JEE Main 2022 25th June Morning Shift Question Paper Question Paper with Solutions | JEE Main Mock Test | JeeMate

Q 1PhysicsSection A • MCQvector algebra

+4 / -1

Which of the following relations is true for two unit vector $\widehat A$ and $\widehat B$ making an angle $\theta$ to each other?

A

|\widehat A + \widehat B| = |\widehat A - \widehat B|\tan {\theta \over 2}

B

|\widehat A - \widehat B| = |\widehat A + \widehat B|\tan {\theta \over 2}

C

|\widehat A + \widehat B| = |\widehat A - \widehat B|cos{\theta \over 2}

D

|\widehat A - \widehat B| = |\widehat A + \widehat B|\cos {\theta \over 2}

Q 2PhysicsSection A • MCQrotational motion

+4 / -1

If force $\overrightarrow F = 3\widehat i + 4\widehat j - 2\widehat k$ acts on a particle position vector $2\widehat i + \widehat j + 2\widehat k$ then, the torque about the origin will be :

A

3\widehat i + 4\widehat j - 2\widehat k

B

- 10\widehat i + 10\widehat j + 5\widehat k

C

10\widehat i + 5\widehat j - 10\widehat k

D

10\widehat i + \widehat j - 5\widehat k

Q 3PhysicsSection A • MCQwave optics

+4 / -1

The two light beams having intensities I and 9I interfere to produce a fringe pattern on a screen. The phase difference between the beams is $\pi$ /2 at point P and $\pi$ at point Q. Then the difference between the resultant intensities at P and Q will be :

A

2 I

B

6 I

C

5 I

D

7 I

Q 4PhysicsSection A • MCQdual nature of radiation

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Given below are two statements :

Statement I : Davisson-Germer experiment establishes the wave nature of electrons.

Statement II : If electrons have wave nature, they can interfere and show diffraction.

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

A

Both Statement I and Statement II are true.

B

Both Statement I and Statement II are false.

C

Statement I is true but Statement II is false.

D

Statement I is false but Statement II is true.

Q 5PhysicsSection A • MCQray optics

+4 / -1

The difference of speed of light in the two media A and B (v_A $-$ v_B) is 2.6 $\times$ 10⁷ m/s. If the refractive index of medium B is 1.47, then the ratio of refractive index of medium B to medium A is : (Given : speed of light in vacuum c = 3 $\times$ 10⁸ ms^{$-$ 1})

A

1.303

B

1.318

C

1.13

D

0.12

Q 6PhysicsSection B • Numericalwork power and energy

+4 / -1

A uniform chain of 6 m length is placed on a table such that a part of its length is hanging over the edge of the table. The system is at rest. The co-efficient of static friction between the chain and the surface of the table is 0.5, the maximum length of the chain hanging from the table is ___________ m.

Answer Format:Enter numerical value rounded to the nearest integer

Q 7PhysicsSection A • MCQelectromagnetic waves

+4 / -1

The electric field in an electromagnetic wave is given by E = 56.5 sin $\omega$ (t $-$ x/c) NC^{$-$ 1}. Find the intensity of the wave if it is propagating along x-axis in the free space.

(Given : $\varepsilon$ ₀ = 8.85 $\times$ 10^{$-$ 12}C²N^{$-$ 1}m^{$-$ 2})

A

5.65 Wm^{$-$ 2}

B

4.24 Wm^{$-$ 2}

C

1.9

\times

10^{$-$ 7} Wm^{$-$ 2}

D

56.5 Wm^{$-$ 2}

Q 8PhysicsSection A • MCQmagnetics

+4 / -1

A long straight wire with a circular cross-section having radius R, is carrying a steady current I. The current I is uniformly distributed across this cross-section. Then the variation of magnetic field due to current I with distance r (r < R) from its centre will be :

A

B

\propto

r²

B

\propto

r

C

B

\propto

{1 \over {{r^2}}}

D

B

\propto

{1 \over {{r}}}

Q 9PhysicsSection A • MCQalternating current

+4 / -1

If wattless current flows in the AC circuit, then the circuit is :

A

Purely Resistive circuit

B

Purely Inductive circuit

C

LCR series circuit

D

RC series circuit only

Q 10PhysicsSection A • MCQelectrostatics

+4 / -1

In the figure, a very large plane sheet of positive charge is shown. P₁ and P₂ are two points at distance l and 2l from the charge distribution. If $\sigma$ is the surface charge density, then the magnitude of electric fields E₁ and E₂ at P₁ and P₂ respectively are :

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

A

{E_1} = \sigma /{\varepsilon _0},\,{E_2} = \sigma /2{\varepsilon _0}

B

{E_1} = 2\sigma /{\varepsilon _0},\,{E_2} = \sigma /{\varepsilon _0}

C

{E_1} = {E_2} = \sigma /2{\varepsilon _0}

D

{E_1} = {E_2} = \sigma /{\varepsilon _0}

Q 11PhysicsSection A • MCQunits and measurements

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If $Z = {{{A^2}{B^3}} \over {{C^4}}}$ , then the relative error in Z will be :

A

{{\Delta A} \over A} + {{\Delta B} \over B} + {{\Delta C} \over C}

B

{{2\Delta A} \over A} + {{3\Delta B} \over B} - {{4\Delta C} \over C}

C

{{2\Delta A} \over A} + {{3\Delta B} \over B} + {{4\Delta C} \over C}

D

{{\Delta A} \over A} + {{\Delta B} \over B} - {{\Delta C} \over C}

Q 12PhysicsSection A • MCQgravitation

+4 / -1

The height of any point P above the surface of earth is equal to diameter of earth. The value of acceleration due to gravity at point P will be : (Given g = acceleration due to gravity at the surface of earth).

