JEE Main 2022 29th July Evening Shift Question Paper Question Paper with Solutions | JEE Main Mock Test | JeeMate

Q 1PhysicsSection A • MCQlaws of motion

+4 / -1

Two bodies of masses $m_{1}=5 \mathrm{~kg}$ and $m_{2}=3 \mathrm{~kg}$ are connected by a light string going over a smooth light pulley on a smooth inclined plane as shown in the figure. The system is at rest. The force exerted by the inclined plane on the body of mass $\mathrm{m}_{1}$ will be : [Take $\left.\mathrm{g}=10 \mathrm{~ms}^{-2}\right]$

qImage-motion-of-connected-bodies-laws-of-motion-mechanics-Jeemate

A

30 N

B

40 N

C

50 N

D

60 N

Q 2PhysicsSection A • MCQunits and measurements

+4 / -1

Match List I with List II.

	List I		List II
A.	Torque	I.	Nms $^{ - 1}$
B.	Stress	II.	J kg $^{ - 1}$
C.	Latent Heat	III.	Nm
D.	Power	IV.	Nm $^{ - 2}$

Choose the correct answer from the options given below :

A

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

B

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

C

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

D

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

Q 3PhysicsSection A • MCQmagnetic properties of matter

+4 / -1

The vertical component of the earth's magnetic field is $6 \times 10^{-5} \mathrm{~T}$ at any place where the angle of dip is $37^{\circ}$ . The earth's resultant magnetic field at that place will be $\left(\right.$ Given $\left.\tan 37^{\circ}=\frac{3}{4}\right)$

A

8 \times 10^{-5} \mathrm{~T}

B

6 \times 10^{-5} \mathrm{~T}

C

5 \times 10^{-4} \mathrm{~T}

D

1 \times 10^{-4} \mathrm{~T}

Q 4PhysicsSection B • Numericalproperties of matter

+4 / -1

The velocity of a small ball of mass $0.3 \mathrm{~g}$ and density $8 \mathrm{~g} / \mathrm{cc}$ when dropped in a container filled with glycerine becomes constant after some time. If the density of glycerine is $1.3 \mathrm{~g} / \mathrm{cc}$ , then the value of viscous force acting on the ball will be $x \times 10^{-4} \mathrm{~N}$ , The value of $x$ is _________. [use $\left.\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}\right]$

Answer Format:Enter numerical value rounded to the nearest integer

Q 5PhysicsSection A • MCQmotion

+4 / -1

A juggler throws balls vertically upwards with same initial velocity in air. When the first ball reaches its highest position, he throws the next ball. Assuming the juggler throws n balls per second, the maximum height the balls can reach is

A

g/2n

B

g/n

C

2gn

D

g/2n²

Q 6PhysicsSection B • Numericalproperties of matter

+4 / -1

A tube of length $50 \mathrm{~cm}$ is filled completely with an incompressible liquid of mass $250 \mathrm{~g}$ and closed at both ends. The tube is then rotated in horizontal plane about one of its ends with a uniform angular velocity $x \sqrt{F} \,\mathrm{rad} \,\mathrm{s}^{-1}$ . If $\mathrm{F}$ be the force exerted by the liquid at the other end then the value of $x$ will be __________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 7PhysicsSection A • MCQmotion

+4 / -1

A ball is released from a height h. If $t_{1}$ and $t_{2}$ be the time required to complete first half and second half of the distance respectively. Then, choose the correct relation between $t_{1}$ and $t_{2}$ .

A

t_{1}=(\sqrt{2}) t_{2}

B

t_{1}=(\sqrt{2}-1) t_{2}

C

t_{2}=(\sqrt{2}+1) t_{1}

D

t_{2}=(\sqrt{2}-1) t_{1}

Q 8PhysicsSection B • Numericalsimple harmonic motion

+4 / -1

The metallic bob of simple pendulum has the relative density 5. The time period of this pendulum is $10 \mathrm{~s}$ . If the metallic bob is immersed in water, then the new time period becomes $5 \sqrt{x}$ s. The value of $x$ will be ________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 9PhysicsSection B • Numericalelectronic devices

+4 / -1

A $8 \mathrm{~V}$ Zener diode along with a series resistance $\mathrm{R}$ is connected across a $20 \mathrm{~V}$ supply (as shown in the figure). If the maximum Zener current is $25 \mathrm{~mA}$ , then the minimum value of R will be _______ $\Omega$ .

qImage-semiconductor-and-p-n-junction-diode-electronic-devices-modern-physics-Jeemate

Answer Format:Enter numerical value rounded to the nearest integer

Q 10PhysicsSection B • Numericalelectromagnetic waves

+4 / -1

Nearly 10% of the power of a $110 \mathrm{~W}$ light bulb is converted to visible radiation. The change in average intensities of visible radiation, at a distance of $1 \mathrm{~m}$ from the bulb to a distance of $5 \mathrm{~m}$ is $a \times 10^{-2} \mathrm{~W} / \mathrm{m}^{2}$ . The value of 'a' will be _________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 11PhysicsSection B • Numericalcommunication systems

+4 / -1

A modulating signal $2 \sin \left(6.28 \times 10^{6}\right) t$ is added to the carrier signal $4 \sin \left(12.56 \times 10^{9}\right) t$ for amplitude modulation. The combined signal is passed through a non-linear square law device. The output is then passed through a band pass filter. The bandwidth of the output signal of band pass filter will be _________ $\mathrm{MHz}$ .

Answer Format:Enter numerical value rounded to the nearest integer

Q 12PhysicsSection A • MCQcurrent electricity

+4 / -1

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

Assertion A: Alloys such as constantan and manganin are used in making standard resistance coils.

Reason R: Constantan and manganin have very small value of temperature coefficient of resistance.

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

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

C

A is true but R is false.

D

A is false but R is true.

