JEE Main 2023 1st February Morning Shift Question Paper Question Paper with Solutions | JEE Main Mock Test | JeeMate

Q 1PhysicsSection B • Numericalsimple harmonic motion

+4 / -1

The amplitude of a particle executing SHM is $3 \mathrm{~cm}$ . The displacement at which its kinetic energy will be $25 \%$ more than the potential energy is: __________ $\mathrm{cm}$

Answer Format:Enter numerical value rounded to the nearest integer

Q 2PhysicsSection A • MCQmagnetics

+4 / -1

Find the magnetic field at the point $\mathrm{P}$ in figure. The curved portion is a semicircle connected to two long straight wires.

qImage-biot-savarts-law-and-magnetic-field-due-to-current-carrying-wire-magnetics-electricity-Jeemate

A

\frac{\mu_{0} i}{2 r}\left(\frac{1}{2}+\frac{1}{2 \pi}\right)

B

\frac{\mu_{\mathrm{o}} i}{2 r}\left(1+\frac{2}{\pi}\right)

C

\frac{\mu_{0} \dot{i}}{2 r}\left(\frac{1}{2}+\frac{1}{\pi}\right)

D

\frac{\mu_{0} i}{2 r}\left(1+\frac{1}{\pi}\right)

Q 3PhysicsSection B • Numericalalternating current

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A series LCR circuit is connected to an ac source of $220 \mathrm{~V}, 50 \mathrm{~Hz}$ . The circuit contain a resistance $\mathrm{R}=100 ~\Omega$ and an inductor of inductive reactance $\mathrm{X}_{\mathrm{L}}=79.6 ~\Omega$ . The capacitance of the capacitor needed to maximize the average rate at which energy is supplied will be _________ $\mu \mathrm{F}$ .

Answer Format:Enter numerical value rounded to the nearest integer

Q 4PhysicsSection A • MCQproperties of matter

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The average kinetic energy of a molecule of the gas is

A

proportional to volume

B

dependent on the nature of the gas

C

proportional to absolute temperature

D

proportional to pressure

Q 5PhysicsSection A • MCQproperties of matter

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A mercury drop of radius $10^{-3}~\mathrm{m}$ is broken into 125 equal size droplets. Surface tension of mercury is $0.45~\mathrm{Nm}^{-1}$ . The gain in surface energy is :

A

28\times10^{-5}~\mathrm{J}

B

17.5\times10^{-5}~\mathrm{J}

C

5\times10^{-5}~\mathrm{J}

D

2.26\times10^{-5}~\mathrm{J}

Q 6PhysicsSection A • MCQlaws of motion

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A block of mass $5 \mathrm{~kg}$ is placed at rest on a table of rough surface. Now, if a force of $30 \mathrm{~N}$ is applied in the direction parallel to surface of the table, the block slides through a distance of $50 \mathrm{~m}$ in an interval of time $10 \mathrm{~s}$ . Coefficient of kinetic friction is (given, $g=10 \mathrm{~ms}^{-2}$ ):

A

0.25

B

0.75

C

0.60

D

0.50

Q 7PhysicsSection B • Numericalray optics

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A thin cylindrical rod of length $10 \mathrm{~cm}$ is placed horizontally on the principle axis of a concave mirror of focal length $20 \mathrm{~cm}$ . The rod is placed in a such a way that mid point of the rod is at $40 \mathrm{~cm}$ from the pole of mirror. The length of the image formed by the mirror will be $\frac{x}{3} \mathrm{~cm}$ . The value of $x$ is _____________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 8PhysicsSection A • MCQmotion

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An object moves with speed $v_1,v_2$ and $v_3$ along a line segment AB, BC and CD respectively as shown in figure. Where AB = BC and AD = 3AB, then average speed of the object will be:

qImage-non-uniform-motion-in-one-dimension-motion-mechanics-Jeemate

A

{{{v_1}{v_2}{v_3}} \over {3({v_1}{v_2} + {v_2}{v_3} + {v_3}{v_1})}}

B

{{({v_1} + {v_2} + {v_3})} \over 3}

C

{{({v_1} + {v_2} + {v_3})} \over {3{v_1}{v_2}{v_3}}}

D

{{3{v_1}{v_2}{v_3}} \over {({v_1}{v_2} + {v_2}{v_3} + {v_3}{v_1})}}

Q 9PhysicsSection A • MCQcommunication systems

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Which of the following frequencies does not belong to FM broadcast ?

A

106 ~\mathrm{MHz}

B

89 ~\mathrm{MHz}

C

64 ~\mathrm{MHz}

D

99 ~\mathrm{MHz}

Q 10PhysicsSection B • Numericalproperties of matter

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A certain pressure ' $\mathrm{P}$ ' is applied to 1 litre of water and 2 litre of a liquid separately. Water gets compressed to $0.01 \%$ whereas the liquid gets compressed to $0.03 \%$ . The ratio of Bulk modulus of water to that of the liquid is $\frac{3}{x}$ . The value of $x$ is ____________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 11PhysicsSection A • MCQelectrostatics

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Let $\sigma$ be the uniform surface charge density of two infinite thin plane sheets shown in figure. Then the electric fields in three different region $E_{I}, E_{I I}$ and $E_{I I I}$ are:

qImage-electric-field-and-electric-field-intensity-electrostatics-electricity-Jeemate

A

\vec{E}_{I}=0, \vec{E}_{I I}=\frac{\sigma}{\epsilon_{0}} \hat{n}, E_{I I I}=0

B

\vec{E}_{I}=\frac{\sigma}{2 \epsilon_{0}} \hat{n}, \vec{E}_{I I}=0, \vec{E}_{I I I}=\frac{\sigma}{2 \epsilon_{0}} \hat{n}

C

\vec{E}_{I}=-\frac{\sigma}{\epsilon_{0}} \hat{n}, \vec{E}_{I I}=0, \vec{E}_{I I I}=\frac{\sigma}{\epsilon_{0}} \hat{n}