A

g/2

B

g/4

C

g/3

D

g/9

Q 13PhysicsSection A • MCQray optics

+4 / -1

A light wave travelling linearly in a medium of dielectric constant 4, incidents on the horizontal interface separating medium with air. The angle of incidence for which the total intensity of incident wave will be reflected back into the same medium will be :

(Given : relative permeability of medium $\mu$ _r = 1)

A

10

^\circ

B

20

^\circ

C

30

^\circ

D

60

^\circ

Q 14PhysicsSection A • MCQelectronic devices

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The photodiode is used to detect the optical signals. These diodes are preferably operated in reverse biased mode because :

A

fractional change in majority carriers produce higher forward bias current

B

fractional change in majority carriers produce higher reverse bias current

C

fractional change in minority carriers produce higher forward bias current

D

fractional change in minority carriers produce higher reverse bias current

Q 15PhysicsSection A • MCQvector algebra

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$\overrightarrow A$ is a vector quantity such that $|\overrightarrow A |$ = non-zero constant. Which of the following expression is true for $\overrightarrow A$ ?

A

\overrightarrow A \,.\,\overrightarrow A = 0

B

\overrightarrow A \times \overrightarrow A < 0

C

\overrightarrow A \times \overrightarrow A = 0

D

\overrightarrow A \times \overrightarrow A > 0

Q 16PhysicsSection A • MCQproperties of matter

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The terminal velocity (v_t) of the spherical rain drop depends on the radius (r) of the spherical rain drop as :

A

r^1/2

B

r

C

r²

D

r³

Q 17PhysicsSection A • MCQcommunication systems

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A signal of 100 THz frequency can be transmitted with maximum efficiency by :

A

Coaxial cable

B

Optical fibre

C

Twisted pair of copper wires

D

Water

Q 18PhysicsSection A • MCQalternating current

+4 / -1

Match List-I with List-II.

	List - I		List -II
(A)	AC generator	(I)	Detects the presence of current in the circuit
(B)	Galvanometer	(II)	Converts mechanical energy into electrical energy
(C)	Transformer	(III)	Works on the principle of resonance in AC circuit
(D)	Metal detector	(IV)	Changes an alternating voltage for smaller or greater value

Choose the correct answer from the options given below :

A

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

B

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

C

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

D

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

Official Correct Answer:Option (A)

Step-by-Step Solution:

AC generator	$\to$	Converts mechanical energy into electrical energy
Galvanometer	$\to$	Detects the presence of current in the circuit
Transformer	$\to$	Change AC voltage for smaller or greater value
Metal detector	$\to$	Works on the principle of resonance in AC circuit

Q 19PhysicsSection A • MCQatoms and nuclei

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The ratio for the speed of the electron in the 3^rd orbit of He⁺ to the speed of the electron in the 3^rd orbit of hydrogen atom will be :

A

1 : 1

B

1 : 2

C

4 : 1

D

2 : 1

Q 20PhysicsSection B • Numericalcapacitor

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The equivalent capacitance between points A and B in below shown figure will be __________ $\mu$ F.

qImage-combination-of-capacitors-capacitor-electricity-Jeemate

Answer Format:Enter numerical value rounded to the nearest integer

Q 21PhysicsSection B • Numericalwork power and energy

+4 / -1

A 0.5 kg block moving at a speed of 12 ms^{$-$ 1} compresses a spring through a distance 30 cm when its speed is halved. The spring constant of the spring will be _______________ Nm^{$-$ 1}.

Answer Format:Enter numerical value rounded to the nearest integer

Q 22PhysicsSection B • Numericalcurrent electricity

+4 / -1

The total current supplied to the circuit as shown in figure by the 5 V battery is ____________ A.

qImage-combination-of-resistances-current-electricity-electricity-Jeemate

Answer Format:Enter numerical value rounded to the nearest integer

Q 23PhysicsSection B • Numericalcurrent electricity

+4 / -1

A resistor develops 300 J of thermal energy in 15 s, when a current of 2 A is passed through it. If the current increases to 3 A, the energy developed in 10 s is ____________ J.

Answer Format:Enter numerical value rounded to the nearest integer

Q 24PhysicsSection B • Numericallaws of motion

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A force on an object of mass 100 g is $\left( {10\widehat i + 5\widehat j} \right)$ N. The position of that object at t = 2 s is $\left( {a\widehat i + b\widehat j} \right)$ m after starting from rest. The value of ${a \over b}$ will be ___________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 25PhysicsSection A • MCQheat and thermodynamics

+4 / -1

The relation between root mean square speed (v_rms) and most probable sped (v_p) for the molar mass M of oxygen gas molecule at the temperature of 300 K will be :

A

{v_{rms}} = \sqrt {{2 \over 3}} {v_p}

B

{v_{rms}} = \sqrt {{3 \over 2}} {v_p}

C

{v_{rms}} = {v_p}

D

{v_{rms}} = \sqrt {{1 \over 3}} {v_p}

Q 26PhysicsSection A • MCQcurrent electricity

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A teacher in his physics laboratory allotted an experiment to determine the resistance (G) of a galvanometer. Students took the observations for ${1 \over 3}$ deflection in the galvanometer. Which of the below is true for measuring value of G?

A

{1 \over 3}

deflection method cannot be used for determining the resistance of the galvanometer.

B

{1 \over 3}

deflection method can be used and in this case the G equals to twice the value of shunt resistances.

C

{1 \over 3}

deflection method can be used and in this case, the G equals to three times the value of shunt resistances.

D

{1 \over 3}

deflection method can be used and in this case the G value equals to the shunt resistances.

Q 27PhysicsSection B • Numericalwaves

+4 / -1

The first overtone frequency of an open organ pipe is equal to the fundamental frequency of a closed organ pipe. If the length of the closed organ pipe is 20 cm. The length of the open organ pipe is _____________ cm.

Answer Format:Enter numerical value rounded to the nearest integer

Q 28PhysicsSection B • Numericalelectromagnetic induction

+4 / -1

The current in a coil of self inductance 2.0 H is increasing according to I = 2 sin(t²) A. The amount of energy spent during the period when current changes from 0 to 2 A is ____________ J.