Q 13PhysicsSection A • MCQheat and thermodynamics

+4 / -1

The root mean square speed of smoke particles of mass $5 \times 10^{-17} \mathrm{~kg}$ in their Brownian motion in air at NTP is approximately. [Given $\mathrm{k}=1.38 \times 10^{-23} \mathrm{JK}^{-1}$ ]

A

60 \mathrm{~mm} \mathrm{~s}^{-1}

B

12 \mathrm{~mm} \mathrm{~s}^{-1}

C

15 \mathrm{~mm} \mathrm{~s}^{-1}

D

36 \mathrm{~mm} \mathrm{~s}^{-1}

Q 14PhysicsSection B • Numericalproperties of matter

+4 / -1

The speed of a transverse wave passing through a string of length $50 \mathrm{~cm}$ and mass $10 \mathrm{~g}$ is $60 \mathrm{~ms}^{-1}$ . The area of cross-section of the wire is $2.0 \mathrm{~mm}^{2}$ and its Young's modulus is $1.2 \times 10^{11} \mathrm{Nm}^{-2}$ . The extension of the wire over its natural length due to its tension will be $x \times 10^{-5} \mathrm{~m}$ . The value of $x$ is __________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 15PhysicsSection A • MCQalternating current

+4 / -1

A circuit element $\mathrm{X}$ when connected to an a.c. supply of peak voltage $100 \mathrm{~V}$ gives a peak current of $5 \mathrm{~A}$ which is in phase with the voltage. A second element $\mathrm{Y}$ when connected to the same a.c. supply also gives the same value of peak current which lags behind the voltage by $\frac{\pi}{2}$ . If $\mathrm{X}$ and $\mathrm{Y}$ are connected in series to the same supply, what will be the rms value of the current in ampere?

A

\frac{10}{\sqrt{2}}

B

\frac{5}{\sqrt{2}}

C

5 \sqrt{2}

D

\frac{5}{2}

Q 16PhysicsSection A • MCQcenter of mass

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If momentum of a body is increased by 20%, then its kinetic energy increases by

A

36%

B

40%

C

44%

D

48%

Q 17PhysicsSection A • MCQwave optics

+4 / -1

An unpolarised light beam of intensity $2 I_{0}$ is passed through a polaroid P and then through another polaroid Q which is oriented in such a way that its passing axis makes an angle of $30^{\circ}$ relative to that of P. The intensity of the emergent light is

A

\frac{\mathrm{I}_{0}}{4}

B

\frac{\mathrm{I}_{0}}{2}

C

\frac{3 I_{0}}{4}

D

\frac{3 \mathrm{I}_{0}}{2}

Q 18PhysicsSection A • MCQray optics

+4 / -1

Light enters from air into a given medium at an angle of $45^{\circ}$ with interface of the air-medium surface. After refraction, the light ray is deviated through an angle of $15^{\circ}$ from its original direction. The refractive index of the medium is:

A

1.732

B

1.333

C

1.414

D

2.732

Q 19PhysicsSection B • Numericalalternating current

+4 / -1

A capacitor of capacitance 500 $\mu$ F is charged completely using a dc supply of 100 V. It is now connected to an inductor of inductance 50 mH to form an LC circuit. The maximum current in LC circuit will be _______ A.

Answer Format:Enter numerical value rounded to the nearest integer

Q 20PhysicsSection A • MCQatoms and nuclei

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Read the following statements :

(A) Volume of the nucleus is directly proportional to the mass number.

(B) Volume of the nucleus is independent of mass number.

(C) Density of the nucleus is directly proportional to the mass number.

(D) Density of the nucleus is directly proportional to the cube root of the mass number.

(E) Density of the nucleus is independent of the mass number.

Choose the correct option from the following options.

A

(A) and (D) only.

B

(A) and (E) only.

C

(B) and (E) only.

D

(A) and (C) only.

Q 21PhysicsSection B • Numericalproperties of matter

+4 / -1

A metal wire of length $0.5 \mathrm{~m}$ and cross-sectional area $10^{-4} \mathrm{~m}^{2}$ has breaking stress $5 \times 10^{8} \,\mathrm{Nm}^{-2}$ . A block of $10 \mathrm{~kg}$ is attached at one end of the string and is rotating in a horizontal circle. The maximum linear velocity of block will be _________ $\mathrm{ms}^{-1}$ .

Answer Format:Enter numerical value rounded to the nearest integer

Q 22PhysicsSection A • MCQcurrent electricity

+4 / -1

A $1 \mathrm{~m}$ long wire is broken into two unequal parts $\mathrm{X}$ and $\mathrm{Y}$ . The $\mathrm{X}$ part of the wire is streched into another wire W. Length of $W$ is twice the length of $X$ and the resistance of $\mathrm{W}$ is twice that of $\mathrm{Y}$ . Find the ratio of length of $\mathrm{X}$ and $\mathrm{Y}$ .

A

1 : 4

B

1 : 2

C

4 : 1

D

2 : 1

Q 23PhysicsSection A • MCQmagnetics

+4 / -1

A wire X of length $50 \mathrm{~cm}$ carrying a current of $2 \mathrm{~A}$ is placed parallel to a long wire $\mathrm{Y}$ of length $5 \mathrm{~m}$ . The wire $\mathrm{Y}$ carries a current of $3 \mathrm{~A}$ . The distance between two wires is $5 \mathrm{~cm}$ and currents flow in the same direction. The force acting on the wire $\mathrm{Y}$ is

qImage-force-and-torque-on-current-carrying-conductor-magnetics-electricity-Jeemate

A

1.2 \times 10^{-5} \mathrm{~N}

directed towards wire

\mathrm{X}

.

B

1.2 \times 10^{-4} \mathrm{~N}

directed away from wire

\mathrm{X}

.

C

1.2 \times 10^{-4} \mathrm{~N}

directed towards wire

\mathrm{X}

.

D

2.4 \times 10^{-5} \mathrm{~N}

directed towards wire

\mathrm{X}

.

