NTA JEE Main 25th July 2022 Shift 2

NTA JEE Main 25th July 2022 Shift 2 - Question 1

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The maximum error in the measurement of resistance, current and time for which current flows in an electrical circuit are $$1\%$$, $$2\%$$ and $$3\%$$ respectively. The maximum percentage error in the detection of the dissipated heat will be:

A

$$2$$

B

$$4$$

C

$$6$$

D

$$8$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 2

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A ball is projected from the ground with a speed $$15 \text{ m s}^{-1}$$ at an angle $$\theta$$ with horizontal so that its range and maximum height are equal, then $$\tan \theta$$ will be equal to

A

$$\dfrac{1}{4}$$

B

$$\dfrac{1}{2}$$

C

$$2$$

D

$$4$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 3

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For a free body diagram shown in the figure, the four forces are applied in the '$$x$$' and '$$y$$' directions. What additional force must be applied and at what angle with positive $$x$$-axis so that the net acceleration of body is zero?

A

$$\sqrt{2} \text{ N}, 45°$$

B

$$\sqrt{2} \text{ N}, 135°$$

C

$$\dfrac{2}{\sqrt{3}} \text{ N}, 30°$$

D

$$2 \text{ N}, 45°$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 4

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A bag of sand of mass $$9.8 \text{ kg}$$ is suspended by a rope. A bullet of $$200 \text{ g}$$ travelling with speed $$10 \text{ m s}^{-1}$$ gets embedded in it, then loss of kinetic energy will be

A

$$4.9 \text{ J}$$

B

$$9.8 \text{ J}$$

C

$$14.7 \text{ J}$$

D

$$19.6 \text{ J}$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 5

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Two billiard balls of mass $$0.05 \text{ kg}$$ each moving in opposite directions with $$10 \text{ m s}^{-1}$$ collide and rebound with the same speed. If the time duration of contact is $$t = 0.005 \text{ s}$$, then what is the force exerted on the ball due to each other?

A

$$100 \text{ N}$$

B

$$200 \text{ N}$$

C

$$300 \text{ N}$$

D

$$400 \text{ N}$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 6

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The length of a seconds pendulum at a height $$h = 2R$$ from earth surface will be: (Given: $$R$$ = Radius of earth and acceleration due to gravity at the surface of earth $$g = \pi^2 \text{ m s}^{-2}$$)

A

$$\dfrac{2}{9} \text{ m}$$

B

$$\dfrac{4}{9} \text{ m}$$

C

$$\dfrac{8}{9} \text{ m}$$

D

$$\dfrac{1}{9} \text{ m}$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 7

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An object is taken to a height above the surface of earth at a distance $$\dfrac{5}{4}R$$ from the centre of the earth. Where radius of earth, $$R = 6400 \text{ km}$$. The percentage decrease in the weight of the object will be

A

$$36\%$$

B

$$50\%$$

C

$$64\%$$

D

$$25\%$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 8

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A drop of liquid of density $$\rho$$ is floating half immersed in a liquid of density $$\sigma$$ and surface tension $$7.5 \times 10^{-4} \text{ N cm}^{-1}$$. The radius of drop in cm will be: (Take: $$g = 10 \text{ m s}^{-2}$$)

A

$$\dfrac{15}{\sqrt{2\rho - \sigma}}$$

B

$$\dfrac{15}{\sqrt{\rho - \sigma}}$$

C

$$\dfrac{3}{2\sqrt{\rho - \sigma}}$$

D

$$\dfrac{3}{20\sqrt{2\rho - \sigma}}$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 9

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Let $$\eta_1$$ is the efficiency of an engine at $$T_1 = 447°C$$ and $$T_2 = 147°C$$ while $$\eta_2$$ is the efficiency at $$T_1 = 947°C$$ and $$T_2 = 47°C$$. The ratio $$\dfrac{\eta_1}{\eta_2}$$ will be

A

$$0.41$$

B

$$0.56$$

C

$$0.73$$

D

$$0.70$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 10

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Sound travels in a mixture of two moles of helium and $$n$$ moles of hydrogen. If rms speed of gas molecules in the mixture is $$\sqrt{2}$$ times the speed of sound, then the value of $$n$$ will be

A

$$1$$

B

$$2$$

C

$$3$$

D

$$4$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 11

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Capacitance of an isolated conducting sphere of radius $$R_1$$ becomes $$n$$ times when it is enclosed by a concentric conducting sphere of radius $$R_2$$ connected to earth. The ratio of their radii  $$\left(\dfrac{R_2}{R_1}\right)$$ is:

A

$$\dfrac{n}{n-1}$$

B

$$\dfrac{2n}{2n+1}$$

C

$$\dfrac{n+1}{n}$$

D

$$\dfrac{2n+1}{n}$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 12

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A current of $$15 \text{ mA}$$ flows in the circuit as shown in figure. The value of potential difference between the points $$A$$ and $$B$$ will be

A

$$50 \text{ V}$$

B

$$75 \text{ V}$$

C

$$150 \text{ V}$$

D

$$275 \text{ V}$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 13

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An electron with energy $$0.1 \text{ keV}$$ moves at right angle to the earth's magnetic field of $$1 \times 10^{-4} \text{ Wb m}^{-2}$$. The frequency of revolution of the electron will be (Take mass of electron $$= 9.0 \times 10^{-31} \text{ kg}$$)

A

$$1.6 \times 10^5 \text{ Hz}$$

B

$$5.6 \times 10^5 \text{ Hz}$$

C

$$2.8 \times 10^6 \text{ Hz}$$

D

$$1.8 \times 10^6 \text{ Hz}$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 14

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The electric current in a circular coil of 2 turns produces a magnetic induction $$B_1$$ at its centre. The coil is unwound and is rewound into a circular coil of 5 turns and the same current produces a magnetic induction $$B_2$$ at its centre. The ratio of $$\dfrac{B_2}{B_1}$$ is:

A

$$\dfrac{5}{2}$$

B

$$\dfrac{25}{4}$$

C

$$\dfrac{5}{4}$$

D

$$\dfrac{25}{2}$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 15

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When you walk through a metal detector carrying a metal object in your pocket, it raises an alarm. This phenomenon works on

A

Electromagnetic induction

B

Resonance in ac circuits

C

Mutual induction in ac circuits

D

Interference of electromagnetic waves

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NTA JEE Main 25th July 2022 Shift 2 - Question 16

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Light wave travelling in air along $$x$$-direction is given by $$E_y = 540 \sin \pi \times 10^4(x - ct) \text{ V m}^{-1}$$. Then, the peak value of magnetic field of wave will be (Given $$c = 3 \times 10^8 \text{ m s}^{-1}$$)

A

$$18 \times 10^{-7} \text{ T}$$

B

$$54 \times 10^{-7} \text{ T}$$

C

$$54 \times 10^{-8} \text{ T}$$

D

$$18 \times 10^{-8} \text{ T}$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 17

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For an object placed at a distance $$2.4 \text{ m}$$ from a lens, a sharp focused image is observed on a screen placed at a distance $$12 \text{ cm}$$ from the lens. A glass plate of refractive index $$1.5$$ and thickness $$1 \text{ cm}$$ is introduced between lens and screen such that the glass plate plane faces parallel to the screen. By what distance should the object be shifted so that a sharp focused image is observed again on the screen?

A

$$0.8 \text{ m}$$

B

$$3.2 \text{ m}$$

C

$$1.2 \text{ m}$$

D

$$5.6 \text{ m}$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 18

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The ratio of wavelengths of proton and deuteron accelerated by potential $$V_p$$ and $$V_d$$ is $$1:\sqrt{2}$$. Then, the ratio of $$V_p$$ to $$V_d$$ will be

A

$$1:1$$

B

$$\sqrt{2}:1$$

C

$$2:1$$

D

$$4:1$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 19

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Hydrogen atom from excited state comes to the ground by emitting a photon of wavelength $$\lambda$$. The value of principal quantum number $$n$$ of the excited state will be: ($$R$$ : Rydberg constant)

A

$$\sqrt{\dfrac{\lambda R}{\lambda - 1}}$$

B

$$\sqrt{\dfrac{\lambda R}{\lambda R - 1}}$$

C

$$\sqrt{\dfrac{\lambda}{\lambda R - 1}}$$

D

$$\sqrt{\dfrac{\lambda R^2}{\lambda R - 1}}$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 20

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In AM modulation, a signal is modulated on a carrier wave such that maximum and minimum amplitude are found to be $$6 \text{ V}$$ and $$2 \text{ V}$$ respectively. The modulation index is

A

$$100\%$$

B

$$80\%$$

C

$$60\%$$

D

$$50\%$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 21

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A particle is moving in a straight line such that its velocity is increasing at $$5 \text{ m s}^{-1}$$ per meter. The acceleration of the particle is ______ $$\text{m s}^{-2}$$ at a point where its velocity is $$20 \text{ m s}^{-1}$$.