Answer Format:Enter numerical value rounded to the nearest integer

Q 29PhysicsSection B • Numericalheat and thermodynamics

+4 / -1

A steam engine intakes 50 g of steam at 100 $^\circ$ C per minute and cools it down to 20 $^\circ$ C. If latent heat of vaporization of steam is 540 cal g^{$-$ 1}, then the heat rejected by the steam engine per minute is __________ $\times$ 10³ cal.

(Given : specific heat capacity of water : 1 cal g^{$-$ 1} $^\circ$ C^{$-$ 1})

Answer Format:Enter numerical value rounded to the nearest integer

Q 30PhysicsSection B • Numericalproperties of matter

+4 / -1

The velocity of upper layer of water in a river is 36 kmh^{$-$ 1}. Shearing stress between horizontal layers of water is 10^{$-$ 3} Nm^{$-$ 2}. Depth of the river is __________ m. (Co-efficient of viscosity of water is 10^{$-$ 2} Pa.s)

Answer Format:Enter numerical value rounded to the nearest integer

Q 31ChemistrySection A • MCQhaloalkanes and haloarenes

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The major product in the reaction

qImage-haloalkanes-haloalkanes-and-haloarenes-organic-chemistry-Jeemate

A

t-Butyl ethyl ether

B

2,2-Dimethyl butane

C

2-Methyl pent-1-ene

D

2-Methyl prop-1-ene

Q 32ChemistrySection B • Numericalsome basic concepts of chemistry

+4 / -1

1 L aqueous solution of H₂SO₄ contains 0.02 m mol H₂SO₄. 50% of this solution is diluted with deionized water to give 1 L solution (A). In solution (A), 0.01 m mol of H₂SO₄ are added. Total m mols of H₂SO₄ in the final solution is ___________ $\times$ 10³ m mols.

Answer Format:Enter numerical value rounded to the nearest integer

Q 33ChemistrySection A • MCQsurface chemistry

+4 / -1

Incorrect statement for Tyndall effect is :

A

The refractive indices of the dispersed phase and the dispersion medium differ greatly in magnitude.

B

The diameter of the dispersed particles is much smaller than the wavelength of the light used.

C

During projection of movies in the cinemas hall, Tyndall effect is noticed.

D

It is used to distinguish a true solution from a colloidal solution.

Q 34ChemistrySection B • Numericalchemical equilibrium

+4 / -1

The standard free energy change ( $\Delta$ G $^\circ$ ) for 50% dissociation of N₂O₄ into NO₂ at 27 $^\circ$ C and 1 atm pressure is $-$ x J mol^{$-$ 1}. The value of x is ___________. (Nearest Integer)

[Given : R = 8.31 J K^{$-$ 1} mol^{$-$ 1}, log 1.33 = 0.1239 ln 10 = 2.3]

Answer Format:Enter numerical value rounded to the nearest integer

Q 35ChemistrySection A • MCQchemical bonding and molecular structure

+4 / -1

Number of electron deficient molecules among the following

PH₃, B₂H₆, CCl₄, NH₃, LiH and BCl₃ is

A

0

B

1

C

2

D

3

Q 36ChemistrySection A • MCQd and f block elements

+4 / -1

Cerium (IV) has a noble gas configuration. Which of the following is correct statement about it?

A

It will not prefer to undergo redox reactions.

B

It will prefer to gain electron and act as an oxidizing agent.

C

It will prefer to give away an electron and behave as reducing agent.

D

It acts as both, oxidizing and reducing agent.

Q 37ChemistrySection A • MCQatomic structure

+4 / -1

The pair, in which ions are isoelectronic with AI³⁺ is :

A

Br^$-$ and Be²⁺

B

Cl^$-$ and Li⁺

C

S^{2 $-$} and K⁺

D

O^{2 $-$} and Mg²⁺

Q 38ChemistrySection B • Numericalelectrochemistry

+4 / -1

In a cell, the following reactions take place

$\matrix{ {F{e^{2 + }} \to F{e^{3 + }} + {e^ - }} & {E_{F{e^{3 + }}/F{e^{2 + }}}^o = 0.77\,V} \cr {2{I^ - } \to {I_2} + 2{e^ - }} & {E_{{I_2}/{I^ - }}^o = 0.54\,V} \cr }$

The standard electrode potential for the spontaneous reaction in the cell is x $\times$ 10^{$-$ 2} V 298 K. The value of x is ____________. (Nearest Integer)

Answer Format:Enter numerical value rounded to the nearest integer

Q 39ChemistrySection A • MCQsurface chemistry

+4 / -1

Using very little soap while washing clothes, does not serve the purpose of cleaning of clothes, because :

A

soap particles remain floating in water as ions.

B

the hydrophobic part of soap is not able to take away grease.

C

the micelles are not formed due to concentration of soap, below its CMC value.

D

colloidal structure of soap in water is completely disturbed.

Q 40ChemistrySection A • MCQp block elements

+4 / -1

White precipitate of AgCl dissolves in aqueous ammonia solution due to formation of :

A

[Ag(NH₃)₄]Cl₂

B

[Ag(Cl)₂(NH₃)₂]

C

[Ag(NH₃)₂]Cl

D

[Ag(NH₃)Cl]Cl

Q 41ChemistrySection A • MCQbasics of organic chemistry

+4 / -1

In the following structures, which on is having staggered conformation with maximum dihedral angle?

A

B

C

D

Q 42ChemistrySection A • MCQchemical bonding and molecular structure

+4 / -1

Bonding in which of the following diatomic molecule(s) become(s) stronger, on the basis of MO Theory, by removal of an electron?