Q 24PhysicsSection A • MCQheat and thermodynamics

+4 / -1

A thermodynamic system is taken from an original state D to an intermediate state E by the linear process shown in the figure. Its volume is then reduced to the original volume from E to F by an isobaric process. The total work done by the gas from D to E to F will be

qImage-thermodynamics-process-heat-and-thermodynamics-mechanics-Jeemate

A

-

450 J

B

450 J

C

900 J

D

1350 J

Q 25PhysicsSection B • Numericalatoms and nuclei

+4 / -1

Two radioactive materials A and B have decay constants $25 \lambda$ and $16 \lambda$ respectively. If initially they have the same number of nuclei, then the ratio of the number of nuclei of B to that of A will be "e" after a time $\frac{1}{a \lambda}$ . The value of a is _________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 26PhysicsSection A • MCQdual nature of radiation

+4 / -1

An $\alpha$ particle and a proton are accelerated from rest through the same potential difference. The ratio of linear momenta acquired by above two particles will be:

A

\sqrt2

: 1

B

2

\sqrt2

: 1

C

4

\sqrt2

: 1

D

8 : 1

Q 27PhysicsSection A • MCQrotational motion

+4 / -1

The torque of a force $5 \hat{i}+3 \hat{j}-7 \hat{k}$ about the origin is $\tau$ . If the force acts on a particle whose position vector is $2 i+2 j+k$ , then the value of $\tau$ will be

A

11 \hat{i}+19 \hat{j}-4 \hat{k}

B

-11 \hat{i}+9 \hat{j}-16 \hat{k}

C

-17 \hat{i}+19 \hat{j}-4 \hat{k}

D

17 \hat{i}+9 \hat{j}+16 \hat{k}

Q 28PhysicsSection A • MCQcapacitor

+4 / -1

Two identical thin metal plates has charge $q_{1}$ and $q_{2}$ respectively such that $q_{1}>q_{2}$ . The plates were brought close to each other to form a parallel plate capacitor of capacitance C. The potential difference between them is :

A

\frac{\left(q_{1}+q_{2}\right)}{C}

B

\frac{\left(q_{1}-q_{2}\right)}{C}

C

\frac{\left(q_{1}-q_{2}\right)}{2 C}

D

\frac{2\left(q_{1}-q_{2}\right)}{C}

Q 29PhysicsSection A • MCQelectrostatics

+4 / -1

Two identical metallic spheres $\mathrm{A}$ and $\mathrm{B}$ when placed at certain distance in air repel each other with a force of $\mathrm{F}$ . Another identical uncharged sphere $\mathrm{C}$ is first placed in contact with $\mathrm{A}$ and then in contact with $\mathrm{B}$ and finally placed at midpoint between spheres A and B. The force experienced by sphere C will be:

A

3F/2

B

3F/4

C

F

D

2F

Q 30PhysicsSection A • MCQgravitation

+4 / -1

An object of mass $1 \mathrm{~kg}$ is taken to a height from the surface of earth which is equal to three times the radius of earth. The gain in potential energy of the object will be [If, $\mathrm{g}=10 \mathrm{~ms}^{-2}$ and radius of earth $=6400 \mathrm{~km}$ ]

A

48 MJ

B

24 MJ

C

36 MJ

D

12 MJ

Q 31ChemistrySection A • MCQsome basic concepts of chemistry

+4 / -1

$\begin{aligned} &\mathrm{C}(\mathrm{s})+\mathrm{O}_{2}(\mathrm{~g}) \rightarrow \mathrm{CO}_{2}(\mathrm{~g})+400 \mathrm{~kJ} \\ &\mathrm{C}(\mathrm{s})+\frac{1}{2} \mathrm{O}_{2}(\mathrm{~g}) \rightarrow \mathrm{CO}(\mathrm{g})+100 \mathrm{~kJ} \end{aligned}$

When coal of purity 60% is allowed to burn in presence of insufficient oxygen, 60% of carbon is converted into 'CO' and the remaining is converted into ' $\mathrm{CO}_{2}$ '. The heat generated when $0.6 \mathrm{~kg}$ of coal is burnt is _________.

A

1600 kJ

B

3200 kJ

C

4400 kJ

D

6600 kJ

Q 32ChemistrySection A • MCQp block elements

+4 / -1

In liquation process used for tin (Sn), the metal :

A

is reacted with acid.

B

is dissolved in water.

C

is brought to molten form which is made to flow on a slope.

D

is fused with NaOH

Q 33ChemistrySection A • MCQalcohols phenols and ethers

+4 / -1

When ethanol is heated with conc. $\mathrm{H}_{2} \mathrm{SO}_{4}$ , a gas is produced. The compound formed, when this gas is treated with cold dilute aqueous solution of Baeyer's reagent, is

A

formaldehyde

B

formic acid

C

glycol

D

ethanoic acid

Q 34ChemistrySection A • MCQpolymers

+4 / -1

Which of the following is NOT a natural polymer?

A

Protein

B

Starch

C

Rubber

D

Rayon

Q 35ChemistrySection B • Numericalgaseous state

+4 / -1

'x' g of molecular oxygen $\left(\mathrm{O}_{2}\right)$ is mixed with $200 \mathrm{~g}$ of neon (Ne). The total pressure of the non-reactive mixture of $\mathrm{O}_{2}$ and Ne in the cylinder is 25 bar. The partial pressure of Ne is 20 bar at the same temperature and volume. The value of 'x' is _________.

[Given: Molar mass of $\mathrm{O}_{2}=32 \mathrm{~g} \mathrm{~mol}^{-1}$ .

Molar mass of $\mathrm{Ne}=20 \mathrm{~g} \mathrm{~mol}^{-1}$ ]

Answer Format:Enter numerical value rounded to the nearest integer

Q 36ChemistrySection A • MCQcompounds containing nitrogen

+4 / -1

The Hinsberg reagent is

A

B

C

D

Q 37ChemistrySection A • MCQredox reactions

+4 / -1

A compound 'X' is a weak acid and it exhibits colour change at pH close to the equivalence point during neutralization of NaOH with $\mathrm{CH}_{3} \mathrm{COOH}$ . Compound 'X' exists in ionized form in basic medium. The compound 'X' is

A

methyl orange

B

methyl red

C

phenolphthalein

D

erichrome Black T

Q 38ChemistrySection A • MCQhaloalkanes and haloarenes

+4 / -1

Compound 'A' undergoes following sequence of reactions to give compound 'B'.

The correct structure and chirality of compound 'B' is

[where Et is $\left.-\mathrm{C}_{2} \mathrm{H}_{5}\right]$

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

A

B

C

D

Q 39ChemistrySection A • MCQsurface chemistry

+4 / -1

Given below are the critical temperatures of some of the gases:

Gas	Critical temperature (K)
He	5.2
CH $_4$	190.0
CO $_2$	304.2
NH $_3$	405.5

The gas showing least adsorption on a definite amount of charcoal is

A

He

B

CH

_4

C

CO

_2

D

NH

_3

Q 40ChemistrySection A • MCQbiomolecules

+4 / -1

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

Assertion A: Amylose is insoluble in water.

Reason R: Amylose is a long linear molecule with more than 200 glucose units.

