JEE Main 2023 31st January Morning Shift Question Paper Question Paper with Solutions | JEE Main Mock Test | JeeMate

Q 1PhysicsSection A • MCQcurrent electricity

+4 / -1

The drift velocity of electrons for a conductor connected in an electrical circuit is $\mathrm{V}_{\mathrm{d}}$ . The conductor in 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

\frac{V_{d}}{4}

B

\mathrm{V}_{\mathrm{d}}

C

2 \mathrm{~V}_{\mathrm{d}}

D

\frac{V_{d}}{2}

Q 2PhysicsSection B • Numericalalternating current

+4 / -1

An inductor of $0.5 ~\mathrm{mH}$ , a capacitor of $20 ~\mu \mathrm{F}$ and resistance of $20 ~\Omega$ are connected in series with a $220 \mathrm{~V}$ 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 ___________

Answer Format:Enter numerical value rounded to the nearest integer

Q 3PhysicsSection A • MCQelectrostatics

+4 / -1

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

A

B

C

D

Q 4PhysicsSection B • Numericalelectromagnetic waves

+4 / -1

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 $=3 \times 10^{8} \mathrm{~m} \mathrm{~s}^{-1}$ and for the given medium $\mu_{\mathrm{r}}=1$ )

Answer Format:Enter numerical value rounded to the nearest integer

Q 5PhysicsSection A • MCQdual nature of radiation

+4 / -1

If a source of electromagnetic radiation having power $15 \mathrm{~kW}$ produces $10^{16}$ photons per second, the radiation belongs to a part of spectrum is.

(Take Planck constant $h=6 \times 10^{-34} \mathrm{Js}$ )

A

Gamma rays

B

Radio waves

C

Micro waves

D

Ultraviolet rays

Q 6PhysicsSection A • MCQheat and thermodynamics

+4 / -1

The pressure of a gas changes linearly with volume from $\mathrm{A}$ to $\mathrm{B}$ 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

qImage-zeroth-and-first-law-of-thermodynamics-heat-and-thermodynamics-mechanics-Jeemate

A

6 J

B

4.5 J

C

zero

D

-

4.5 J

Q 7PhysicsSection B • Numericalproperties of matter

+4 / -1

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

(Take $\mathrm{g=10~m~s^{-2}}$ )

Answer Format:Enter numerical value rounded to the nearest integer

Q 8PhysicsSection B • Numericalsimple harmonic motion

+4 / -1

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

qImage-some-systems-of-executing-shm-simple-harmonic-motion-mechanics-Jeemate

Answer Format:Enter numerical value rounded to the nearest integer

Q 9PhysicsSection B • Numericalrotational motion

+4 / -1

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

Answer Format:Enter numerical value rounded to the nearest integer

Q 10PhysicsSection A • MCQgravitation

+4 / -1

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 $\mathrm{3 R}$ above earth surface. Where $\mathrm{R}$ is Radius of earth (Take $\mathrm{R}=6400 \mathrm{~km}$ ). The depth $\mathrm{d}$ is equal to

A

5260 km

B

2560 km

C

640 km

D

4800 km

Q 11PhysicsSection A • MCQmotion

+4 / -1

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

A

\frac{u}{g}

B

\frac{2u}{g}

C

\frac{u}{2g}

D

\frac{\sqrt3u}{g}

Q 12PhysicsSection B • Numericalelectrostatics

+4 / -1

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

qImage-electric-flux-and-gauss-law-electrostatics-electricity-Jeemate

Answer Format:Enter numerical value rounded to the nearest integer

Q 13PhysicsSection B • Numericalmotion

+4 / -1

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

The speed of the river water is ___________ $\mathrm{km} ~\mathrm{h}^{-1}$

Answer Format:Enter numerical value rounded to the nearest integer

Q 14PhysicsSection B • Numericalcurrent electricity

+4 / -1

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 ___________ $\Omega$ .

Answer Format:Enter numerical value rounded to the nearest integer

Q 15PhysicsSection A • MCQwave optics

+4 / -1

Two polaroide $\mathrm{A}$ and $\mathrm{B}$ are placed in such a way that the pass-axis of polaroids are perpendicular to each other. Now, another polaroid $\mathrm{C}$ is placed between $\mathrm{A}$ and $\mathrm{B}$ bisecting angle between them. If intensity of unpolarized light is $\mathrm{I}_{0}$ then intensity of transmitted light after passing through polaroid $\mathrm{B}$ will be:

A

\frac{I_{0}}{4}

B

\frac{I_{0}}{8}

C

Zero

D

\frac{I_{0}}{2}

Q 16PhysicsSection B • Numericalwork power and energy

+4 / -1

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

Answer Format:Enter numerical value rounded to the nearest integer

Q 17PhysicsSection A • MCQatoms and nuclei

+4 / -1

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

neutron has larger rest mass than proton

C

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

D

proton is a charged particle

Q 18PhysicsSection A • MCQheat and thermodynamics

+4 / -1

The correct relation between $\gamma = {{{c_p}} \over {{c_v}}}$ and temperature T is :

