NTA JEE Main 28th July 2022 Shift 1

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

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The dimensions of $$\frac{B^2}{\mu_0}$$ will be (if $$\mu_0$$: permeability of free space and $$B$$: magnetic field)

A

$$[ML^2T^{-2}]$$

B

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

C

$$[ML^{-1}T^{-2}]$$

D

$$[MLL^{-2}A^{-1}]$$

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NTA JEE Main 28th July 2022 Shift 1 - Question 2

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A NCC parade is going at a uniform speed of 9 km h$$^{-1}$$ under a mango tree on which a monkey is sitting at a height of 19.6 m. At any particular instant, the monkey drops a mango. A cadet will receive the mango whose distance from the tree at time of drop is: (Given $$g = 9.8 \ m s^{-2}$$)

A

5 m

B

10 m

C

19.8 m

D

24.5 m

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NTA JEE Main 28th July 2022 Shift 1 - Question 3

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A balloon has mass of 10 g in air. The air escapes from the balloon at a uniform rate with velocity 4.5 cm s$$^{-1}$$. If the balloon shrinks in 5 s completely. Then, the average force acting on that balloon will be (in dyne).

A

3

B

9

C

12

D

18

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NTA JEE Main 28th July 2022 Shift 1 - Question 4

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In two different experiments, an object of mass 5 kg moving with a speed of 25 ms$$^{-1}$$ hits two different walls and comes to rest within (i) 3 second, (ii) 5 seconds, respectively. Choose the correct option out of the following:

A

Impulse and average force acting on the object will be same for both the cases.

B

Impulse will be same for both the cases but the average force will be different.

C

Average force will be same for both the cases but the impulse will be different.

D

Average force and impulse will be different for both the cases.

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NTA JEE Main 28th July 2022 Shift 1 - Question 5

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If the radius of earth shrinks by 2% while its mass remains same. The acceleration due to gravity on the earth's surface will approximately

A

decrease by 2%

B

decrease by 4%

C

increase by 2%

D

increase by 4%

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NTA JEE Main 28th July 2022 Shift 1 - Question 6

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The force required to stretch a wire of cross-section 1 cm$$^2$$ to double its length will be: (Given Young's modulus of the wire $$= 2 \times 10^{11}$$ N m$$^{-2}$$)

A

$$1 \times 10^7$$ N

B

$$1.5 \times 10^7$$ N

C

$$2 \times 10^7$$ N

D

$$2.5 \times 10^7$$ N

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NTA JEE Main 28th July 2022 Shift 1 - Question 7

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A Carnot engine has efficiency of 50%. If the temperature of sink is reduced by 40°C, its efficiency increases by 30%. The temperature of the source will be:

A

166.7 K

B

255.1 K

C

266.7 K

D

367.7 K

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NTA JEE Main 28th July 2022 Shift 1 - Question 8

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Given below are two statements:
Statement I: The average momentum of a molecule in a sample of an ideal gas depends on temperature.
Statement II: The rms speed of oxygen molecules in a gas is $$v$$. If the temperature is doubled and the oxygen molecules dissociate into oxygen atoms, the rms speed will become $$2v$$.
In the light of the above statements, choose the correct answer from the options given below:

A

Both Statement I and Statement II are true

B

Both Statement I and Statement II are false

C

Statement I is true but Statement II is false

D

Statement I is false but Statement II is true

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NTA JEE Main 28th July 2022 Shift 1 - Question 9

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In the wave equation $$y = 0.5 \sin\frac{2\pi}{\lambda}(400t - x)$$ m, the velocity of the wave will be:

A

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

B

$$200\sqrt{2}$$ m s$$^{-1}$$

C

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

D

$$400\sqrt{2}$$ m s$$^{-1}$$

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NTA JEE Main 28th July 2022 Shift 1 - Question 10

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Two capacitors, each having capacitance $$40 \mu F$$ are connected in series. The space between one of the capacitors is filled with dielectric material of dielectric constant $$K$$ such that the equivalence capacitance of the system became $$24 \mu F$$. The value of $$K$$ will be:

A

1.5

B

2.5

C

1.2

D

3

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NTA JEE Main 28th July 2022 Shift 1 - Question 11

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A wire of resistance $$R_1$$ is drawn out so that its length is increased by twice of its original length. The ratio of new resistance to original resistance is:

A

9 : 1

B

1 : 9

C

4 : 1

D

3 : 1

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NTA JEE Main 28th July 2022 Shift 1 - Question 12

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The current sensitivity of a galvanometer can be increased by:
(A) decreasing the number of turns
(B) increasing the magnetic field
(C) decreasing the area of the coil
(D) decreasing the torsional constant of the spring
Choose the most appropriate answer from the options given below:

A

(B) and (C) only

B

(C) and (D) only

C

(A) and (C) only

D

(B) and (D) only

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NTA JEE Main 28th July 2022 Shift 1 - Question 13