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

A

Both

\mathbf{A}

and

\mathbf{R}

are correct and

\mathbf{R}

is the correct explanation of

\mathbf{A}

.

B

Both

\mathbf{A}

and

\mathbf{R}

are correct but

\mathbf{R}

is NOT the correct explanation of

\mathbf{A}

.

C

\mathbf{A}

is correct but

\mathbf{R}

is not correct

D

\mathbf{A}

is not correct but

\mathbf{R}

is correct.

Q 41ChemistrySection A • MCQsome basic concepts of chemistry

+4 / -1

Consider the reaction

$4 \mathrm{HNO}_{3}(1)+3 \mathrm{KCl}(\mathrm{s}) \rightarrow \mathrm{Cl}_{2}(\mathrm{~g})+\mathrm{NOCl}(\mathrm{g})+2 \mathrm{H}_{2} \mathrm{O}(\mathrm{g})+3 \mathrm{KNO}_{3}(\mathrm{~s})$

The amount of $\mathrm{HNO}_{3}$ required to produce $110.0 \mathrm{~g}$ of $\mathrm{KNO}_{3}$ is

(Given: Atomic masses of $\mathrm{H}, \mathrm{O}, \mathrm{N}$ and $\mathrm{K}$ are $1,16,14$ and 39, respectively.)

A

32.2 g

B

69.4 g

C

91.5 g

D

162.5 g

Q 42ChemistrySection A • MCQp block elements

+4 / -1

Given below are two statements.

Statement I : Stannane is an example of a molecular hydride.

Statement II : Stannane is a planar molecule.

In the light of the above statement, choose the most appropriate answer from the options given below.

A

Both Statement I and Statement II are 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 43ChemistrySection A • MCQcoordination compounds

+4 / -1

Octahedral complexes of copper(II) undergo structural distortion (Jahn-Teller). Which one of the given copper (II) complexes will show the maximum structural distortion? (en - ethylenediamine; $\mathrm{H}_{2} \mathrm{~N}_{-} \mathrm{CH}_{2}-\mathrm{CH}_{2}-\mathrm{NH}_{2}$ )

A

\left[\mathrm{Cu}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right] \mathrm{SO}_{4}

B

\left[\mathrm{Cu}(\mathrm{en})\left(\mathrm{H}_{2} \mathrm{O}\right)_{4}\right] \mathrm{SO}_{4}

C

cis-

\left[\mathrm{Cu}\left(\mathrm{en})_{2} \mathrm{Cl}_{2}\right]\right.

D

trans-

\left[\mathrm{Cu}\left(\mathrm{en})_{2} \mathrm{Cl}_{2}\right]\right.

Q 44ChemistrySection A • MCQionic equilibrium

+4 / -1

$200 \mathrm{~mL}$ of $0.01 \,\mathrm{M} \,\mathrm{HCl}$ is mixed with $400 \mathrm{~mL}$ of $0.01 \,\mathrm{M} \,\mathrm{H}_{2} \mathrm{SO}_{4}$ . The $\mathrm{pH}$ of the mixture is _________.

Given: $\log {2}=0.30, \log 3=0.48, \log 5=0.70, \log 7=0.84, \log 11=1.04$

A

1.14

B

1.78

C

2.34

D

3.02

Q 45ChemistrySection B • Numericalchemical bonding and molecular structure

+4 / -1

Consider, $\mathrm{PF}_{5}, \mathrm{BrF}_{5}, \mathrm{PCl}_{3}, \mathrm{SF}_{6},\left[\mathrm{ICl}_{4}\right]^{-}, \mathrm{ClF}_{3}$ and $\mathrm{IF}_{5}$ .

Amongst the above molecule(s)/ion(s), the number of molecule(s)/ion(s) having $\mathrm{sp}^{3}\mathrm{~d}^{2}$ hybridisation is __________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 46ChemistrySection A • MCQatomic structure

+4 / -1

Given below are the quantum numbers for 4 electrons.

A. $\mathrm{n}=3,l=2, \mathrm{~m}_{1}=1, \mathrm{~m}_{\mathrm{s}}=+1 / 2$

B. $\mathrm{n}=4,l=1, \mathrm{~m}_{1}=0, \mathrm{~m}_{\mathrm{s}}=+1 / 2$

C. $\mathrm{n}=4,l=2, \mathrm{~m}_{1}=-2, \mathrm{~m}_{\mathrm{s}}=-1 / 2$

D. $\mathrm{n}=3,l=1, \mathrm{~m}_{1}=-1, \mathrm{~m}_{\mathrm{s}}=+1 / 2$

The correct order of increasing energy is

A

D < B < A < C

B

D < A < B < C

C

B < D < A < C

D

B < D < C < A

Q 47ChemistrySection A • MCQs block elements

+4 / -1

Portland cement contains 'X' to enhance the setting time. What is 'X'?

A

\mathrm{CaSO}_{4} \cdot \frac{1}{2} \mathrm{H}_{2} \mathrm{O}

B

\mathrm{CaSO}_{4} \cdot 2 \mathrm{H}_{2} \mathrm{O}

C

\mathrm{CaSO}_{4}

D

\mathrm{CaCO}_{3}

Q 48ChemistrySection A • MCQd and f block elements

+4 / -1

Which of the following $3\mathrm{d}$ -metal ion will give the lowest enthalpy of hydration $\left(\Delta_{\text {hyd }} \mathrm{H}\right)$ when dissolved in water ?

A

\mathrm{Cr}^{2+}

B

\operatorname{Mn}^{2+}

C

\mathrm{Fe}^{2+}

D

\mathrm{Co}^{2+}

Q 49ChemistrySection A • MCQp block elements

+4 / -1

Dinitrogen is a robust compound, but reacts at high altitudes to form oxides. The oxide of nitrogen that can damage plant leaves and retard photosynthesis is :

A

\mathrm{NO}

B

\mathrm{NO}_{3}^{-}

C

\mathrm{NO}_{2}

D

\mathrm{NO}_{2}^{-}

Q 50ChemistrySection A • MCQp block elements

+4 / -1

When borax is heated with $\mathrm{CoO}$ on a platinum loop, blue coloured bead formed is largely due to :

A

\mathrm{B}_{2} \mathrm{O}_{3}

B

\mathrm{Co}\left(\mathrm{BO}_{2}\right)_{2}

C

\mathrm{CoB}_{4} \mathrm{O}_{7}

D

\mathrm{Co}\left[\mathrm{B}_{4} \mathrm{O}_{5}(\mathrm{OH})_{4}\right]

Q 51ChemistrySection A • MCQbasics of organic chemistry

+4 / -1

Given below are two statements.