A

\gamma \propto T

B

\gamma \propto {1 \over {\sqrt T }}

C

\gamma \propto {1 \over T}

D

\gamma \propto T^\circ

Q 19PhysicsSection A • MCQmagnetics

+4 / -1

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

(Given : Permeability of vacuum $\mu_{0}=4 \pi \times 10^{-7} \mathrm{NA}^{-2}$ )

A

\frac{5}{16}

B

125

C

\frac{32}{5}

D

12.5

Q 20PhysicsSection B • Numericalatoms and nuclei

+4 / -1

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 $\frac{x}{32}$ . The value of $x$ is __________.

qImage-bohrs-model-and-hydrogen-spectrum-atoms-and-nuclei-modern-physics-Jeemate

Answer Format:Enter numerical value rounded to the nearest integer

Q 21PhysicsSection A • MCQcenter of mass

+4 / -1

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

A

\frac{200 m v}{t}

B

\frac{100 m v}{t}

C

\frac{m v}{100 t}

D

200 m v t

Q 22PhysicsSection A • MCQalternating current

+4 / -1

If $\mathrm{R}, \mathrm{X}_{\mathrm{L}}$ , and $\mathrm{X}_{\mathrm{C}}$ represent resistance, inductive reactance and capacitive reactance. Then which of the following is dimensionless :

A

\frac{R}{X_{L} X_{C}}

B

R X_{L} X_{C}

C

\frac{R}{\sqrt{X_{L} X_{C}}}

D

R \frac{X_{L}}{X_{C}}

Q 23PhysicsSection A • MCQsimple harmonic motion

+4 / -1

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

A

9.75 J

B

37.5 J

C

18.75 J

D

12.5 J

Q 24PhysicsSection A • MCQlaws of motion

+4 / -1

As shown in figure, a $70 \mathrm{~kg}$ garden roller is pushed with a force of $\vec{F}=200 \mathrm{~N}$ at an angle of $30^{\circ}$ with horizontal. The normal reaction on the roller is

(Given $\mathrm{g=10~m~s^{-2}}$ )

qImage-equilibrium-of-a-particle-laws-of-motion-mechanics-Jeemate

A

800 N

B

600 N

C

200

\sqrt3

N

D

800

\sqrt2

N

Q 25PhysicsSection A • MCQelectronic devices

+4 / -1

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

A

\mathrm{n}_{\mathrm{e}}

decreases, resistance increases

B

Both

\mathrm{n}_{\mathrm{e}}

and resistance increase

C

\mathrm{n}_{\mathrm{e}}

increases, resistance decreases

D

Both

\mathrm{n}_{\mathrm{e}}

and resistance decrease

Q 26PhysicsSection A • MCQcommunication systems

+4 / -1

The amplitude of $15 \sin (1000 \pi \mathrm{t})$ is modulated by $10 \sin (4 \pi \mathrm{t})$ signal. The amplitude modulated signal contains frequency (ies) of

A. $500 \mathrm{~Hz}$

B. $2 \mathrm{~Hz}$

C. $250 \mathrm{~Hz}$

D. $498 \mathrm{~Hz}$

E. $502 \mathrm{~Hz}$

Choose the correct answer from the options given below:

A

B only

B

A only

C

A, D and E only

D

A and B only

Q 27PhysicsSection A • MCQmagnetic properties of matter

+4 / -1

A bar magnet with a magnetic moment $5.0 \mathrm{Am}^{2}$ is placed in parallel position relative to a magnetic field of $0.4 \mathrm{~T}$ . The amount of required work done in turning the magnet from parallel to antiparallel position relative to the field direction is _____________.

A

zero

B

1 J

C

2 J

D

4 J

Q 28PhysicsSection A • MCQproperties of matter

+4 / -1

If 1000 droplets of water of surface tension $0.07 \mathrm{~N} / \mathrm{m}$ , having same radius $1 \mathrm{~mm}$ each, combine to from a single drop. In the process the released surface energy is :-

$\left( {\mathrm{Take}\,\pi = {{22} \over 7}} \right)$

A

7 .92 \times 10^{-4} \mathrm{~J}

B

7 .92 \times 10^{-6} \mathrm{~J}

C

8 .8 \times 10^{-5} \mathrm{~J}

D

9 .68 \times 10^{-4} \mathrm{~J}

Q 29PhysicsSection A • MCQmotion

+4 / -1

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

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

B

Both will reach the earth's surface simultaneously.

C

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

D

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

Q 30PhysicsSection A • MCQdual nature of radiation

+4 / -1

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 show 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 not correct but R is correct.

B

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

C

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

D

A is correct but R is not correct

Q 31ChemistrySection B • Numericalgaseous state

+4 / -1

The total pressure of a mixture of non-reacting gases $\mathrm{X}(0.6 \mathrm{~g})$ and $\mathrm{Y}(0.45 \mathrm{~g})$ in a vessel is $740 \mathrm{~mm}$ of $\mathrm{Hg}$ .