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NTA JEE Main 25th July 2022 Shift 2 - Question 22

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Three identical spheres each of mass $$M$$ are placed at the corners of a right angled triangle with mutually perpendicular sides equal to $$3 \text{ m}$$ each. Taking point of intersection of mutually perpendicular sides as origin, the magnitude of position vector of centre of mass of the system will be $$\sqrt{x}$$ m. The value of $$x$$ is ______.

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NTA JEE Main 25th July 2022 Shift 2 - Question 23

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A block of ice of mass $$120 \text{ g}$$ at temperature $$0°C$$ is put in $$300 \text{ g}$$ of water at $$25°C$$. The $$x$$ g of ice melts as the temperature of the water reaches $$0°C$$. The value of $$x$$ is ______.
[Use: Specific heat capacity of water $$= 4200 \text{ J kg}^{-1} \text{ K}^{-1}$$, Latent heat of ice $$= 3.5 \times 10^5 \text{ J kg}^{-1}$$]

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NTA JEE Main 25th July 2022 Shift 2 - Question 24

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Two waves executing simple harmonic motion travelling in the same direction with same amplitude and frequency are superimposed. The resultant amplitude is equal to the $$\sqrt{3}$$ times of amplitude of individual motions. The phase difference between the two motions is ______ (degree).

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NTA JEE Main 25th July 2022 Shift 2 - Question 25

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Two parallel plate capacitors of capacity $$C$$ and $$3C$$ are connected in parallel combination and charged to a potential difference $$18 \text{ V}$$. The battery is then disconnected and the space between the plates of the capacitor of capacity $$C$$ is completely filled with a material of dielectric constant $$9$$. The final potential difference across the combination of capacitors will be ______ V.

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NTA JEE Main 25th July 2022 Shift 2 - Question 26

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In a potentiometer arrangement, a cell of emf $$1.20 \text{ V}$$ gives a balance point at $$36 \text{ cm}$$ length of wire. This cell is now replaced by another cell of emf $$1.80 \text{ V}$$. The difference in balancing length of potentiometer wire in above conditions will be ______ cm.

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NTA JEE Main 25th July 2022 Shift 2 - Question 27

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Magnetic flux (in weber) in a closed circuit of resistance $$20 \text{ }\Omega$$ varies with time $$t$$ (s) as $$\phi = 8t^2 - 9t + 5$$. The magnitude of the induced current at $$t = 0.25 \text{ s}$$ will be ______ mA.

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NTA JEE Main 25th July 2022 Shift 2 - Question 28

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A convex lens of focal length $$20 \text{ cm}$$ is placed in front of convex mirror with principal axis coinciding each other. The distance between the lens and mirror is $$10 \text{ cm}$$. A point object is placed on principal axis at a distance of $$60 \text{ cm}$$ from the convex lens. The image formed by combination coincides the object itself. The focal length of the convex mirror is ______ cm.

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NTA JEE Main 25th July 2022 Shift 2 - Question 29

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$$\dfrac{x}{x+4}$$ is the ratio of energies of photons produced due to transition of an electron of hydrogen atom from its (i) third permitted energy level to the second level and (ii) the highest permitted energy level to the second permitted level. The value of $$x$$ will be ______.

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NTA JEE Main 25th July 2022 Shift 2 - Question 30

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Two ideal diodes are connected in the network as shown in figure. The equivalent resistance between $$A$$ and $$B$$ is ______ $$\Omega$$.

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NTA JEE Main 25th July 2022 Shift 2 - Question 31

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The first ionization enthalpies of $$Be$$, $$B$$, $$N$$ and $$O$$ follow the order

A

$$B < Be < O < N$$

B

$$O < N < B < Be$$

C

$$Be < B < N < O$$

D

$$B < Be < N < O$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 32

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Match List I with List II

Choose the correct answer from the options given below

A

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

B

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

C

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

D

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

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NTA JEE Main 25th July 2022 Shift 2 - Question 33

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$$K_{a1}$$, $$K_{a2}$$ and $$K_{a3}$$ are the respective ionization constants for the following reactions (a), (b) and (c).
(a) $$H_2C_2O_4 \rightleftharpoons H^+ + HC_2O_4^-$$
(b) $$HC_2O_4^- \rightleftharpoons H^+ + C_2O_4^{2-}$$
(c) $$H_2C_2O_4 \rightleftharpoons 2H^+ + C_2O_4^{2-}$$
The relationship between $$K_{a1}$$, $$K_{a2}$$ and $$K_{a3}$$ is given as