Statement I : The compound qImage-isomerism-basics-of-organic-chemistry-organic-chemistry-Jeemate is optically active.

Statement II : qImage-isomerism-basics-of-organic-chemistry-organic-chemistry-Jeemate is mirror image of above compound A.

In the light of the above statement, choose the most appropriate answer from the options given below.

A

Both Statement I and Statement II are correct.

B

Both Statement I and Statement II are incorrect.

C

Statement I is correct but Statement II is incorrect.

D

Statement I is incorrect but Statement II is correct.

Q 52ChemistrySection A • MCQaldehydes ketones and carboxylic acids

+4 / -1

Correct structure of $\gamma$ -methylcyclohexane carbaldehyde is

A

B

C

D

Q 53ChemistrySection B • Numericalelectrochemistry

+4 / -1

For a cell, $\mathrm{Cu}(\mathrm{s})\left|\mathrm{Cu}^{2+}(0.001 \,\mathrm{M}) \| \mathrm{Ag}^{+}(0.01 \,\mathrm{M})\right| \mathrm{Ag}(\mathrm{s})$

the cell potential is found to be $0.43 \mathrm{~V}$ at $298 \mathrm{~K}$ . The magnitude of standard electrode potential for $\mathrm{Cu}^{2+} / \mathrm{Cu}$ is _________ $\times 10^{-2} \mathrm{~V}$ .

[Given : $E_{A{g^ + }/Ag}^\Theta$ = 0.80 V and ${{2.303RT} \over F}$ = 0.06 V]

Answer Format:Enter numerical value rounded to the nearest integer

Q 54ChemistrySection B • Numericalchemical kinetics and nuclear chemistry

+4 / -1

Assuming $1 \,\mu \mathrm{g}$ of trace radioactive element X with a half life of 30 years is absorbed by a growing tree. The amount of X remaining in the tree after 100 years is ______ $\times\, 10^{-1} \mu \mathrm{g}$ .

[Given : ln 10 = 2.303; log 2 = 0.30]

Answer Format:Enter numerical value rounded to the nearest integer

Q 55ChemistrySection B • Numericalsolutions

+4 / -1

$1.80 \mathrm{~g}$ of solute A was dissolved in $62.5 \mathrm{~cm}^{3}$ of ethanol and freezing point of the solution was found to be $155.1 \mathrm{~K}$ . The molar mass of solute A is ________ g $\mathrm{mol}^{-1}$ .

[Given : Freezing point of ethanol is 156.0 K.

Density of ethanol is 0.80 g cm^{$-$ 3}.

Freezing point depression constant of ethanol is 2.00 K kg mol^{$-$ 1}]

Answer Format:Enter numerical value rounded to the nearest integer

Q 56ChemistrySection B • Numericalsome basic concepts of chemistry

+4 / -1

A 1.84 mg sample of polyhydric alcoholic compound 'X' of molar mass 92.0 g/mol gave 1.344 mL of $\mathrm{H}_{2}$ gas at STP. The number of alcoholic hydrogens present in compound 'X' is ________.

Answer Format:Enter numerical value rounded to the nearest integer

Official Correct Answer:6

Step-by-Step Solution:

STP conditions define the volume of 1 mole of any gas as $22.4$ liters or $22400$ mL. The volume of hydrogen gas evolved here is $1.344$ mL, so we can calculate the number of moles of hydrogen gas ( $H_2$ ) using the formula:

$n = \frac{V}{V_m}$

where:

$n$ is the number of moles,
$V$ is the volume of the gas, and
$V_m$ is the molar volume of the gas.

Substituting in the given values:

$n = \frac{1.344 \text{ mL}}{22400 \text{ mL/mol}} = 5.995 \times 10^{-5} \text{ mol}$

Each molecule of $H_2$ contains $2$ atoms of hydrogen. Therefore, the number of moles of hydrogen atoms is twice the number of moles of hydrogen gas:

$n_H = 2 \times n = 2 \times 5.995 \times 10^{-5} \text{ mol} = 1.199 \times 10^{-4} \text{ mol}$

This number of moles of hydrogen represents the number of moles of alcoholic hydrogen atoms in the $1.84$ mg sample of the compound 'X'.

We can determine the number of moles of compound 'X' in the sample by dividing the mass of the sample by the molar mass of the compound:

$n_X = \frac{m}{M}$

where:

$n_X$ is the number of moles of 'X',
$m$ is the mass of 'X', and
$M$ is the molar mass of 'X'.

Substituting in the given values:

$n_X = \frac{1.84 \text{ mg}}{92 \text{ g/mol}} = \frac{1.84 \times 10^{-3} \text{ g}}{92 \text{ g/mol}} = 2.0 \times 10^{-5} \text{ mol}$

The number of alcoholic hydrogens per molecule of 'X' is then given by the ratio of the number of moles of hydrogen to the number of moles of 'X':

$\frac{n_H}{n_X} = \frac{1.199 \times 10^{-4} \text{ mol}}{2.0 \times 10^{-5} \text{ mol}} = 6$

So, compound 'X' contains $6$ alcoholic hydrogens.

Q 57ChemistrySection B • Numericalcoordination compounds

+4 / -1

Sum of oxidation state (magnitude) and coordination number of cobalt in $\mathrm{Na}\left[\mathrm{Co}(\mathrm{bpy}) \mathrm{Cl}_{4}\right]$ is _________.

qImage-coordination-number-coordination-compounds-inorganic-chemistry-Jeemate

Answer Format:Enter numerical value rounded to the nearest integer

Q 58ChemistrySection B • Numericalp block elements

+4 / -1

Consider the following sulphur based oxoacids.

$\mathrm{H}_{2} \mathrm{SO}_{3}, \mathrm{H}_{2} \mathrm{SO}_{4}, \mathrm{H}_{2} \mathrm{S}_{2} \mathrm{O}_{8}$ and $\mathrm{H}_{2}\mathrm{S}_{2} \mathrm{O}_{7}$ .