D

\vec{E}_{I}=\frac{2 \sigma}{\epsilon_{0}} \hat{n}, \vec{E}_{I I}=0, \vec{E}_{I I I}=\frac{2 \sigma}{\epsilon_{0}} \hat{n}

Q 12PhysicsSection B • Numericalrotational motion

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A solid cylinder is released from rest from the top of an inclined plane of inclination $30^{\circ}$ and length $60 \mathrm{~cm}$ . If the cylinder rolls without slipping, its speed upon reaching the bottom of the inclined plane is __________ $\mathrm{ms}^{-1}$ . (Given $\mathrm{g}=10 \mathrm{~ms}^{-2}$ )

qImage-combined-translational-and-rotational-motion-rotational-motion-mechanics-Jeemate

Answer Format:Enter numerical value rounded to the nearest integer

Q 13PhysicsSection A • MCQalternating current

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Match List - I with List - II :

	List I		List II
A.	AC generator	I.	Presence of both L and C
B.	Transformer	II.	Electromagnetic Induction
C.	Resonance phenomenon to occur	III.	Quality factor
D.	Sharpness of resonance	IV.	Mutual Induction

Choose the correct answer from the options given below :

A

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

B

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

C

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

D

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

Q 14PhysicsSection A • MCQcurrent electricity

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The equivalent resistance between $A$ and $B$ of the network shown in figure;

qImage-wheatstone-bridge-current-electricity-electricity-Jeemate

A

21 R

B

\frac{8}{3}

R

C

11

\frac{2}{3}

R

D

14 R

Q 15PhysicsSection A • MCQatoms and nuclei

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The mass of proton, neutron and helium nucleus are respectively $1.0073~u,1.0087~u$ and $4.0015~u$ . The binding energy of helium nucleus is :

A

28.4~\mathrm{MeV}

B

56.8~\mathrm{MeV}

C

7.1~\mathrm{MeV}

D

14.2~\mathrm{MeV}

Q 16PhysicsSection A • MCQheat and thermodynamics

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$\left(P+\frac{a}{V^{2}}\right)(V-b)=R T$ represents the equation of state of some gases. Where $P$ is the pressure, $V$ is the volume, $T$ is the temperature and $a, b, R$ are the constants. The physical quantity, which has dimensional formula as that of $\frac{b^{2}}{a}$ , will be:

A

Energy density

B

Bulk modulus

C

Modulus of rigidity

D

Compressibility

Q 17PhysicsSection B • Numericalcurrent electricity

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In an experiment to find emf of a cell using potentiometer, the length of null point for a cell of emf $1.5 \mathrm{~V}$ is found to be $60 \mathrm{~cm}$ . If this cell is replaced by another cell of emf E, the length-of null point increases by $40 \mathrm{~cm}$ . The value of $E$ is $\frac{x}{10} V$ . The value of $x$ is ____________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 18PhysicsSection A • MCQelectromagnetic waves

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Match List I with List II :

	List I		List II
A.	Microwaves	I.	Radio active decay of the nucleus
B.	Gamma rays	II.	Rapid acceleration and deceleration of electron in aerials
C.	Radio waves	III.	Inner shell electrons
D.	X-rays	IV.	Klystron valve

Choose the correct answer from the options given below :

A

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

B

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

C

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

D

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

Q 19PhysicsSection A • MCQgravitation

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Given below are two statements:

Statement I: Acceleration due to gravity is different at different places on the surface of earth.

Statement II: Acceleration due to gravity increases as we go down below the earth's surface.

In the light of the above statements, choose the correct answer from the options given below

A

Both Statement I and Statement II are true

B

Both Statement I and Statement II are false

C

Statement I is false but Statement II is true

D

Statement I is true but Statement II is false

Q 20PhysicsSection A • MCQwave optics

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' $n$ ' polarizing sheets are arranged such that each makes an angle $45^{\circ}$ with the preceeding sheet. An unpolarized light of intensity I is incident into this arrangement. The output intensity is found to be $I / 64$ . The value of $n$ will be:

A

4

B

5

C

3

D

6

Q 21PhysicsSection A • MCQdual nature of radiation

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A proton moving with one tenth of velocity of light has a certain de Broglie wavelength of $\lambda$ . An alpha particle having certain kinetic energy has the same de-Brogle wavelength $\lambda$ . The ratio of kinetic energy of proton and that of alpha particle is:

A

1 : 4

B

2 : 1

C

4 : 1

D

1 : 2

Q 22PhysicsSection A • MCQelectronic devices

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Match List I with List II:

	List I		List II
A.	Intrinsic semiconductor	I.	Fermi-level near the valence bond
B.	n-type semiconductor	II.	Fermi-level in the middle of valence and conduction band.
C.	p-type semiconductor	III.	Fermi-level near the conduction band
D.	Metals	IV.	Fermi-level inside the conduction band

Choose the correct answer from the options given below :

A

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

B

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

C

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

D

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

Q 23PhysicsSection B • Numericalwork power and energy

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A small particle moves to position $5 \hat{i}-2 \hat{j}+\hat{k}$ from its initial position $2 \hat{i}+3 \hat{j}-4 \hat{k}$ under the action of force $5 \hat{i}+2 \hat{j}+7 \hat{k} \mathrm{~N}$ . The value of work done will be __________ J.

Answer Format:Enter numerical value rounded to the nearest integer

Q 24PhysicsSection A • MCQmotion

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A child stands on the edge of the cliff $10 \mathrm{~m}$ above the ground and throws a stone horizontally with an initial speed of $5 \mathrm{~ms}^{-1}$ . Neglecting the air resistance, the speed with which the stone hits the ground will be $\mathrm{ms}^{-1}$ (given, $g=10 \mathrm{~ms}^{-2}$ ).

