	0.4	$$M \times 10^{-3}$$
IV	0.1	0.2	$$2 \times 10^{-3}$$

Examine the data of table and calculate ratio of numerical values of M and L.

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NTA JEE Main 29th July 2022 Shift 1 - Question 59

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$$[Fe(CN)_6]^{3-}$$ should be an inner orbital complex. Ignoring the pairing energy, the value of crystal field stabilization energy for this complex is $$(-) \Delta_0$$ _____ .

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NTA JEE Main 29th July 2022 Shift 1 - Question 60

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In a linear tetrapeptide (Constituted with different amino acids), (number of amino acids) - (number of peptide bonds) is

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NTA JEE Main 29th July 2022 Shift 1 - Question 61

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If $$z = 2 + 3i$$, then $$z^5 + \bar{z}^5$$ is equal to:

A

244

B

224

C

245

D

265

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NTA JEE Main 29th July 2022 Shift 1 - Question 62

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If $$\frac{1}{(20-a)(40-a)} + \frac{1}{(40-a)(60-a)} + \ldots + \frac{1}{(180-a)(200-a)} = \frac{1}{256}$$, then the maximum value of $$a$$ is

A

198

B

202

C

212

D

218

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NTA JEE Main 29th July 2022 Shift 1 - Question 63

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Let the circumcentre of a triangle with vertices $$A(a, 3)$$, $$B(b, 5)$$ and $$C(a, b)$$, $$ab > 0$$ be $$P(1, 1)$$. If the line AP intersects the line BC at the point $$Q(k_1, k_2)$$, then $$k_1 + k_2$$ is equal to

A

2

B

$$\frac{4}{7}$$

C

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

D

4

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NTA JEE Main 29th July 2022 Shift 1 - Question 64

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Let a line L pass through the point of intersection of the lines $$bx + 10y - 8 = 0$$ and $$2x - 3y = 0$$, $$b \in \mathbb{R} - \{\frac{4}{3}\}$$. If the line L also passes through the point (1, 1) and touches the circle $$17(x^2 + y^2) = 16$$, then the eccentricity of the ellipse $$\frac{x^2}{5} + \frac{y^2}{b^2} = 1$$ is

A

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

B

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

C

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

D

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

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NTA JEE Main 29th July 2022 Shift 1 - Question 65

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Let the focal chord of the parabola $$P: y^2 = 4x$$ along the line $$L: y = mx + c, m > 0$$ meet the parabola at the points M and N. Let the line L be a tangent to the hyperbola $$H: x^2 - y^2 = 4$$. If O is the vertex of P and F is the focus of H on the positive x-axis, then the area of the quadrilateral OMFN is

A

$$2\sqrt{6}$$

B

$$2\sqrt{14}$$

C

$$4\sqrt{6}$$

D

$$4\sqrt{14}$$

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NTA JEE Main 29th July 2022 Shift 1 - Question 66

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If $$\lim_{x \to 0} \frac{\alpha e^x + \beta e^{-x} + \gamma \sin x}{x \sin^2 x} = \frac{2}{3}$$, where $$\alpha, \beta, \gamma \in \mathbb{R}$$, then which of the following is NOT correct?

A

$$\alpha^2 + \beta^2 + \gamma^2 = 6$$

B

$$\alpha\beta + \beta\gamma + \gamma\alpha + 1 = 0$$

C

$$\alpha\beta^2 + \beta\gamma^2 + \gamma\alpha^2 + 3 = 0$$

D

$$\alpha^2 - \beta^2 + \gamma^2 = 4$$

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NTA JEE Main 29th July 2022 Shift 1 - Question 67

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The statement $$(p \wedge q) \Rightarrow (p \wedge r)$$ is equivalent to

A

$$q \Rightarrow (p \wedge r)$$

B

$$p \Rightarrow (p \wedge r)$$

C

$$(p \wedge r) \Rightarrow (p \wedge q)$$

D

$$(p \wedge q) \Rightarrow r$$

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NTA JEE Main 29th July 2022 Shift 1 - Question 68

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The angle of elevation of the top of a tower from a point A due north of it is $$\alpha$$ and from a point B at a distance of 9 units due west of A is $$\cos^{-1}\left(\frac{3}{\sqrt{13}}\right)$$. If the distance of the point B from the tower is 15 units, then $$\cot\alpha$$ is equal to

A

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

B

$$\frac{9}{5}$$

C

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

D

$$\frac{7}{3}$$

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NTA JEE Main 29th July 2022 Shift 1 - Question 69

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Let R be a relation from the set $$\{1, 2, 3, \ldots, 60\}$$ to itself such that $$R = \{(a, b) : b = pq$$, where $$p, q \geq 3$$ are prime numbers$$\}$$. Then, the number of elements in R is