(A) NO

(B) N₂

(C) O₂

(D) C₂

(E) B₂

Choose the most appropriate answer from the options given below :

A

(A), (B), (C) only

B

(B), (C), (E) only

C

(A), (C) only

D

(D) only

Q 43ChemistrySection A • MCQalcohols phenols and ethers

+4 / -1

The intermediate X, in the reaction :

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

A

B

C

D

Q 44ChemistrySection A • MCQalcohols phenols and ethers

+4 / -1

In the following reaction :

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

The compounds A and B respectively are :

A

B

C

D

Q 45ChemistrySection B • Numericalchemical kinetics and nuclear chemistry

+4 / -1

For a given chemical reaction

$\gamma$ ₁A + $\gamma$ ₂B $\to$ $\gamma$ ₃C + $\gamma$ ₄D

Concentration of C changes from 10 mmol dm^{$-$ 3} to 20 mmol dm^{$-$ 3} in 10 seconds. Rate of appearance of D is 1.5 times the rate of disappearance of B which is twice the rate of disappearance A. The rate of appearance of D has been experimentally determined to be 9 mmol dm^{$-$ 3} s^{$-$ 1}. Therefore, the rate of reaction is _____________ mmol dm^{$-$ 3} s^{$-$ 1}. (Nearest Integer)

Answer Format:Enter numerical value rounded to the nearest integer

Official Correct Answer:1

Step-by-Step Solution:

Rate

=\frac{1}{r_{1}}\left(\frac{-d[A]}{d t}\right)=\frac{1}{r_{2}}\left(\frac{-d[B]}{d t}\right)=\frac{1}{r_{3}}\left(\frac{-d[C]}{d t}\right)

=\frac{1}{r_{4}}\left(\frac{d[D]}{d t}\right)

\frac{\mathrm{d}[\mathrm{D}]}{\mathrm{dt}}=\frac{\mathrm{r}_{4}}{\mathrm{r}_{2}}\left(\frac{-\mathrm{d}[\mathrm{B}]}{\mathrm{dt}}\right)

\frac{r_{4}}{r_{2}}=\frac{3}{2}

\frac{-\mathrm{d}[\mathrm{B}]}{\mathrm{dt}}=\frac{\mathrm{r}_{2}}{\mathrm{r}_{1}}\left(\frac{-\mathrm{d}[\mathrm{A}]}{\mathrm{dt}}\right) \Rightarrow \frac{\mathrm{r}_{2}}{\mathrm{r}_{1}}=2

r_{2}=2 r_{1}

\begin{aligned} &\mathrm{r}_{4}=1.5 \mathrm{r}_{2}=3 \mathrm{r}_{1} \\ &\frac{\mathrm{d}[\mathrm{C}]}{\mathrm{dt}}=1 \mathrm{~m} \cdot \mathrm{mol} ~\mathrm{dm}^{-3} \mathrm{sec}^{-1} \\ &\frac{\mathrm{d}[\mathrm{D}]}{\mathrm{dt}}=9 \mathrm{~m} \cdot \mathrm{mol}~ \mathrm{dm}^{-3} \mathrm{sec}^{-1} \\ &\frac{\mathrm{d}[\mathrm{D}]}{\mathrm{dt}}=\frac{\mathrm{r}_{4}}{r_{3}} \cdot \frac{\mathrm{d}[\mathrm{C}]}{\mathrm{dt}} \Rightarrow \frac{\mathrm{r}_{4}}{\mathrm{r}_{3}}=9 \\ &\mathrm{r}_{4}=9 \mathrm{r}_{3}=3 \mathrm{r}_{1} \\ &\Rightarrow \mathrm{r}_{1}=3 \mathrm{r}_{3} \\ &\begin{array}{rr} 3 \mathrm{r}_{3} \mathrm{~A}+6 \mathrm{r}_{3} \mathrm{~B} \rightarrow \mathrm{r}_{3} \mathrm{C}+9 \mathrm{r}_{3} \mathrm{D} \end{array} \\ &\begin{aligned} \therefore \text { rate of reaction } &=\frac{1}{9} \times 9 \mathrm{~m} \cdot \mathrm{mol}~ \mathrm{dm}^{-3} \mathrm{sec}^{-1} \\ &=1 \mathrm{~m} \cdot \mathrm{mol}~ \mathrm{dm}^{-3} \mathrm{sec}^{-1} \end{aligned} \end{aligned}

Q 46ChemistrySection B • Numericalsome basic concepts of chemistry

+4 / -1

Number of grams of bromine that will completely react with 5.0 g of pent-1-ene is ___________ $\times$ 10^{$-$ 2} g. (Atomic mass of Br = 80 g/mol) [Nearest Integer]

Answer Format:Enter numerical value rounded to the nearest integer

Q 47ChemistrySection A • MCQchemistry in everyday life

+4 / -1

Which one of the following is an example of artificial sweetner?

A

Bithional

B

Alitame

C

Salvarsan

D

Lactose

Q 48ChemistrySection B • Numericalatomic structure

+4 / -1

The longest wavelength of light that can be used for the ionisation of lithium atom (Li) in its ground state is x $\times$ 10^{$-$ 8} m. The value of x is ___________. (Nearest Integer).

(Given : Energy of the electron in the first shell of the hydrogen atom is $-$ 2.2 $\times$ 10^{$-$ 18} J ; h = 6.63 $\times$ 10^{$-$ 34} Js and c = 3 $\times$ 10⁸ ms^{$-$ 1})

Answer Format:Enter numerical value rounded to the nearest integer

Q 49ChemistrySection A • MCQbasics of organic chemistry

+4 / -1

Phenol on reaction with dilute nitric acid, gives two products. Which method will be most efficient for large scale separation?

A

Chromatographic separation

B

Fractional Crystallisation

C

Steam distillation

D

Sublimation

Q 50ChemistrySection B • Numericalsolid state

+4 / -1

The distance between Na⁺ and Cl^$-$ ions in solid NaCl of density 43.1 g cm^{$-$ 3} is _______________ $\times$ 10^{$-$ 10} m. (Nearest Integer)

(Given : N_A = 6.02 $\times$ 10²³ mol^{$-$ 1})

Answer Format:Enter numerical value rounded to the nearest integer

Q 51ChemistrySection A • MCQbasics of organic chemistry

+4 / -1

The IUPAC name of ethylidene chloride is :

A

1-Chloroethene

B

1-Chloroethyne

C

1,2-Dichloroethane

D

1,1-Dichloroethane

Q 52ChemistrySection A • MCQhydrocarbons

+4 / -1

The product formed in the following reaction.