A

20

B

25

C

30

D

15

Q 25PhysicsSection B • Numericalmagnetics

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A charge particle of $2 ~\mu \mathrm{C}$ accelerated by a potential difference of $100 \mathrm{~V}$ enters a region of uniform magnetic field of magnitude $4 ~\mathrm{mT}$ at right angle to the direction of field. The charge particle completes semicircle of radius $3 \mathrm{~cm}$ inside magnetic field. The mass of the charge particle is __________ $\times 10^{-18} \mathrm{~kg}$

Answer Format:Enter numerical value rounded to the nearest integer

Q 26PhysicsSection A • MCQgravitation

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If earth has a mass nine times and radius twice to that of a planet P. Then $\frac{v_{e}}{3} \sqrt{x} \mathrm{~ms}^{-1}$ will be the minimum velocity required by a rocket to pull out of gravitational force of $\mathrm{P}$ , where $v_{e}$ is escape velocity on earth. The value of $x$ is

A

1

B

3

C

2

D

18

Q 27PhysicsSection A • MCQwaves

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A steel wire with mass per unit length $7.0 \times 10^{-3} \mathrm{~kg} \mathrm{~m}^{-1}$ is under tension of $70 \mathrm{~N}$ . The speed of transverse waves in the wire will be:

A

10 \mathrm{~m} / \mathrm{s}

B

50 \mathrm{~m} / \mathrm{s}

C

100 \mathrm{~m} / \mathrm{s}

D

200 \pi\mathrm{~m} / \mathrm{s}

Q 28PhysicsSection B • Numericalelectrostatics

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Two equal positive point charges are separated by a distance $2 a$ . The distance of a point from the centre of the line joining two charges on the equatorial line (perpendicular bisector) at which force experienced by a test charge $\mathrm{q}_{0}$ becomes maximum is $\frac{a}{\sqrt{x}}$ . The value of $x$ is __________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 29PhysicsSection A • MCQheat and thermodynamics

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A sample of gas at temperature $T$ is adiabatically expanded to double its volume. The work done by the gas in the process is $\left(\mathrm{given}, \gamma=\frac{3}{2}\right)$ :

A

W=T R[\sqrt{2}-2]

B

W=\frac{T}{R}[\sqrt{2}-2]

C

W=\frac{R}{T}[2-\sqrt{2}]

D

W=R T[2-\sqrt{2}]

Q 30PhysicsSection B • Numericalatoms and nuclei

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A light of energy $12.75 ~\mathrm{eV}$ is incident on a hydrogen atom in its ground state. The atom absorbs the radiation and reaches to one of its excited states. The angular momentum of the atom in the excited state is $\frac{x}{\pi} \times 10^{-17} ~\mathrm{eVs}$ . The value of $x$ is ___________ (use $h=4.14 \times 10^{-15} ~\mathrm{eVs}, c=3 \times 10^{8} \mathrm{~ms}^{-1}$ ).

Answer Format:Enter numerical value rounded to the nearest integer

Q 31ChemistrySection B • Numericalsolutions

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25 mL of an aqueous solution of KCl was found to require 20 mL of 1 M $\mathrm{AgNO_3}$ solution when titrated using $\mathrm{K_2CrO_4}$ as an indicator. What is the depression in freezing point of KCl solution of the given concentration? _________ (Nearest integer).

(Given : $\mathrm{K_f=2.0~K~kg~mol^{-1}}$ )

Assume 1) 100% ionization and 2) density of the aqueous solution as 1 g mL $^{-1}$

Answer Format:Enter numerical value rounded to the nearest integer

Q 32ChemistrySection A • MCQbasics of organic chemistry

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Resonance in carbonate ion $\left(\mathrm{CO}_{3}{ }^{2-}\right)$ is

qImage-electron-displacement-effect-basics-of-organic-chemistry-organic-chemistry-Jeemate

Which of the following is true?

A

It is possible to identify each structure individually by some physical or chemical method.

B

All these structures are in dynamic equilibrium with each other.

C

Each structure exists for equal amount of time.

D

\mathrm{CO}_{3}{ }^{2-}

has a single structure i.e., resonance hybrid of the above three structures.

Q 33ChemistrySection A • MCQhydrogen

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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 : Hydrogen is an environment friendly fuel.

Reason R : Atomic number of hydrogen is 1 and it is a very light element.

In the light of the above statements, choose the correct answer from the options given below :

A

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

B

A is false but R is true

C

A is true but R is false

D

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

Q 34ChemistrySection B • Numericalbasics of organic chemistry

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The total number of chiral compound/s from the following is ______________.

qImage-isomerism-basics-of-organic-chemistry-organic-chemistry-Jeemate

Answer Format:Enter numerical value rounded to the nearest integer

Q 35ChemistrySection A • MCQbiomolecules

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The correct representation in six membered pyranose form for the following sugar [X] is

qImage-carbohydrates-biomolecules-organic-chemistry-Jeemate

A

B

C

D

Q 36ChemistrySection A • MCQenvironmental chemistry

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How can photochemical smog be controlled?

A

By using catalytic convertors in the automobiles/industry.

B

By complete combustion of fuel.

C

By using tall chimneys.

D

By using catalyst.

Official Correct Answer:Option (A)

Step-by-Step Solution:

Option A, By using catalytic converters in the automobiles/industry, is the most appropriate answer to control photochemical smog.

Photochemical smog is a type of air pollution that forms in urban areas when pollutants from automobile and industrial emissions, such as nitrogen oxides (NOx) and volatile organic compounds (VOCs), react in the presence of sunlight.

Catalytic converters are devices that are installed in the exhaust systems of automobiles and in some industrial processes to remove pollutants from exhaust gases. They work by using a catalyst to convert harmful pollutants, such as NOx and VOCs, into less harmful substances before they are released into the atmosphere. This can significantly reduce the formation of photochemical smog.

Option B, By complete combustion of fuel, is not a complete or sufficient solution to control photochemical smog. Complete combustion of fuel, which refers to the complete oxidation of the fuel to produce carbon dioxide and water, can reduce the amount of unburned hydrocarbons and carbon monoxide in the exhaust gases.

However, even with complete combustion, other pollutants, such as nitrogen oxides (NOx), can still be produced and contribute to the formation of photochemical smog. NOx emissions are a result of high temperatures and pressures in the combustion process and are not easily prevented through complete combustion alone.

Additionally, while complete combustion can reduce the amount of certain pollutants, it does not address other sources of pollution, such as industrial emissions, that can contribute to the formation of photochemical smog.