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As shown in the figure, a metallic rod of linear density 0.45 kg m$$^{-1}$$ is lying horizontally on a smooth incline plane which makes an angle of 45° with the horizontal. The minimum current flowing in the rod required to keep it stationary, when 0.15 T magnetic field is acting on it in the vertical upward direction, will be (Use $$g = 10$$ m s$$^{-2}$$)

A

30 A

B

15 A

C

10 A

D

3 A

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NTA JEE Main 28th July 2022 Shift 1 - Question 14

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The equation of current in a purely inductive circuit is $$5\sin(49\pi t - 30°)$$. If the inductance is 30 mH then the equation for the voltage across the inductor, will be

A

$$1.47\sin(49\pi t - 30°)$$

B

$$1.47\sin(49\pi t + 60°)$$

C

$$23.1\sin(49\pi t - 30°)$$

D

$$23.1\sin(49\pi t + 60°)$$

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NTA JEE Main 28th July 2022 Shift 1 - Question 15

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As shown in the figure, after passing through the medium 1, the speed of light $$v_2$$ in medium 2 will be: (Given $$c = 3 \times 10^8$$ m s$$^{-1}$$)

A

$$1.0 \times 10^8$$ m s$$^{-1}$$

B

$$0.5 \times 10^8$$ m s$$^{-1}$$

C

$$1.5 \times 10^8$$ m s$$^{-1}$$

D

$$3.0 \times 10^8$$ m s$$^{-1}$$

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NTA JEE Main 28th July 2022 Shift 1 - Question 16

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In normal adjustment, for a refracting telescope, the distance between objective and eye piece is 30 cm. The focal length of the objective, when the angular magnification of the telescope is 2, will be:

A

20 cm

B

30 cm

C

10 cm

D

15 cm

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NTA JEE Main 28th July 2022 Shift 1 - Question 17

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The equation $$\lambda = \frac{1.227}{x}$$ nm can be used to find the de-Broglie wavelength of an electron. In this equation $$x$$ stands for:
Where, $$m$$ = mass of electron, $$P$$ = momentum of electron, $$K$$ = Kinetic energy of electron, $$V$$ = Accelerating potential in volts for electron

A

$$\sqrt{mK}$$

B

$$\sqrt{P}$$

C

$$\sqrt{K}$$

D

$$\sqrt{V}$$

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NTA JEE Main 28th July 2022 Shift 1 - Question 18

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The half life period of a radioactive substance is 60 days. The time taken for $$\frac{7}{8}$$th of its original mass to disintegrate will be:

A

120 days

B

130 days

C

180 days

D

20 days

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NTA JEE Main 28th July 2022 Shift 1 - Question 19

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Identify the solar cell characteristics from the following options:

A

B

C

D

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NTA JEE Main 28th July 2022 Shift 1 - Question 20

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In the case of amplitude modulation to avoid distortion the modulation index $$\mu$$ should be:

A

$$\mu \leq 1$$

B

$$\mu \geq 1$$

C

$$\mu = 2$$

D

$$\mu = 0$$

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NTA JEE Main 28th July 2022 Shift 1 - Question 21

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If the projection of $$2\hat{i} + 4\hat{j} - 2\hat{k}$$ on $$\hat{i} + 2\hat{j} + \alpha\hat{k}$$ is zero. Then, the value of $$\alpha$$ will be

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NTA JEE Main 28th July 2022 Shift 1 - Question 22

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A block of mass 'm' (as shown in figure) moving with kinetic energy E compresses a spring through a distance 25 cm when, its speed is halved. The value of spring constant of used spring will be $$nE$$ N m$$^{-1}$$ for $$n$$ = _____.

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NTA JEE Main 28th July 2022 Shift 1 - Question 23

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Four identical discs each of mass 'M' and diameter 'a' are arranged in a small plane as shown in figure. If the moment of inertia of the system about OO' is $$\frac{x}{4}Ma^2$$. Then, the value of $$x$$ will be _____.

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NTA JEE Main 28th July 2022 Shift 1 - Question 24

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The diameter of an air bubble which was initially 2 mm, rises steadily through a solution of density 1750 kg/m$$^3$$ at the rate of 0.35 cm/s. Coefficient of viscosity of the solution is _____ Poise. (Assume mass of the bubble to be negligible) (Answer in Poise to the nearest integer)

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NTA JEE Main 28th July 2022 Shift 1 - Question 25

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The frequency of echo will be _____ Hz if the train blowing a whistle of frequency 320 Hz is moving with a velocity of 36 km h$$^{-1}$$ towards a hill from which an echo is heard by the train driver. Velocity of sound in air is 330 m s$$^{-1}$$.

