NTA JEE Main 31st January 2023 Shift 1

NTA JEE Main 31st January 2023 Shift 1 - Question 1

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If $$R$$, $$X_L$$ and $$X_C$$ represent resistance, inductive reactance and capacitive reactance. Then which of the following is dimensionless:

A

$$R \; X_L \; X_C$$

B

$$\dfrac{R}{\sqrt{X_L X_C}}$$

C

$$\dfrac{R}{X_L X_C}$$

D

$$R \dfrac{X_L}{X_C}$$

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NTA JEE Main 31st January 2023 Shift 1 - Question 2

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The initial speed of a projectile fired from ground is $$u$$. At the highest point during its motion, the speed of projectile is $$\dfrac{\sqrt{3}}{2}u$$. The time of flight of the projectile is:

A

$$\dfrac{u}{2g}$$

B

$$\dfrac{u}{g}$$

C

$$\dfrac{2u}{g}$$

D

$$\dfrac{\sqrt{3}u}{g}$$

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NTA JEE Main 31st January 2023 Shift 1 - Question 3

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As shown in figure, a 70 kg garden roller is pushed with a force of $$\vec{F} = 200$$ N at an angle of 30° with horizontal. The normal reaction on the roller is (Given $$g = 10$$ m s$$^{-2}$$)

A

$$800\sqrt{2}$$ N

B

600 N

C

800 N

D

$$200\sqrt{3}$$ N

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NTA JEE Main 31st January 2023 Shift 1 - Question 4

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100 balls each of mass $$m$$ moving with speed $$v$$ simultaneously strike a wall normally and reflected back with same speed, in time $$t$$ s. The total force exerted by the balls on the wall is

A

$$\dfrac{100mv}{t}$$

B

$$\dfrac{200mv}{t}$$

C

$$200 \; mvt$$

D

$$\dfrac{mv}{100t}$$

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NTA JEE Main 31st January 2023 Shift 1 - Question 5

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At a certain depth $$d$$ below surface of earth, value of acceleration due to gravity becomes four times that of its value at a height $$3R$$ above earth surface. Where $$R$$ is Radius of earth (Take $$R = 6400$$ km). The depth $$d$$ is equal to

A

5260 km

B

640 km

C

2560 km

D

4800 km

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NTA JEE Main 31st January 2023 Shift 1 - Question 6

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Spherical insulating ball and a spherical metallic ball of same size and mass are dropped from the same height. Choose the correct statement out of the following {Assume negligible air friction}

A

Time taken by them to reach the earth's surface will be independent of the properties of their materials

B

Insulating ball will reach the earth's surface earlier than the metal ball

C

Both will reach the earth's surface simultaneously

D

Metal ball will reach the earth's surface earlier than the insulating ball

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NTA JEE Main 31st January 2023 Shift 1 - Question 7

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If 1000 droplets of water of surface tension 0.07 N m$$^{-1}$$, having same radius 1 mm each, combine to form a single drop. In the process the released surface energy is- (Take $$\pi = \dfrac{22}{7}$$)

A

$$7.92 \times 10^{-6}$$ J

B

$$7.92 \times 10^{-4}$$ J

C

$$9.68 \times 10^{-4}$$ J

D

$$8.8 \times 10^{-5}$$ J

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NTA JEE Main 31st January 2023 Shift 1 - Question 8

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The pressure of a gas changes linearly with volume from $$A$$ to $$B$$ as shown in figure. If no heat is supplied to or extracted from the gas then change in the internal energy of the gas will be

A

6 J

B

Zero

C

-4.5 J

D

4.5 J

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NTA JEE Main 31st January 2023 Shift 1 - Question 9

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The correct relation between $$\gamma = \dfrac{C_P}{C_V}$$ and temperature $$T$$ is:

A

$$\gamma \propto \dfrac{1}{\sqrt{T}}$$

B

$$\gamma \propto T^0$$

C

$$\gamma \propto \dfrac{1}{T}$$

D

$$\gamma \propto T$$

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NTA JEE Main 31st January 2023 Shift 1 - Question 10

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The maximum potential energy of a block executing simple harmonic motion is 25 J. A is amplitude of oscillation. At $$\dfrac{A}{2}$$, the kinetic energy of the block is

A

37.5 J

B

9.75 J

C

18.75 J

D

12.5 J

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NTA JEE Main 31st January 2023 Shift 1 - Question 11

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Which of the following correctly represents the variation of electric potential $$(V)$$ of a charged spherical conductor of radius $$(R)$$ with radial distance $$(r)$$ from the centre?

