NTA JEE Main 8th April 2023 Shift 2

NTA JEE Main 8th April 2023 Shift 2 - Question 1

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Match List I with List II

	List I		List II
A.	Torque	I.	$$M L^{-2} T^{-2}$$
B.	Stress	II.	$$M L^2 T^{-2}$$
C.	Pressure gradient	III.	$$M L^{-1} T^{-1}$$
D.	Coefficient of viscosity	IV.	$$M L^{-1} T^{-2}$$

Choose the correct answer from the options given below:

A

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

B

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

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 8th April 2023 Shift 2 - Question 2

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Given below are two statements:
Statement I: Area under velocity-time graph gives the distance travelled by the body in a given time.
Statement II: Area under acceleration-time graph is equal to the change in velocity in the given time.
In the light of given 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 correct but Statement II is false

D

Statement I is incorrect but Statement II is true

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NTA JEE Main 8th April 2023 Shift 2 - Question 3

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The trajectory of projectile, projected from the ground is given by $$y = x - \frac{x^2}{20}$$. Where $$x$$ and $$y$$ are measured in meter. The maximum height attained by the projectile will be.

A

200 m

B

10 m

C

5 m

D

$$10\sqrt{2}$$ m

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NTA JEE Main 8th April 2023 Shift 2 - Question 4

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A bullet of mass 0.1 kg moving horizontally with speed 400 m s$$^{-1}$$ hits a wooden block of mass 3.9 kg kept on a horizontal rough surface. The bullet gets embedded into the block and moves 20 m before coming to rest. The coefficient of friction between the block and the surface is

A

0.90

B

0.50

C

0.65

D

0.25

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NTA JEE Main 8th April 2023 Shift 2 - Question 5

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The orbital angular momentum of a satellite is $$\mathbf{L}$$, when it is revolving in a circular orbit at height h from earth surface. If the distance of satellite from the earth centre is increased by eight times to its initial value, then the new angular momentum will be

A

8 L

B

9 L

C

4 L

D

3 L

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NTA JEE Main 8th April 2023 Shift 2 - Question 6

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The acceleration due to gravity at height $$h$$ above the earth if $$h \ll R$$ (Radius of earth) is given by

A

$$g' = g\left(1 - \frac{h^2}{2R^2}\right)$$

B

$$g' = g\left(1 - \frac{h}{2R}\right)$$

C

$$g' = g\left(1 - \frac{2h}{R}\right)$$

D

$$g' = g\left(1 - \frac{2h^2}{R^2}\right)$$

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NTA JEE Main 8th April 2023 Shift 2 - Question 7

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A hydraulic automobile lift is designed to lift vehicles of mass 5000 kg. The area of cross section of the cylinder carrying load is 250 cm$$^2$$. The maximum pressure the smaller piston would have to bear is [Assume $$g = 10$$ m s$$^{-2}$$]

A

$$20 \times 10^6$$ Pa

B

$$2 \times 10^5$$ Pa

C

$$200 \times 10^6$$ Pa

D

$$2 \times 10^6$$ Pa

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NTA JEE Main 8th April 2023 Shift 2 - Question 8

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Work done by a Carnot engine operating between temperatures 127$$^\circ$$C and 27$$^\circ$$C is 2 kJ. The amount of heat transferred to the engine by the reservoir is:

A

8 kJ

B

2.67 kJ

C

2 kJ

D

4 kJ

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NTA JEE Main 8th April 2023 Shift 2 - Question 9

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The temperature at which the kinetic energy of oxygen molecules becomes double than its value at 27$$^\circ$$C is

A

927$$^\circ$$C

B

327$$^\circ$$C

C

1227$$^\circ$$C

D

627$$^\circ$$C

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NTA JEE Main 8th April 2023 Shift 2 - Question 10

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For particle $$P$$ revolving round the centre $$O$$ with radius of circular path $$r$$ and angular velocity $$\omega$$, as shown in below figure, the projection of $$OP$$ on the $$x$$-axis at time $$t$$ is

A

$$x(t) = r\cos\left(\omega t - \frac{\pi}{6}\right)$$

B

$$x(t) = r\cos\left(\omega t + \frac{\pi}{6}\right)$$

C

$$x(t) = r\sin\left(\omega t + \frac{\pi}{6}\right)$$

D

$$x(t) = r\cos(\omega t)$$

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NTA JEE Main 8th April 2023 Shift 2 - Question 11