The partial pressure of the gas $\mathrm{X}$ is _______ $\mathrm{mm}$ of $\mathrm{Hg}$ . (Nearest Integer)

(Given : molar mass $\mathrm{X}=20$ and $\mathrm{Y}=45 \mathrm{~g} \mathrm{~mol}^{-1}$ )

Answer Format:Enter numerical value rounded to the nearest integer

Q 32ChemistrySection A • MCQd and f block elements

+4 / -1

The correct order of basicity of oxides of vanadium is :

A

\mathrm{V_2O_3 > V_2O_5 > V_2O_4}

B

\mathrm{V_2O_3 > V_2O_4 > V_2O_5}

C

\mathrm{V_2O_5 > V_2O_4 > V_2O_3}

D

\mathrm{V_2O_4 > V_2O_3 > V_2O_5}

Q 33ChemistrySection B • Numericalredox reactions

+4 / -1

The oxidation state of phosphorus in hypophosphoric acid is + _____________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 34ChemistrySection A • MCQchemical bonding and molecular structure

+4 / -1

Match List I with List II

	List I		List II
A.	$\mathrm{XeF_4}$	I.	See-saw
B.	$\mathrm{SF_4}$	II.	Square-planar
C.	$\mathrm{NH_{4}^{+}}$	III.	Bent T-shaped
D.	$\mathrm{BrF_3}$	IV.	Tetrahedral

Choose the correct answer from the options given below :

A

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

B

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

C

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

D

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

Q 35ChemistrySection B • Numericalsome basic concepts of chemistry

+4 / -1

On complete combustion, $0.492 \mathrm{~g}$ of an organic compound gave $0.792 \mathrm{~g}$ of $\mathrm{CO}_{2}$ . The % of carbon in the organic compound is ___________ (Nearest integer)

Answer Format:Enter numerical value rounded to the nearest integer

Q 36ChemistrySection A • MCQbasics of organic chemistry

+4 / -1

Match items of column I and II

	Column I (Mixture of compounds)		Column II (Separation Technique)
A.	$\mathrm{H_2O/CH_2Cl_2}$	i.	Crystalization
B.		ii.	Differential solvent extraction
C.	Kerosene / Naphthalene	iii.	Column chromatography
D.	$\mathrm{C_6H_{12}O_6/NaCl}$	iv.	Fractional Distillation

Correct match is

A

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

B

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

C

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

D

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

Q 37ChemistrySection A • MCQhaloalkanes and haloarenes

+4 / -1

The correct order of melting points of dichlorobenzenes is

A

B

C

D

Q 38ChemistrySection A • MCQd and f block elements

+4 / -1

$\mathrm{Nd^{2+}}$ = __________

A

\mathrm{4f^4 6s^2}

B

\mathrm{4f^3}

C

\mathrm{4f^4}

D

\mathrm{4f^2 6s^2}

Q 39ChemistrySection A • MCQsurface chemistry

+4 / -1

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

qImage-colloids-surface-chemistry-physical-chemistry-Jeemate

A

d

B

a

C

c

D

b

Q 40ChemistrySection B • Numericalcompounds containing nitrogen

+4 / -1

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

qImage-aromatic-amines-compounds-containing-nitrogen-organic-chemistry-Jeemate

Answer Format:Enter numerical value rounded to the nearest integer

Q 41ChemistrySection B • Numericalsome basic concepts of chemistry

+4 / -1

Zinc reacts with hydrochloric acid to give hydrogen and zinc chloride. The volume of hydrogen gas produced at STP from the reaction of $11.5 \mathrm{~g}$ of zinc with excess $\mathrm{HCl}$ is __________ L (Nearest integer)

(Given : Molar mass of $\mathrm{Zn}$ is $65.4 \mathrm{~g} \mathrm{~mol}^{-1}$ and Molar volume of $\mathrm{H}_{2}$ at $\mathrm{STP}=22.7 \mathrm{~L}$ )

Answer Format:Enter numerical value rounded to the nearest integer

Q 42ChemistrySection A • MCQalcohols phenols and ethers

+4 / -1

An organic compound 'A' with emperical formula $\mathrm{C}_{6} \mathrm{H}_{6} \mathrm{O}$ 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

Q 43ChemistrySection B • Numericalelectrochemistry

+4 / -1

The logarithm of equilibrium constant for the reaction $\mathrm{Pd}^{2+}+4 \mathrm{Cl}^{-} \rightleftharpoons \mathrm{PdCl}_{4}^{2-}$ is ___________ (Nearest integer)

Given : $\frac{2.303 R \mathrm{~T}}{\mathrm{~F}}=0.06 \mathrm{~V}$

$\mathrm{Pd}_{(\mathrm{aq})}^{2+}+2 \mathrm{e}^{-} \rightleftharpoons \mathrm{Pd}(\mathrm{s}) \quad \mathrm{E}^{\ominus}=0.83 \mathrm{~V}$

$\begin{aligned} & \mathrm{PdCl}_{4}^{2-}(\mathrm{aq})+2 \mathrm{e}^{-} \rightleftharpoons \mathrm{Pd}(\mathrm{s})+4 \mathrm{Cl}^{-}(\mathrm{aq}) \mathrm{E}^{\ominus}=0.65 \mathrm{~V} \end{aligned}$