A

B

C

D

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NTA JEE Main 31st January 2023 Shift 1 - Question 12

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The drift velocity of electrons for a conductor connected in an electrical circuit is $$V_d$$. The conductor is now replaced by another conductor with same material and same length but double the area of cross-section. The applied voltage remains same. The new drift velocity of electrons will be

A

$$V_d$$

B

$$\dfrac{V_d}{2}$$

C

$$\dfrac{V_d}{4}$$

D

$$2V_d$$

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NTA JEE Main 31st January 2023 Shift 1 - Question 13

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A bar magnet with a magnetic moment 5.0 A m$$^2$$ is placed in parallel position relative to a magnetic field of 0.4 T. The amount of required work done in turning the magnet from parallel to antiparallel position relative to the field direction is ________.

A

4 J

B

1 J

C

2 J

D

Zero

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NTA JEE Main 31st January 2023 Shift 1 - Question 14

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A rod with circular cross-section area 2 cm$$^2$$ and length 40 cm is wound uniformly with 400 turns of an insulated wire. If a current of 0.4 A flows in the wire windings, the total magnetic flux produced inside the windings is $$4\pi \times 10^{-6}$$ Wb. The relative permeability of the rod is
(Given: Permeability of vacuum $$\mu_0 = 4\pi \times 10^{-7}$$ N A$$^{-2}$$)

A

12.5

B

$$\dfrac{32}{5}$$

C

125

D

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

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NTA JEE Main 31st January 2023 Shift 1 - Question 15

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Two polaroids $$A$$ and $$B$$ are placed in such a way that the pass-axis of polaroids are perpendicular to each other. Now, another polaroid $$C$$ is placed between $$A$$ and $$B$$ bisecting angle between them. If intensity of unpolarised light is $$I_0$$ then intensity of transmitted light after passing through polaroid $$B$$ will be:

A

$$\dfrac{I_0}{4}$$

B

$$\dfrac{I_0}{2}$$

C

$$\dfrac{I_0}{8}$$

D

Zero

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NTA JEE Main 31st January 2023 Shift 1 - Question 16

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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: The beam of electrons shows wave nature and exhibit interference and diffraction.
Reason R: Davisson Germer Experimentally verified the wave nature of electrons.
In the light of the above statements. Choose the most appropriate answer from the options given below:

A

A is correct but R is not correct

B

A is not correct but R is correct

C

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

D

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

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NTA JEE Main 31st January 2023 Shift 1 - Question 17

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If a source of electromagnetic radiation having power 15 kW produces $$10^{16}$$ photons per second, the radiation belongs to a part of spectrum is:
(Take Planck constant $$h = 6 \times 10^{-34}$$ J s)

A

Micro waves

B

Ultraviolet rays

C

Gamma rays

D

Radio waves

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NTA JEE Main 31st January 2023 Shift 1 - Question 18

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A free neutron decays into a proton but a free proton does not decay into neutron. This is because

A

neutron is an uncharged particle

B

proton is a charged particle

C

neutron is a composite particle made of a proton and an electron

D

neutron has larger rest mass than proton

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NTA JEE Main 31st January 2023 Shift 1 - Question 19

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The effect of increase in temperature on the number of electrons in conduction band $$(n_e)$$ and resistance of a semiconductor will be as:

A

Both $$n_e$$ and resistance decrease

B

Both $$n_e$$ and resistance increase

C

$$n_e$$ increases, resistance decreases

D

$$n_e$$ decreases, resistance increases

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NTA JEE Main 31st January 2023 Shift 1 - Question 20

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The amplitude of 15sin(1000$$\pi t$$) is modulated by 10sin(4$$\pi t$$) signal. The amplitude modulated signal contains frequency(ies) of
(A) 500 Hz
(B) 2 Hz
(C) 250 Hz
(D) 498 Hz
(E) 502 Hz
Choose the correct answer from the options given below:

A

A only

B

A, D and E only

C

B only

D

A and B only

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NTA JEE Main 31st January 2023 Shift 1 - Question 21

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The speed of a swimmer is 4 km h$$^{-1}$$ in still water. If the swimmer makes his strokes normal to the flow of river of width 1 km, he reaches a point 750 m down the stream on the opposite bank. The speed of the river water is ______ km h$$^{-1}$$.

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NTA JEE Main 31st January 2023 Shift 1 - Question 22

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A lift of mass $$M = 500$$ kg is descending with speed of 2 m s$$^{-1}$$. Its supporting cable begins to slip thus allowing it to fall with a constant acceleration of 2 m s$$^{-2}$$. The kinetic energy of the lift at the end of fall through to a distance of 6 m will be ______ kJ.

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NTA JEE Main 31st January 2023 Shift 1 - Question 23

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A solid sphere of mass 1 kg rolls without slipping on a plane surface. Its kinetic energy is $$7 \times 10^{-3}$$ J. The speed of the centre of mass of the sphere is ______ cm s$$^{-1}$$.

