JEE Main 2021 27th August Morning Shift Question Paper Question Paper with Solutions | JEE Main Mock Test | JeeMate

Q 1PhysicsSection A • MCQrotational motion

+4 / -1

Moment of inertia of a square plate of side l about the axis passing through one of the corner and perpendicular to the plane of square plate is given by :

A

{{M{l^2}} \over 6}

B

{M{l^2}}

C

{{M{l^2}} \over {12}}

D

{2 \over 3}M{l^2}

Q 2PhysicsSection B • Numericalheat and thermodynamics

+4 / -1

A rod CD of thermal resistance 10.0 KW^{$-$ 1} is joined at the middle of an identical rod AB as shown in figure. The end A, B and D are maintained at 200

^\circ

C, 100

^\circ

C and 125

^\circ

C respectively. The heat current in CD is P watt. The value of P is ................. .

qImage-heat-transfer-heat-and-thermodynamics-mechanics-Jeemate

Answer Format:Enter numerical value rounded to the nearest integer

Q 3PhysicsSection A • MCQheat and thermodynamics

+4 / -1

An ideal gas is expanding such that PT³ = constant. The coefficient of volume expansion of the gas is :

A

{1 \over T}

B

{2 \over T}

C

{4 \over T}

D

{3 \over T}

Q 4PhysicsSection B • Numericalcurrent electricity

+4 / -1

First, a set of n equal resistors of 10

\Omega

each are connected in series to a battery of emf 20V and internal resistance 10

\Omega

. A current I is observed to flow. Then, the n resistors are connected in parallel to the same battery. It is observed that the current is increased 20 times, then the value of n is ............... .

Answer Format:Enter numerical value rounded to the nearest integer

Q 5PhysicsSection A • MCQelectronic devices

+4 / -1

For a transistor in CE mode to be used as an amplifier, it must be operated in :

A

Both cut-off and Saturation

B

Saturation region only

C

Cut-off region only

D

The active region only

Q 6PhysicsSection A • MCQcapacitor

+4 / -1

Calculate the amount of charge on capacitor of 4

\mu

F. The internal resistance of battery is 1

\Omega

:

qImage-rc-circuit-capacitor-electricity-Jeemate

A

8

\mu

C

B

zero

C

16

\mu

C

D

4

\mu

C

Q 7PhysicsSection A • MCQmagnetics

+4 / -1

Two ions of masses 4 amu and 16 amu have charges +2e and +3e respectively. These ions pass through the region of constant perpendicular magnetic field. The kinetic energy of both ions is same. Then :

A

lighter ion will be deflected less than heavier ion

B

lighter ion will be deflected more than heavier ion

C

both ions will be deflected equally

D

no ion will be deflected

Q 8PhysicsSection B • Numericalwork power and energy

+4 / -1

Two persons A and B perform same amount of work in moving a body through a certain distance d with application of forces acting at angle 45

^\circ

and 60

^\circ

with the direction of displacement respectively. The ratio of force applied by person A to the force applied by person B is

{1 \over {\sqrt x }}

. The value of x is .................... .

Answer Format:Enter numerical value rounded to the nearest integer

Q 9PhysicsSection A • MCQcircular motion

+4 / -1

A huge circular arc of length 4.4 ly subtends an angle '4s' at the centre of the circle. How long it would take for a body to complete 4 revolution if its speed is 8 AU per second?

Given : 1 ly = 9.46

\times

10¹⁵ m

1 AU = 1.5

\times

10¹¹ m

A

4.1

\times

10⁸ s

B

4.5

\times

10¹⁰ s

C

3.5

\times

10⁶ s

D

7.2

\times

10⁸ s

Q 10PhysicsSection A • MCQsimple harmonic motion

+4 / -1

The variation of displacement with time of a particle executing free simple harmonic motion is shown in the figure.

qImage-simple-harmonic-motion-simple-harmonic-motion-mechanics-Jeemate

qImage-simple-harmonic-motion-simple-harmonic-motion-mechanics-Jeemate

The potential energy U(x) versus time (t) plot of the particle is correctly shown in figure :

A

B

C

D

Q 11PhysicsSection A • MCQelectromagnetic induction

+4 / -1

A bar magnet is passing through a conducting loop of radius R with velocity

\upsilon

. The radius of the bar magnet is such that it just passes through the loop. The induced e.m.f. in the loop can be represented by the approximate curve :

qImage-motional-emf-and-eddy-current-electromagnetic-induction-electricity-Jeemate

qImage-motional-emf-and-eddy-current-electromagnetic-induction-electricity-Jeemate

A

B

C

D

Q 12PhysicsSection B • Numericalmagnetics

+4 / -1

A uniform conducting wire of length is 24a, and resistance R is wound up as a current carrying coil in the shape of an equilateral triangle of side 'a' and then in the form of a square of side 'a'. The coil is connected to a voltage source V₀. The ratio of magnetic moment of the coils in case of equilateral triangle to that for square is 1 :

\sqrt y

where y is ................. .