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

A

B

C

D

Q 53ChemistrySection A • MCQs block elements

+4 / -1

Which one of the following alkaline earth metal ions has the highest ionic mobility in its aqueous solution?

A

Be²⁺

B

Mg²⁺

C

Ca²⁺

D

Sr²⁺

Q 54ChemistrySection B • Numericalsome basic concepts of chemistry

+4 / -1

The number of N atoms in 681 g of C₇H₅N₃O₆ is x $\times$ 10²¹. The value of x is (N_A = 6.02 $\times$ 10²³ mol^{$-$ 1}) (Nearest Integer)

Answer Format:Enter numerical value rounded to the nearest integer

Q 55ChemistrySection A • MCQenvironmental chemistry

+4 / -1

The eutrophication of water body results in :

A

loss of Biodiversity.

B

breakdown of organic matter.

C

increase in biodiversity.

D

decrease in BOD.

Q 56ChemistrySection B • Numericalthermodynamics

+4 / -1

The standard entropy change for the reaction

4Fe(s) + 3O₂(g) $\to$ 2Fe₂O₃(s) is $-$ 550 J K^{$-$ 1} at 298 K.

[Given : The standard enthalpy change for the reaction is $-$ 165 kJ mol^{$-$ 1}]. The temperature in K at which the reaction attains equilibrium is _____________. (Nearest Integer)

Answer Format:Enter numerical value rounded to the nearest integer

Q 57ChemistrySection A • MCQd and f block elements

+4 / -1

Among the following, which is the strongest oxidizing agent?

A

Mn³⁺

B

Fe³⁺

C

Ti³⁺

D

Cr³⁺

Q 58ChemistrySection A • MCQisolation of elements

+4 / -1

Leaching of gold with dilute aqueous solution of NaCN in presence of oxygen gives complex [A], which on reaction with zinc forms the elemental gold and another complex [B]. [A] and [B], respectively are :

A

{[Au{(CN)_4}]^ - }

and

{[Zn{(CN)_2}{(OH)_2}]^{2 - }}

B

{[Au{(CN)_2}]^ - }

and

{[Zn{(OH)_4}]^{2 - }}

C

{[Au{(CN)_2}]^{ - }}

and

{[Zn{(CN)_4}]^{2 - }}

D

{[Au{(CN)_4}]^{2 - }}

and

{[Zn{(CN)_6}]^{4 - }}

Q 59ChemistrySection B • Numericalcoordination compounds

+4 / -1

If [Cu(H₂O)₄]²⁺ absorbs a light of wavelength 600 nm for d-d transition, then the value of octahedral crystal field splitting energy for [Cu(H₂O)₆]²⁺ will be ____________ $\times$ 10^{$-$ 21} J. [Nearest Integer]

(Given : h = 6.63 $\times$ 10^{$-$ 34} Js and c = 3.08 $\times$ 10⁸ ms^{$-$ 1})

Answer Format:Enter numerical value rounded to the nearest integer

Q 60ChemistrySection A • MCQcompounds containing nitrogen

+4 / -1

The reaction of qImage-aliphatic-amines-compounds-containing-nitrogen-organic-chemistry-Jeemate with bromine and KOH gives RNH₂ as the end product. Which one of the following is the intermediate product formed in this reation?

A

B

R

-

NH

-

Br

C

R

-

N = C = O

D

Q 61MathematicsSection A • MCQcircle

+4 / -1

Let a circle C touch the lines

{L_1}:4x - 3y + {K_1} = 0

and

{L_2} = 4x - 3y + {K_2} = 0

,

{K_1},{K_2} \in R

. If a line passing through the centre of the circle C intersects L₁ at

( - 1,2)

and L₂ at

(3, - 6)

, then the equation of the circle C is :

A

{(x - 1)^2} + {(y - 2)^2} = 4

B

{(x + 1)^2} + {(y - 2)^2} = 4

C

{(x - 1)^2} + {(y + 2)^2} = 16

D

{(x - 1)^2} + {(y - 2)^2} = 16

Q 62MathematicsSection A • MCQfunctions

+4 / -1

Let f : N $\to$ R be a function such that $f(x + y) = 2f(x)f(y)$ for natural numbers x and y. If f(1) = 2, then the value of $\alpha$ for which

$\sum\limits_{k = 1}^{10} {f(\alpha + k) = {{512} \over 3}({2^{20}} - 1)}$

holds, is :

A

2

B

3

C

4

D

6

Q 63MathematicsSection A • MCQmatrices and determinants

+4 / -1

Let A be a 3 $\times$ 3 real matrix such that

$A\left( {\matrix{ 1 \cr 1 \cr 0 \cr } } \right) = \left( {\matrix{ 1 \cr 1 \cr 0 \cr } } \right);A\left( {\matrix{ 1 \cr 0 \cr 1 \cr } } \right) = \left( {\matrix{ { - 1} \cr 0 \cr 1 \cr } } \right)$ and $A\left( {\matrix{ 0 \cr 0 \cr 1 \cr } } \right) = \left( {\matrix{ 1 \cr 1 \cr 2 \cr } } \right)$ .

If $X = {({x_1},{x_2},{x_3})^T}$ and I is an identity matrix of order 3, then the system $(A - 2I)X = \left( {\matrix{ 4 \cr 1 \cr 1 \cr } } \right)$ has :

A

no solution

B

infinitely many solutions

C

unique solution

D

exactly two solutions

Official Correct Answer:Option (B)

Step-by-Step Solution:

Let $A = \left[ {\matrix{ a & b & c \cr d & e & f \cr g & h & i \cr } } \right]$

$A = \left[ {\matrix{ 1 \cr 1 \cr 0 \cr } } \right] = \left[ {\matrix{ 1 \cr 1 \cr 0 \cr } } \right] \Rightarrow \left[ {\matrix{ a & b & c \cr d & e & f \cr g & h & i \cr } } \right]\left[ {\matrix{ 1 \cr 1 \cr 0 \cr } } \right] = \left[ {\matrix{ 1 \cr 1 \cr 0 \cr } } \right] \Rightarrow \matrix{ {a + b = 1} \cr {d + e = 1} \cr {g + h = 0} \cr }$