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NTA JEE Main 31st January 2023 Shift 1 - Question 24

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A thin rod having a length of 1 m and area of cross-section $$3 \times 10^{-6}$$ m$$^2$$ is suspended vertically from one end. The rod is cooled from 210°C to 160°C. After cooling, a mass $$M$$ is attached at the lower end of the rod such that the length of rod again becomes 1 m. Young's modulus and coefficient of linear expansion of the rod are $$2 \times 10^{11}$$ N m$$^{-2}$$ and $$2 \times 10^{-5}$$ K$$^{-1}$$, respectively. The value of $$M$$ is ______ kg. (Take $$g = 10$$ m s$$^{-2}$$)

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NTA JEE Main 31st January 2023 Shift 1 - Question 25

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In the figure given below, a block of mass $$M = 490$$ g placed on a frictionless table is connected with two springs having same spring constant ($$K = 2$$ N m$$^{-1}$$). If the block is horizontally displaced through $$X$$ m then the number of complete oscillations it will make in $$14\pi$$ seconds will be ______.

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NTA JEE Main 31st January 2023 Shift 1 - Question 26

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Expression for an electric field is given by $$\vec{E} = 4000 \; x^2 \; \hat{i}$$ V m$$^{-1}$$. The electric flux through the cube of side 20 cm when placed in electric field (as shown in the figure) is ______ V cm.

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NTA JEE Main 31st January 2023 Shift 1 - Question 27

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Two identical cells, when connected either in parallel or in series gives same current in an external resistance 5 $$\Omega$$. The internal resistance of each cell will be ______ $$\Omega$$.

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NTA JEE Main 31st January 2023 Shift 1 - Question 28

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An inductor of 0.5 mH, a capacitor of 20 $$\mu$$F and resistance of 20 $$\Omega$$ are connected in series with a 220 V ac source. If the current is in phase with the emf, the amplitude of current of the circuit is $$\sqrt{x}$$ A. The value of $$x$$ is-

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NTA JEE Main 31st January 2023 Shift 1 - Question 29

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In a medium the speed of light wave decreases to 0.2 times to its speed in free space. The ratio of relative permittivity to the refractive index of the medium is $$x$$:1. The value of $$x$$ is ______.
(Given speed of light in free space $$= 3 \times 10^8$$ m s$$^{-1}$$ and for the given medium $$\mu_r = 1$$)

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NTA JEE Main 31st January 2023 Shift 1 - Question 30

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For hydrogen atom, $$\lambda_1$$ and $$\lambda_2$$ are the wavelengths corresponding to the transitions 1 and 2 respectively as shown in figure. The ratio of $$\lambda_1$$ and $$\lambda_2$$ is $$\dfrac{x}{32}$$. The value of $$x$$ is ______.

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NTA JEE Main 31st January 2023 Shift 1 - Question 31

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Which transition in the hydrogen spectrum would have the same wavelength as the Balmer type transition from n = 4 to n = 2 of He$$^+$$ spectrum

A

n = 2 to n = 1

B

n = 1 to n = 3

C

n = 1 to n = 2

D

n = 3 to n = 4

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NTA JEE Main 31st January 2023 Shift 1 - Question 32

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The correct increasing order of the ionic radii is

A

$$Cl^- < Ca^{2+} < K^+ < S^{2-}$$

B

$$K^+ < S^{2-} < Ca^{2+} < Cl^-$$

C

$$S^{2-} < Cl^- < Ca^{2+} < K^+$$

D

$$Ca^{2+} < K^+ < Cl^- < S^{2-}$$

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NTA JEE Main 31st January 2023 Shift 1 - Question 33

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Match List I with List II
List I   List II
A. XeF$$_4$$   I. See-saw
B. SF$$_4$$   II. Square planar
C. NH$$_4^+$$   III. Bent T-shaped
D. BrF$$_3$$   IV. Tetrahedral
Choose the correct answer from the options given below:

A

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

B

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

C

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

D

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

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NTA JEE Main 31st January 2023 Shift 1 - Question 34

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$$H_2O_2$$ acts as a reducing agent in

A

$$2NaOCl + H_2O_2 \rightarrow 2NaCl + H_2O + O_2$$

B

$$2Fe^{2+} + 2H^+ + H_2O_2 \rightarrow 2Fe^{3+} + 2H_2O$$

C

$$Mn^{2+} + 2H_2O_2 \rightarrow MnO_2 + 2H_2O$$

D

$$Na_2S + 4H_2O_2 \rightarrow Na_2SO_4 + 4H_2O$$

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NTA JEE Main 31st January 2023 Shift 1 - Question 35

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Match items of column I and II

A

A - (iii), B - (iv), C - (ii), D - (i)

B

A - (ii), B - (iii), C - (iv), D - (i)

C

A - (i), B - (iii), C - (ii), D - (iv)

D

A - (ii), B - (iv), C - (i), D - (iii)