Answer Format:Enter numerical value rounded to the nearest integer

Q 13PhysicsSection A • MCQelectromagnetic waves

+4 / -1

Electric field in a plane electromagnetic wave is given by E = 50 sin(500x

-

10

\times

10¹⁰ t) V/m The velocity of electromagnetic wave in this medium is :

(Given C = speed of light in vacuum)

A

{3 \over 2}

C

B

C

{2 \over 3}

C

D

{C \over 2}

Q 14PhysicsSection B • Numericalwaves

+4 / -1

Two cars X and Y are approaching each other with velocities 36 km/h and 72 km/h respectively. The frequency of a whistle sound as emitted by a passenger in car X, heard by the passenger in car Y is 1320 Hz. If the velocity of sound in air is 340 m/s, the actual frequency of the whistle sound produced is .................. Hz.

Answer Format:Enter numerical value rounded to the nearest integer

Q 15PhysicsSection A • MCQcurrent electricity

+4 / -1

Five identical cells each of internal resistance 1

\Omega

and emf 5V are connected in series and in parallel with an external resistance 'R'. For what value of 'R', current in series and parallel combination will remain the same?

A

1

\Omega

B

25

\Omega

C

5

\Omega

D

10

\Omega

Q 16PhysicsSection A • MCQatoms and nuclei

+4 / -1

There are 10¹⁰ radioactive nuclei in a given radioactive element, its half-life time is 1 minute. How many nuclei will remain after 30 seconds?

\left( {\sqrt 2 = 1.414} \right)

A

2

\times

10¹⁰

B

7

\times

10⁹

C

10⁵

D

4

\times

10¹⁰

Q 17PhysicsSection B • Numericalelectronic devices

+4 / -1

A circuit is arranged as shown in figure. The output voltage V₀ is equal to ................... V.

qImage-transistors-electronic-devices-modern-physics-Jeemate

Answer Format:Enter numerical value rounded to the nearest integer

Q 18PhysicsSection A • MCQelectrostatics

+4 / -1

A uniformly charged disc of radius R having surface charge density

\sigma

is placed in the xy plane with its center at the origin. Find the electric field intensity along the z-axis at a distance Z from origin :-

A

E = {\sigma \over {2{\varepsilon _0}}}\left( {1 - {Z \over {{{({Z^2} + {R^2})}^{1/2}}}}} \right)

B

E = {\sigma \over {2{\varepsilon _0}}}\left( {1 + {Z \over {{{({Z^2} + {R^2})}^{1/2}}}}} \right)

C

E = {{2{\varepsilon _0}} \over \sigma }\left( {{1 \over {{{({Z^2} + {R^2})}^{1/2}}}} + Z} \right)

D

E = {\sigma \over {2{\varepsilon _0}}}\left( {{1 \over {({Z^2} + {R^2})}} + {1 \over {{Z^2}}}} \right)

Q 19PhysicsSection A • MCQproperties of matter

+4 / -1

In Millikan's oil drop experiment, what is viscous force acting on an uncharged drop of radius 2.0

\times

10^{$-$ 5} m and density 1.2

\times

10³ kgm^{$-$ 3} ? Take viscosity of liquid = 1.8

\times

10^{$-$ 5} Nsm^{$-$ 2}. (Neglect buoyancy due to air).

A

3.8

\times

10^{$-$ 11} N

B

3.9

\times

10^{$-$ 10} N

C

1.8

\times

10^{$-$ 10} N

D

5.8

\times

10^{$-$ 10} N

Q 20PhysicsSection B • Numericalcommunication systems

+4 / -1

A transmitting antenna has a height of 320 m and that of receiving antenna is 2000 m. The maximum distance between them for satisfactory communication in line of sight mode is 'd'. The value of 'd' is ................. km.

Answer Format:Enter numerical value rounded to the nearest integer

Q 21PhysicsSection A • MCQunits and measurements

+4 / -1

Which of the following is not a dimensionless quantity?

A

Relative magnetic permeability (

\mu

_r)

B

Power factor

C

Permeability of free space (

\mu

₀)

D

Quality factor

Q 22PhysicsSection B • Numericalmotion

+4 / -1

If the velocity of a body related to displacement x is given by

\upsilon = \sqrt {5000 + 24x}

m/s, then the acceleration of the body is .................... m/s².