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Electric potential at a point $$P$$ due to a point charge of $$5 \times 10^{-9}$$ C is 50 V. The distance of $$P$$ from the point charge is:
(Assume, $$\frac{1}{4\pi\varepsilon_0} = 9 \times 10^9$$ N m$$^2$$ C$$^{-2}$$)

A

9 cm

B

3 cm

C

0.9 cm

D

90 cm

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NTA JEE Main 8th April 2023 Shift 2 - Question 12

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The equivalent resistance between A and B as shown in figure is:

A

10 k$$\Omega$$

B

5 k$$\Omega$$

C

20 k$$\Omega$$

D

30 k$$\Omega$$

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NTA JEE Main 8th April 2023 Shift 2 - Question 13

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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: Electromagnets are made of soft iron.
Reason R: Soft iron has high permeability and low retentivity.
In the light of above statements, choose the most appropriate answer from the options given below.

A

A is not correct but R is correct

B

A is correct but R is not correct

C

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

D

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

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NTA JEE Main 8th April 2023 Shift 2 - Question 14

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An emf of 0.08 V is induced in a metal rod of length 10 cm held normal to a uniform magnetic field of 0.4 T, when move with a velocity of:

A

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

B

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

C

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

D

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

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NTA JEE Main 8th April 2023 Shift 2 - Question 15

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The waves emitted when a metal target is bombarded with high energy electrons are

A

Microwaves

B

Infrared rays

C

X-rays

D

Radio Waves

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NTA JEE Main 8th April 2023 Shift 2 - Question 16

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The width of fringe is 2 mm on the screen in a double slit experiment for the light of wavelength of 400 nm. The width of the fringe for the light of wavelength 600 nm will be:

A

4 mm

B

2 mm

C

1.33 mm

D

3 mm

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NTA JEE Main 8th April 2023 Shift 2 - Question 17

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In photoelectric effect
A. The photocurrent is proportional to the intensity of the incident radiation.
B. Maximum kinetic energy with which photoelectrons are emitted depends on the intensity of incident light.
C. Max K.E. with which photoelectrons are emitted depends on the frequency of incident light.
D. The emission of photoelectrons require a minimum threshold intensity of incident radiation.
E. Max K.E. of the photoelectrons is independent of the frequency of the incident light.
Choose the correct option from the options given below:

A

A and B only

B

A and E only

C

A and C only

D

B and C only

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NTA JEE Main 8th April 2023 Shift 2 - Question 18

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A radio active material is reduced to $$\frac{1}{8}$$ of its original amount in 3 days. If $$8 \times 10^{-3}$$ kg of the material is left after 5 days the initial amount of the material is

A

40 g

B

32 g

C

64 g

D

256 g

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NTA JEE Main 8th April 2023 Shift 2 - Question 19

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For a given transistor amplifier circuit in CE configuration $$V_{CC} = 1$$ V, $$R_C = 1$$ k$$\Omega$$, $$R_b = 100$$ k$$\Omega$$ and $$\beta = 100$$. Value of base current $$I_b$$ is

A

$$I_b = 1.0$$ $$\mu$$A

B

$$I_b = 0.1$$ $$\mu$$A

C

$$I_b = 100$$ $$\mu$$A

D

$$I_b = 10$$ $$\mu$$A

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NTA JEE Main 8th April 2023 Shift 2 - Question 20

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The power radiated from a linear antenna of length $$l$$ is proportional to (Given, $$\lambda$$ = Wavelength of wave):

A

$$\frac{l}{\lambda}$$

B

$$\left(\frac{l}{\lambda}\right)^2$$

C

$$\frac{l}{\lambda^2}$$

D

$$\frac{l^2}{\lambda}$$

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NTA JEE Main 8th April 2023 Shift 2 - Question 21

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A body of mass 5 kg is moving with a momentum of 10 kg m s$$^{-1}$$. Now a force of 2 N acts on the body in the direction of its motion for 5 s. The increase in the Kinetic energy of the body is _______ J.