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NTA JEE Main 31st January 2023 Shift 1 - Question 36

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Choose the correct set of reagents for the following conversion
trans Ph - CH = CH - CH$$_3$$ $$\rightarrow$$ cis Ph - CH = CH - CH$$_3$$

A

Br$$_2$$, alc KOH, NaNH$$_2$$, NaLiqNH$$_3$$

B

Br$$_2$$, alc KOH, NaNH$$_2$$, H$$_2$$ Lindlar Catalyst

C

Br$$_2$$, aq KOH, NaNH$$_2$$, H$$_2$$ Lindlar Catalyst

D

Br$$_2$$, aq KOH, NaNH$$_2$$, NaLiqNH$$_3$$

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NTA JEE Main 31st January 2023 Shift 1 - Question 37

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Which one of the following statements is correct for electrolysis of brine solution?

A

Cl$$_2$$ is formed at cathode

B

O$$_2$$ is formed at cathode

C

H$$_2$$ is formed at anode

D

OH$$^-$$ is formed at cathode

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NTA JEE Main 31st January 2023 Shift 1 - Question 38

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Adding surfactants in non polar solvent, the micelles structure will look like
(a)

(b)

(c)

(d)

A

(b)

B

(c)

C

(a)

D

(d)

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NTA JEE Main 31st January 2023 Shift 1 - Question 39

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The methods NOT involved in concentration of ore are
(A) Liquation
(B) Leaching
(C) Electrolysis
(D) Hydraulic washing
(E) Froth floatation
Choose the correct answer from the options given below:

A

B, D and C only

B

C, D and E only

C

A and C only

D

B, D and E only

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NTA JEE Main 31st January 2023 Shift 1 - Question 40

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Identify X, Y and Z in the following reaction. (Equation not balanced)
$$ClO^. + NO_2 \rightarrow X \xrightarrow{H_2O} Y + Z$$

A

X = ClONO$$_2$$, Y = HOCl, Z = NO$$_2$$

B

X = ClNO$$_2$$, Y = HCl, Z = HNO$$_3$$

C

X = ClONO$$_2$$, Y = HOCl, Z = HNO$$_3$$

D

X = ClNO$$_3$$, Y = Cl$$_2$$, Z = NO$$_2$$

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NTA JEE Main 31st January 2023 Shift 1 - Question 41

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When Cu$$^{2+}$$ ion is treated with KI, a white precipitate, X appears in solution. The solution is titrated with sodium thiosulphate, the compound Y is formed. X and Y respectively are

A

X = Cu$$_2$$I$$_2$$, Y = Na$$_2$$S$$_4$$O$$_5$$

B

X = Cu$$_2$$I$$_2$$, Y = Na$$_2$$S$$_4$$O$$_6$$

C

X = CuI$$_2$$, Y = Na$$_2$$S$$_4$$O$$_3$$

D

X = CuI$$_2$$, Y = Na$$_2$$S$$_4$$O$$_6$$

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NTA JEE Main 31st January 2023 Shift 1 - Question 42

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The correct order of basicity of oxides of vanadium is

A

$$V_2O_3 > V_2O_4 > V_2O_5$$

B

$$V_2O_3 > V_2O_5 > V_2O_4$$

C

$$V_2O_5 > V_2O_4 > V_2O_3$$

D

$$V_2O_4 > V_2O_3 > V_2O_5$$

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NTA JEE Main 31st January 2023 Shift 1 - Question 43

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Nd$$^{2+}$$ = ______

A

$$4f^2 6s^2$$

B

$$4f^4$$

C

$$4f^3$$

D

$$4f^4 6s^2$$

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NTA JEE Main 31st January 2023 Shift 1 - Question 44

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Cobalt chloride when dissolved in water forms pink colored complex X which has octahedral geometry. This solution on treating with conc HCl forms deep blue complex, Y which has a Z geometry. X, Y and Z, respectively, are

A

X = CoH$$_2$$O$$_6^{2+}$$, Y = CoCl$$_4^{2-}$$, Z = Tetrahedral

B

X = CoH$$_2$$O$$_6^{2+}$$, Y = CoCl$$_6^{3-}$$, Z = Octahedral

C

X = CoH$$_2$$O$$_6^{3+}$$, Y = CoCl$$_6^{3-}$$, Z = Octahedral

D

X = CoH$$_2$$O$$_4$$Cl$$_2^+$$, Y = CoCl$$_4^{2-}$$, Z = Tetrahedral

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NTA JEE Main 31st January 2023 Shift 1 - Question 45

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The correct order of melting point of dichlorobenzenes is

A

B

C

D

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NTA JEE Main 31st January 2023 Shift 1 - Question 46

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An organic compound 'A' with empirical formula C$$_6$$H$$_6$$O gives sooty flame on burning. Its reaction with bromine solution in low polarity solvent results in high yield of B. B is

A

B

C

D

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NTA JEE Main 31st January 2023 Shift 1 - Question 47

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Propanal reacts with methanal according to the following sequence:

$$\mathrm{Propanal+Methanal\xrightarrow{(i)\ dil.\ NaOH}\xrightarrow{(ii)\ \Delta}\xrightarrow{(iii)\ NaCN}\xrightarrow{(iv)\ H_3O^+}B}$$

The molecular formula of product $$B$$ is:

$$C_5H_8O_3$$

Identify the correct statement regarding product $$B$$.