Answer Format:Enter numerical value rounded to the nearest integer

Q 44ChemistrySection A • MCQd and f block elements

+4 / -1

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

A

$\mathrm{X=CuI_2}$	$\mathrm{Y=Na_2S_4O_6}$

B

$\mathrm{X=Cu_2I_2}$	$\mathrm{Y=Na_2S_4O_6}$

C

$\mathrm{X=CuI_2}$	$\mathrm{Y=Na_2S_2O_3}$

D

$\mathrm{X=Cu_2I_2}$	$\mathrm{Y=Na_2S_4O_5}$

Q 45ChemistrySection A • MCQcoordination compounds

+4 / -1

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

A

\mathrm{X}=\left[\mathrm{Co}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{2+}, \mathrm{Y}=\left[\mathrm{CoCl}_{4}\right]^{2-}, \mathrm{Z}=

Tetrahedral

B

\mathrm{X}=\left[\mathrm{Co}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{2+}, \mathrm{Y}=\left[\mathrm{CoCl}_{6}\right]^{3-}, \mathrm{Z}=

Octahedral

C

\mathrm{X}=\left[\mathrm{Co}\left(\mathrm{H}_{2} \mathrm{O}\right)_{4} \mathrm{Cl}_{2}\right]^{+}, \mathrm{Y}=\left[\mathrm{CoCl}_{4}\right]^{2-}, \mathrm{Z}=

Tetrahedral

D

\mathrm{X}=\left[\mathrm{Co}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{3+}, \mathrm{Y}=\left[\mathrm{CoCl}_{6}\right]^{3-}, \mathrm{Z}=

Octahedral

Q 46ChemistrySection A • MCQhydrogen

+4 / -1

$\mathrm{H}_{2} \mathrm{O}_{2}$ acts as a reducing agent in :

A

\mathrm{Mn}^{2+}+2 \mathrm{H}_{2} \mathrm{O}_{2} \rightarrow \mathrm{MnO}_{2}+2 \mathrm{H}_{2} \mathrm{O}

B

2 \mathrm{Fe}^{2+}+2 \mathrm{H}^{+}+\mathrm{H}_{2} \mathrm{O}_{2} \rightarrow 2 \mathrm{Fe}^{3+}+2 \mathrm{H}_{2} \mathrm{O}

C

\mathrm{Na}_{2} \mathrm{S}+4 \mathrm{H}_{2} \mathrm{O}_{2} \rightarrow \mathrm{Na}_{2} \mathrm{SO}_{4}+4 \mathrm{H}_{2} \mathrm{O}

D

2 \mathrm{NaOCl}+\mathrm{H}_{2} \mathrm{O}_{2} \rightarrow 2 \mathrm{NaCl}+\mathrm{H}_{2} \mathrm{O}+\mathrm{O}_{2}

Q 47ChemistrySection A • MCQhydrocarbons

+4 / -1

Choose the correct set of reagents for the following conversion.

trans $\left(\mathrm{Ph}-\mathrm{CH}=\mathrm{CH}-\mathrm{CH}_{3}\right) \rightarrow \operatorname{cis}\left(\mathrm{Ph}-\mathrm{CH}=\mathrm{CH}-\mathrm{CH}_{3}\right)$

A

\mathrm{Br_2,alc\cdot KOH,NaNH_2,H_2}

Lindlar Catalyst

B

\mathrm{Br_2,aq\cdot KOH,NaNH_2,Na}

(Liq NH

_3

)

C

\mathrm{Br_2,alc\cdot KOH,NaNH_2,Na}

(Liq NH

_3

)

D

\mathrm{Br_2,aq\cdot KOH,NaNH_2,H_2}

Lindlar Catalyst

Q 48ChemistrySection B • Numericalchemical equilibrium

+4 / -1

For reaction : $\mathrm{SO}_{2}(\mathrm{~g})+\frac{1}{2} \mathrm{O}_{2}(\mathrm{~g}) \rightleftharpoons \mathrm{SO}_{3}(\mathrm{~g})$

$\mathrm{K}_{\mathrm{p}}=2 \times 10^{12}$ at $27^{\circ} \mathrm{C}$ and $1 \mathrm{~atm}$ pressure. The $\mathrm{K}_{\mathrm{c}}$ for the same reaction is ____________ $\times 10^{13}$ . (Nearest integer)

(Given $\mathrm{R}=0.082 \mathrm{~L} \mathrm{~atm} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$ )

Answer Format:Enter numerical value rounded to the nearest integer

Q 49ChemistrySection A • MCQelectrochemistry

+4 / -1

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

A

\mathrm{O}_{2}

is formed at cathode

B

\mathrm{H}_{2}

is formed at anode

C

\mathrm{Cl}_{2}

is formed at cathode

D

\mathrm{OH}^{-}

is formed at cathode

Q 50ChemistrySection A • MCQcompounds containing nitrogen

+4 / -1

qImage-aromatic-amines-compounds-containing-nitrogen-organic-chemistry-Jeemate

Consider the above reaction and identify the product B.