Therefore, while complete combustion can be an important step in reducing emissions and controlling photochemical smog, it is not a sufficient solution on its own and must be combined with other methods, such as the use of catalytic converters, to effectively address this issue.

Option C, using tall chimneys, is not a direct way to control photochemical smog. Tall chimneys can help to disperse pollutants into the atmosphere, but do not necessarily prevent their formation in the first place.

Option D, By using a catalyst, is not a complete or sufficient solution to control photochemical smog. While catalysts can be used to convert harmful pollutants into less harmful substances, they need to be part of a larger system, such as a catalytic converter, to effectively reduce emissions.

Using a catalyst alone, without the proper design, engineering, and installation, is unlikely to be effective in controlling photochemical smog. It is important to note that the goal is to reduce the overall emissions of pollutants, such as nitrogen oxides (NOx) and volatile organic compounds (VOCs), which are the precursors to photochemical smog.

Therefore, while catalysts can be a useful tool in controlling photochemical smog, they must be used in combination with other methods, such as the use of catalytic converters in automobiles and industry, to be most effective.

Q 37ChemistrySection A • MCQcoordination compounds

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A solution of $\mathrm{FeCl_3}$ when treated with $\mathrm{K_4[Fe(CN)_6]}$ gives a prussium blue precipitate due to the formation of :

A

\mathrm{Fe[Fe(CN)_{6}]}

B

\mathrm{Fe_{4}[Fe(CN)_{6}]_{3}}

C

\mathrm{Fe_{3}[Fe(CN)_{6}]_{2}}

D

\mathrm{K[Fe_{2}(CN)_{6}]}

Q 38ChemistrySection A • MCQalcohols phenols and ethers

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Identify the incorrect option from the following

A

B

C

D

Q 39ChemistrySection A • MCQd and f block elements

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Highest oxidation state of Mn is exhibited in $\mathrm{Mn_2O_7}$ . The correct statements about $\mathrm{Mn_2O_7}$ are

(A) Mn is tetrahedrally surrounded by oxygen atoms.

(B) Mn is octahedrally surrounded by oxygen atoms.

(C) Contains Mn-O-Mn bridge.

(D) Contains Mn-Mn bond.

Choose the correct answer from the options given below :

A

A and C only

B

A and D only

C

B and C only

D

B and D only

Q 40ChemistrySection B • Numericalsome basic concepts of chemistry

+4 / -1

The density of $3 \mathrm{M}$ solution of $\mathrm{NaCl}$ is $1.0 \mathrm{~g} \mathrm{~mL}^{-1}$ . Molality of the solution is ____________ $\times 10^{-2} \mathrm{~m}$ . (Nearest integer).

Given: Molar mass of $\mathrm{Na}$ and $\mathrm{Cl}$ is $23$ and $35.5 \mathrm{~g} \mathrm{~mol}^{-1}$ respectively.

Answer Format:Enter numerical value rounded to the nearest integer

Q 41ChemistrySection B • Numericalthermodynamics

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At $25^{\circ} \mathrm{C}$ , the enthalpy of the following processes are given :

$\mathrm{H_2(g)+O_2(g)}$	$\to$	$2\mathrm{OH(g)}$	$\mathrm{\Delta H^\circ=78~kJ~mol^{-1}}$
$\mathrm{H_2(g)+\frac{1}{2}O_2(g)}$	$\to$	$\mathrm{H_2O(g)}$	$\mathrm{\Delta H^\circ=-242~kJ~mol^{-1}}$
$\mathrm{H_2(g)}$	$\to$	$\mathrm{2H(g)}$	$\mathrm{\Delta H^\circ=436~kJ~mol^{-1}}$
$\frac{1}{2}\mathrm{O_2(g)}$	$\to$	$\mathrm{O(g)}$	$\mathrm{\Delta H^\circ=249~kJ~mol^{-1}}$

What would be the value of X for the following reaction ? _____________ (Nearest integer)

$\mathrm{H_2O(g)\to H(g)+OH(g)~\Delta H^\circ=X~kJ~mol^{-1}}$

Answer Format:Enter numerical value rounded to the nearest integer

Q 42ChemistrySection B • Numericalredox reactions

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Sum of oxidation states of bromine in bromic acid and perbromic acid is ___________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 43ChemistrySection A • MCQchemistry in everyday life

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Match List I with List II

	List I		List II
A.	Tranquilizers	I.	Anti blood clotting
B.	Aspirin	II.	Salvarsan
C.	Antibiotic	III.	antidepressant drugs
D.	Antiseptic	IV.	soframicine

Choose the correct answer from the options given below:

A

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

B

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

C

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

D

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

Q 44ChemistrySection B • Numericalchemical kinetics and nuclear chemistry

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A and B are two substances undergoing radioactive decay in a container. The half life of A is 15 min and that of B is 5 min. If the initial concentration of B is 4 times that of A and they both start decaying at the same time, how much time will it take for the concentration of both of them to be same? _____________ min.

Answer Format:Enter numerical value rounded to the nearest integer

Q 45ChemistrySection B • Numericalaldehydes ketones and carboxylic acids

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Number of isomeric compounds with molecular formula $\mathrm{C}_{9} \mathrm{H}_{10} \mathrm{O}$ which (i) do not dissolve in $\mathrm{NaOH}$ (ii) do not dissolve in $\mathrm{HCl}$ . (iii) do not give orange precipitate with 2,4-DNP (iv) on hydrogenation give identical compound with molecular formula $\mathrm{C}_{9} \mathrm{H}_{12} \mathrm{O}$ is ____________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 46ChemistrySection B • Numericalelectrochemistry

+4 / -1

At what pH, given half cell $\mathrm{MnO_{4}^{-}(0.1~M)~|~Mn^{2+}(0.001~M)}$ will have electrode potential of 1.282 V? ___________ (Nearest Integer)

Given $\mathrm{E_{MnO_4^ - |M{n^{2 + }}}^o}=1.54~\mathrm{V},\frac{2.303\mathrm{RT}}{\mathrm{F}}=0.059\mathrm{V}$

Answer Format:Enter numerical value rounded to the nearest integer

Q 47ChemistrySection B • Numericalatomic structure

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Electrons in a cathode ray tube have been emitted with a velocity of 1000 m s $^{-1}$ . The number of following statements which is/are $\underline {\mathrm{true}}$ about the emitted radiation is ____________.