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NTA JEE Main 28th July 2022 Shift 1 - Question 26

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Two electric dipoles of dipole moments $$1.2 \times 10^{-30}$$ Cm and $$2.4 \times 10^{-30}$$ Cm are placed in two different uniform electric fields of strengths $$5 \times 10^4$$ N C$$^{-1}$$ and $$15 \times 10^4$$ N C$$^{-1}$$ respectively. The ratio of maximum torque experienced by the electric dipoles will be $$\frac{1}{x}$$. The value of $$x$$ is _____.

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NTA JEE Main 28th July 2022 Shift 1 - Question 27

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As shown in the figure, a potentiometer wire of resistance 20 $$\Omega$$ and length 300 cm is connected with resistance box (R.B.) and a standard cell of emf 4 V. For a resistance 'R' of resistance box introduced into the circuit, the null point for a cell of 20 mV is found to be 60 cm. The value of 'R' is _____ $$\Omega$$.

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

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The frequencies at which the current amplitude in an LCR series circuit becomes $$\frac{1}{\sqrt{2}}$$ times its maximum value, are 212 rad s$$^{-1}$$ and 232 rad s$$^{-1}$$. The value of resistance in the circuit is $$R = 5 \ \Omega$$. The self inductance in the circuit is _____ mH.

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

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In a Young's double slit experiment, a laser light of 560 nm produces an interference pattern with consecutive bright fringes' separation of 7.2 mm. Now another light is used to produce an interference pattern with consecutive bright fringes' separation of 8.1 mm. The wavelength of second light is _____ nm.

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NTA JEE Main 28th July 2022 Shift 1 - Question 30

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A freshly prepared radioactive source of half life 2 hours 30 minutes emits radiation which is 64 times the permissible safe level. The minimum time, after which it would be possible to work safely with source, will be _____ hours.

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NTA JEE Main 28th July 2022 Shift 1 - Question 31

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Identify the incorrect statement from the following.

A

A circular path around the nucleus in which an electron moves is proposed as Bohr's orbit.

B

An orbital is the one electron wave function

C

The existence of Bohr's orbits is supported by hydrogen spectrum.

D

Atomic orbital is characterised by the quantum numbers n and l only

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NTA JEE Main 28th July 2022 Shift 1 - Question 32

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In which of the following pairs, electron gain enthalpies of constituent elements are nearly the same or identical?
(A) Rb and Cs
(B) Na and K
(C) Ar and Kr
(D) I and At
Choose the correct answer from the options given below

A

A & B only

B

B & C only

C

A & C only

D

C & D only

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NTA JEE Main 28th July 2022 Shift 1 - Question 33

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Which of the following relation is not correct?

A

$$\Delta H = \Delta U - P\Delta V$$

B

$$\Delta U = q + W$$

C

$$\Delta S_{sys} + \Delta S_{surr} \geq 0$$

D

$$\Delta G = \Delta H - T\Delta S$$

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NTA JEE Main 28th July 2022 Shift 1 - Question 34

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The metal salts formed during softening of hard water using Clark's method are

A

$$Ca(OH)_2$$ and $$Mg(OH)_2$$

B

$$CaCO_3$$ and $$Mg(OH)_2$$

C

$$Ca(OH)_2$$ and $$MgCO_3$$

D

$$CaCO_3$$ and $$MgCO_3$$

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NTA JEE Main 28th July 2022 Shift 1 - Question 35

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For kinetic study of the reaction of iodide ion with $$H_2O_2$$ at room temperature:
(A) Always use freshly prepared starch solution.
(B) Always keep the concentration of sodium thiosulphate solution less than that of KI solution.
(C) Record the time immediately after the appearance of blue colour.
(D) Record the time immediately before the appearance of blue colour.
(E) Always keep the concentration of sodium thiosulphate solution more than that of KI solution.
Choose the correct answer from the options given below

A

A, B, C only

B

A, D, E only

C

D, E only

D

A, B, E only

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NTA JEE Main 28th July 2022 Shift 1 - Question 36

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Which of the following statement is incorrect?

A

Low solubility of LiF in water is due to its small hydration enthalpy.

B

$$KO_2$$ is paramagnetic.

C

Solution of sodium in liquid ammonia is conducting in nature.

D

Sodium metal has higher density than potassium metal

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NTA JEE Main 28th July 2022 Shift 1 - Question 37

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Match List-I with List-II.

List-I	List-II
A	I. Spiro compound
B	II. Aromatic compound
C	III. Non-planar Heterocyclic compound
D	IV. Bicyclo compound

A

A-II, B-I, C-IV, D-III

B

A-IV, B-III, C-I, D-II

C

A-III, B-IV, C-I, D-II

D

A-IV, B-III, C-II, D-I

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NTA JEE Main 28th July 2022 Shift 1 - Question 38

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Among the following marked proton of which compound shows lowest pKa value?

A

B

C

D

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NTA JEE Main 28th July 2022 Shift 1 - Question 39

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Choose the correct option for the following reactions.

A

'A' and 'B' are both Markovnikov addition products.