A

$$K_{a3} = K_{a1} + K_{a2}$$

B

$$K_{a3} = \dfrac{K_{a1}}{K_{a2}}$$

C

$$K_{a3} = K_{a1} - K_{a2}$$

D

$$K_{a3} = K_{a1} \times K_{a2}$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 34

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In base vs. Acid titration, at the end point methyl orange is present as

A

quinonoid form

B

heterocyclic form

C

phenolic form

D

benzenoid form

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NTA JEE Main 25th July 2022 Shift 2 - Question 35

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High purity ($$> 99.95\%$$) dihydrogen is obtained by

A

reaction of zinc with aqueous alkali

B

electrolysis of warm aqueous barium hydroxide solution between nickel electrodes

C

electrolysis of acidified water using platinum electrodes

D

reaction of zinc with dilute acid

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NTA JEE Main 25th July 2022 Shift 2 - Question 36

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The correct order of density is

A

$$Be > Mg > Ca > Sr$$

B

$$Sr > Ca > Mg > Be$$

C

$$Sr > Be > Mg > Ca$$

D

$$Be > Sr > Mg > Ca$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 37

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Arrange the following in decreasing acidic strength.

A

$$A > B > C > D$$

B

$$B > A > C > D$$

C

$$D > C > A > B$$

D

$$D > C > B > A$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 38

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Major product of the following reaction is

A

B

C

D

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NTA JEE Main 25th July 2022 Shift 2 - Question 39

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Match List I with List II.

Choose the correct answer from the options given below

A

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

B

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

C

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

D

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

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NTA JEE Main 25th July 2022 Shift 2 - Question 40

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Two solutions A and B are prepared by dissolving $$1 \text{ g}$$ of non-volatile solutes X and Y respectively in $$1 \text{ kg}$$ of water. The ratio of depression in freezing points for A and B is found to be $$1:4$$. The ratio of molar masses of X and Y is

A

$$1:4$$

B

$$1:0.25$$

C

$$1:0.20$$

D

$$1:5$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 41

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The molar conductivity of a conductivity cell filled with $$10$$ moles of $$20 \text{ mL NaCl}$$ solution is $$\Lambda_{m1}$$ and that of $$20$$ moles of another identical cell having $$80 \text{ mL NaCl}$$ solution is $$\Lambda_{m2}$$. The conductivities exhibited by these two cells are same. The relationship between $$\Lambda_{m2}$$ and $$\Lambda_{m1}$$ is

A

$$\Lambda_{m2} = 2\Lambda_{m1}$$

B

$$\Lambda_{m2} = \dfrac{\Lambda_{m1}}{2}$$

C

$$\Lambda_{m2} = \Lambda_{m1}$$

D

$$\Lambda_{m2} = 4\Lambda_{m1}$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 42

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For micelle formation, which of the following statements are correct?
(A) Micelle formation is an exothermic process.
(B) Micelle formation is an endothermic process.
(C) The entropy change is positive.
(D) The entropy change is negative.

A

A and D only

B

A and C only

C

B and C only

D

B and D only

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NTA JEE Main 25th July 2022 Shift 2 - Question 43

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Given below are two statements.
Statement I: Pig iron is obtained by heating cast iron with scrap iron.
Statement II: Pig iron has a relatively lower carbon content than that of cast iron.
In the light of the above statements, choose the correct answer from the options given below.

A

Both Statement I and Statement II are correct

B

Both Statement I and Statement II are not correct

C

Statement I is correct but Statement II is not correct

D

Statement I is not correct but Statement II is correct

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NTA JEE Main 25th July 2022 Shift 2 - Question 44

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The total number of acidic oxides from the following list is:
$$NO$$, $$N_2O$$, $$B_2O_3$$, $$N_2O_5$$, $$CO$$, $$SO_3$$, $$P_4O_{10}$$

A

$$5$$

B

$$4$$

C

$$3$$

D

$$6$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 45

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The correct order of energy of absorption for the following metal complexes is
$$A: [Ni(en)_3]^{2+}$$, $$B: [Ni(NH_3)_6]^{2+}$$, $$C: [Ni(H_2O)_6]^{2+}$$

A

$$C < B < A$$

B

$$B < C < A$$

C

$$C < A < B$$

D

$$A < C < B$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 46

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What is the major product of the following reaction?