A

optically active and adds one mole of bromine

B

racemic mixture and is neutral

C

racemic mixture and gives a gas with saturated NaHCO$$_3$$ solution

D

optically active alcohol and is neutral

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NTA JEE Main 31st January 2023 Shift 1 - Question 48

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A

B

C

D

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NTA JEE Main 31st January 2023 Shift 1 - Question 49

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Which of the following artificial sweeteners has the highest sweetness value in comparison to cane sugar?

A

Aspartame

B

Sucralose

C

Alitame

D

Saccharin

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NTA JEE Main 31st January 2023 Shift 1 - Question 50

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A protein 'X' with molecular weight of 70,000 u, on hydrolysis gives amino acids. One of these amino acid is

A

B

C

D

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NTA JEE Main 31st January 2023 Shift 1 - Question 51

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On complete combustion, 0.492 g of an organic compound gave 0.792 g of CO$$_2$$. The % of carbon in the organic compound is (Nearest integer)

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NTA JEE Main 31st January 2023 Shift 1 - Question 52

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Zinc reacts with hydrochloric acid to give hydrogen and zinc chloride. The volume of hydrogen gas produced at STP from the reaction of 11.5 g of zinc with excess HCl is L (Nearest integer)
(Given: Molar mass of Zn is 65.4 g mol$$^{-1}$$ and Molar volume of H$$_2$$ at STP = 22.7 L)

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NTA JEE Main 31st January 2023 Shift 1 - Question 53

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The enthalpy change for the conversion of $$\dfrac{1}{2}$$Cl$$_2$$(g) to Cl$$^-$$(aq) is (-) ______ kJmol$$^{-1}$$ (Nearest integer)
Given: $$\Delta_{dis}H^0_{Cl_2(g)} = 240$$ kJmol$$^{-1}$$
$$\Delta_{eg}H^o_{Cl(g)} = -350$$ kJmol$$^{-1}$$
$$\Delta_{hyd}H^o_{Cl^-(g)} = -380$$ kJmol$$^{-1}$$

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NTA JEE Main 31st January 2023 Shift 1 - Question 54

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For reaction: $$SO_2(g) + \dfrac{1}{2}O_2(g) \rightleftharpoons SO_3(g)$$ $$K_P = 2 \times 10^{12}$$ at 27°C and 1 atm pressure. The $$K_c$$ for the same reaction is ______ $$\times 10^{13}$$. (Nearest integer)
(Given $$R = 0.082$$ L atm K$$^{-1}$$ mol$$^{-1}$$)

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NTA JEE Main 31st January 2023 Shift 1 - Question 55

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The total pressure of a mixture of non-reacting gases X(0.6 g) and Y(0.45 g) in a vessel is 740 mm of Hg. The partial pressure of the gas X is mm of Hg. (Nearest Integer)
(Given: molar mass X = 20 and Y = 45 g mol$$^{-1}$$)

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NTA JEE Main 31st January 2023 Shift 1 - Question 56

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At 27°C, a solution containing 2.5 g of solute in 250.0 mL of solution exerts an osmotic pressure of 400 Pa. The molar mass of the solute is g mol$$^{-1}$$ (Nearest integer)
(Given: R = 0.083 L bar$$^{-1}$$ mol$$^{-1}$$)

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NTA JEE Main 31st January 2023 Shift 1 - Question 57

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The logarithm of equilibrium constant for the reaction $$Pd^{2+} + 4Cl^- \rightleftharpoons PdCl_4^{2-}$$ is (Nearest integer)
Given: $$\dfrac{2.303RT}{F} = 0.06$$ V
$$Pd^{2+}_{(aq)} + 2e^- \rightleftharpoons Pd(s)$$ $$E^o = 0.83$$ V
$$PdCl_4^{2-}(aq) + 2e^- \rightleftharpoons Pd(s) + 4Cl^-(aq)$$ $$
$$E^o = 0.65$$ V

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NTA JEE Main 31st January 2023 Shift 1 - Question 58

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$$A \rightarrow B$$
The rate constants of the above reaction at 200 K and 300 K are 0.03 min$$^{-1}$$ and 0.05 min$$^{-1}$$ respectively. The activation energy for the reaction is J (Nearest integer)
(Given: ln 10 = 2.3, R = 8.3 J K$$^{-1}$$ mol$$^{-1}$$, log 5 = 0.70, log 3 = 0.48, log 2 = 0.30)

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NTA JEE Main 31st January 2023 Shift 1 - Question 59

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The oxidation state of phosphorus in hypophosphoric acid is

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NTA JEE Main 31st January 2023 Shift 1 - Question 60

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How many of the transformation given below would result in aromatic amines?