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NTA JEE Main 8th April 2023 Shift 2 - Question 22

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A hollow spherical ball of uniform density rolls up a curved surface with an initial velocity 3 m s$$^{-1}$$ (as shown in figure). Maximum height with respect to the initial position covered by it will be _______ cm.
(take, $$g = 10$$ m s$$^{-2}$$)

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NTA JEE Main 8th April 2023 Shift 2 - Question 23

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A steel rod of length 1 m and cross-sectional area $$10^{-4}$$ m$$^2$$ is heated from 0$$^\circ$$C to 200$$^\circ$$C without being allowed to extend or bend. The compressive tension produced in the rod is _______ $$\times 10^4$$ N. (Given Young's modulus of steel $$= 2 \times 10^{11}$$ N m$$^{-2}$$, coefficient of linear expansion $$= 10^{-5}$$ K$$^{-1}$$)

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NTA JEE Main 8th April 2023 Shift 2 - Question 24

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A guitar string of length 90 cm vibrates with a fundamental frequency of 120 Hz. The length of the string producing a fundamental of 180 Hz will be _______ cm.

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NTA JEE Main 8th April 2023 Shift 2 - Question 25

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A 600 pF capacitor is charged by 200 V supply. It is then disconnected from the supply and is connected to another uncharged 600 pF capacitor. Electrostatic energy lost in the process is _______ $$\mu$$J.

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NTA JEE Main 8th April 2023 Shift 2 - Question 26

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The number density of free electrons in copper is nearly $$8 \times 10^{28}$$ m$$^{-3}$$. A copper wire has its area of cross-section $$= 2 \times 10^{-6}$$ m$$^2$$ and is carrying a current of 3.2 A. The drift speed of the electrons is _______ $$\times 10^{-6}$$ m s$$^{-1}$$.

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NTA JEE Main 8th April 2023 Shift 2 - Question 27

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The ratio of magnetic field at the centre of a current carrying coil of radius $$r$$ to the magnetic field at distance $$r$$ from the centre of coil on its axis is $$\sqrt{x}$$ : 1. The value of $$x$$ is _____.

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NTA JEE Main 8th April 2023 Shift 2 - Question 28

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A series combination of resistor of resistance 100 $$\Omega$$, inductor of inductance 1 H and capacitor of capacitance 6.25 $$\mu$$F is connected to an ac source. The quality factor of the circuit will be _____.

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NTA JEE Main 8th April 2023 Shift 2 - Question 29

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Two transparent media having refractive indices 1.0 and 1.5 are separated by a spherical refracting surface of radius of curvature 30 cm. The centre of curvature of surface is towards denser medium and a point object is placed on the principal axis in rarer medium at a distance of 15 cm from the pole of the surface. The distance of image from the pole of the surface is _______ cm.

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NTA JEE Main 8th April 2023 Shift 2 - Question 30

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The ratio of wavelength of spectral lines $$H_\alpha$$ and $$H_\beta$$ in the Balmer series is $$\frac{x}{20}$$. The value of $$x$$ is _____.

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NTA JEE Main 8th April 2023 Shift 2 - Question 31

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Which of the following have same number of significant figures?
(A) 0.00253
(B) 1.0003
(C) 15.0
(D) 163
Choose the correct answer from the options given below

A

A, B and C only

B

C and D only

C

B and C only

D

A, C and D only

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NTA JEE Main 8th April 2023 Shift 2 - Question 32

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Henry Moseley studied characteristic X-ray spectra of elements. The graph which represents his observation correctly is
Given $$\nu$$ = Frequency of X-ray emitted, $$Z$$ = Atomic number

A

B

C

D

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NTA JEE Main 8th April 2023 Shift 2 - Question 33

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The number of atomic orbitals from the following having 5 radial nodes is
7s, 7p, 6s, 8p, 8d

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NTA JEE Main 8th April 2023 Shift 2 - Question 34

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The number of species from the following carrying a single lone pair on central atom Xenon is
$$XeF_5^+$$, $$XeO_3$$, $$XeO_2F_2$$, $$XeF_5^-$$, $$XeO_3F_2$$, $$XeOF_4$$, $$XeF_4$$

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NTA JEE Main 8th April 2023 Shift 2 - Question 35

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Arrange the following gases in increasing order of van der Waals constant 'a'
(A) Ar
(B) CH$$_4$$
(C) H$$_2$$O
(D) C$$_6$$H$$_6$$
Choose the correct option from the following.

A

D, C, B and A

B

B, C, D and A

C

C, D, B and A

D

A, B, C and D

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NTA JEE Main 8th April 2023 Shift 2 - Question 36

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The incorrect statements from the following is:
A. The electrical work that a reaction can perform at constant pressure and temperature is equal to the reaction Gibbs energy.
B. $$E^\circ_{cell}$$ is dependent on the pressure.
C. $$\frac{dE_{cell}}{dT} = \frac{\Delta_r S}{nF}$$
D. A cell is operating reversibly if the cell potential is exactly balanced by an opposing source of potential difference.