B

'A' is Markovnikov product and 'B' is anti-Markovnikov product.

C

'A' and 'B' are both anti-Markovnikov products.

D

'B' is Markovnikov and 'A' is anti-Markovnikov product.

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NTA JEE Main 28th July 2022 Shift 1 - Question 40

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Identify the correct statement for the below given transformation.

A

A - $$CH_3CH_2CH=CH-CH_3$$, B - $$CH_3CH_2CH_2CH=CH_2$$, Saytzeff products

B

A - $$CH_3CH_2CH=CH-CH_3$$, B - $$CH_3CH_2CH_2CH=CH_2$$, Hofmann products

C

A - $$CH_3CH_2CH_2CH=CH_2$$, B - $$CH_3CH_2CH=CHCH_3$$, Hofmann products

D

A - $$CH_3CH_2CH_2CH=CH_2$$, B - $$CH_3CH_2CH=CHCH_3$$, Saytzeff products

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NTA JEE Main 28th July 2022 Shift 1 - Question 41

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Given below are two statements:
Statement I: In polluted water values of both dissolved oxygen and BOD are very low.
Statement II: Eutrophication results in decrease in the amount of dissolved oxygen.
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 true

B

Both Statement I and Statement II are false

C

Statement I is true but Statement II is false

D

Statement I is false but Statement II is true

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NTA JEE Main 28th July 2022 Shift 1 - Question 42

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Match List-I with List-II.

List-I	List-II
A. $$Cd(s) + 2Ni(OH)_3(s) \to CdO(s) + 2Ni(OH)_2(s) + H_2O(l)$$	I. Primary battery
B. $$Zn(Hg) + HgO(s) \to ZnO(s) + Hg(l)$$	II. Discharging of secondary battery
C. $$2PbSO_4(s) + 2H_2O(l) \to Pb(s) + PbO_2(s) + 2H_2SO_4(aq)$$	III. Fuel cell
D. $$2H_2(g) + O_2(g) \to 2H_2O(l)$$	IV. Charging of secondary battery

A

A-I, B-II, C-III, D-IV

B

A-IV, B-I, C-IV, D-III

C

A-II, B-I, C-IV, D-III

D

A-II, B-I, C-III, D-IV

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NTA JEE Main 28th July 2022 Shift 1 - Question 43

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Which of the reaction is suitable for concentrating ore by leaching process?

A

$$2Cu_2S + 3O_2 \to 2Cu_2O + 2SO_2$$

B

$$Fe_3O_4 + CO \to 3FeO + CO_2$$

C

$$Al_2O_3 + 2NaOH + 3H_2O \to 2Na[Al(OH)_4]$$

D

$$Al_2O_3 + 6Mg \to 6MgO + 4Al$$

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NTA JEE Main 28th July 2022 Shift 1 - Question 44

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Match List-I with List-II.

List-I	List-II
A. $$4NH_3(g) + 5O_2(g) \to 4NO(g) + 6H_2O(g)$$	I. NO(g)
B. $$N_2(g) + 3H_2(g) \to 2NH_3(g)$$	II. $$H_2SO_4(l)$$
C. $$C_{12}H_{22}O_{11}(aq) + H_2O(l) \to C_6H_{12}O_6 (Glucose) + C_6H_{12}O_6 (Fructose)$$	III. Pt(s)
D. $$2SO_2(g) + O_2(g) \to 2SO_3(g)$$	IV. Fe(s)

A

A-II, B-III, C-I, D-IV

B

A-III, B-II, C-I, D-IV

C

A-III, B-IV, C-II, D-I

D

A-III, B-II, C-IV, D-I

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NTA JEE Main 28th July 2022 Shift 1 - Question 45

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Match List-I with List-II, match the gas evolved during each reaction.

List-I	List-II
A. $$(NH_4)_2Cr_2O_7 \xrightarrow{\Delta}$$	I. $$H_2$$
B. $$KMnO_4 + HCl \to$$	II. $$N_2$$
C. $$Al + NaOH + H_2O \to$$	III. $$O_2$$
D. $$NaNO_3 \xrightarrow{\Delta}$$	IV. $$Cl_2$$

Choose the correct answer from the options given below

A

A-II, B-III, C-I, D-IV

B

A-III, B-I, C-IV, D-II

C

A-II, B-IV, C-I, D-III

D

A-III, B-IV, C-I, D-II

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NTA JEE Main 28th July 2022 Shift 1 - Question 46

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Which of the following has least tendency to liberate $$H_2$$ from mineral acids?

A

Cu

B

Mn

C

Ni

D

Zn

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NTA JEE Main 28th July 2022 Shift 1 - Question 47

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Identify the major product A and B for the below given reaction sequence.