A

B

C

D

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NTA JEE Main 25th July 2022 Shift 2 - Question 47

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The correct structure of C is

A

$$CH_3-CH_2-CH_2-CH_3$$

B

C

D

$$CH_3-CH_2-CH=CH_3$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 48

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Match List-I with List-II

Choose the correct answer from the options given below

A

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

B

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

C

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

D

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

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NTA JEE Main 25th July 2022 Shift 2 - Question 49

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Some drugs bind to a site other than the active site of an enzyme. This site is known as

A

non-active site

B

allosteric site

C

competitive site

D

therapeutic site

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NTA JEE Main 25th July 2022 Shift 2 - Question 50

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Glycosidic linkage between $$C_1$$ of $$\alpha$$-glucose and $$C_2$$ of $$\beta$$-fructose is found in

A

Maltose

B

Sucrose

C

Lactose

D

Amylose

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NTA JEE Main 25th July 2022 Shift 2 - Question 51

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$$56.0 \text{ L}$$ of nitrogen gas is mixed with excess of hydrogen gas and it is found that $$20 \text{ L}$$ of ammonia gas is produced. The volume of unused nitrogen gas is found to be ______ L.

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NTA JEE Main 25th July 2022 Shift 2 - Question 52

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When the excited electron of a H atom from $$n = 5$$ drops to the ground state, the maximum number of emission lines observed are ______.

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NTA JEE Main 25th July 2022 Shift 2 - Question 53

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The sum of number of lone pairs of electrons present on the central atoms of $$XeO_3$$, $$XeOF_4$$ and $$XeF_6$$ is ______.

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NTA JEE Main 25th July 2022 Shift 2 - Question 54

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A sealed flask with a capacity of $$2 \text{ dm}^3$$ contains $$11 \text{ g}$$ of propane gas. The flask is so weak that it will burst if the pressure becomes $$2 \text{ MPa}$$. The minimum temperature at which the flask will burst is ______ °C. [Nearest integer]
(Given: $$R = 8.3 \text{ J K}^{-1} \text{ mol}^{-1}$$. Atomic masses of C and H are $$12u$$ and $$1u$$ respectively.) (Assume that propane behaves as an ideal gas.)

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NTA JEE Main 25th July 2022 Shift 2 - Question 55

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While performing a thermodynamics experiment, a student made the following observations:
$$HCl + NaOH \rightarrow NaCl + H_2O \quad \Delta H = -57.3 \text{ kJ mol}^{-1}$$
$$CH_3COOH + NaOH \rightarrow CH_3COONa + H_2O \quad \Delta H = -55.3 \text{ kJ mol}^{-1}$$
The enthalpy of ionization of $$CH_3COOH$$ as calculated by the student is ______ $$\text{kJ mol}^{-1}$$.

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NTA JEE Main 25th July 2022 Shift 2 - Question 56

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The separation of two coloured substances was done by paper chromatography. The distances travelled by solvent front, substance A and substance B from the base line are $$3.25 \text{ cm}$$, $$2.08 \text{ cm}$$ and $$1.05 \text{ cm}$$ respectively. The ratio of $$R_f$$ values of A to B is ______ (Answer the nearest integer).

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NTA JEE Main 25th July 2022 Shift 2 - Question 57

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The total number of monobromo derivatives formed by the alkanes with molecular formula $$C_5H_{12}$$ is ______ (excluding stereo isomers).

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NTA JEE Main 25th July 2022 Shift 2 - Question 58

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For the decomposition of azomethane $$CH_3N_2CH_3(g) \rightarrow CH_3CH_3(g) + N_2$$, a first order reaction, the variation in partial pressure with time at $$600 \text{ K}$$ is given as shown.

The half life of the reaction is ______ $$\times 10^{-5}$$ s.

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NTA JEE Main 25th July 2022 Shift 2 - Question 59

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The spin-only magnetic moment value of $$M^{3+}$$ ion (in gaseous state) from the pairs $$Cr^{3+}/Cr^{2+}$$, $$Mn^{3+}/Mn^{2+}$$, $$Fe^{3+}/Fe^{2+}$$ and $$Co^{3+}/Co^{2+}$$ that has negative standard electrode potential, is ______ B.M.

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NTA JEE Main 25th July 2022 Shift 2 - Question 60

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A sample of $$4.5 \text{ mg}$$ of an unknown monohydric alcohol, $$R-OH$$ was added to methylmagnesium iodide. A gas is evolved and is collected and its volume measured to be $$3.1 \text{ mL}$$. The molecular weight of the unknown alcohol is ______ g/mol.