Given : $\mathrm{h=6\times10^{-34}~J~s,m_e=9\times10^{-31}~kg}$ .

(A) The de-Broglie wavelength of the electron emitted is 666.67 nm.

(B) The characteristic of electrons emitted depend upon the material of the electrodes of the cathode ray tube.

(C) The cathode rays start from cathode and move towards anode.

(D) The nature of the emitted electrons depends on the nature of the gas present in cathode ray tube.

Answer Format:Enter numerical value rounded to the nearest integer

Q 48ChemistrySection A • MCQp block elements

+4 / -1

Given below are two statements:

Statement I : Chlorine can easily combine with oxygen to form oxides; and the product has a tendency to explode.

Statement II : Chemical reactivity of an element can be determined by its reaction with oxygen and halogens.

In the light of the above statements, choose the correct answer from the options given below :

A

Statement I is true but Statement II is false

B

Both the Statements I and II are false

C

Statement I is false but Statement II is true

D

Both the Statements I and II are true

Q 49ChemistrySection A • MCQsolid state

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Which of the following represents the lattice structure of $\mathrm{A_{0.95}O}$ containing $\mathrm{A^{2+},A^{3+}}$ and $\mathrm{O^{2-}}$ ions?

qImage-defects-in-crystal-solid-state-physical-chemistry-Jeemate

A

B only

B

A and B only

C

A only

D

B and C only

Q 50ChemistrySection A • MCQs block elements

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Match List I with List II :

	List I		List II
A.	Slaked lime	I.	$\mathrm{NaOH}$
B.	Dead burnt plaster	II.	$\mathrm{Ca(OH)_2}$
C.	Caustic soda	III.	$\mathrm{Na_2CO_3\cdot 10H_2O}$
D.	Washing soda	IV.	$\mathrm{CaSO_4}$

Choose the correct answer from the options given below :

A

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

B

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

C

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

D

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

Q 51ChemistrySection A • MCQisolation of elements

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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: In an Ellingham diagram, the oxidation of carbon to carbon monoxide shows a negative slope with respect to temperature.

Reason R: CO tends to get decomposed at higher temperature.

In the light of the above statements, choose the correct answer from the options given below

A

A is correct but R is not correct

B

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

C

A is not correct but R is correct

D

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

Q 52ChemistrySection A • MCQbiomolecules

+4 / -1

Match List I with List II

	List I		List II
	Test		Functional group / Class of Compound
A.	Molisch's Test	I.	Peptide
B.	Biuret Test	II.	Carbohydrate
C.	Carbylamine Test	III.	Primary amine
D.	Schiff's Test	IV.	Aldehyde

Choose the correct answer from the options given below :

A

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

B

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

C

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

D

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

Q 53ChemistrySection A • MCQhydrocarbons

+4 / -1

But-2-yne is reacted separately with one mole of Hydrogen as shown below :

qImage-reaction-of-alkynes-hydrocarbons-organic-chemistry-Jeemate

A. A is more soluble than B.

B. The boiling point & melting point of A are higher and lower than B respectively.

C. A is more polar than B because dipole moment of A is zero.

D. $\mathrm{Br_2}$ adds easily to B than A.

Identify the incorrect statements from the options given below :

A

A and B only

B

A, C & D only

C

B, C & D only

D

C and D only

Q 54ChemistrySection B • Numericalchemical equilibrium

+4 / -1

(i) $\mathrm{X}(\mathrm{g}) \rightleftharpoons \mathrm{Y}(\mathrm{g})+\mathrm{Z}(\mathrm{g}) \quad \mathrm{K}_{\mathrm{p} 1}=3$

(ii) $\mathrm{A}(\mathrm{g}) \rightleftharpoons 2 \mathrm{~B}(\mathrm{g}) \quad \mathrm{K}_{\mathrm{p} 2}=1$

If the degree of dissociation and initial concentration of both the reactants $\mathrm{X}(\mathrm{g})$ and $\mathrm{A}(\mathrm{g})$ are equal, then the ratio of the total pressure at equilibrium $\left(\frac{p_{1}}{p_{2}}\right)$ is equal to $\mathrm{x}: 1$ . The value of $\mathrm{x}$ is _____________ (Nearest integer)

Answer Format:Enter numerical value rounded to the nearest integer

Q 55ChemistrySection A • MCQcoordination compounds

+4 / -1

Which of the following complex will show largest splitting of d-orbitals?

A

[\mathrm{Fe(C_2O_4)_3]^{3-}}

B

[\mathrm{Fe(CN)_6]^{3-}}

C

[\mathrm{Fe(NH_3)_6]^{3+}}

D

[\mathrm{FeF_6]^{3-}}

Q 56ChemistrySection A • MCQs block elements

+4 / -1

Choose the correct statement(s) :

A. Beryllium oxide is purely acidic in nature.

B. Beryllium carbonate is kept in the atmosphere of $\mathrm{CO_2}$ .

C. Beryllium sulphate is readily soluble in water.

D. Beryllium shows anomalous behaviour.

Choose the correct answer from the options given below :

A

A only

B

B, C and D only

C

A, B and C only

D

A and B only

Q 57ChemistrySection A • MCQalcohols phenols and ethers

+4 / -1

Decreasing order of dehydration of the following alcohols is

qImage-properties-preparation-and-uses-of-phenols-alcohols-phenols-and-ethers-organic-chemistry-Jeemate

A

d > b > c > a

B

b > d > c > a

C

a > d > b > c

D

b > a > d > c

Q 58ChemistrySection A • MCQsurface chemistry

+4 / -1

Given below are two statements: One is labelled as Assertion A and the other is labelled as Reason R

Assertion A: Amongst $\mathrm{He}, \mathrm{Ne}, \mathrm{Ar}$ and $\mathrm{Kr}$ ; 1 $\mathrm{g}$ of activated charcoal adsorbs more of $\mathrm{Kr}$ .