A

B

C

D

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NTA JEE Main 28th July 2022 Shift 1 - Question 48

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Terylene polymer is obtained by condensation of

A

Ethane-1,2-diol and Benzene-1,3 dicarboxylic acid

B

Propane-1,2-diol and Benzene-1,4 dicarboxylic acid

C

Ethane-1,2-diol and Benzene-1,4 dicarboxylic acid

D

Ethane-1,2-diol and Benzene-1,2 dicarboxylic acid

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NTA JEE Main 28th July 2022 Shift 1 - Question 49

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Statements about Enzyme Inhibitor Drugs are given below:
(A) There are Competitive and Non-competitive inhibitor drugs.
(B) These can bind at the active sites and allosteric sites.
(C) Competitive Drugs are allosteric site blocking drugs.
(D) Non-competitive Drugs are active site blocking drugs.
Choose the correct answer from the options given below:

A

A, D only

B

A, C only

C

A, B only

D

A, B, C only

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NTA JEE Main 28th July 2022 Shift 1 - Question 50

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For the below given cyclic hemiacetal X, the correct pyranose structure is

A

B

C

D

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NTA JEE Main 28th July 2022 Shift 1 - Question 51

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In the given reaction, $$X + Y + 3Z \rightleftharpoons XYZ_3$$. If one mole of each of X and Y with 0.05 mol of Z gives compound $$XYZ_3$$. (Given: Atomic masses of X, Y and Z are 10, 20 and 30 amu, respectively). The yield of $$XYZ_3$$ is _____ g.

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NTA JEE Main 28th July 2022 Shift 1 - Question 52

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On complete combustion of 0.492 g of an organic compound containing C, H and O, 0.7938 g of $$CO_2$$ and 0.4428 g of $$H_2O$$ was produced. The % composition of oxygen in the compound is _____ (Nearest Integer)

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NTA JEE Main 28th July 2022 Shift 1 - Question 53

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The number of paramagnetic species among the following is $$B_2, Li_2, C_2, C_2^-, O_2^{2-}, O_2^+$$ and $$He_2^+$$

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NTA JEE Main 28th July 2022 Shift 1 - Question 54

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$$K_a$$ for butyric acid ($$C_3H_7COOH$$) is $$2 \times 10^{-5}$$. The pH of 0.2 M solution of butyric acid is _____ $$\times 10^{-1}$$. (Nearest integer) [Given $$\log 2 = 0.30$$]

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NTA JEE Main 28th July 2022 Shift 1 - Question 55

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An element M crystallises in a body centred cubic unit cell with a cell edge of 300 pm. The density of the element is 6.0 g cm$$^{-3}$$. The number of atoms present in 180 g of the element is _____ $$\times 10^{23}$$. (Nearest integer)

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NTA JEE Main 28th July 2022 Shift 1 - Question 56

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150 g of acetic acid was contaminated with 10.2 g ascorbic acid $$(C_6H_8O_6)$$ to lower down its freezing point by $$x \times 10^{-1}$$ °C. The value of $$x$$ is _____ (Nearest integer). [Given $$K_f = 3.9$$ K kg mol$$^{-1}$$; Molar mass of ascorbic acid = 176 g mol$$^{-1}$$]

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NTA JEE Main 28th July 2022 Shift 1 - Question 57

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For the given first order reaction $$A \to B$$ the half life of the reaction is 0.3010 min. The ratio of the initial concentration of reactant to the concentration of reactant at time 2.0 min will be equal to _____. (Nearest integer)

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NTA JEE Main 28th July 2022 Shift 1 - Question 58

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The number of inter halogens from the following having square pyramidal structure is
$$ClF_3, IF_7, BrF_5, BrF_3, I_2Cl_6, IF_5, ClF, ClF_5$$

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NTA JEE Main 28th July 2022 Shift 1 - Question 59

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The disproportionation of $$MnO_4^{2-}$$ in acidic medium resulted in the formation of two manganese compounds A and B. If the oxidation state of Mn in B is smaller than that of A, then the spin-only magnetic moment $$\mu$$ value of B in BM is _____ (Nearest integer)

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NTA JEE Main 28th July 2022 Shift 1 - Question 60

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Total number of relatively more stable isomer(s) possible for octahedral complex $$[Cu(en)_2(SCN)_2]$$ will be _____

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NTA JEE Main 28th July 2022 Shift 1 - Question 61

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Let $$S_1 = \{z_1 \in \mathbb{C} : |z_1 - 3| = \frac{1}{2}\}$$ and $$S_2 = \{z_2 \in \mathbb{C} : |z_2 - |z_2 + 1|| = |z_2 + |z_2 - 1||\}$$. Then, for $$z_1 \in S_1$$ and $$z_2 \in S_2$$, the least value of $$|z_2 - z_1|$$ is

A

0

B

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

C

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

D

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

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NTA JEE Main 28th July 2022 Shift 1 - Question 62

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If the minimum value of $$f(x) = \frac{5x^2}{2} + \frac{\alpha}{x^5}, x \gt 0$$, is 14, then the value of $$\alpha$$ is equal to