A

600

B

660

C

540

D

720

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NTA JEE Main 29th July 2022 Shift 1 - Question 70

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Let A and B be two $$3 \times 3$$ non-zero real matrices such that AB is a zero matrix. Then

A

The system of linear equations $$AX = 0$$ has a unique solution

B

The system of linear equations $$AX = 0$$ has infinitely many solutions

C

B is an invertible matrix

D

adj(A) is an invertible matrix

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NTA JEE Main 29th July 2022 Shift 1 - Question 71

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The number of points, where the function $$f: \mathbb{R} \to \mathbb{R}$$, $$f(x) = |x - 1|\cos|x - 2|\sin|x - 1| + (x - 3)|x^2 - 5x + 4|$$, is NOT differentiable, is

A

1

B

2

C

3

D

4

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NTA JEE Main 29th July 2022 Shift 1 - Question 72

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Let $$f(x) = 3(x^2 - 2)^3 + 4$$, $$x \in \mathbb{R}$$. Then which of the following statements are true?
P: $$x = 0$$ is a point of local minima of f
Q: $$x = \sqrt{2}$$ is a point of inflection of f
R: $$f'$$ is increasing for $$x > \sqrt{2}$$

A

Only P and Q

B

Only P and R

C

Only Q and R

D

All P, Q and R

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NTA JEE Main 29th July 2022 Shift 1 - Question 73

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The integral $$\int_0^{\pi/2} \frac{1}{3 + 2\sin x + \cos x} dx$$ is equal to:

A

$$\tan^{-1}(2)$$

B

$$\tan^{-1}(2) - \frac{\pi}{4}$$

C

$$\frac{1}{2}\tan^{-1}(2) - \frac{\pi}{8}$$

D

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

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NTA JEE Main 29th July 2022 Shift 1 - Question 74

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If $$f(\alpha) = \int_1^\alpha \frac{\log_{10} t}{1+t} dt$$, $$\alpha > 0$$, then $$f(e^3) + f(e^{-3})$$ is equal to

A

9

B

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

C

$$\frac{9}{\log_e(10)}$$

D

$$\frac{9}{2\log_e(10)}$$

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NTA JEE Main 29th July 2022 Shift 1 - Question 75

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The area of the region $$\{(x, y) : |x - 1| \leq y \leq \sqrt{5 - x^2}\}$$ is equal to

A

$$\frac{5}{2}\sin^{-1}\left(\frac{3}{5}\right) - \frac{1}{2}$$

B

$$\frac{5\pi}{4} - \frac{3}{2}$$

C

$$\frac{3\pi}{4} + \frac{3}{2}$$

D

$$\frac{5\pi}{4} - \frac{1}{2}$$

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NTA JEE Main 29th July 2022 Shift 1 - Question 76

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Let the solution curve $$y = y(x)$$ of the differential equation $$(1 + e^{2x})\left(\frac{dy}{dx} + y\right) = 1$$ pass through the point $$\left(0, \frac{\pi}{2}\right)$$. Then, $$\lim_{x \to \infty} e^x y(x)$$ is equal to

A

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

B

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

C

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

D

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

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NTA JEE Main 29th July 2022 Shift 1 - Question 77

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Let $$\vec{a} = 3\hat{i} + \hat{j}$$ and $$\vec{b} = \hat{i} + 2\hat{j} + \hat{k}$$. Let $$\vec{c}$$ be a vector satisfying $$\vec{a} \times (\vec{b} \times \vec{c}) = \vec{b} + \lambda\vec{c}$$. If $$\vec{b}$$ and $$\vec{c}$$ are non-parallel, then the value of $$\lambda$$ is

A

-5

B

5

C

1

D

-1

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NTA JEE Main 29th July 2022 Shift 1 - Question 78

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Let $$\hat{a}$$ and $$\hat{b}$$ be two unit vectors such that the angle between them is $$\frac{\pi}{4}$$. If $$\theta$$ is the angle between the vectors $$(\hat{a} + \hat{b})$$ and $$(\hat{a} + 2\hat{b} + 2(\hat{a} \times \hat{b}))$$ then the value of $$164\cos^2\theta$$ is equal to

A

$$90 + 27\sqrt{2}$$

B

$$45 + 18\sqrt{2}$$

C

$$90 + 3\sqrt{2}$$

D

$$54 + 90\sqrt{2}$$

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NTA JEE Main 29th July 2022 Shift 1 - Question 79

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If the foot of the perpendicular from the point $$A(-1, 4, 3)$$ on the plane $$P: 2x + my + nz = 4$$, is $$\left(-2, \frac{7}{2}, \frac{3}{2}\right)$$, then the distance of the point A from the plane P, measured parallel to a line with direction ratios 3, -1, -4, is equal to