A

B

C

D

Q 51ChemistrySection B • Numericalchemical kinetics and nuclear chemistry

+4 / -1

A $\to$ 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 : $\mathrm{ln10=2.3}$

$\mathrm{R=8.3~J~K^{-1}~mol^{-1}}$

$\mathrm{\log5=0.70}$

$\mathrm{\log3=0.48}$

$\mathrm{\log2=0.30}$ )

Answer Format:Enter numerical value rounded to the nearest integer

Q 52ChemistrySection A • MCQenvironmental chemistry

+4 / -1

Identify X, Y and Z in the following reaction. (Equation not balanced)

qImage-air-pollution-environmental-chemistry-inorganic-chemistry-Jeemate

A

\mathrm{X}=\mathrm{ClONO}_{2}, \mathrm{Y}=\mathrm{HOCl}, \mathrm{Z}=\mathrm{NO}_{2}

B

\mathrm{X}=\mathrm{ClNO}_{3}, \mathrm{Y}=\mathrm{Cl}_{2}, \mathrm{Z}=\mathrm{NO}_{2}

C

\mathrm{X}=\mathrm{ClONO}_{2}, \mathrm{Y}=\mathrm{HOCl}, \mathrm{Z}=\mathrm{HNO}_{3}

D

\mathrm{X}=\mathrm{ClNO}_{2}, \mathrm{Y}=\mathrm{HCl}, \mathrm{Z}=\mathrm{HNO}_{3}

Q 53ChemistrySection B • Numericalthermodynamics

+4 / -1

The enthalpy change for the conversion of $\frac{1}{2} \mathrm{Cl}_{2}(\mathrm{~g})$ to $\mathrm{Cl}^{-}$ (aq) is ( $-$ ) ___________ $\mathrm{kJ} \mathrm{mol}^{-1}$ (Nearest integer)

Given : $\Delta_{\mathrm{dis}} \mathrm{H}_{\mathrm{Cl}_{2(\mathrm{~g})}^{\theta}}^{\ominus}=240 \mathrm{~kJ} \mathrm{~mol}^{-1}, \Delta_{\mathrm{eg}} \mathrm{H}_{\mathrm{Cl_{(g)}}}^{\ominus}=-350 \mathrm{~kJ} \mathrm{~mol}^{-1}$ ,

${\mathrm{\Delta _{hyd}}H_{Cl_{(g)}^ - }^\Theta = - 380}$ $\mathrm{kJ~mol^{-1}}$

Answer Format:Enter numerical value rounded to the nearest integer

Q 54ChemistrySection A • MCQperiodic classification of elements

+4 / -1

The correct increasing order of the ionic radii is

A

\mathrm{Cl^- < Ca^{2+} < K^+ < S^{2-}}

B

\mathrm{K^+ < S^{2-} < Ca^{2+} < Cl^-}

C

\mathrm{Ca^{2+} < K^+ < Cl^- < S^{2-}}

D

\mathrm{S^{2-} < Cl^- < Ca^{2+} < K^+}

Q 55ChemistrySection A • MCQaldehydes ketones and carboxylic acids

+4 / -1

Consider the following reaction

qImage-chemical-reactions-for-aldehydes-and-ketones-aldehydes-ketones-and-carboxylic-acids-organic-chemistry-Jeemate

The correct statement for product B is. It is

A

racemic mixture and is neutral

B

optically active and adds one mole of bromine

C

racemic mixture and gives a gas with saturated

\mathrm{NaHCO}_{3}

solution

D

optically active alcohol and is neutral

Q 56ChemistrySection A • MCQbiomolecules

+4 / -1

A protein ' $\mathrm{X}$ ' with molecular weight of $70,000 \mathrm{~u}$ , on hydrolysis gives amino acids. One of these amino acid is

A

B

C

D

Q 57ChemistrySection B • Numericalsolutions

+4 / -1

At $27^{\circ} \mathrm{C}$ , a solution containing $2.5 \mathrm{~g}$ of solute in $250.0 \mathrm{~mL}$ of solution exerts an osmotic pressure of $400 \mathrm{~Pa}$ . The molar mass of the solute is ___________ $\mathrm{g} \mathrm{~mol}^{-1}$ (Nearest integer)

(Given : $\mathrm{R}=0.083 \mathrm{~L} \mathrm{~bar} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$ )