$A = \left[ {\matrix{ 1 \cr 0 \cr 1 \cr } } \right] = \left[ {\matrix{ { - 1} \cr 0 \cr 1 \cr } } \right] \Rightarrow \left[ {\matrix{ a & b & c \cr d & e & f \cr g & h & i \cr } } \right]\left[ {\matrix{ 1 \cr 0 \cr 1 \cr } } \right] = \left[ {\matrix{ { - 1} \cr 0 \cr 1 \cr } } \right] \Rightarrow \matrix{ {a + c = - 1} \cr {d + f = 1} \cr {g + i = 0} \cr }$

$A = \left[ {\matrix{ 0 \cr 0 \cr 1 \cr } } \right] = \left[ {\matrix{ 1 \cr 1 \cr 2 \cr } } \right] \Rightarrow \left[ {\matrix{ a & b & c \cr d & e & f \cr g & h & i \cr } } \right]\left[ {\matrix{ 0 \cr 0 \cr 1 \cr } } \right] = \left[ {\matrix{ 1 \cr 1 \cr 2 \cr } } \right] \Rightarrow \matrix{ {c = 1} \cr {f = 1} \cr {i = 2} \cr }$

Solving will get

$a = - 2,\,b = 3,\,c = 1,\,d = - 1,\,e = 2,\,f = 1,\,g = - 1,\,h = 1,\,i = 2$

$A = \left[ {\matrix{ { - 2} & 3 & 1 \cr { - 1} & 2 & 1 \cr { - 1} & 1 & 2 \cr } } \right] \Rightarrow A = 2I = \left[ {\matrix{ { - 4} & 3 & 1 \cr { - 1} & 0 & 1 \cr { - 1} & 1 & 0 \cr } } \right]$

$(A - 2I)x = \left[ {\matrix{ 4 \cr 1 \cr 1 \cr } } \right]$

$\Rightarrow - 4{x_1} + 3{x_2} + {x_3} = 4$ ..... (i)

$- {x_1} + {x_3} = 1$ ...... (ii)

$- {x_1} + {x_2} = 1$ ...... (iii)

So 3(iii) + (ii) = (i)

$\therefore$ Infinite solution

Q 64MathematicsSection A • MCQ3d geometry

+4 / -1

Let Q be the mirror image of the point P(1, 0, 1) with respect to the plane S : x + y + z = 5. If a line L passing through (1, $-$ 1, $-$ 1), parallel to the line PQ meets the plane S at R, then QR² is equal to :

A

2

B

5

C

7

D

11

Q 65MathematicsSection B • Numericalcircle

+4 / -1

Let the abscissae of the two points P and Q be the roots of $2{x^2} - rx + p = 0$ and the ordinates of P and Q be the roots of ${x^2} - sx - q = 0$ . If the equation of the circle described on PQ as diameter is $2({x^2} + {y^2}) - 11x - 14y - 22 = 0$ , then $2r + s - 2q + p$ is equal to __________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 66MathematicsSection B • Numericalvector algebra

+4 / -1

Let $\theta$ be the angle between the vectors $\overrightarrow a$ and $\overrightarrow b$ , where $|\overrightarrow a | = 4,$ $|\overrightarrow b | = 3$ and $\theta \in \left( {{\pi \over 4},{\pi \over 3}} \right)$ . Then ${\left| {\left( {\overrightarrow a - \overrightarrow b } \right) \times \left( {\overrightarrow a + \overrightarrow b } \right)} \right|^2} + 4{\left( {\overrightarrow a \,.\,\overrightarrow b } \right)^2}$ is equal to __________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 67MathematicsSection B • Numericalpermutations and combinations

+4 / -1

The number of 3-digit odd numbers, whose sum of digits is a multiple of 7, is _____________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 68MathematicsSection A • MCQdifferential equations

+4 / -1

Let $g:(0,\infty ) \to R$ be a differentiable function such that

$\int {\left( {{{x(\cos x - \sin x)} \over {{e^x} + 1}} + {{g(x)\left( {{e^x} + 1 - x{e^x}} \right)} \over {{{({e^x} + 1)}^2}}}} \right)dx = {{x\,g(x)} \over {{e^x} + 1}} + c}$ , for all x > 0, where c is an arbitrary constant. Then :

A

g is decreasing in

\left( {0,{\pi \over 4}} \right)

B

g' is increasing in

\left( {0,{\pi \over 4}} \right)

C

g + g' is increasing in

\left( {0,{\pi \over 2}} \right)

D

g

-

g' is increasing in

\left( {0,{\pi \over 2}} \right)

Official Correct Answer:Option (D)

Step-by-Step Solution:

\int\left(\frac{x(\cos x-\sin x)}{e^x+1}+\frac{g(x)\left(e^x+1-x e^x\right)}{\left(e^x+1\right)^2}\right) d x=\frac{x g(x)}{e^x+1}+c

On differentiating both sides w.r.t.

\mathrm{x}

, we get

\begin{aligned} &\left(\frac{x(\cos x-\sin x)}{e^x+1}+\frac{g(x)\left(e^x+1-x e^x\right.}{\left(e^x+1\right)^2}\right) \\ &=\frac{\left(e^x+1\right)\left(g(x)+x g^{\prime}(x)\right)-e^x \cdot x \cdot g(x)}{\left(e^x+1\right)^2} \\ &\left(e^x+1\right) x(\cos x-\sin x)+g(x)\left(e^x+1-x e^x\right) \\ &=\left(e^x+1\right)\left(g(x)+x g^{\prime}(x)\right)-e^x \cdot x \cdot g(x) \\ &\Rightarrow g g^{\prime}(x)=\cos x-\sin x \\ &\Rightarrow g(x)=\sin x+\cos x+C \end{aligned}