A

1

B

$$\sqrt{26}$$

C

$$2\sqrt{2}$$

D

$$\sqrt{14}$$

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NTA JEE Main 29th July 2022 Shift 1 - Question 80

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Let $$S = \{1, 2, 3, \ldots, 2022\}$$. Then the probability, that a randomly chosen number n from the set S such that $$HCF(n, 2022) = 1$$, is

A

$$\frac{128}{1011}$$

B

$$\frac{166}{1011}$$

C

$$\frac{127}{337}$$

D

$$\frac{112}{337}$$

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NTA JEE Main 29th July 2022 Shift 1 - Question 81

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Let $$S = \{4, 6, 9\}$$ and $$T = \{9, 10, 11, \ldots, 1000\}$$. If $$A = \{a_1 + a_2 + \ldots + a_k : k \in \mathbb{N}, a_1, a_2, a_3, \ldots, a_k \in S\}$$, then the sum of all the elements in the set $$T - A$$ is equal to _______

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NTA JEE Main 29th July 2022 Shift 1 - Question 82

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Let $$a_1, a_2, a_3, \ldots$$ be an A.P. If $$\sum_{r=1}^{\infty} \frac{a_r}{2^r} = 4$$, then $$4a_2$$ is equal to ______

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NTA JEE Main 29th July 2022 Shift 1 - Question 83

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If $$\frac{1}{2 \times 3 \times 4} + \frac{1}{3 \times 4 \times 5} + \frac{1}{4 \times 5 \times 6} + \ldots + \frac{1}{100 \times 101 \times 102} = \frac{k}{101}$$, then $$34k$$ is equal to _______

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NTA JEE Main 29th July 2022 Shift 1 - Question 84

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Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of $$\left(\sqrt[4]{2} + \frac{1}{\sqrt[4]{3}}\right)^n$$, in the increasing powers of $$\frac{1}{\sqrt[4]{3}}$$ be $$\sqrt[4]{6} : 1$$. If the sixth term from the beginning is $$\frac{\alpha}{\sqrt[4]{3}}$$, then $$\alpha$$ is equal to _______

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NTA JEE Main 29th July 2022 Shift 1 - Question 85

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Let $$S = \{\theta \in (0, 2\pi) : 7\cos^2\theta - 3\sin^2\theta - 2\cos^2(2\theta) = 2\}$$. Then the sum of roots of all the equations $$x^2 - 2(\tan^2\theta + \cot^2\theta)x + 6\sin^2\theta = 0$$, $$\theta \in S$$, is _______

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NTA JEE Main 29th July 2022 Shift 1 - Question 86

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Let the mirror image of a circle $$c_1: x^2 + y^2 - 2x - 6y + \alpha = 0$$ in line $$y = x + 1$$ be $$c_2: 5x^2 + 5y^2 + 10gx + 10fy + 38 = 0$$. If r is the radius of circle $$c_2$$, then $$\alpha + 6r^2$$ is equal to ______

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NTA JEE Main 29th July 2022 Shift 1 - Question 87

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Let the mean and the variance of 20 observations $$x_1, x_2, \ldots, x_{20}$$ be 15 and 9, respectively. For $$\alpha \in \mathbb{R}$$, if the mean of $$(x_1 + \alpha)^2, (x_2 + \alpha)^2, \ldots, (x_{20} + \alpha)^2$$ is 178, then the square of the maximum value of $$\alpha$$ is equal to _______

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NTA JEE Main 29th July 2022 Shift 1 - Question 88

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The number of matrices of order $$3 \times 3$$, whose entries are either 0 or 1 and the sum of all the entries is a prime number, is _______

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NTA JEE Main 29th July 2022 Shift 1 - Question 89

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Let p and p+2 be prime numbers and let $$\Delta = \begin{vmatrix} p! & (p+1)! & (p+2)! \\ (p+1)! & (p+2)! & (p+3)! \\ (p+2)! & (p+3)! & (p+4)! \end{vmatrix}$$
Then the sum of the maximum values of $$\alpha$$ and $$\beta$$, such that $$p^\alpha$$ and $$(p+2)^\beta$$ divide $$\Delta$$, is _______

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NTA JEE Main 29th July 2022 Shift 1 - Question 90

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Let a line with direction ratios $$a, -4a, -7$$ be perpendicular to the lines with direction ratios $$3, -1, 2b$$ and $$b, a, -2$$. If the point of intersection of the line $$\frac{x+1}{a^2+b^2} = \frac{y-2}{a^2-b^2} = \frac{z}{1}$$ and the plane $$x - y + z = 0$$ is $$(\alpha, \beta, \gamma)$$, then $$\alpha + \beta + \gamma$$ is equal to ________

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