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NTA JEE Main 8th April 2023 Shift 2 - Question 37

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Given below are two statements:
Statement-I: Methyl orange is a weak acid.
Statement-II: The benzenoid form of methyl orange is more intense/deeply coloured than the quinonoid form.
In the light of the above statement, choose the most appropriate answer from the options given below:

A

Both Statement-I and Statement-II are incorrect

B

Statement-I is incorrect but Statement-II is correct

C

Both Statement-I and Statement-II are correct

D

Statement-I is correct but Statement-II is incorrect

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NTA JEE Main 8th April 2023 Shift 2 - Question 38

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Given below are two statements:
Statement I: In redox titration, the indicators used are sensitive to change in pH of the solution.
Statement II: In acid-base titration, the indicators used are sensitive to change in oxidation potential.
In the light of the above statements, choose the most appropriate answer from the options given below

A

Statement I is correct but Statement II is incorrect

B

Both Statement I and Statement II are incorrect

C

Statement I is incorrect but Statement II is correct

D

Both Statement I and Statement II are correct

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NTA JEE Main 8th April 2023 Shift 2 - Question 39

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Which of the following can reduce decomposition of H$$_2$$O$$_2$$ on exposure to light

A

Urea

B

Alkali

C

Glass containers

D

Dust

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NTA JEE Main 8th April 2023 Shift 2 - Question 40

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For a good quality cement, the ratio of lime to the total of the oxides of Si, Al and Fe should be as close as to

A

4

B

1

C

2

D

3

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NTA JEE Main 8th April 2023 Shift 2 - Question 41

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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: Sodium is about 30 times as abundant as potassium in the oceans.
Reason R: Potassium is bigger in size than sodium.
In the light of the 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

A is true but R is false

C

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

D

Both A and R are false

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NTA JEE Main 8th April 2023 Shift 2 - Question 42

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The descending order of acidity for the following carboxylic acid is-
(A) CH$$_3$$COOH
(B) F$$_3$$C-COOH
(C) ClCH$$_2$$-COOH
(D) FCH$$_2$$-COOH
(E) BrCH$$_2$$-COOH
Choose the correct answer from the options given below:

A

B > C > D > E > A

B

E > D > B > A > C

C

B > D > C > E > A

D

D > B > A > E > C

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NTA JEE Main 8th April 2023 Shift 2 - Question 43

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The correct IUPAC nomenclature for the following compound is:

A

2-Methyl-5-oxohexanoic acid

B

2-Formyl-5-methylhexan-6-oic acid

C

5-Methyl-2-oxohexan-6-oic acid

D

5-Formyl-2-methylhexanoic acid

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NTA JEE Main 8th April 2023 Shift 2 - Question 44

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Which of these reactions is not a part of breakdown of ozone in stratosphere?

A

B

C

D

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NTA JEE Main 8th April 2023 Shift 2 - Question 45

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The correct reaction profile diagram for a positive catalyst reaction.

A

B

C

D

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NTA JEE Main 8th April 2023 Shift 2 - Question 46

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The statement/s which are true about antagonists from the following is/are:
A. They bind to the receptor site.
B. Get transferred inside the cell for their action.
C. Inhibit the natural communication of the body.
D. Mimic the natural messenger.
Choose the correct option from the options given below:

A

B only

B

A, C and D

C

A and C

D

A and B

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NTA JEE Main 8th April 2023 Shift 2 - Question 47

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In Hall-Heroult process, the following is used for reducing Al$$_2$$O$$_3$$:-

A

Magnesium

B

Na$$_3$$AlF$$_6$$

C

Graphite

D

CaF$$_2$$

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NTA JEE Main 8th April 2023 Shift 2 - Question 48

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Match List-I with List-II

	LIST-I Coordination Complex		LIST-II Number of unpaired electrons
A.	$$[Cr(CN)_6]^{3-}$$	I.	0
B.	$$[Fe(H_2O)_6]^{2+}$$	II.	3
C.	$$[Co(NH_3)_6]^{3+}$$	III.	2
D.	$$[Ni(NH_3)_6]^{2+}$$	IV.	4

Choose the correct answer from the options given below:

A

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

B

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

C

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

D

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

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NTA JEE Main 8th April 2023 Shift 2 - Question 49