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NTA JEE Main 25th July 2022 Shift 2 - Question 61

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For $$z \in \mathbb{C}$$, if the minimum value of $$(|z - 3\sqrt{2}| + |z - p\sqrt{2}i|)$$ is $$5\sqrt{2}$$, then a value of $$p$$ is

A

$$3$$

B

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

C

4

D

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

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NTA JEE Main 25th July 2022 Shift 2 - Question 62

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The sum $$\displaystyle\sum_{n=1}^{21} \dfrac{3}{(4n-1)(4n+3)}$$ is equal to

A

$$\dfrac{7}{87}$$

B

$$\dfrac{7}{29}$$

C

$$\dfrac{14}{87}$$

D

$$\dfrac{21}{29}$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 63

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The remainder when $$(11)^{1011} + (1011)^{11}$$ is divided by $$9$$ is

A

$$1$$

B

$$8$$

C

$$6$$

D

$$4$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 64

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The value of $$2\sin\dfrac{\pi}{22} \sin\dfrac{3\pi}{22} \sin\dfrac{5\pi}{22} \sin\dfrac{7\pi}{22} \sin\dfrac{9\pi}{22}$$ is

A

$$\dfrac{1}{16}$$

B

$$\dfrac{5}{16}$$

C

$$\dfrac{7}{16}$$

D

$$\dfrac{3}{16}$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 65

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Let the point $$P(\alpha, \beta)$$ be at a unit distance from each of the two lines $$L_1: 3x - 4y + 12 = 0$$, and $$L_2: 8x + 6y + 11 = 0$$. If $$P$$ lies below $$L_1$$ and above $$L_2$$, then $$100(\alpha + \beta)$$ is equal to

A

$$-14$$

B

$$42$$

C

$$-22$$

D

$$14$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 66

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The tangents at the points $$A(1, 3)$$ and $$B(1, -1)$$ on the parabola $$y^2 - 2x - 2y = 1$$ meet at the point $$P$$. Then the area (in $$\text{unit}^2$$) of the triangle $$PAB$$ is:

A

$$4$$

B

$$6$$

C

$$7$$

D

$$8$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 67

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If the ellipse $$\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$$ meets the line $$\dfrac{x}{7} + \dfrac{y}{2\sqrt{6}} = 1$$ on the $$x$$-axis and the line $$\dfrac{x}{7} - \dfrac{y}{2\sqrt{6}} = 1$$ on the $$y$$-axis, then the eccentricity of the ellipse is

A

$$\dfrac{5}{7}$$

B

$$\dfrac{2\sqrt{6}}{7}$$

C

$$\dfrac{3}{7}$$

D

$$\dfrac{2\sqrt{5}}{7}$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 68

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Let the foci of the ellipse $$\dfrac{x^2}{16} + \dfrac{y^2}{7} = 1$$ and the hyperbola $$\dfrac{x^2}{144} - \dfrac{y^2}{\alpha} = \dfrac{1}{25}$$ coincide. Then the length of the latus rectum of the hyperbola is:

A

$$\dfrac{32}{9}$$

B

$$\dfrac{18}{5}$$

C

$$\dfrac{27}{4}$$

D

$$\dfrac{27}{10}$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 69

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$$\displaystyle\lim_{x \to \frac{\pi}{4}} \dfrac{8\sqrt{2} - (\cos x + \sin x)^7}{\sqrt{2} - \sqrt{2}\sin 2x}$$ is equal to

A

$$14$$

B

$$7$$

C

$$14\sqrt{2}$$

D

$$7\sqrt{2}$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 70

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Consider the following statements:
$$P$$: Ramu is intelligent.
$$Q$$: Ramu is rich.
$$R$$: Ramu is not honest.
The negation of the statement "Ramu is intelligent and honest if and only if Ramu is not rich" can be expressed as:

A

$$((P \wedge (\sim R)) \wedge Q) \wedge ((\sim Q) \wedge ((\sim P) \vee R))$$

B

$$((P \wedge R) \wedge Q) \vee ((\sim Q) \wedge ((\sim P) \vee (\sim R)))$$

C

$$((P \wedge R) \wedge Q) \wedge ((\sim Q) \wedge ((\sim P) \vee (\sim R)))$$

D

$$((P \wedge (\sim R)) \wedge Q) \vee ((\sim Q) \wedge ((\sim P) \vee R))$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 71