Answer Format:Enter numerical value rounded to the nearest integer

Q 58ChemistrySection A • MCQisolation of elements

+4 / -1

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

A and C only

B

C, D and E only

C

B, D and C only

D

B, D and E only

Q 59ChemistrySection A • MCQchemistry in everyday life

+4 / -1

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

A

Alitame

B

Sucralose

C

Saccharin

D

Aspartame

Q 60ChemistrySection A • MCQatomic structure

+4 / -1

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

A

\mathrm{n}=3

to

\mathrm{n}=4

B

\mathrm{n}=2

to

\mathrm{n}=1

C

\mathrm{n}=1

to

\mathrm{n}=2

D

\mathrm{n}=1

to

\mathrm{n}=3

Q 61MathematicsSection A • MCQquadratic equation and inequalities

+4 / -1

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

A

0

B

1

C

3

D

2

Q 62MathematicsSection B • Numericalbinomial theorem

+4 / -1

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

Answer Format:Enter numerical value rounded to the nearest integer

Q 63MathematicsSection A • MCQprobability

+4 / -1

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

\frac{3}{7}

B

\frac{5}{6}

C

\frac{5}{7}

D

\frac{2}{7}

Q 64MathematicsSection A • MCQsets and relations

+4 / -1

Let $\mathrm{R}$ be a relation on $\mathrm{N} \times \mathbb{N}$ defined by $(a, b) ~\mathrm{R}~(c, d)$ if and only if $a d(b-c)=b c(a-d)$ . Then $\mathrm{R}$ is

A

symmetric and transitive but not reflexive

B

reflexive and symmetric but not transitive

C

transitive but neither reflexive nor symmetric

D

symmetric but neither reflexive nor transitive

Q 65MathematicsSection B • Numericalpermutations and combinations

+4 / -1

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

Answer Format:Enter numerical value rounded to the nearest integer

Q 66MathematicsSection A • MCQmatrices and determinants

+4 / -1

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

A

4094

B

2050

C

6144

D

4097

Q 67MathematicsSection A • MCQdefinite integration

+4 / -1

Let $\alpha \in (0,1)$ and $\beta = {\log _e}(1 - \alpha )$ . Let ${P_n}(x) = x + {{{x^2}} \over 2} + {{{x^3}} \over 3}\, + \,...\, + \,{{{x^n}} \over n},x \in (0,1)$ . Then the integral $\int\limits_0^\alpha {{{{t^{50}}} \over {1 - t}}dt}$ is equal to

A

- \left( {\beta + {P_{50}}\left( \alpha \right)} \right)

B

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

C

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

D

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

Q 68MathematicsSection A • MCQinverse trigonometric functions

+4 / -1

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

A

16

B

\pi

C

16

-

5

\pi

D

0

Q 69MathematicsSection A • MCQdefinite integration

+4 / -1

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

A

\frac{10}{3}-\sqrt{3}+\log _{e} \sqrt{3}

B

\frac{7}{2}-\sqrt{3}-\log _{e} \sqrt{3}

C

\frac{10}{3}-\sqrt{3}-\log _{e} \sqrt{3}

D

-2+3\sqrt{3}+\log _{e} \sqrt{3}

Official Correct Answer:Option (A)

Step-by-Step Solution:

Let I =

\int_\limits{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{(2+3 \sin x)}{\sin x(1+\cos x)} d x

=\int\limits_{\pi / 3}^{\pi / 2} \frac{2}{\sin x(1+\cos x)} d x+\int\limits_{\pi / 3}^{\pi / 2} \frac{3}{1+\cos x} d x

\begin{aligned} =\int\limits_{\pi / 3}^{\pi / 2} \frac{2(1-\cos x)}{\sin x\left(1-\cos ^2 x\right)} & d x +\int\limits_{\pi / 3}^{\pi / 2} \frac{3}{1+\cos x} d x \end{aligned}

=\int\limits_{\pi / 3}^{\pi / 2} \frac{2(1-\cos x)}{\sin ^3 x} d x+\int\limits_{\pi / 3}^{\pi / 2} \frac{3}{2 \cos ^2 \frac{x}{2}} d x

\begin{array}{r} =\int\limits_{\pi / 3}^{\pi / 2} \frac{2}{\sin ^3 x} d x-\int\limits_{\pi / 3}^{\pi / 2} 2 \cot x \operatorname{cosec}^2 x d x +\int\limits_{\pi / 3}^{\pi / 2} \frac{3}{2} \sec ^2 \frac{x}{2} d x \end{array}

=2 I_1-2 I_2+\frac{3}{2} I_3

Now,

\begin{aligned} I_1 & =\int\limits_{\pi / 3}^{\pi / 2} \operatorname{cosec}^3 x d x \\ & =\int\limits_{\pi / 3}^{\pi / 2} \sqrt{1+\cot ^2 x} \operatorname{cosec}^2 x d x \end{aligned}

Put

\cot x=t \Rightarrow-\operatorname{cosec}^2 x d x=d t

When, $x=\frac{\pi}{3} \Rightarrow t=\frac{1}{\sqrt{3}} \text { and } x=\frac{\pi}{2} \Rightarrow t=0$

\begin{aligned} \therefore & I_1=-\int\limits_{\frac{1}{\sqrt{3}}}^0 \sqrt{1+t^2} d t \\ = & \int\limits_0^{\frac{1}{\sqrt{3}}} \sqrt{1+t^2} d t \\ = & {\left[\frac{t}{2} \sqrt{1+t^2}+\frac{1}{2} \log \left[t+\sqrt{1+t^2}\right]\right]_0^{\frac{1}{\sqrt{3}}} } \\ = & \frac{1}{3}+\frac{1}{2} \log \sqrt{3} \end{aligned}