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NTA JEE Main 31st January 2023 Shift 1 - Question 61

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The number of real roots of the equation $$\sqrt{x^2 - 4x + 3} + \sqrt{x^2 - 9} = \sqrt{4x^2 - 14x + 6}$$, is:

A

0

B

1

C

3

D

2

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NTA JEE Main 31st January 2023 Shift 1 - Question 62

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For all $$z \in C$$ on the curve $$C_1$$: $$|z| = 4$$, let the locus of the point $$z + \dfrac{1}{z}$$ be the curve $$C_2$$. Then

A

the curves $$C_1$$ and $$C_2$$ intersect at 4 points

B

the curves $$C_1$$ lies inside $$C_2$$

C

the curves $$C_1$$ and $$C_2$$ intersect at 2 points

D

the curves $$C_2$$ lies inside $$C_1$$

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NTA JEE Main 31st January 2023 Shift 1 - Question 63

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If the sum and product of four positive consecutive terms of a G.P., are 126 and 1296, respectively, then the sum of common ratios of all such GPs is

A

7

B

$$\dfrac{9}{2}$$

C

3

D

14

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NTA JEE Main 31st January 2023 Shift 1 - Question 64

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Let a circle $$C_1$$ be obtained on rolling the circle $$x^2 + y^2 - 4x - 6y + 11 = 0$$ upwards 4 units on the tangent T to it at the point (3, 2). Let $$C_2$$ be the image of $$C_1$$ in T. Let $$A$$ and $$B$$ be the centers of circles $$C_1$$ and $$C_2$$ respectively, and $$M$$ and $$N$$ be respectively the feet of perpendiculars drawn from $$A$$ and $$B$$ on the x-axis. Then the area of the trapezium AMNB is:

A

$$2(2 + \sqrt{2})$$

B

$$4(1 + \sqrt{2})$$

C

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

D

$$2(1 + \sqrt{2})$$

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NTA JEE Main 31st January 2023 Shift 1 - Question 65

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If the maximum distance of normal to the ellipse $$\dfrac{x^2}{4} + \dfrac{y^2}{b^2} = 1, b < 2$$, from the origin is 1, then the eccentricity of the ellipse is:

A

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

B

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

C

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

D

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

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NTA JEE Main 31st January 2023 Shift 1 - Question 66

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Consider:
S1: $$p \Rightarrow q \lor p \land \sim q$$ is a tautology.
S2: $$\sim p \Rightarrow \sim q \land \sim p \lor q$$ is a contradiction.
Then

A

only S2 is correct

B

both S1 and S2 are correct

C

both S1 and S2 are wrong

D

only S1 is correct

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NTA JEE Main 31st January 2023 Shift 1 - Question 67

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Let $$R$$ be a relation on $$N \times N$$ defined by $$a, b R c, d$$ if and only if $$ad(b - c) = bc(a - d)$$. Then $$R$$ is

A

symmetric but neither reflexive nor transitive

B

transitive but neither reflexive nor symmetric

C

reflexive and symmetric but not transitive

D

symmetric and transitive but not reflexive

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NTA JEE Main 31st January 2023 Shift 1 - Question 68

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Let $$A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 4 & -1 \\ 0 & 12 & -3 \end{pmatrix}$$. Then the sum of the diagonal elements of the matrix $$(A + I)^{11}$$ is equal to:

A

6144

B

4094

C

4097

D

2050

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NTA JEE Main 31st January 2023 Shift 1 - Question 69

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For the system of linear equations
$$x + y + z = 6$$
$$\alpha x + \beta y + 7z = 3$$
$$x + 2y + 3z = 14$$
which of the following is NOT true?

A

If $$\alpha = \beta = 7$$, then the system has no solution

B

If $$\alpha = \beta$$ and $$\alpha \neq 7$$ then the system has a unique solution.

C

There is a unique point $$(\alpha, \beta)$$ on the line $$x + 2y + 18 = 0$$ for which the system has infinitely many solutions

D

For every point $$(\alpha, \beta) \neq (7, 7)$$ on the line $$x - 2y + 7 = 0$$, the system has infinitely many solutions.