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The correct order of reactivity of following haloarenes towards nucleophilic substitution with aqueous NaOH is:

A

A > B > D > C

B

C > A > D > B

C

D > C > B > A

D

D > B > A > C

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NTA JEE Main 8th April 2023 Shift 2 - Question 50

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A compound 'X' when treated with phthalic anhydride in presence of concentrated H$$_2$$SO$$_4$$ yields 'Y'. 'Y' is used as an acid/base indicator. 'X' and 'Y' are respectively

A

Anisole, methyl orange

B

Salicylaldehyde, Phenolphthalein

C

Toludine, Phenolphthalein

D

Carbolic acid, Phenolphthalein

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NTA JEE Main 8th April 2023 Shift 2 - Question 51

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For complete combustion of ethene,
$$C_2H_4(g) + 3O_2(g) \to 2CO_2(g) + 2H_2O(l)$$
the amount of heat produced as measured in bomb calorimeter is 1406 kJ mol$$^{-1}$$ at 300 K. The minimum value of T$$\Delta$$S needed to reach equilibrium is ($$-$$) kJ. (Nearest integer)
Given: R = 8.3 J K$$^{-1}$$ mol$$^{-1}$$

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NTA JEE Main 8th April 2023 Shift 2 - Question 52

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The solubility product of BaSO$$_4$$ is $$1 \times 10^{-10}$$ at 298 K. The solubility of BaSO$$_4$$ in 0.1 M K$$_2$$SO$$_4$$(aq) solution is _______ $$\times 10^{-9}$$ g L$$^{-1}$$ (nearest integer).
Given: Molar mass of BaSO$$_4$$ is 233 g mol$$^{-1}$$

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NTA JEE Main 8th April 2023 Shift 2 - Question 53

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If the boiling points of two solvents X and Y (having same molecular weights) are in the ratio 2:1 and their enthalpy of vaporizations are in the ratio 1:2, then the boiling point elevation constant of X is m times the boiling point elevation constant of Y. The value of m is (nearest integer).

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NTA JEE Main 8th April 2023 Shift 2 - Question 54

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Coagulating value of the electrolytes AlCl$$_3$$ and NaCl for As$$_2$$S$$_3$$ are 0.09 and 50.04 respectively. The coagulating power of AlCl$$_3$$ is x times the coagulating power of NaCl. The value of x is _____.

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NTA JEE Main 8th April 2023 Shift 2 - Question 55

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The ratio of sigma and $$\pi$$ bonds present in pyrophosphoric acid is _____.

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NTA JEE Main 8th April 2023 Shift 2 - Question 56

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The observed magnetic moment of the complex $$[Mn(NCS)_6]^{x-}$$ is 6.06 BM. The numerical value of x is _____.

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NTA JEE Main 8th April 2023 Shift 2 - Question 57

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The sum of oxidation state of the metals in Fe(CO)$$_5$$, VO$$^{2+}$$ and WO$$_3$$ is _____.

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NTA JEE Main 8th April 2023 Shift 2 - Question 58

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Major product 'P' formed in the following reaction is:

A

B

C

D

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NTA JEE Main 8th April 2023 Shift 2 - Question 59

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The product (P) formed from the following multistep reaction is:

A

B

C

D

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NTA JEE Main 8th April 2023 Shift 2 - Question 60

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Match List I with List II

	List I Natural amino acid		List II One Letter Code
(A)	Glutamic acid	(I)	Q
(B)	Glutamine	(II)	W
(C)	Tyrosine	(III)	E
(D)	Tryptophan	(IV)	Y

Choose the correct answer from the options given below:

A

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

B

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

C

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

D

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

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NTA JEE Main 8th April 2023 Shift 2 - Question 61

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Let $$m$$ and $$n$$ be the numbers of real roots of the quadratic equations $$x^2 - 12x + [x] + 31 = 0$$ and $$x^2 - 5|x + 2| - 4 = 0$$ respectively, where $$[x]$$ denotes the greatest integer $$\leq x$$. Then $$m^2 + mn + n^2$$ is equal to

A

9

B

12

C

6

D

15

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NTA JEE Main 8th April 2023 Shift 2 - Question 62

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Let $$A = \left\{\theta \in (0, 2\pi) : \frac{1 + 2i\sin\theta}{1 - i\sin\theta} \text{ is purely imaginary}\right\}$$. Then the sum of the elements in $$A$$ is