A

32

B

64

C

128

D

256

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NTA JEE Main 28th July 2022 Shift 1 - Question 63

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Consider the sequence $$a_1, a_2, a_3, \ldots$$ such that $$a_1 = 1, a_2 = 2$$ and $$a_{n+2} = \frac{2}{a_{n+1}} + a_n$$ for $$n = 1, 2, 3, \ldots$$. If $$\frac{a_1 + \frac{1}{a_2}}{a_3} \cdot \frac{a_2 + \frac{1}{a_3}}{a_4} \cdot \frac{a_3 + \frac{1}{a_4}}{a_5} \cdots \frac{a_{30} + \frac{1}{a_{31}}}{a_{32}} = 2^\alpha \cdot {}^{61}C_{31}$$ then $$\alpha$$ is equal to

A

-30

B

-31

C

-60

D

-61

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NTA JEE Main 28th July 2022 Shift 1 - Question 64

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The remainder when $$7^{2022} + 3^{2022}$$ is divided by 5 is

A

0

B

2

C

3

D

4

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NTA JEE Main 28th July 2022 Shift 1 - Question 65

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For $$t \in (0, 2\pi)$$, if ABC is an equilateral triangle with vertices $$A(\sin t, -\cos t)$$, $$B(\cos t, \sin t)$$ and $$C(a, b)$$ such that its orthocentre lies on a circle with centre $$(1, \frac{1}{3})$$, then $$a^2 - b^2$$ is equal to

A

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

B

8

C

$$\frac{77}{9}$$

D

$$\frac{80}{9}$$

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NTA JEE Main 28th July 2022 Shift 1 - Question 66

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Let C be the centre of the circle $$x^2 + y^2 - x + 2y = \frac{11}{4}$$ and P be a point on the circle. A line passes through the point C, makes an angle of $$\frac{\pi}{4}$$ with the line CP and intersects the circle at the points Q and R. Then the area of the triangle PQR (in unit$$^2$$) is

A

2

B

$$2\sqrt{2}$$

C

$$8\sin\frac{\pi}{8}$$

D

$$8\cos\frac{\pi}{8}$$

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NTA JEE Main 28th July 2022 Shift 1 - Question 67

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If the tangents drawn at the points P and Q on the parabola $$y^2 = 2x - 3$$ intersect at the point $$R(0, 1)$$, then the orthocentre of the triangle PQR is

A

(0, 1)

B

(2, -1)

C

(6, 3)

D

(2, 1)

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NTA JEE Main 28th July 2022 Shift 1 - Question 68

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Let the operations $$*, \odot \in \{\wedge, \vee\}$$. If $$(p * q) \odot (p \odot \sim q)$$ is a tautology, then the ordered pair $$(*, \odot)$$ is

A

$$(\vee, \wedge)$$

B

$$(\vee, \vee)$$

C

$$(\wedge, \wedge)$$

D

$$(\wedge, \vee)$$

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NTA JEE Main 28th July 2022 Shift 1 - Question 69

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For $$\alpha \in \mathbb{N}$$, consider a relation R on $$\mathbb{N}$$ given by $$R = \{(x, y) : 3x + \alpha y$$ is a multiple of 7$$\}$$. The relation R is an equivalence relation if and only if

A

$$\alpha = 14$$

B

$$\alpha$$ is a multiple of 4

C

4 is the remainder when $$\alpha$$ is divided by 10

D

4 is the remainder when $$\alpha$$ is divided by 7

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NTA JEE Main 28th July 2022 Shift 1 - Question 70

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Let the matrix $$A = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$ and the matrix $$B_0 = A^{49} + 2A^{98}$$. If $$B_n = \text{Adj}(B_{n-1})$$ for all $$n \geq 1$$, then $$\det(B_4)$$ is equal to

A

$$3^{28}$$

B

$$3^{30}$$

C

$$3^{32}$$

D

$$3^{36}$$

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NTA JEE Main 28th July 2022 Shift 1 - Question 71

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Considering the principal values of the inverse trigonometric functions, the sum of all the solutions of the equation $$\cos^{-1}(x) - 2\sin^{-1}(x) = \cos^{-1}(2x)$$ is equal to

A

0

B

1

C

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

D

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

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NTA JEE Main 28th July 2022 Shift 1 - Question 72

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Let $$\alpha, \beta$$ and $$\gamma$$ be three positive real numbers. Let $$f(x) = \alpha x^5 + \beta x^3 + \gamma x$$, $$x \in \mathbb{R}$$ and $$g: \mathbb{R} \to \mathbb{R}$$ be such that $$g(f(x)) = x$$ for all $$x \in \mathbb{R}$$. If $$a_1, a_2, a_3, \ldots, a_n$$ be in arithmetic progression with mean zero, then the value of $$f\left(g\left(\frac{1}{n}\sum_{i=1}^{n} f(a_i)\right)\right)$$ is equal to