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If the mean deviation about median for the numbers $$3, 5, 7, 2k, 12, 16, 21, 24$$ arranged in the ascending order, is $$6$$ then the median is

A

$$11.5$$

B

$$10.5$$

C

$$12$$

D

$$11$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 72

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The number of real values of $$\lambda$$, such that the system of linear equations
$$2x - 3y + 5z = 9$$
$$x + 3y - z = -18$$
$$3x - y + (\lambda^2 - |\lambda|)z = 16$$
has no solutions, is

A

$$0$$

B

$$1$$

C

$$2$$

D

$$4$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 73

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The number of bijective functions $$f(\{1, 3, 5, 7, \ldots, 99\}) \to \{2, 4, 6, 8, \ldots, 100\}$$ if $$f(3) > f(5) > f(7) \ldots > f(99)$$ is

A

$$^{50}C_1$$

B

$$^{50}C_2$$

C

$$\dfrac{50!}{2}$$

D

$$^{50}C_3 \times 3!$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 74

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$$\displaystyle\lim_{n \to \infty} \dfrac{1}{2n}\left(\dfrac{1}{\sqrt{1 - \frac{1}{2n}}} + \dfrac{1}{\sqrt{1 - \frac{2}{2n}}} + \dfrac{1}{\sqrt{1 - \frac{3}{2n}}} + \ldots + \dfrac{1}{\sqrt{1 - \frac{2n-1}{2n}}}\right)$$ is equal to

A

$$\dfrac{1}{2}$$

B

$$1$$

C

$$2$$

D

$$-2$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 75

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Let $$[t]$$ denote the greatest integer less than or equal to $$t$$. Then the value of the integral $$\displaystyle\int_{-3}^{101} ([\sin(\pi x)] + e^{[\cos(2\pi x)]}) dx$$ is equal to

A

$$\dfrac{52(1-e)}{e}$$

B

$$\dfrac{52}{e}$$

C

$$\dfrac{52(2+e)}{e}$$

D

$$\dfrac{104}{e}$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 76

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Let a smooth curve $$y = f(x)$$ be such that the slope of the tangent at any point $$(x, y)$$ on it is directly proportional to $$\left(\dfrac{-y}{x}\right)$$. If the curve passes through the points $$(1, 2)$$ and $$(8, 1)$$, then $$\left|y\left(\dfrac{1}{8}\right)\right|$$ is equal to

A

$$2\log_e 2$$

B

$$4$$

C

$$1$$

D

$$4\log_e 2$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 77

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Let $$\vec{a} = \hat{i} - \hat{j} + 2\hat{k}$$ and let $$\vec{b}$$ be a vector such that $$\vec{a} \times \vec{b} = 2\hat{i} - \hat{k}$$ and $$\vec{a} \cdot \vec{b} = 3$$. Then the projection of $$\vec{b}$$ on the vector $$\vec{a} - \vec{b}$$ is:

A

$$\dfrac{2}{\sqrt{21}}$$

B

$$\dfrac{2\sqrt{3}}{7}$$

C

$$\dfrac{2}{3}\sqrt{\dfrac{7}{3}}$$

D

$$\dfrac{2}{3}$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 78

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A plane $$E$$ is perpendicular to the two planes $$2x - 2y + z = 0$$ and $$x - y + 2z = 4$$, and passes through the point $$P(1, -1, 1)$$. If the distance of the plane $$E$$ from the point $$Q(a, a, 2)$$ is $$3\sqrt{2}$$, then $$(PQ)^2$$ is equal to

A

$$9$$

B

$$12$$

C

$$21$$

D

$$33$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 79

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The shortest distance between the lines $$\dfrac{x+7}{-6} = \dfrac{y-6}{7} = z$$ and $$\dfrac{7-x}{2} = y - 2 = z - 6$$ is

A

$$2\sqrt{29}$$

B

$$1$$

C

$$\sqrt{37}$$

D

$$\sqrt{29}$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 80

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If $$A$$ and $$B$$ are two events such that $$P(A) = \dfrac{1}{3}$$, $$P(B) = \dfrac{1}{5}$$ and $$P(A \cup B) = \dfrac{1}{2}$$, then $$P\left(\dfrac{A}{B'}\right) + P\left(\dfrac{B}{A'}\right)$$ is equal to

A

$$\dfrac{3}{4}$$

B

$$\dfrac{5}{8}$$

C

$$\dfrac{5}{4}$$

D

$$\dfrac{7}{8}$$

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NTA JEE Main 25th July 2022 Shift 2 - Question 81

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Let $$f(x)$$ be a quadratic polynomial with leading coefficient $$1$$ such that $$f(0) = p, p \neq 0$$, and $$f(1) = \dfrac{1}{3}$$. If the equations $$f(x) = 0$$ and $$fofofof(x) = 0$$ have a common real root, then $$f(-3)$$ is equal to ______.