A

$$4\pi$$

B

$$3\pi$$

C

$$\pi$$

D

$$2\pi$$

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NTA JEE Main 8th April 2023 Shift 2 - Question 63

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If the number of words, with or without meaning, which can be made using all the letters of the word MATHEMATICS in which C and S do not come together, is $$(6!)k$$ then $$k$$ is equal to

A

2835

B

5670

C

1890

D

945

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NTA JEE Main 8th April 2023 Shift 2 - Question 64

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Let $$a_n$$ be $$n^{th}$$ term of the series $$5 + 8 + 14 + 23 + 35 + 50 + \ldots$$ and $$S_n = \sum_{k=1}^{n} a_k$$. Then $$S_{30} - a_{40}$$ is equal to

A

11310

B

11260

C

11290

D

11280

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NTA JEE Main 8th April 2023 Shift 2 - Question 65

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The absolute difference of the coefficients of $$x^{10}$$ and $$x^7$$ in the expansion of $$\left(2x^2 + \frac{1}{2x}\right)^{11}$$ is equal to

A

$$13^3 - 13$$

B

$$11^3 - 11$$

C

$$10^3 - 10$$

D

$$12^3 - 12$$

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NTA JEE Main 8th April 2023 Shift 2 - Question 66

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$$25^{190} - 19^{190} - 8^{190} + 2^{190}$$ is divisible by

A

neither 14 nor 34

B

14 but not by 34

C

34 but not by 14

D

both 14 and 34

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NTA JEE Main 8th April 2023 Shift 2 - Question 67

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The value of $$36(4\cos^2 9^\circ - 1)(4\cos^2 27^\circ - 1)(4\cos^2 81^\circ - 1)(4\cos^2 243^\circ - 1)$$ is

A

54

B

18

C

27

D

36

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NTA JEE Main 8th April 2023 Shift 2 - Question 68

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Let $$A(0, 1)$$, $$B(1, 1)$$ and $$C(1, 0)$$ be the mid-points of the sides of a triangle with incentre at the point $$D$$. If the focus of the parabola $$y^2 = 4ax$$ passing through $$D$$ is $$\left(\alpha + \beta\sqrt{2}, 0\right)$$, where $$\alpha$$ and $$\beta$$ are rational numbers, then $$\frac{\alpha}{\beta^2}$$ is equal to

A

8

B

12

C

6

D

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

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NTA JEE Main 8th April 2023 Shift 2 - Question 69

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Let $$O$$ be the origin and $$OP$$ and $$OQ$$ be the tangents to the circle $$x^2 + y^2 - 6x + 4y + 8 = 0$$ at the points $$P$$ and $$Q$$ on it. If the circumcircle of the triangle $$OPQ$$ passes through the point $$\left(\alpha, \frac{1}{2}\right)$$, then a value of $$\alpha$$ is

A

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

B

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

C

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

D

1

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NTA JEE Main 8th April 2023 Shift 2 - Question 70

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If $$\alpha > \beta > 0$$ are the roots of the equation $$ax^2 + bx + 1 = 0$$, and
$$\lim_{x \to \frac{1}{\alpha}} \left(\frac{1 - \cos(x^2 + bx + a)}{2(1 - \alpha x)^2}\right)^{\frac{1}{2}} = \frac{1}{k}\left(\frac{1}{\beta} - \frac{1}{\alpha}\right)$$, then $$k$$ is equal to

A

$$2\beta$$

B

$$\alpha$$

C

$$2\alpha$$

D

$$\beta$$

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NTA JEE Main 8th April 2023 Shift 2 - Question 71

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The negation of $$(p \wedge (-q)) \vee (-p)$$ is equivalent to

A

$$p \wedge (-q)$$

B

$$p \wedge q$$

C

$$p \vee (q \vee (-p))$$

D

$$p \wedge (q \wedge (-p))$$

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NTA JEE Main 8th April 2023 Shift 2 - Question 72

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Let the mean and variance of 12 observations be $$\frac{9}{2}$$ and 4 respectively. Later on, it was observed that two observations were considered as 9 and 10 instead of 7 and 14 respectively. If the correct variance is $$\frac{m}{n}$$, where $$m$$ and $$n$$ are coprime, then $$m + n$$ is equal to

A

315

B

316

C

314

D

317

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NTA JEE Main 8th April 2023 Shift 2 - Question 73