Reason R: The critical volume $\mathrm{V}_{\mathrm{c}}\left(\mathrm{cm}^{3} \mathrm{~mol}^{-1}\right)$ and critical pressure $\mathrm{P}_{\mathrm{c}}$ (atm) is highest for Krypton but the compressibility factor at critical point $\mathrm{Z}_{\mathrm{c}}$ is lowest for Krypton.

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 false but R is true

C

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

D

A is true but R is false

Q 59ChemistrySection A • MCQcompounds containing nitrogen

+4 / -1

In the following reaction, 'A' is

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

A

B

C

D

Q 60ChemistrySection A • MCQcoordination compounds

+4 / -1

Which of the following are the example of double salt?

A. $\mathrm{FeSO}_{4} \cdot\left(\mathrm{NH}_{4}\right)_{2} \mathrm{SO}_{4} \cdot 6 \mathrm{H}_{2} \mathrm{O}$

B. $\mathrm{CuSO}_{4}\cdot 4 \mathrm{NH}_{3} \cdot \mathrm{H}_{2} \mathrm{O}$

C. $\mathrm{K}_{2} \mathrm{SO}_{4} \cdot \mathrm{Al}_{2}\left(\mathrm{SO}_{4}\right)_{3} \cdot 24 \mathrm{H}_{2} \mathrm{O}$

D. $\mathrm{Fe}(\mathrm{CN})_{2}\cdot4 \mathrm{KCN}$

Choose the correct answer :

A

A, B and D only

B

B and D only

C

A and B only

D

A and C only

Official Correct Answer:Option (D)

Step-by-Step Solution:

A=\mathrm{FeSO}_{4} \cdot\left(\mathrm{NH}_{4}\right)_{2} \mathrm{SO}_{4} \cdot 6 \mathrm{H}_{2} \mathrm{O}

- double salt

B.

\mathrm{CuSO}_{4} \cdot 4 \mathrm{NH}_{3} \cdot \mathrm{H}_{2} \mathrm{O}

=\left[\mathrm{Cu}\left(\mathrm{NH}_{3}\right)_{4}\right] \mathrm{SO}_{4} \cdot \mathrm{H}_{2} \mathrm{O}

- complex salt

C.

\mathrm{K}_{2} \mathrm{SO}_{4} \cdot \mathrm{Al}_{2}\left(\mathrm{SO}_{4}\right)_{3} \cdot 24 \mathrm{H}_{2} \mathrm{O}

- double salt

D.

\mathrm{Fe}(\mathrm{CN})_{2} \cdot 4 \mathrm{KCN}

=

\mathrm{K}_{4}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]

- complex salt

Option A,

\mathrm{FeSO}_{4} \cdot\left(\mathrm{NH}_{4}\right)_{2} \mathrm{SO}_{4} \cdot 6 \mathrm{H}_{2} \mathrm{O}

, is a double salt because it is formed by the combination of two different salts: iron(II) sulfate (

\mathrm{FeSO}_{4}

) and ammonium sulfate (

\left(\mathrm{NH}_{4}\right)_{2} \mathrm{SO}_{4}

).

Option B,

\mathrm{CuSO}_{4}\cdot 4 \mathrm{NH}_{3} \cdot \mathrm{H}_{2} \mathrm{O}

, is a complex salt because it contains a central copper ion coordinated to several ammonia molecules.

In this compound, the copper(II) ion (Cu²⁺) acts as the central metal ion, while the four ammonia molecules (NH₃) act as ligands, coordinating to the copper ion through their lone pairs of electrons. The sulfate ion (SO₄^2-) and water molecules (H₂O) are not involved in the coordination sphere and simply crystallize along with the complex.

Option C,

\mathrm{K}_{2} \mathrm{SO}_{4} \cdot \mathrm{Al}_{2}\left(\mathrm{SO}_{4}\right)_{3} \cdot 24 \mathrm{H}_{2} \mathrm{O}

, is also a double salt, formed by the combination of two different salts: potassium sulfate (

\mathrm{K}_{2} \mathrm{SO}_{4}

) and aluminum sulfate (

\mathrm{Al}_{2}\left(\mathrm{SO}_{4}\right)_{3}

).

The 24 water molecules in the formula are not directly involved in the salt formation but instead function to stabilize the crystal lattice by forming hydrogen bonds with the sulfate ions and other water molecules.

Option D,

\mathrm{Fe}(\mathrm{CN})_{2}\cdot4 \mathrm{KCN}

, is not an example of a double salt. It is a complex salt, formed by the coordination of iron(II) ions to cyanide ligands and potassium ions.

Q 61MathematicsSection A • MCQpermutations and combinations

+4 / -1

The value of $\frac{1}{1 ! 50 !}+\frac{1}{3 ! 48 !}+\frac{1}{5 ! 46 !}+\ldots .+\frac{1}{49 ! 2 !}+\frac{1}{51 ! 1 !}$ is :

A

\frac{2^{51}}{50 !}

B

\frac{2^{51}}{51 !}

C

\frac{2^{50}}{50 !}

D

\frac{2^{50}}{51 !}

Q 62MathematicsSection A • MCQ3d geometry

+4 / -1

The shortest distance between the lines

${{x - 5} \over 1} = {{y - 2} \over 2} = {{z - 4} \over { - 3}}$ and

${{x + 3} \over 1} = {{y + 5} \over 4} = {{z - 1} \over { - 5}}$ is :

A

7\sqrt 3

B

5\sqrt 3

C

4\sqrt 3

D

6\sqrt 3

Q 63MathematicsSection A • MCQmatrices and determinants

+4 / -1

Let $S$ denote the set of all real values of $\lambda$ such that the system of equations

$\lambda x+y+z=1$

$x+\lambda y+z=1$

$x+y+\lambda z=1$

is inconsistent, then $\sum_\limits{\lambda \in S}\left(|\lambda|^{2}+|\lambda|\right)$ is equal to