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NTA JEE Main 31st January 2023 Shift 1 - Question 70

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If $$\sin^{-1}\dfrac{\alpha}{17} + \cos^{-1}\dfrac{4}{5} - \tan^{-1}\dfrac{77}{36} = 0$$, $$0 < \alpha < 13$$, then $$\sin^{-1}\sin\alpha + \cos^{-1}\cos\alpha$$ is equal to

A

$$\pi$$

B

16

C

0

D

$$16 - 5\pi$$

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NTA JEE Main 31st January 2023 Shift 1 - Question 71

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Let $$y = fx$$ represent a parabola with focus $$(-\dfrac{1}{2}, 0)$$ and directrix $$y = -\dfrac{1}{2}$$. Then
$$S = \{x \in \mathbb{R}: \tan^{-1}(\sqrt{fx}) + \sin^{-1}(\sqrt{fx+1}) = \dfrac{\pi}{2}\}$$:

A

contains exactly two elements

B

contains exactly one element

C

is an infinite set

D

is an empty set

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NTA JEE Main 31st January 2023 Shift 1 - Question 72

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If the domain of the function $$f(x) = \dfrac{x}{1 + x^2}$$, where $$x$$ is greatest integer $$\le x$$, is $$[2, 6)$$, then its range is

A

$$\dfrac{5}{26}, \dfrac{2}{5}, \dfrac{9}{29}, \dfrac{27}{109}, \dfrac{18}{89}, \dfrac{9}{53}$$

B

$$\dfrac{5}{26}, \dfrac{2}{5}$$

C

$$\dfrac{5}{37}, \dfrac{2}{5}, \dfrac{9}{29}, \dfrac{27}{109}, \dfrac{18}{89}, \dfrac{9}{53}$$

D

$$\dfrac{5}{37}, \dfrac{2}{5}$$

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NTA JEE Main 31st January 2023 Shift 1 - Question 73

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Let $$y = f(x) = \sin^3\left(\frac{\pi}{3} \cos\left(\frac{\pi}{3\sqrt{2}}(-4x^3 + 5x^2 + 1)^{\frac{3}{2}}\right)\right)$$. Then, at $$x = 1$$,

A

$$2y' + \sqrt{3}\pi^2 y = 0$$

B

$$2y' + 3\pi^2 y = 0$$

C

$$\sqrt{2}y' - 3\pi^2 y = 0$$

D

$$y' + 3\pi^2 y = 0$$

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NTA JEE Main 31st January 2023 Shift 1 - Question 74

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A wire of length 20 m is to be cut into two pieces. A piece of length $$\ell_1$$ is bent to make a square of area $$A_1$$ and the other piece of length $$\ell_2$$ is made into a circle of area $$A_2$$. If $$2A_1 + 3A_2$$ is minimum then $$\pi\ell_1 : \ell_2$$ is equal to:

A

6 : 1

B

3 : 1

C

1 : 6

D

4 : 1

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NTA JEE Main 31st January 2023 Shift 1 - Question 75

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Let $$\alpha \in (0, 1)$$ and $$\beta = \log_e(1 - \alpha)$$. Let $$P_n(x) = x + \dfrac{x^2}{2} + \dfrac{x^3}{3} + \ldots + \dfrac{x^n}{n}$$, $$x \in (0, 1)$$. Then the integral $$\int_0^{\alpha} \dfrac{t^{50}}{1-t} dt$$ is equal to

A

$$\beta - P_{50}\alpha$$

B

$$-\beta - P_{50}(\alpha)$$

C

$$-\beta + P_{50}(\alpha)$$

D

$$\beta + P_{50}\alpha$$

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NTA JEE Main 31st January 2023 Shift 1 - Question 76

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The value of $$\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \dfrac{2 + 3\sin x}{\sin x(1 + \cos x)} dx$$ is equal to

A

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

B

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

C

$$\dfrac{10}{3} - \sqrt{3} + \log_e \sqrt{3}$$

D

$$\dfrac{10}{3} - \sqrt{3} - \log_e \sqrt{3}$$

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NTA JEE Main 31st January 2023 Shift 1 - Question 77

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Let a differentiable function $$f$$ satisfy $$f(x) + \int_3^x \dfrac{f(t)}{t} dt = \sqrt{x+1}$$, $$x \ge 3$$. Then $$12f(8)$$ is equal to:

A

34

B

19

C

17

D

1

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NTA JEE Main 31st January 2023 Shift 1 - Question 78

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Let $$\vec{a} = 2\hat{i} + \hat{j} + \hat{k}$$, and $$\vec{b}$$ and $$\vec{c}$$ be two nonzero vectors such that $$\vec{a} + \vec{b} + \vec{c} = \vec{a} + \vec{b} - \vec{c}$$ and $$\vec{b} \cdot \vec{c} = 0$$. Consider the following two statements:
A: $$\vec{a} + \lambda\vec{c} \ge \vec{a}$$ for all $$\lambda \in \mathbb{R}$$.
B: $$\vec{a}$$ and $$\vec{c}$$ are always parallel.