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Let $$A = \{1, 2, 3, 4, 5, 6, 7\}$$. Then the relation $$R = \{(x, y) \in A \times A : x + y = 7\}$$ is

A

an equivalence relation

B

symmetric but neither reflexive nor transitive

C

transitive but neither symmetric nor reflexive

D

reflexive but neither symmetric nor transitive

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NTA JEE Main 8th April 2023 Shift 2 - Question 74

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If $$A = \begin{bmatrix} 1 & 5 \\ \lambda & 10 \end{bmatrix}$$, $$A^{-1} = \alpha A + \beta I$$ and $$\alpha + \beta = -2$$, then $$4\alpha^2 + \beta^2 + \lambda^2$$ is equal to:

A

12

B

19

C

14

D

10

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NTA JEE Main 8th April 2023 Shift 2 - Question 75

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Let $$S$$ be the set of all values of $$\theta \in [-\pi, \pi]$$ for which the system of linear equations
$$x + y + \sqrt{3}z = 0$$
$$-x + (\tan\theta)y + \sqrt{7}z = 0$$
$$x + y + (\tan\theta)z = 0$$
has non-trivial solution. Then $$\frac{120}{\pi}\sum_{\theta \in S} \theta$$ is equal to

A

20

B

40

C

30

D

10

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NTA JEE Main 8th April 2023 Shift 2 - Question 76

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If domain of the function $$\log_e\left(\frac{6x^2 + 5x + 1}{2x - 1}\right) + \cos^{-1}\left(\frac{2x^2 - 3x + 4}{3x - 5}\right)$$ is $$(\alpha, \beta) \cup (\gamma, \delta)$$, then $$18(\alpha^2 + \beta^2 + \gamma^2 + \delta^2)$$ is equal to

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NTA JEE Main 8th April 2023 Shift 2 - Question 77

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The integral $$\int\left(\left(\frac{x}{2}\right)^x + \left(\frac{2}{x}\right)^x\right) \log_2 x \, dx$$ is equal to

A

$$\left(\frac{x}{2}\right)^x + \left(\frac{2}{x}\right)^x + C$$

B

$$\left(\frac{x}{2}\right)^x - \left(\frac{2}{x}\right)^x + C$$

C

$$\left(\frac{x}{2}\right)^x \log_2\left(\frac{x}{2}\right) + C$$

D

$$\left(\frac{x}{2}\right)^x \log_2\left(\frac{2}{x}\right) + C$$

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NTA JEE Main 8th April 2023 Shift 2 - Question 78

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Let the vectors $$\vec{u}_1 = \hat{i} + \hat{j} + a\hat{k}$$, $$\vec{u}_2 = \hat{i} + b\hat{j} + \hat{k}$$, and $$\vec{u}_3 = c\hat{i} + \hat{j} + \hat{k}$$ be coplanar. If the vectors $$\vec{v}_1 = (a+b)\hat{i} + c\hat{j} + c\hat{k}$$, $$\vec{v}_2 = a\hat{i} + (b+c)\hat{j} + a\hat{k}$$ and $$\vec{v}_3 = b\hat{i} + b\hat{j} + (c+a)\hat{k}$$ are also coplanar, then $$6(a + b + c)$$ is equal to

A

0

B

4

C

12

D

6

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NTA JEE Main 8th April 2023 Shift 2 - Question 79

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Let $$P$$ be the plane passing through the line $$\frac{x-1}{1} = \frac{y-2}{-3} = \frac{z+5}{7}$$ and the point $$(2, 4, -3)$$. If the image of the point $$(-1, 3, 4)$$ in the plane $$P$$ is $$(\alpha, \beta, \gamma)$$, then $$\alpha + \beta + \gamma$$ is equal to

A

10

B

12

C

9

D

11

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NTA JEE Main 8th April 2023 Shift 2 - Question 80

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If the probability that the random variable $$X$$ takes values $$x$$ is given by $$P(X = x) = k(x + 1)3^{-x}$$, $$x = 0, 1, 2, 3, \ldots$$, where $$k$$ is a constant, then $$P(X \geq 2)$$ is equal to

A

$$\frac{7}{27}$$

B

$$\frac{7}{18}$$

C

$$\frac{11}{18}$$

D

$$\frac{20}{27}$$

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NTA JEE Main 8th April 2023 Shift 2 - Question 81

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Let $$0 < z < y < x$$ be three real numbers such that $$\frac{1}{x}, \frac{1}{y}, \frac{1}{z}$$ are in an arithmetic progression and $$x, \sqrt{2}y, z$$ are in a geometric progression. If $$xy + yz + zx = \frac{3}{\sqrt{2}}xyz$$, then $$3(x + y + z)^2$$ is equal to _____.