A

12

B

2

C

4

D

6

Q 64MathematicsSection A • MCQdefinite integration

+4 / -1

$\mathop {\lim }\limits_{n \to \infty } \left[ {{1 \over {1 + n}} + {1 \over {2 + n}} + {1 \over {3 + n}}\, + \,...\, + \,{1 \over {2n}}} \right]$ is equal to

A

0

B

{\log _e}2

C

{\log _e}\left( {{2 \over 3}} \right)

D

{\log _e}\left( {{3 \over 2}} \right)

Q 65MathematicsSection A • MCQsequences and series

+4 / -1

The sum of 10 terms of the series

${1 \over {1 + {1^2} + {1^4}}} + {2 \over {1 + {2^2} + {2^4}}} + {3 \over {1 + {3^2} + {3^4}}}\, + \,....$ is

A

{{58} \over {111}}

B

{{56} \over {111}}

C

{{55} \over {111}}

D

{{59} \over {111}}

Q 66MathematicsSection A • MCQinverse trigonometric functions

+4 / -1

Let $S$ be the set of all solutions of the equation $\cos ^{-1}(2 x)-2 \cos ^{-1}\left(\sqrt{1-x^{2}}\right)=\pi, x \in\left[-\frac{1}{2}, \frac{1}{2}\right]$ . Then $\sum_\limits{x \in S} 2 \sin ^{-1}\left(x^{2}-1\right)$ is equal to :

A

\pi-2 \sin ^{-1}\left(\frac{\sqrt{3}}{4}\right)

B

\pi-\sin ^{-1}\left(\frac{\sqrt{3}}{4}\right)

C

\frac{-2 \pi}{3}

D

None

Q 67MathematicsSection A • MCQsets and relations

+4 / -1

Let $R$ be a relation on $\mathbb{R}$ , given by $R=\{(a, b): 3 a-3 b+\sqrt{7}$ is an irrational number $\}$ . Then $R$ is

A

an equivalence relation

B

reflexive and symmetric but not transitive

C

reflexive and transitive but not symmetric

D

reflexive but neither symmetric nor transitive

Q 68MathematicsSection A • MCQprobability

+4 / -1

In a binomial distribution $B(n,p)$ , the sum and the product of the mean and the variance are 5 and 6 respectively, then $6(n+p-q)$ is equal to :

A

52

B

50

C

51

D

53

Q 69MathematicsSection B • Numericalvector algebra

+4 / -1

Let $\vec{v}=\alpha \hat{i}+2 \hat{j}-3 \hat{k}, \vec{w}=2 \alpha \hat{i}+\hat{j}-\hat{k}$ and $\vec{u}$ be a vector such that $|\vec{u}|=\alpha>0$ . If the minimum value of the scalar triple product $\left[ {\matrix{ {\overrightarrow u } & {\overrightarrow v } & {\overrightarrow w } \cr } } \right]$ is $-\alpha \sqrt{3401}$ , and $|\vec{u} \cdot \hat{i}|^{2}=\frac{m}{n}$ where $m$ and $n$ are coprime natural numbers, then $m+n$ is equal to ____________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 70MathematicsSection B • Numericaldifferentiation

+4 / -1

If $f(x)=x^{2}+g^{\prime}(1) x+g^{\prime \prime}(2)$ and $g(x)=f(1) x^{2}+x f^{\prime}(x)+f^{\prime \prime}(x)$ , then the value of $f(4)-g(4)$ is equal to ____________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 71MathematicsSection A • MCQ3d geometry

+4 / -1

Let the image of the point $P(2,-1,3)$ in the plane $x+2 y-z=0$ be $Q$ .

Then the distance of the plane $3 x+2 y+z+29=0$ from the point $Q$ is :

A

2\sqrt{14}

B

\frac{22\sqrt2}{7}

C

\frac{24\sqrt2}{7}

D

3\sqrt{14}

Q 72MathematicsSection A • MCQdifferential equations

+4 / -1

If $y=y(x)$ is the solution curve of the differential equation

$\frac{d y}{d x}+y \tan x=x \sec x, 0 \leq x \leq \frac{\pi}{3}, y(0)=1$ , then $y\left(\frac{\pi}{6}\right)$ is equal to

A

\frac{\pi}{12}-\frac{\sqrt{3}}{2} \log _{e}\left(\frac{2 \sqrt{3}}{e}\right)

B

\frac{\pi}{12}+\frac{\sqrt{3}}{2} \log _{e}\left(\frac{2 \sqrt{3}}{e}\right)

C

\frac{\pi}{12}+\frac{\sqrt{3}}{2} \log _{e}\left(\frac{2}{e \sqrt{3}}\right)

D

\frac{\pi}{12}-\frac{\sqrt{3}}{2} \log _{e}\left(\frac{2}{e \sqrt{3}}\right)

Q 73MathematicsSection A • MCQquadratic equation and inequalities

+4 / -1

Let $S = \left\{ {x:x \in \mathbb{R}\,\mathrm{and}\,{{(\sqrt 3 + \sqrt 2 )}^{{x^2} - 4}} + {{(\sqrt 3 - \sqrt 2 )}^{{x^2} - 4}} = 10} \right\}$ . Then $n(S)$ is equal to

A

6

B

4

C

0

D

2

Q 74MathematicsSection A • MCQdifferentiation

+4 / -1

Let $f(x) = 2x + {\tan ^{ - 1}}x$ and $g(x) = {\log _e}(\sqrt {1 + {x^2}} + x),x \in [0,3]$ . Then

A

there exists

\widehat x \in [0,3]

such that

f'(\widehat x) < g'(\widehat x)

B

there exist

0 < {x_1} < {x_2} < 3

such that

f(x) < g(x),\forall x \in ({x_1},{x_2})

C

\min f'(x) = 1 + \max g'(x)

D

\max f(x) > \max g(x)

Q 75MathematicsSection A • MCQstraight lines and pair of straight lines

+4 / -1

The combined equation of the two lines $ax+by+c=0$ and $a'x+b'y+c'=0$ can be written as

$(ax+by+c)(a'x+b'y+c')=0$ .