\mathrm{g}(\mathrm{x})

is increasing in

(0, \pi / 4)

g "(x)=-\sin x-\cos x<0

\Rightarrow \mathrm{g}^{\prime}(\mathrm{x})

is decreasing function

let

h(x)=g(x)+g^{\prime}(x)=2 \cos x+C

\Rightarrow \mathrm{h}^{\prime}(\mathrm{x})=\mathrm{g}^{\prime}(\mathrm{x})+\mathrm{g}^{\prime \prime}(\mathrm{x})=-2 \sin \mathrm{x}<0

\Rightarrow \mathrm{h}

is decreasing

let

\phi(\mathrm{x})=\mathrm{g}(\mathrm{x})-\mathrm{g}^{\prime}(\mathrm{x})=2 \sin \mathrm{x}+\mathrm{C}

\Rightarrow \phi^{\prime}(\mathrm{x})=\mathrm{g}^{\prime}(\mathrm{x})-\mathrm{g}{ }^{\prime \prime}(\mathrm{x})=2 \cos \mathrm{x}>0

\Rightarrow \phi

is increasing

Hence option D is correct.

Q 69MathematicsSection A • MCQdefinite integration

+4 / -1

The value of $\int\limits_0^\pi {{{{e^{\cos x}}\sin x} \over {(1 + {{\cos }^2}x)({e^{\cos x}} + {e^{ - \cos x}})}}dx}$ is equal to:

A

{{{\pi ^2}} \over 4}

B

{{{\pi ^2}} \over 2}

C

{\pi \over 4}

D

{\pi \over 2}

Q 70MathematicsSection B • Numericaltrigonometric functions and equations

+4 / -1

The number of values of x in the interval $\left( {{\pi \over 4},{{7\pi } \over 4}} \right)$ for which

$14\cos e{c^2}x - 2{\sin ^2}x = 21 - 4{\cos ^2}x$ holds, is ____________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 71MathematicsSection A • MCQdifferentiation

+4 / -1

Let f : R $\to$ R be defined as $f(x) = {x^3} + x - 5$ . If g(x) is a function such that $f(g(x)) = x,\forall 'x' \in R$ , then g'(63) is equal to ________________.

A

{1 \over {49}}

B

{3 \over {49}}

C

{43 \over {49}}

D

{91 \over {49}}

Q 72MathematicsSection A • MCQdifferential equations

+4 / -1

If the solution curve $y = y(x)$ of the differential equation ${y^2}dx + ({x^2} - xy + {y^2})dy = 0$ , which passes through the point (1, 1) and intersects the line $y = \sqrt 3 x$ at the point $(\alpha ,\sqrt 3 \alpha )$ , then value of ${\log _e}(\sqrt 3 \alpha )$ is equal to :

A

{\pi \over 3}

B

{\pi \over 2}

C

{\pi \over 12}

D

{\pi \over 6}

Q 73MathematicsSection A • MCQmatrices and determinants

+4 / -1

Let $A = \left[ {\matrix{ 0 & { - 2} \cr 2 & 0 \cr } } \right]$ . If M and N are two matrices given by $M = \sum\limits_{k = 1}^{10} {{A^{2k}}}$ and $N = \sum\limits_{k = 1}^{10} {{A^{2k - 1}}}$ then MN² is :

A

a non-identity symmetric matrix

B

a skew-symmetric matrix

C

neither symmetric nor skew-symmetric matrix

D

an identity matrix

Q 74MathematicsSection B • Numericalsequences and series

+4 / -1

For a natural number n, let ${\alpha _n} = {19^n} - {12^n}$ . Then, the value of ${{31{\alpha _9} - {\alpha _{10}}} \over {57{\alpha _8}}}$ is ___________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 75MathematicsSection A • MCQfunctions

+4 / -1

Let $f:R \to R$ and $g:R \to R$ be two functions defined by $f(x) = {\log _e}({x^2} + 1) - {e^{ - x}} + 1$ and $g(x) = {{1 - 2{e^{2x}}} \over {{e^x}}}$ . Then, for which of the following range of $\alpha$ , the inequality $f\left( {g\left( {{{{{(\alpha - 1)}^2}} \over 3}} \right)} \right) > f\left( {g\left( {\alpha -{5 \over 3}} \right)} \right)$ holds ?

A

(2, 3)

B

(

-

2,

-

1)

C

(1, 2)

D

(

-

1, 1)

Q 76MathematicsSection A • MCQproperties of triangles

+4 / -1

Let a, b and c be the length of sides of a triangle ABC such that ${{a + b} \over 7} = {{b + c} \over 8} = {{c + a} \over 9}$ . If r and R are the radius of incircle and radius of circumcircle of the triangle ABC, respectively, then the value of ${R \over r}$ is equal to :

A

{5 \over 2}

B

2

C

{3 \over 2}

D

1

Q 77MathematicsSection A • MCQvector algebra

+4 / -1

Let $\overrightarrow a = {a_1}\widehat i + {a_2}\widehat j + {a_3}\widehat k$ ${a_i} > 0$ , $i = 1,2,3$ be a vector which makes equal angles with the coordinate axes OX, OY and OZ. Also, let the projection of $\overrightarrow a$ on the vector $3\widehat i + 4\widehat j$ be 7. Let $\overrightarrow b$ be a vector obtained by rotating $\overrightarrow a$ with 90 $^\circ$ . If $\overrightarrow a$ , $\overrightarrow b$ and x-axis are coplanar, then projection of a vector $\overrightarrow b$ on $3\widehat i + 4\widehat j$ is equal to:

A

\sqrt 7

B

\sqrt 2

C

2

D

7

Q 78MathematicsSection B • Numericalfunctions

+4 / -1

Let $f:R \to R$ be a function defined by

$f(x) = {\left( {2\left( {1 - {{{x^{25}}} \over 2}} \right)(2 + {x^{25}})} \right)^{{1 \over {50}}}}$ . If the function $g(x) = f(f(f(x))) + f(f(x))$ , then the greatest integer less than or equal to g(1) is ____________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 79MathematicsSection A • MCQprobability