I_2=\int\limits_{\pi / 3}^{\pi / 2} \cot x \operatorname{cosec}^2 x d x

Put

\cot x=t \Rightarrow \operatorname{cosec}^2 x d x=-d t

When,

x=\frac{\pi}{3}

\Rightarrow t=\frac{1}{\sqrt{3}}

and

x=\frac{\pi}{2} \Rightarrow t=0

\therefore \begin{aligned} I_2 & =-\int\limits_{\frac{1}{\sqrt{3}}}^0 t d t \\ & =\int\limits_0^{\frac{1}{\sqrt{3}} t} t d t=\left[\frac{t^2}{2}\right]_0^{\frac{1}{\sqrt{3}}}=\frac{1}{6} \end{aligned}

\begin{aligned} & I_3=\int\limits_{\frac{\pi}{3}}^{\frac{\pi}{2}} \sec ^2 \frac{x}{2} d x \\ = & 2\left(\tan \frac{\pi}{4}-\tan \frac{\pi}{6}\right)=2\left(1-\frac{1}{\sqrt{3}}\right) \end{aligned}

\begin{aligned} & \therefore \quad I=2 I_1-2 I_2+\frac{3}{2} I_3 \\ & =2\left(\frac{1}{3}+\frac{1}{2} \log \sqrt{3}\right)-2\left(\frac{1}{6}\right) +\frac{3}{2}\left(2\left(1-\frac{1}{\sqrt{3}}\right)\right) \end{aligned}

\begin{aligned} & =\frac{2}{3}+\log \sqrt{3}-\frac{1}{3}+3-\sqrt{3} \\ & =\frac{10}{3}-\sqrt{3}+\log _e \sqrt{3} \end{aligned}

Q 70MathematicsSection B • Numerical3d geometry

+4 / -1

Let the line $L: \frac{x-1}{2}=\frac{y+1}{-1}=\frac{z-3}{1}$ intersect the plane $2 x+y+3 z=16$ 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 $P Q R$ , then $\alpha^{2}$ is equal to __________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 71MathematicsSection A • MCQparabola

+4 / -1

Let $\mathrm{y}=f(x)$ represent a parabola with focus $\left(-\frac{1}{2}, 0\right)$ and directrix $y=-\frac{1}{2}$ . Then

$S=\left\{x \in \mathbb{R}: \tan ^{-1}(\sqrt{f(x)})+\sin ^{-1}(\sqrt{f(x)+1})=\frac{\pi}{2}\right\}$ :

A

is an empty set

B

contains exactly one element

C

contains exactly two elements

D

is an infinite set

Q 72MathematicsSection B • Numericalsequences and series

+4 / -1

Let $a_{1}, a_{2}, \ldots, a_{n}$ be in A.P. If $a_{5}=2 a_{7}$ and $a_{11}=18$ , then

$12\left(\frac{1}{\sqrt{a_{10}}+\sqrt{a_{11}}}+\frac{1}{\sqrt{a_{11}}+\sqrt{a_{12}}}+\ldots+\frac{1}{\sqrt{a_{17}}+\sqrt{a_{18}}}\right)$ is equal to ____________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 73MathematicsSection B • Numericalstatistics

+4 / -1

If the variance of the frequency distribution

$x_i$	2	3	4	5	6	7	8
Frequency $f_i$	3	6	16	$\alpha$	9	5	6

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

Answer Format:Enter numerical value rounded to the nearest integer

Q 74MathematicsSection A • MCQ3d geometry

+4 / -1

Let the shortest distance between the lines

$L: \frac{x-5}{-2}=\frac{y-\lambda}{0}=\frac{z+\lambda}{1}, \lambda \geq 0$ 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

\alpha-2 \gamma=19

D

2 \alpha-\gamma=9

Q 75MathematicsSection B • Numericalbinomial theorem

+4 / -1

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 $\beta x^{-\alpha}, \beta \in \mathbb{N}$ . Then $\alpha$ is equal to ___________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 76MathematicsSection A • MCQfunctions

+4 / -1

If the domain of the function

f(x)=\frac{[x]}{1+x^{2}}

, where

[x]

is greatest integer

\leq x

, is

[2,6)

, then its range is

A

\left(\frac{5}{37}, \frac{2}{5}\right]-\left\{\frac{9}{29}, \frac{27}{109}, \frac{18}{89}, \frac{9}{53}\right\}

B

\left(\frac{5}{37}, \frac{2}{5}\right]

C

\left(\frac{5}{26}, \frac{2}{5}\right]

D

\left(\frac{5}{26}, \frac{2}{5}\right]-\left\{\frac{9}{29}, \frac{27}{109}, \frac{18}{89}, \frac{9}{53}\right\}

Q 77MathematicsSection B • Numericalpermutations and combinations

+4 / -1

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 __________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 78MathematicsSection A • MCQmatrices and determinants

+4 / -1

For the system of linear equations

$x+y+z=6$

$\alpha x+\beta y+7 z=3$

$x+2 y+3 z=14$

which of the following is NOT true ?