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NTA JEE Main 8th April 2023 Shift 2 - Question 82

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The ordinates of the points $$P$$ and $$Q$$ on the parabola with focus $$(3, 0)$$ and directrix $$x = -3$$ are in the ratio 3 : 1. If $$R(\alpha, \beta)$$ is the point of intersection of the tangents to the parabola at $$P$$ and $$Q$$, then $$\frac{\beta^2}{\alpha}$$ is equal to _____.

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NTA JEE Main 8th April 2023 Shift 2 - Question 83

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Let $$R = \{a, b, c, d, e\}$$ and $$S = \{1, 2, 3, 4\}$$. Total number of onto functions $$f : R \to S$$ such that $$f(a) \neq 1$$, is equal to _____.

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NTA JEE Main 8th April 2023 Shift 2 - Question 84

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Let k and m be positive real numbers such that the function $$f(x) = \begin{cases} 3x^2 + k\sqrt{x + 1}, & 0 < x < 1 \\ mx^2 + k^2, & x \geq 1 \end{cases}$$ is differentiable for all $$x > 0$$. Then $$\frac{8f'(8)}{f'(\frac{1}{8})}$$ is equal to _____.

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NTA JEE Main 8th April 2023 Shift 2 - Question 85

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Let $$[t]$$ denote the greatest integer function. If $$\int_0^{2.4} [x^2] dx = \alpha + \beta\sqrt{2} + \gamma\sqrt{3} + \delta\sqrt{5}$$, then $$\alpha + \beta + \gamma + \delta$$ is equal to _____.

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NTA JEE Main 8th April 2023 Shift 2 - Question 86

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Let the area enclosed by the lines $$x + y = 2$$, $$y = 0$$, $$x = 0$$ and the curve $$f(x) = \min\left\{x^2 + \frac{3}{4}, 1 + [x]\right\}$$ where $$[x]$$ denotes the greatest integer $$\leq x$$, be $$A$$. Then the value of $$12A$$ is _____.

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NTA JEE Main 8th April 2023 Shift 2 - Question 87

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Let the solution curve $$x = x(y)$$, $$0 < y < \frac{\pi}{2}$$, of the differential equation $$(\log_e(\cos y))^2 \cos y \, dx - (1 + 3x \log_e(\cos y)) \sin y \, dy = 0$$ satisfy $$x\left(\frac{\pi}{3}\right) = \frac{1}{2\log_e 2}$$. If $$x\left(\frac{\pi}{6}\right) = \frac{1}{\log_e m - \log_e n}$$, where $$m$$ and $$n$$ are coprime, then $$mn$$ is equal to _____.

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NTA JEE Main 8th April 2023 Shift 2 - Question 88

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The area of the quadrilateral $$ABCD$$ with vertices $$A(2, 1, 1)$$, $$B(1, 2, 5)$$, $$C(-2, -3, 5)$$ and $$D(1, -6, -7)$$ is equal to _____.

A

48

B

$$8 \sqrt{38}$$

C

54

D

$$9 \sqrt{38}$$

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NTA JEE Main 8th April 2023 Shift 2 - Question 89

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For $$a, b \in \mathbb{Z}$$ and $$|a - b| \leq 10$$, let the angle between the plane $$P: ax + y - z = b$$ and the line $$L: x - 1 = a - y = z + 1$$ be $$\cos^{-1}\left(\frac{1}{3}\right)$$. If the distance of the point $$(6, -6, 4)$$ from the plane $$P$$ is $$3\sqrt{6}$$, then $$a^4 + b^2$$ is equal to _____.

A

32

B

85

C

25

D

48

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NTA JEE Main 8th April 2023 Shift 2 - Question 90

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Let $$P_1$$ be the plane $$3x - y - 7z = 11$$ and $$P_2$$ be the plane passing through the points $$(2, -1, 0)$$, $$(2, 0, -1)$$, and $$(5, 1, 1)$$. If the foot of the perpendicular drawn from the point $$(7, 4, -1)$$ on the line of intersection of the planes $$P_1$$ and $$P_2$$ is $$(\alpha, \beta, \gamma)$$, then $$\alpha + \beta + \gamma$$ is equal to _____.

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