A

0

B

3

C

9

D

27

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NTA JEE Main 28th July 2022 Shift 1 - Question 73

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Considering only the principal values of the inverse trigonometric functions, the domain of the function $$f(x) = \cos^{-1}\left(\frac{x^2 - 4x + 2}{x^2 + 3}\right)$$ is

A

$$\left(-\infty, \frac{1}{4}\right]$$

B

$$\left[-\frac{1}{4}, \infty\right)$$

C

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

D

$$\left(-\infty, \frac{1}{3}\right]$$

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NTA JEE Main 28th July 2022 Shift 1 - Question 74

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The minimum value of the twice differentiable function $$f(x) = \int_0^x e^{x-t} f'(t) dt - (x^2 - x + 1)e^x$$, $$x \in \mathbb{R}$$, is

A

$$-\frac{2}{\sqrt{e}}$$

B

$$-2\sqrt{e}$$

C

$$-\sqrt{e}$$

D

$$\frac{2}{\sqrt{e}}$$

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NTA JEE Main 28th July 2022 Shift 1 - Question 75

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Let the solution curve of the differential equation $$x dy = (\sqrt{x^2 + y^2} + y) dx$$, $$x > 0$$, intersect the line $$x = 1$$ at $$y = 0$$ and the line $$x = 2$$ at $$y = \alpha$$. Then the value of $$\alpha$$ is

A

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

B

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

C

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

D

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

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NTA JEE Main 28th July 2022 Shift 1 - Question 76

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If $$y = y(x)$$, $$x \in \left(0, \frac{\pi}{2}\right)$$ be the solution curve of the differential equation $$\sin^2(2x)\frac{dy}{dx} + (8\sin^2(2x) + 2\sin(4x))y = 2e^{-4x}(2\sin(2x) + \cos(2x))$$, with $$y\left(\frac{\pi}{4}\right) = e^{-\pi}$$, then $$y\left(\frac{\pi}{6}\right)$$ is equal to

A

$$\frac{2}{\sqrt{3}}e^{-\frac{2\pi}{3}}$$

B

$$\frac{2}{\sqrt{3}}e^{\frac{2\pi}{3}}$$

C

$$\frac{1}{\sqrt{3}}e^{-\frac{2\pi}{3}}$$

D

$$\frac{1}{\sqrt{3}}e^{\frac{2\pi}{3}}$$

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NTA JEE Main 28th July 2022 Shift 1 - Question 77

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Let the vectors $$\vec{a} = (1+t)\hat{i} + (1-t)\hat{j} + \hat{k}$$, $$\vec{b} = (1-t)\hat{i} + (1+t)\hat{j} + 2\hat{k}$$ and $$\vec{c} = t\hat{i} - t\hat{j} + \hat{k}$$, $$t \in \mathbb{R}$$ be such that for $$\alpha, \beta, \gamma \in \mathbb{R}$$, $$\alpha\vec{a} + \beta\vec{b} + \gamma\vec{c} = \vec{0} \Rightarrow \alpha = \beta = \gamma = 0$$. Then, the set of all values of $$t$$ is

A

a non-empty finite set

B

equal to $$\mathbb{N}$$

C

equal to $$\mathbb{R} - \{0\}$$

D

equal to $$\mathbb{R}$$

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NTA JEE Main 28th July 2022 Shift 1 - Question 78

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Let a vector $$\vec{a}$$ has magnitude 9. Let a vector $$\vec{b}$$ be such that for every $$(x, y) \in \mathbb{R} \times \mathbb{R} - \{(0,0)\}$$, the vector $$x\vec{a} + y\vec{b}$$ is perpendicular to the vector $$6y\vec{a} - 18x\vec{b}$$. Then the value of $$|\vec{a} \times \vec{b}|$$ is equal to

A

$$9\sqrt{3}$$

B

$$27\sqrt{3}$$

C

9

D

81

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NTA JEE Main 28th July 2022 Shift 1 - Question 79

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The foot of the perpendicular from a point on the circle $$x^2 + y^2 = 1, z = 0$$ to the plane $$2x + 3y + z = 6$$ lies on which one of the following curves?