Amongst these oxoacids, the number of those with peroxo $(\mathrm{O}-\mathrm{O})$ bond is ________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 59ChemistrySection B • Numericalchemistry in everyday life

+4 / -1

The number of chlorine atoms in bithionol is __________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 60ChemistrySection B • Numericalaldehydes ketones and carboxylic acids

+4 / -1

The number of stereoisomers formed in a reaction of $(±)\mathrm{Ph}(\mathrm{C}=\mathrm{O}) \mathrm{C}(\mathrm{OH})(\mathrm{CN}) \mathrm{Ph}$ with $\mathrm{HCN}$ is ___________.

$\left[\right.$ where $\mathrm{Ph}$ is $-\mathrm{C}_{6} \mathrm{H}_{5}$ ]

Answer Format:Enter numerical value rounded to the nearest integer

Q 61MathematicsSection B • Numericalapplication of derivatives

+4 / -1

If the tangent to the curve $y=x^{3}-x^{2}+x$ at the point $(a, b)$ is also tangent to the curve $y = 5{x^2} + 2x - 25$ at the point (2, $-$ 1), then $|2a + 9b|$ is equal to __________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 62MathematicsSection B • Numericalprobability

+4 / -1

The sum and product of the mean and variance of a binomial distribution are 82.5 and 1350 respectively. Then the number of trials in the binomial distribution is ____________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 63MathematicsSection B • Numericalcircle

+4 / -1

$\text { Let } S=\left\{(x, y) \in \mathbb{N} \times \mathbb{N}: 9(x-3)^{2}+16(y-4)^{2} \leq 144\right\}$ and $T=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}:(x-7)^{2}+(y-4)^{2} \leq 36\right\}$ . Then $n(S \cap T)$ is equal to __________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 64MathematicsSection A • MCQvector algebra

+4 / -1

Let $\vec{a}, \vec{b}, \vec{c}$ be three coplanar concurrent vectors such that angles between any two of them is same. If the product of their magnitudes is 14 and $(\vec{a} \times \vec{b}) \cdot(\vec{b} \times \vec{c})+(\vec{b} \times \vec{c}) \cdot(\vec{c} \times \vec{a})+(\vec{c} \times \vec{a}) \cdot(\vec{a} \times \vec{b})=168$ , then $|\vec{a}|+|\vec{b}|+|\vec{c}|$ is equal to :

A

10

B

14

C

16

D

18

Official Correct Answer:Option (C)

Step-by-Step Solution:

|\vec{a}||\vec{b}||\vec{c}|=14

\begin{aligned} & \vec{a} \wedge \vec{b}=\vec{b} \wedge \vec{c}=\vec{c} \wedge \vec{a}=\theta=\frac{2 \pi}{3} \\ & \vec{a} \cdot \vec{b}=-\frac{1}{2}|\vec{a}||\vec{b}| \\ & \vec{b} \cdot \vec{c}=-\frac{1}{2}|\vec{b}||\vec{c}| \\ & \vec{c} \cdot \vec{a}=-\frac{1}{2}|\vec{c}||\vec{a}| \end{aligned}

Now,

\begin{aligned} & \begin{aligned} &(\vec{a} \times \vec{b}) \cdot(\vec{b} \times \vec{c})+(\vec{b} \times \vec{c}) \cdot(\vec{c} \times \vec{a})+(\vec{c} \times \vec{a}) \cdot(\vec{a} \times \vec{b}) \\ &=168 \quad\quad...(i)\\ &(\vec{a} \times \vec{b}) \cdot(\vec{b} \times \vec{c})=(\vec{a} \cdot \vec{b})(\vec{b} \cdot \vec{c})-(\vec{a} \cdot \vec{c})|\vec{b}|^{2} \\ &= \frac{1}{4}|\vec{b}|^{2}|\vec{a}||\vec{c}|+\frac{1}{2}|\vec{a}||\vec{b}|^{2}|\vec{c}| \\ &= \frac{3}{4}|\vec{a}||\vec{b}|^{2}|\vec{c}| \quad\quad...(ii) \end{aligned} \end{aligned}

Similarly

(\vec{b} \times \vec{c}) \cdot(\vec{c} \times \vec{a})=\frac{3}{4}|\vec{a}||\vec{b}||\vec{c}|^{2} \quad\quad...(iii)

(\vec{c} \times \vec{a}) \cdot(\vec{a} \times \vec{b})=\frac{3}{4}|\vec{a}|^{2}|\vec{b}||\vec{c}| \quad\quad...(iv)

Substitute (ii), (iii), (iv) in (i)

\frac{3}{4}|\vec{a}||\vec{b}||\vec{c}|[|\vec{a}|+|\vec{b}|+|\vec{c}|]=168

\frac{3}{4} \times 14[|\vec{a}|+|\vec{b}|+|\vec{c}|]=168

|\vec{a}|+|\vec{b}|+|\vec{c}|=16

Q 65MathematicsSection A • MCQdifferential equations

+4 / -1

If the solution curve of the differential equation $\frac{d y}{d x}=\frac{x+y-2}{x-y}$ passes through the points $(2,1)$ and $(\mathrm{k}+1,2), \mathrm{k}>0$ , then

A

2 \tan ^{-1}\left(\frac{1}{k}\right)=\log _{e}\left(k^{2}+1\right)

B

\tan ^{-1}\left(\frac{1}{k}\right)=\log _{e}\left(k^{2}+1\right)

C

2 \tan ^{-1}\left(\frac{1}{k+1}\right)=\log _{e}\left(k^{2}+2 k+2\right)

D

2 \tan ^{-1}\left(\frac{1}{k}\right)=\log _{e}\left(\frac{k^{2}+1}{k^{2}}\right)

Q 66MathematicsSection A • MCQcomplex numbers

+4 / -1

Let $\mathrm{S}=\{z=x+i y:|z-1+i| \geq|z|,|z|<2,|z+i|=|z-1|\}$ . Then the set of all values of $x$ , for which $w=2 x+i y \in \mathrm{S}$ for some $y \in \mathbb{R}$ , is :

A

\left(-\sqrt{2}, \frac{1}{2 \sqrt{2}}\right]

B

\left(-\frac{1}{\sqrt{2}}, \frac{1}{4}\right]

C

\left(-\sqrt{2}, \frac{1}{2}\right]