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NTA JEE Main 25th July 2022 Shift 2 - Question 82

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If the circles $$x^2 + y^2 + 6x + 8y + 16 = 0$$ and $$x^2 + y^2 + 2(3 - \sqrt{3})x + 2(4 - \sqrt{6})y = k + 6\sqrt{3} + 8\sqrt{6}$$, $$k > 0$$, touch internally at the point $$P(\alpha, \beta)$$, then $$(\alpha + \sqrt{3})^2 + (\beta + \sqrt{6})^2$$ is equal to ______.

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NTA JEE Main 25th July 2022 Shift 2 - Question 83

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Let $$A = \{1, 2, 3, 4, 5, 6, 7\}$$. Define $$B = \{T \subseteq A :$$ either $$1 \notin T$$ or $$2 \in T\}$$ and $$C = \{T \subseteq A :$$ the sum of all the elements of $$T$$ is a prime number $$\}$$. Then the number of elements in the set $$B \cup C$$ is ______.

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NTA JEE Main 25th July 2022 Shift 2 - Question 84

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Let $$A = \begin{pmatrix} 1 & a & a \\ 0 & 1 & b \\ 0 & 0 & 1 \end{pmatrix}$$, $$a, b \in \mathbb{R}$$. If for some $$n \in \mathbb{N}$$, $$A^n = \begin{pmatrix} 1 & 48 & 2160 \\ 0 & 1 & 96 \\ 0 & 0 & 1 \end{pmatrix}$$ then $$n + a + b$$ is equal to ______.

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NTA JEE Main 25th July 2022 Shift 2 - Question 85

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Let $$x = \sin(2\tan^{-1}\alpha)$$ and $$y = \sin\left(\dfrac{1}{2}\tan^{-1}\dfrac{4}{3}\right)$$. If $$S = \{\alpha \in \mathbb{R} : y^2 = 1 - x\}$$, then $$\displaystyle\sum_{\alpha \in S} 16\alpha^3$$ is equal to ______.

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NTA JEE Main 25th July 2022 Shift 2 - Question 86

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The sum of the maximum and minimum values of the function $$f(x) = |5x - 7| + [x^2 + 2x]$$ in the interval $$\left[\dfrac{5}{4}, 2\right]$$, where $$[t]$$ is the greatest integer $$\le t$$, is ______.

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NTA JEE Main 25th July 2022 Shift 2 - Question 87

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Let the area enclosed by the $$x$$-axis, and the tangent and normal drawn to the curve $$4x^3 - 3xy^2 + 6x^2 - 5xy - 8y^2 + 9x + 14 = 0$$ at the point $$(-2, 3)$$ be $$A$$. Then $$8A$$ is equal to ______.

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NTA JEE Main 25th July 2022 Shift 2 - Question 88

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Let $$f$$ be a twice differentiable function on $$\mathbb{R}$$. If $$f'(0) = 4$$ and $$f(x) + \displaystyle\int_0^x (x-t)f'(t) dt = (e^{2x} + e^{-2x})\cos 2x + \dfrac{2}{a}x$$, then $$(2a+1)^5a^2$$ is equal to ______.

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NTA JEE Main 25th July 2022 Shift 2 - Question 89

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Let $$a_n = \displaystyle\int_{-1}^{n} \left(1 + \dfrac{x}{2} + \dfrac{x^2}{3} + \ldots + \dfrac{x^{n-1}}{n}\right) dx$$ for every $$n \in \mathbb{N}$$. Then the sum of all the elements of the set $$\{n \in \mathbb{N} : a_n \in (2, 30)\}$$ is ______.

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NTA JEE Main 25th July 2022 Shift 2 - Question 90

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Let $$y = y(x)$$ be the solution of the differential equation $$\dfrac{dy}{dx} = \dfrac{4y^3 + 2yx^2}{3xy^2 + x^3}$$, $$y(1) = 1$$. If for some $$n \in \mathbb{N}$$, $$y(2) \in [n-1, n)$$, then $$n$$ is equal to ______.

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