+4 / -1

Let E₁ and E₂ be two events such that the conditional probabilities $P({E_1}|{E_2}) = {1 \over 2}$ , $P({E_2}|{E_1}) = {3 \over 4}$ and $P({E_1} \cap {E_2}) = {1 \over 8}$ . Then :

A

P({E_1} \cap {E_2}) = P({E_1})\,.\,P({E_2})

B

P(E{'_1} \cap E{'_2}) = P(E{'_1})\,.\,P(E{_2})

C

P({E_1} \cap E{'_2}) = P({E_1})\,.\,P({E_2})

D

P(E{'_1} \cap {E_2}) = P({E_1})\,.\,P({E_2})

Q 80MathematicsSection A • MCQparabola

+4 / -1

If $y = {m_1}x + {c_1}$ and $y = {m_2}x + {c_2}$ , ${m_1} \ne {m_2}$ are two common tangents of circle ${x^2} + {y^2} = 2$ and parabola y² = x, then the value of $8|{m_1}{m_2}|$ is equal to :

A

3 + 4\sqrt 2

B

- 5 + 6\sqrt 2

C

- 4 + 3\sqrt 2

D

7 + 6\sqrt 2

Q 81MathematicsSection A • MCQmathematical reasoning

+4 / -1

Consider the following two propositions:

$P1: \sim (p \to \sim q)$

$P2:(p \wedge \sim q) \wedge (( \sim p) \vee q)$

If the proposition $p \to (( \sim p) \vee q)$ is evaluated as FALSE, then :

A

P1 is TRUE and P2 is FALSE

B

P1 is FALSE and P2 is TRUE

C

Both P1 and P2 are FALSE

D

Both P1 and P2 are TRUE

Q 82MathematicsSection B • Numericalpermutations and combinations

+4 / -1

Let A be a 3 $\times$ 3 matrix having entries from the set { $-$ 1, 0, 1}. The number of all such matrices A having sum of all the entries equal to 5, is ___________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 83MathematicsSection B • Numericalsequences and series

+4 / -1

The greatest integer less than or equal to the sum of first 100 terms of the sequence ${1 \over 3},{5 \over 9},{{19} \over {27}},{{65} \over {81}},$ ...... is equal to ___________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 84MathematicsSection A • MCQcomplex numbers

+4 / -1

Let a circle C in complex plane pass through the points ${z_1} = 3 + 4i$ , ${z_2} = 4 + 3i$ and ${z_3} = 5i$ . If $z( \ne {z_1})$ is a point on C such that the line through z and z₁ is perpendicular to the line through z₂ and z₃, then $arg(z)$ is equal to :

A

{\tan ^{ - 1}}\left( {{2 \over {\sqrt 5 }}} \right) - \pi

B

{\tan ^{ - 1}}\left( {{{24} \over 7}} \right) - \pi

C

{\tan ^{ - 1}}\left( 3 \right) - \pi

D

{\tan ^{ - 1}}\left( {{3 \over 4}} \right) - \pi

Q 85MathematicsSection A • MCQdifferential equations

+4 / -1

Let $y = y(x)$ be the solution of the differential equation $(x + 1)y' - y = {e^{3x}}{(x + 1)^2}$ , with $y(0) = {1 \over 3}$ . Then, the point $x = - {4 \over 3}$ for the curve $y = y(x)$ is :

A

not a critical point

B

a point of local minima

C

a point of local maxima

D

a point of inflection

Q 86MathematicsSection B • Numerical3d geometry

+4 / -1

Let the lines

${L_1}:\overrightarrow r = \lambda \left( {\widehat i + 2\widehat j + 3\widehat k} \right),\,\lambda \in R$

${L_2}:\overrightarrow r = \left( {\widehat i + 3\widehat j + \widehat k} \right) + \mu \left( {\widehat i + \widehat j + 5\widehat k} \right);\,\mu \in R$ ,

intersect at the point S. If a plane ax + by $-$ z + d = 0 passes through S and is parallel to both the lines L₁ and L₂, then the value of a + b + d is equal to ____________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 87MathematicsSection A • MCQlimits continuity and differentiability

+4 / -1

Let f(x) be a polynomial function such that $f(x) + f'(x) + f''(x) = {x^5} + 64$ . Then, the value of $\mathop {\lim }\limits_{x \to 1} {{f(x)} \over {x - 1}}$ is equal to:

A

-

15

B

-

60

C

60

D

15

Q 88MathematicsSection A • MCQbinomial theorem

+4 / -1

If ${1 \over {2\,.\,{3^{10}}}} + {1 \over {{2^2}\,.\,{3^9}}} + \,\,.....\,\, + \,\,{1 \over {{2^{10}}\,.\,3}} = {K \over {{2^{10}}\,.\,{3^{10}}}}$ , then the remainder when K is divided by 6 is :

A

1

B

2

C

3

D

5

Q 89MathematicsSection A • MCQparabola

+4 / -1

Let $x = 2t$ , $y = {{{t^2}} \over 3}$ be a conic. Let S be the focus and B be the point on the axis of the conic such that $SA \bot BA$ , where A is any point on the conic. If k is the ordinate of the centroid of the $\Delta$ SAB, then $\mathop {\lim }\limits_{t \to 1} k$ is equal to :

A

{{17} \over {18}}

B

{{19} \over {18}}

C

{{11} \over {18}}

D

{{13} \over {18}}

Q 90MathematicsSection B • Numericalbinomial theorem

+4 / -1

Let C_r denote the binomial coefficient of x^r in the expansion of ${(1 + x)^{10}}$ . If for $\alpha$ , $\beta$ $\in$ R, ${C_1} + 3.2{C_2} + 5.3{C_3} +$ ....... upto 10 terms $= {{\alpha \times {2^{11}}} \over {{2^\beta } - 1}}\left( {{C_0} + {{{C_1}} \over 2} + {{{C_2}} \over 3} + \,\,.....\,\,upto\,10\,terms} \right)$ then the value of $\alpha$ + $\beta$ is equal to ___________.

Answer Format:Enter numerical value rounded to the nearest integer

JEE Main 2022 25th June Morning Shift Question Paper with Solutions