D

\left(-\frac{1}{\sqrt{2}}, \frac{1}{2 \sqrt{2}}\right]

Q 67MathematicsSection B • Numericalpermutations and combinations

+4 / -1

The number of natural numbers lying between 1012 and 23421 that can be formed using the digits $2,3,4,5,6$ (repetition of digits is not allowed) and divisible by 55 is _________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 68MathematicsSection B • Numericalbinomial theorem

+4 / -1

$\text { If } \sum\limits_{k=1}^{10} K^{2}\left(10_{C_{K}}\right)^{2}=22000 L \text {, then } L \text { is equal to }$ ________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 69MathematicsSection A • MCQsequences and series

+4 / -1

$\begin{aligned} &\text { Let }\left\{a_{n}\right\}_{n=0}^{\infty} \text { be a sequence such that } a_{0}=a_{1}=0 \text { and } \\ &a_{n+2}=3 a_{n+1}-2 a_{n}+1, \forall n \geq 0 . \end{aligned}$

Then $a_{25} a_{23}-2 a_{25} a_{22}-2 a_{23} a_{24}+4 a_{22} a_{24}$ is equal to

A

483

B

528

C

575

D

624

Q 70MathematicsSection A • MCQtrigonometric functions and equations

+4 / -1

The number of elements in the set $S=\left\{x \in \mathbb{R}: 2 \cos \left(\frac{x^{2}+x}{6}\right)=4^{x}+4^{-x}\right\}$ is :

A

1

B

3

C

0

D

infinite

Q 71MathematicsSection A • MCQinverse trigonometric functions

+4 / -1

The domain of the function $f(x)=\sin ^{-1}\left(\frac{x^{2}-3 x+2}{x^{2}+2 x+7}\right)$ is :

A

[1, \infty)

B

[-1,2]

C

[-1, \infty)

D

(-\infty, 2]

Q 72MathematicsSection B • Numericalquadratic equation and inequalities

+4 / -1

Let $\alpha, \beta(\alpha>\beta)$ be the roots of the quadratic equation $x^{2}-x-4=0 .$ If $P_{n}=\alpha^{n}-\beta^{n}$ , $n \in \mathrm{N}$ , then $\frac{P_{15} P_{16}-P_{14} P_{16}-P_{15}^{2}+P_{14} P_{15}}{P_{13} P_{14}}$ is equal to __________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 73MathematicsSection A • MCQstraight lines and pair of straight lines

+4 / -1

Let $m_{1}, m_{2}$ be the slopes of two adjacent sides of a square of side a such that $a^{2}+11 a+3\left(m_{1}^{2}+m_{2}^{2}\right)=220$ . If one vertex of the square is $(10(\cos \alpha-\sin \alpha), 10(\sin \alpha+\cos \alpha))$ , where $\alpha \in\left(0, \frac{\pi}{2}\right)$ and the equation of one diagonal is $(\cos \alpha-\sin \alpha) x+(\sin \alpha+\cos \alpha) y=10$ , then $72\left(\sin ^{4} \alpha+\cos ^{4} \alpha\right)+a^{2}-3 a+13$ is equal to :

A

119

B

128

C

145

D

155

Q 74MathematicsSection A • MCQmathematical reasoning

+4 / -1

The statement $(p \Rightarrow q) \vee(p \Rightarrow r)$ is NOT equivalent to

A

(p \wedge(\sim r)) \Rightarrow q

B

(\sim q) \Rightarrow((\sim r) \vee p)

C

p \Rightarrow(q \vee r)

D

(p \wedge(\sim q)) \Rightarrow r

Q 75MathematicsSection A • MCQindefinite integration

+4 / -1

For $I(x)=\int \frac{\sec ^{2} x-2022}{\sin ^{2022} x} d x$ , if $I\left(\frac{\pi}{4}\right)=2^{1011}$ , then

A

3^{1010} I\left(\frac{\pi}{3}\right)-I\left(\frac{\pi}{6}\right)=0

B

3^{1010} I\left(\frac{\pi}{6}\right)-I\left(\frac{\pi}{3}\right)=0

C

3^{1011} I\left(\frac{\pi}{3}\right)-I\left(\frac{\pi}{6}\right)=0

D

3^{1011} I\left(\frac{\pi}{6}\right)-I\left(\frac{\pi}{3}\right)=0

Q 76MathematicsSection B • Numericalcircle

+4 / -1

Let $A B$ be a chord of length 12 of the circle $(x-2)^{2}+(y+1)^{2}=\frac{169}{4}$ . If tangents drawn to the circle at points $A$ and $B$ intersect at the point $P$ , then five times the distance of point $P$ from chord $A B$ is equal to __________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 77MathematicsSection A • MCQprobability

+4 / -1

Bag I contains 3 red, 4 black and 3 white balls and Bag II contains 2 red, 5 black and 2 white balls. One ball is transferred from Bag I to Bag II and then a ball is drawn from Bag II. The ball so drawn is found to be black in colour. Then the probability, that the transferred ball is red, is :

A

\frac{4}{9}

B

\frac{5}{18}

C

\frac{1}{6}

D

\frac{3}{10}

Q 78MathematicsSection B • Numericalvector algebra

+4 / -1

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a}+\vec{b}|^{2}=|\vec{a}|^{2}+2|\vec{b}|^{2}, \vec{a} \cdot \vec{b}=3$ and $|\vec{a} \times \vec{b}|^{2}=75$ . Then $|\vec{a}|^{2}$ is equal to __________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 79MathematicsSection A • MCQmatrices and determinants

+4 / -1

Which of the following matrices can NOT be obtained from the matrix $\left[\begin{array}{cc}-1 & 2 \\ 1 & -1\end{array}\right]$ by a single elementary row operation ?

A

\left[\begin{array}{cc}0 & 1 \\ 1 & -1\end{array}\right]

B

\left[\begin{array}{cc}1 & -1 \\ -1 & 2\end{array}\right]

C

\left[\begin{array}{rr}-1 & 2 \\ -2 & 7\end{array}\right]

D

\left[\begin{array}{ll}-1 & 2 \\ -1 & 3\end{array}\right]

Q 80MathematicsSection A • MCQcomplex numbers

+4 / -1

If $z \neq 0$ be a complex number such that $\left|z-\frac{1}{z}\right|=2$ , then the maximum value of $|z|$ is :

A

\sqrt{2}

B

1

C

\sqrt{2}-1

D

\sqrt{2}+1

Q 81MathematicsSection B • Numericalmatrices and determinants

+4 / -1

Let $X=\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$ and $A=\left[\begin{array}{ccc}-1 & 2 & 3 \\ 0 & 1 & 6 \\ 0 & 0 & -1\end{array}\right]$ . For $\mathrm{k} \in N$ , if $X^{\prime} A^{k} X=33$ , then $\mathrm{k}$ is equal to _______.