A

$$(6x+5y-12)^2 + 4(3x+7y-8)^2 = 1, z = 6-2x-3y$$

B

$$(5x+6y-12)^2 + 4(3x+5y-9)^2 = 1, z = 6-2x-3y$$

C

$$(6x+5y-14)^2 + 9(3x+5y-7)^2 = 1, z = 6-2x-3y$$

D

$$(5x+6y-14)^2 + 9(3x+7y-8)^2 = 1, z = 6-2x-3y$$

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NTA JEE Main 28th July 2022 Shift 1 - Question 80

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Out of 60% female and 40% male candidates appearing in an exam, 60% candidates qualify it. The number of females qualifying the exam is twice the number of males qualifying it. A candidate is randomly chosen from the qualified candidates. The probability, that the chosen candidate is a female, is

A

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

B

$$\frac{11}{16}$$

C

$$\frac{23}{32}$$

D

$$\frac{13}{16}$$

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NTA JEE Main 28th July 2022 Shift 1 - Question 81

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The sum of all real values of $$x$$ for which $$\frac{3x^2 - 9x + 17}{x^2 + 3x + 10} = \frac{5x^2 - 7x + 19}{3x^2 + 5x + 12}$$ is equal to

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NTA JEE Main 28th July 2022 Shift 1 - Question 82

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Let S be the set of all passwords which are six to eight characters long, where each character is either an alphabet from {A, B, C, D, E} or a number from {1, 2, 3, 4, 5} with the repetition of characters allowed. If the number of passwords in S whose at least one character is a number from {1, 2, 3, 4, 5} is $$\alpha \times 5^6$$, then $$\alpha$$ is equal to

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NTA JEE Main 28th July 2022 Shift 1 - Question 83

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For $$p, q \in \mathbb{R}$$, consider the real valued function $$f(x) = (x - p)^2 - q$$, $$x \in \mathbb{R}$$ and $$q > 0$$. Let $$a_1, a_2, a_3$$ and $$a_4$$ be in an arithmetic progression with mean $$p$$ and positive common difference. If $$|f(a_i)| = 500$$ for all $$i = 1, 2, 3, 4$$, then the absolute difference between the roots of $$f(x) = 0$$ is

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NTA JEE Main 28th July 2022 Shift 1 - Question 84

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Let $$x_1, x_2, x_3, \ldots, x_{20}$$ be in geometric progression with $$x_1 = 3$$ and the common ratio $$\frac{1}{2}$$. A new data is constructed replacing each $$x_i$$ by $$(x_i - i)^2$$. If $$\bar{x}$$ is the mean of new data, then the greatest integer less than or equal to $$\bar{x}$$ is

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NTA JEE Main 28th July 2022 Shift 1 - Question 85

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For the hyperbola $$H: x^2 - y^2 = 1$$ and the ellipse $$E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$, $$a > b > 0$$, let the
(1) eccentricity of E be reciprocal of the eccentricity of H, and
(2) the line $$y = \sqrt{\frac{5}{2}}x + K$$ be a common tangent of E and H.
Then $$4(a^2 + b^2)$$ is equal to

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NTA JEE Main 28th July 2022 Shift 1 - Question 86

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$$\lim_{x \to 0} \left(\frac{(x+2\cos x)^3 + 2(x+2\cos x)^2 + 3\sin(x+2\cos x)}{(x+2)^3 + 2(x+2)^2 + 3\sin(x+2)}\right)^{\frac{100}{x}}$$ is equal to

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NTA JEE Main 28th July 2022 Shift 1 - Question 87

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Let $$A = \begin{pmatrix} 1 & -1 \\ 2 & \alpha \end{pmatrix}$$ and $$B = \begin{pmatrix} \beta & 1 \\ 1 & 0 \end{pmatrix}$$, $$\alpha, \beta \in \mathbb{R}$$. Let $$\alpha_1$$ be the value of $$\alpha$$ which satisfies $$(A + B)^2 = A^2 + \begin{pmatrix} 2 & 2 \\ 2 & 2 \end{pmatrix}$$ and $$\alpha_2$$ be the value of $$\alpha$$ which satisfies $$(A + B)^2 = B^2$$. Then $$|\alpha_1 - \alpha_2|$$ is equal to

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NTA JEE Main 28th July 2022 Shift 1 - Question 88

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Let $$f: [0, 1] \to \mathbb{R}$$ be a twice differentiable function in (0, 1) such that $$f(0) = 3$$ and $$f(1) = 5$$. If the line $$y = 2x + 3$$ intersects the graph of $$f$$ at only two distinct points in (0, 1), then the least number of points $$x \in (0, 1)$$, at which $$f''(x) = 0$$, is

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NTA JEE Main 28th July 2022 Shift 1 - Question 89

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If $$\int_0^{\sqrt{3}} \frac{15x^3}{\sqrt{(1+x^2)} + \sqrt{(1+x^2)^3}} dx = \alpha\sqrt{2} + \beta\sqrt{3}$$, where $$\alpha, \beta$$ are integers, then $$\alpha + \beta$$ is equal to

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NTA JEE Main 28th July 2022 Shift 1 - Question 90

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Let $$P(-2, -1, 1)$$ and $$Q\left(\frac{56}{17}, \frac{43}{17}, \frac{111}{17}\right)$$ be the vertices of the rhombus PRQS. If the direction ratios of the diagonal RS are $$\alpha, -1, \beta$$, where both $$\alpha$$ and $$\beta$$ are integers of minimum absolute values, then $$\alpha^2 + \beta^2$$ is equal to

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