A

only (B) is correct

B

neither (A) nor (B) is correct

C

only (A) is correct

D

both (A) and (B) are correct.

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NTA JEE Main 31st January 2023 Shift 1 - Question 79

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Let the shortest distance between the lines L: $$\dfrac{x-5}{-2} = \dfrac{y-\lambda}{0} = \dfrac{z+\lambda}{1}$$, $$\lambda \ge 0$$ and L$$_1$$: $$x+1 = y-1 = 4-z$$ be $$2\sqrt{6}$$. If $$(\alpha, \beta, \gamma)$$ lies on L, then which of the following is NOT possible?

A

$$\alpha + 2\gamma = 24$$

B

$$2\alpha + \gamma = 7$$

C

$$2\alpha - \gamma = 9$$

D

$$\alpha - 2\gamma = 19$$

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NTA JEE Main 31st January 2023 Shift 1 - Question 80

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A bag contains 6 balls. Two balls are drawn from it at random and both are found to be black. The probability that the bag contains at least 5 black balls is

A

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

B

$$\dfrac{2}{7}$$

C

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

D

$$\dfrac{5}{6}$$

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NTA JEE Main 31st January 2023 Shift 1 - Question 81

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Let 5 digit numbers be constructed using the digits 0, 2, 3, 4, 7, 9 with repetition allowed, and are arranged in ascending order with serial numbers. Then the serial number of the number 42923 is ______.

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NTA JEE Main 31st January 2023 Shift 1 - Question 82

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Let $$a_1, a_2, \ldots, a_n$$ be in A.P. If $$a_5 = 2a_7$$ and $$a_{11} = 18$$, then $$12\left(\dfrac{1}{\sqrt{a_{10}} + \sqrt{a_{11}}} + \dfrac{1}{\sqrt{a_{11}} + \sqrt{a_{12}}} + \ldots + \dfrac{1}{\sqrt{a_{17}} + \sqrt{a_{18}}}\right)$$ is equal to ______.

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NTA JEE Main 31st January 2023 Shift 1 - Question 83

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Number of 4-digit numbers that are less than or equal to 2800 and either divisible by 3 or by 11, is equal to ______.

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NTA JEE Main 31st January 2023 Shift 1 - Question 84

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Let $$\alpha > 0$$, be the smallest number such that the expansion of $$\left(x^{\frac{2}{3}}+\frac{2}{x^3}\right)^{30}$$ has a term $$\beta x^{-\alpha}$$, $$\beta \in N$$. Then $$\alpha$$ is equal to ______.

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NTA JEE Main 31st January 2023 Shift 1 - Question 85

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The remainder on dividing $$5^{99}$$ by 11 is ______.

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NTA JEE Main 31st January 2023 Shift 1 - Question 86

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If the variance of the frequency distribution

is 3, then $$\alpha$$ is equal to ______.

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NTA JEE Main 31st January 2023 Shift 1 - Question 87

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Let for $$x \in R$$, $$f(x) = \dfrac{x+x}{2}$$ and $$g(x) = \begin{cases} x, & x < 0 \\ x^2, & x \ge 0 \end{cases}$$. Then area bounded by the curve $$y = f \circ g(x)$$ and the lines $$y = 0, 2y - x = 15$$ is equal to ______.

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NTA JEE Main 31st January 2023 Shift 1 - Question 88

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Let $$\vec{a}$$ and $$\vec{b}$$ be two vectors such that $$\vec{a} = \sqrt{14}$$, $$\vec{b} = \sqrt{6}$$ and $$|\vec{a} \times \vec{b}| = \sqrt{48}$$. Then $$(\vec{a} \cdot \vec{b})^2$$ is equal to ______.

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NTA JEE Main 31st January 2023 Shift 1 - Question 89

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Let the line $$L$$: $$\dfrac{x-1}{2} = \dfrac{y+1}{-1} = \dfrac{z-3}{1}$$ intersect the plane $$2x + y + 3z = 16$$ at the point $$P$$. Let the point $$Q$$ be the foot of perpendicular from the point $$R(1, -1, -3)$$ on the line $$L$$. If $$\alpha$$ is the area of triangle $$PQR$$, then $$\alpha^2$$ is equal to ______.

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NTA JEE Main 31st January 2023 Shift 1 - Question 90

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Let $$\theta$$ be the angle between the planes $$P_1 = \vec{r} \cdot (\hat{i} + \hat{j} + 2\hat{k}) = 9$$ and $$P_2 = \vec{r} \cdot (2\hat{i} - \hat{j} + \hat{k}) = 15$$. Let L be the line that meets $$P_2$$ at the point (4, -2, 5) and makes angle $$\theta$$ with the normal of $$P_2$$. If $$\alpha$$ is the angle between L and $$P_2$$ then $$\tan^2\theta \cot^2\alpha$$ is equal to ______.

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