A

If

\alpha=\beta=7

, then the system has no solution

B

For every point

(\alpha, \beta) \neq(7,7)

on the line

x-2 y+7=0

, the system has infinitely many solutions

C

There is a unique point

(\alpha, \beta)

on the line

x+2 y+18=0

for which the system has infinitely many solutions

D

If

\alpha=\beta

and

\alpha \neq 7

, then the system has a unique solution

Q 79MathematicsSection A • MCQvector algebra

+4 / -1

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) $|\vec{a}+\lambda \vec{c}| \geq|\vec{a}|$ for all $\lambda \in \mathbb{R}$ .

(B) $\vec{a}$ and $\vec{c}$ are always parallel.

Then,

A

only (B) is correct

B

both (A) and (B) are correct

C

only (A) is correct

D

neither (A) nor (B) is correct

Q 80MathematicsSection A • MCQmathematical reasoning

+4 / -1

$(\mathrm{S} 1)~(p \Rightarrow q) \vee(p \wedge(\sim q))$ is a tautology

$(\mathrm{S} 2)~((\sim p) \Rightarrow(\sim q)) \wedge((\sim p) \vee q)$ is a contradiction.

Then

A

only (S2) is correct

B

both (S1) and (S2) are correct

C

only (S1) is correct

D

both (S1) and (S2) are wrong

Q 81MathematicsSection A • MCQsequences and series

+4 / -1

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

14

C

3

D

\frac{9}{2}

Q 82MathematicsSection A • MCQcircle

+4 / -1

Let a circle $C_{1}$ be obtained on rolling the circle $x^{2}+y^{2}-4 x-6 y+11=0$ upwards 4 units on the tangent $\mathrm{T}$ to it at the point $(3,2)$ . Let $C_{2}$ be the image of $C_{1}$ in $\mathrm{T}$ . 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\left( {2 + \sqrt 2 } \right)

B

4\left( {1 + \sqrt 2 } \right)

C

3 + 2\sqrt 2

D

2\left( {1 + \sqrt 2 } \right)

Q 83MathematicsSection A • MCQdifferential equations

+4 / -1

Let a differentiable function $f$ satisfy $f(x)+\int_\limits{3}^{x} \frac{f(t)}{t} d t=\sqrt{x+1}, x \geq 3$ . Then $12 f(8)$ is equal to :

A

19

B

34

C

17

D

1

Q 84MathematicsSection B • Numerical3d geometry

+4 / -1

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 $\mathrm{L}$ be the line that meets $P_{2}$ at the point $(4,-2,5)$ and makes an angle $\theta$ with the normal of $P_{2}$ . If $\alpha$ is the angle between $\mathrm{L}$ and $P_{2}$ , then $\left(\tan ^{2} \theta\right)\left(\cot ^{2} \alpha\right)$ is equal to ____________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 85MathematicsSection A • MCQellipse

+4 / -1

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

A

\frac{\sqrt{3}}{4}

B

\frac{1}{2}

C

\frac{1}{\sqrt{2}}

D

\frac{\sqrt{3}}{2}

Q 86MathematicsSection B • Numericalarea under the curves

+4 / -1

Let for $x \in \mathbb{R}$ ,

$f(x)=\frac{x+|x|}{2} \text { and } g(x)=\left\{\begin{array}{cc} x, & x<0 \\ x^{2}, & x \geq 0 \end{array}\right. \text {. }$

Then area bounded by the curve $y=(f \circ g)(x)$ and the lines $y=0,2 y-x=15$ is equal to __________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 87MathematicsSection A • MCQdifferentiation

+4 / -1

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

A

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

B

y^{\prime}+3 \pi^{2} y=0

C

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

D

2 y^{\prime}+3 \pi^{2} y=0

Q 88MathematicsSection A • MCQapplication of derivatives

+4 / -1

A wire of length $20 \mathrm{~m}$ is to be cut into two pieces. A piece of length $l_{1}$ is bent to make a square of area $A_{1}$ and the other piece of length $l_{2}$ is made into a circle of area $A_{2}$ . If $2 A_{1}+3 A_{2}$ is minimum then $\left(\pi l_{1}\right): l_{2}$ is equal to :

A

6 : 1

B

1 : 6

C

4 : 1

D

3 : 1

Q 89MathematicsSection A • MCQcomplex numbers

+4 / -1

For all $z \in C$ on the curve $C_{1}:|z|=4$ , let the locus of the point $z+\frac{1}{z}$ be the curve $\mathrm{C}_{2}$ . Then :

A

the curves

C_{1}

and

C_{2}

intersect at 4 points

B

the curve

C_{2}

lies inside

C_{1}

C

the curve

C_{1}

lies inside

C_{2}

D

the curves

C_{1}

and

C_{2}

intersect at 2 points

Q 90MathematicsSection B • Numericalvector algebra

+4 / -1

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 ___________.

Answer Format:Enter numerical value rounded to the nearest integer

JEE Main 2023 31st January Morning Shift Question Paper with Solutions