Answer Format:Enter numerical value rounded to the nearest integer

Q 82MathematicsSection A • MCQ3d geometry

+4 / -1

If $(2,3,9),(5,2,1),(1, \lambda, 8)$ and $(\lambda, 2,3)$ are coplanar, then the product of all possible values of $\lambda$ is:

A

\frac{21}{2}

B

\frac{59}{8}

C

\frac{57}{8}

D

\frac{95}{8}

Q 83MathematicsSection A • MCQstraight lines and pair of straight lines

+4 / -1

Let $\mathrm{A}(\alpha,-2), \mathrm{B}(\alpha, 6)$ and $\mathrm{C}\left(\frac{\alpha}{4},-2\right)$ be vertices of a $\triangle \mathrm{ABC}$ . If $\left(5, \frac{\alpha}{4}\right)$ is the circumcentre of $\triangle \mathrm{ABC}$ , then which of the following is NOT correct about $\triangle \mathrm{ABC}$ ?

A

area is 24

B

perimeter is 25

C

circumradius is 5

D

inradius is 2

Q 84MathematicsSection A • MCQ3d geometry

+4 / -1

Let $Q$ be the foot of perpendicular drawn from the point $P(1,2,3)$ to the plane $x+2 y+z=14$ . If $R$ is a point on the plane such that $\angle P R Q=60^{\circ}$ , then the area of $\triangle P Q R$ is equal to :

A

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

B

\sqrt{3}

C

2 \sqrt{3}

D

3

Q 85MathematicsSection A • MCQmatrices and determinants

+4 / -1

If the system of equations

$\begin{aligned} &x+y+z=6 \\ &2 x+5 y+\alpha z=\beta \\ &x+2 y+3 z=14 \end{aligned}$

has infinitely many solutions, then $\alpha+\beta$ is equal to

A

8

B

36

C

44

D

48

Q 86MathematicsSection A • MCQdifferential equations

+4 / -1

Let $y=y(x)$ be the solution curve of the differential equation $\frac{d y}{d x}+\left(\frac{2 x^{2}+11 x+13}{x^{3}+6 x^{2}+11 x+6}\right) y=\frac{(x+3)}{x+1}, x>-1$ , which passes through the point $(0,1)$ . Then $y(1)$ is equal to :

A

\frac{1}{2}

B

\frac{3}{2}

C

\frac{5}{2}

D

\frac{7}{2}

Q 87MathematicsSection B • Numericallimits continuity and differentiability

+4 / -1

If $[t]$ denotes the greatest integer $\leq t$ , then the number of points, at which the function $f(x)=4|2 x+3|+9\left[x+\frac{1}{2}\right]-12[x+20]$ is not differentiable in the open interval $(-20,20)$ , is __________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 88MathematicsSection A • MCQbinomial theorem

+4 / -1

$\sum\limits_{r=1}^{20}\left(r^{2}+1\right)(r !)$ is equal to

A

22 !-21 !

B

22 !-2(21 !)

C

21 !-2(20 !)

D

21 !-20

!

Official Correct Answer:Option (B)

Step-by-Step Solution:

Given,

$\sum\limits_{r = 1}^{20} {({r^2} + 1)(r!)}$

Let, $f(r) = ({r^2} + 1)(r!)$

$= ({r^2})(r!) + r!$

$= r(r\,r!) + r!$

$= r[(r + 1 - 1)r!] + r!$

$= r[(r + 1)r! - r!] + r!$

$= r[(r + 1)! - (r!)] + r!$

$= r(r + 1)! - r(r!) + r!$

$= (r + 2 - 2)(r + 1)! - r(r!) + r!$

$= (r + 2)(r + 1)! - 2(r + 1)! - [(r + 1 - 1)(r!)] + r!$

$= (r + 2)! - 2(r + 1)! - (r + 1)! + r! + r!$

$= (r + 2)! - 3(r + 1)! + 2r!$

$= [(r + 2)! - (r + 1)!] - 2[(r + 1)! - r!]$

$\therefore$ $\sum\limits_{r = 1}^{20} {f(r)}$

$= \sum\limits_{r = 1}^{20} {[(r + 2)! - (r + 1)!] - 2\sum\limits_{r = 1}^{20} {[(r + 1)! - r!]} }$

$= [(22! + 21! + 20!\, + \,.....\, + \,4! + 3!) - (21! + 20! + 19!\, + \,....\, + \,3! + 2!] - 2[(21! + 20!\, + \,.....\, + \,3! + 2!) - (20! + 19!\, + \,.....\,1!)]$

$= [(22!) - (2!)] - 2[(21)! - (1!)]$

$= 22! - 2! - 2\,.\,(21)! + 2\,.\,1!$

$= 22! - 2\,.\,(21)!$

Q 89MathematicsSection A • MCQdefinite integration

+4 / -1

If $[t]$ denotes the greatest integer $\leq t$ , then the value of $\int_{0}^{1}\left[2 x-\left|3 x^{2}-5 x+2\right|+1\right] \mathrm{d} x$ is :

A

\frac{\sqrt{37}+\sqrt{13}-4}{6}

B

\frac{\sqrt{37}-\sqrt{13}-4}{6}

C

\frac{-\sqrt{37}-\sqrt{13}+4}{6}

D

\frac{-\sqrt{37}+\sqrt{13}+4}{6}

Q 90MathematicsSection A • MCQlimits continuity and differentiability

+4 / -1

$\text { Let the function } f(x)=\left\{\begin{array}{cl} \frac{\log _{e}(1+5 x)-\log _{e}(1+\alpha x)}{x} & ;\text { if } x \neq 0 \\ 10 & ; \text { if } x=0 \end{array} \text { be continuous at } x=0 .\right.$

Then $\alpha$ is equal to

A

10

B

-

10

C

5

D

-

5

JEE Main 2022 29th July Evening Shift Question Paper with Solutions