The equation of the angle bisectors of the lines represented by the equation $2x^2+xy-3y^2=0$ is :

A

3{x^2} + xy - 2{y^2} = 0

B

{x^2} - {y^2} - 10xy = 0

C

{x^2} - {y^2} + 10xy = 0

D

3{x^2} + 5xy + 2{y^2} = 0

Q 76MathematicsSection B • Numericalsequences and series

+4 / -1

Let $a_{1}=8, a_{2}, a_{3}, \ldots, a_{n}$ be an A.P. If the sum of its first four terms is 50 and the sum of its last four terms is 170 , then the product of its middle two terms is ___________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 77MathematicsSection A • MCQcomplex numbers

+4 / -1

If the center and radius of the circle $\left| {{{z - 2} \over {z - 3}}} \right| = 2$ are respectively $(\alpha,\beta)$ and $\gamma$ , then $3(\alpha+\beta+\gamma)$ is equal to :

A

12

B

10

C

11

D

9

Q 78MathematicsSection A • MCQstraight lines and pair of straight lines

+4 / -1

If the orthocentre of the triangle, whose vertices are (1, 2), (2, 3) and (3, 1) is $(\alpha,\beta)$ , then the quadratic equation whose roots are $\alpha+4\beta$ and $4\alpha+\beta$ , is :

A

x^2-20x+99=0

B

x^2-22x+120=0

C

x^2-19x+90=0

D

x^2-18x+80=0

Q 79MathematicsSection B • Numericalbinomial theorem

+4 / -1

The remainder, when $19^{200}+23^{200}$ is divided by 49 , is ___________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 80MathematicsSection B • Numericalpermutations and combinations

+4 / -1

The number of 3-digit numbers, that are divisible by either 2 or 3 but not divisible by 7, is ____________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 81MathematicsSection A • MCQstatistics

+4 / -1

The mean and variance of 5 observations are 5 and 8 respectively. If 3 observations are 1, 3, 5, then the sum of cubes of the remaining two observations is :

A

1792

B

1216

C

1456

D

1072

Q 82MathematicsSection B • Numericaldefinite integration

+4 / -1

If $\int_\limits{0}^{1}\left(x^{21}+x^{14}+x^{7}\right)\left(2 x^{14}+3 x^{7}+6\right)^{1 / 7} d x=\frac{1}{l}(11)^{m / n}$ where $l, m, n \in \mathbb{N}, m$ and $n$ are coprime then $l+m+n$ is equal to ____________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 83MathematicsSection B • Numericalvector algebra

+4 / -1

$A(2,6,2), B(-4,0, \lambda), C(2,3,-1)$ and $D(4,5,0),|\lambda| \leq 5$ are the vertices of a quadrilateral $A B C D$ . If its area is 18 square units, then $5-6 \lambda$ is equal to __________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 84MathematicsSection A • MCQmathematical reasoning

+4 / -1

The negation of the expression $q \vee \left( {( \sim \,q) \wedge p} \right)$ is equivalent to

A

( \sim \,p) \wedge ( \sim \,q)

B

( \sim \,p) \vee q

C

p \wedge ( \sim \,q)

D

( \sim \,p) \vee ( \sim \,q)

Q 85MathematicsSection A • MCQproperties of triangles

+4 / -1

For a triangle $ABC$ , the value of $\cos 2A + \cos 2B + \cos 2C$ is least. If its inradius is 3 and incentre is M, then which of the following is NOT correct?

A

\overrightarrow {MA} \,.\,\overrightarrow {MB} = - 18

B

\sin 2A + \sin 2B + \sin 2C = \sin A + \sin B + \sin C

C

perimeter of

\Delta ABC

is 18

\sqrt3

D

area of

\Delta ABC

is

{{27\sqrt 3 } \over 2}

Q 86MathematicsSection B • Numericalarea under the curves

+4 / -1

Let $A$ be the area bounded by the curve $y=x|x-3|$ , the $x$ -axis and the ordinates $x=-1$ and $x=2$ . Then $12 A$ is equal to ____________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 87MathematicsSection B • Numericaldefinite integration

+4 / -1

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that $f^{\prime}(x)+f(x)=\int_\limits{0}^{2} f(t) d t$ . If $f(0)=e^{-2}$ , then $2 f(0)-f(2)$ is equal to ____________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 88MathematicsSection B • Numericalpermutations and combinations

+4 / -1

The number of words, with or without meaning, that can be formed using all the letters of the word ASSASSINATION so that the vowels occur together, is ___________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 89MathematicsSection A • MCQdifferential equations

+4 / -1

The area enclosed by the closed curve $\mathrm{C}$ given by the differential equation

$\frac{d y}{d x}+\frac{x+a}{y-2}=0, y(1)=0$ is $4 \pi$ .

Let $P$ and $Q$ be the points of intersection of the curve $\mathrm{C}$ and the $y$ -axis. If normals at $P$ and $Q$ on the curve $\mathrm{C}$ intersect $x$ -axis at points $R$ and $S$ respectively, then the length of the line segment $R S$ is :

A

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

B

2 \sqrt{3}

C

2

D

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

Q 90MathematicsSection A • MCQfunctions

+4 / -1

Let $f(x) = \left| {\matrix{ {1 + {{\sin }^2}x} & {{{\cos }^2}x} & {\sin 2x} \cr {{{\sin }^2}x} & {1 + {{\cos }^2}x} & {\sin 2x} \cr {{{\sin }^2}x} & {{{\cos }^2}x} & {1 + \sin 2x} \cr } } \right|,\,x \in \left[ {{\pi \over 6},{\pi \over 3}} \right]$ . If $\alpha$ and $\beta$ respectively are the maximum and the minimum values of $f$ , then

A

{\alpha ^2} - {\beta ^2} = 4\sqrt 3

B

{\beta ^2} - 2\sqrt \alpha = {{19} \over 4}

C

{\beta ^2} + 2\sqrt \alpha = {{19} \over 4}

D

{\alpha ^2} + {\beta ^2} = {9 \over 2}

JEE Main 2023 1st February Morning Shift Question Paper with Solutions