Answer Format:Enter numerical value rounded to the nearest integer

Q 23PhysicsSection A • MCQray optics

+4 / -1

An object is placed beyond the centre of curvature C of the given concave mirror. If the distance of the object is d₁ from C and the distance of the image formed is d₂ from C, the radius of curvature of this mirror is :

A

{{2{d_1}{d_2}} \over {{d_1} - {d_2}}}

B

{{2{d_1}{d_2}} \over {{d_1} + {d_2}}}

C

{{{d_1}{d_2}} \over {{d_1} + {d_2}}}

D

{{{d_1}{d_2}} \over {{d_1} - {d_2}}}

Q 24PhysicsSection A • MCQray optics

+4 / -1

Find the distance of the image from object O, formed by the combination of lenses in the figure :

qImage-lenses-geometrical-optics-optics-Jeemate

A

75 cm

B

10 cm

C

20 cm

D

infinity

Q 25PhysicsSection A • MCQheat and thermodynamics

+4 / -1

A balloon carries a total load of 185 kg at normal pressure and temperature of 27

^\circ

C. What load will the balloon carry on rising to a height at which the barometric pressure is 45 cm of Hg and the temperature is

-

7

^\circ

C. Assuming the volume constant?

A

181.46 kg

B

214.15 kg

C

219.07 kg

D

123.54 kg

Q 26PhysicsSection A • MCQunits and measurements

+4 / -1

If E and H represents the intensity of electric field and magnetising field respectively, then the unit of E/H will be :

A

ohm

B

mho

C

joule

D

newton

Q 27PhysicsSection A • MCQvector algebra

+4 / -1

The resultant of these forces

\overrightarrow {OP} ,\overrightarrow {OQ} ,\overrightarrow {OR} ,\overrightarrow {OS}

and

\overrightarrow {OT}

is approximately .......... N.

[Take

\sqrt 3 = 1.7

,

\sqrt 2 = 1.4

Given

\widehat i

and

\widehat j

unit vectors along x, y axis]

qImage-vector-projections-vector-algebra-mechanics-Jeemate

A

9.25\widehat i + 5\widehat j

B

3\widehat i + 15\widehat j

C

2.5\widehat i - 14.5\widehat j

D

- 1.5\widehat i - 15.5\widehat j

Q 28PhysicsSection B • Numericalgravitation

+4 / -1

A body of mass (2M) splits into four masses (m, M

-

m, m, M

-

m}, which are rearranged to form a square as shown in the figure. The ratio of

{M \over m}

for which, the gravitational potential energy of the system becomes maximum is x : 1. The value of x is ............ .

qImage-gravitational-potential-and-gravitational-potential-energy-gravitation-mechanics-Jeemate

qImage-gravitational-potential-and-gravitational-potential-energy-gravitation-mechanics-Jeemate

Answer Format:Enter numerical value rounded to the nearest integer

Q 29PhysicsSection A • MCQdual nature of radiation

+4 / -1

In a photoelectric experiment, increasing the intensity of incident light :

A

increases the number of photons incident and also increases the K.E. of the ejected electrons

B

increases the frequency of photons incident and increases the K.E. of the ejected electrons

C

increases the frequency of photons incident and the K.E. of the ejected electrons remains unchanged

D

increases the number of photons incident and the K.E. of the ejected electrons remains unchanged

Q 30PhysicsSection B • Numericalalternating current

+4 / -1

The alternating current is given by

i = \left\{ {\sqrt {42} \sin \left( {{{2\pi } \over T}t} \right) + 10} \right\}A

The r.m.s. value of of this current is ................. A.

Answer Format:Enter numerical value rounded to the nearest integer

Q 31ChemistrySection A • MCQalcohols phenols and ethers

+4 / -1

In the following sequence of reactions, the final product D is :

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

A

B

CH₃

-

CH = CH

-

CH₂

-

CH₂

-

CH₂

-

COOH

C

H₃C

-

CH = CH

-

CH(OH)

-

CH₂

-

CH₂

-

CH₃

D

Q 32ChemistrySection A • MCQhydrogen

+4 / -1

Deuterium resembles hydrogen in properties but :

A

reacts slower than hydrogen

B

reacts vigorously than hydrogen

C

reacts just as hydrogen

D

emits

\beta

⁺ particles

Q 33ChemistrySection A • MCQcompounds containing nitrogen

+4 / -1

The major product of the following reaction is :

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

A

B

C

D

Q 34ChemistrySection A • MCQalcohols phenols and ethers

+4 / -1

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

Assertion (A) : Synthesis of ethyl phenyl ether may be achieved by Williamson synthesis.

Reason (R) : Reaction of bromobenzene with sodium ethoxide yields ethyl phenyl ether.

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

A

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

B

(A) is correct but (R) is not correct

C

(A) is not correct but (R) is correct

D

Both (A) and (R) are correct but (R) is not the correct explanation of (A).

Q 35ChemistrySection A • MCQchemistry in everyday life

+4 / -1

qImage-chemicals-in-medicines-chemistry-in-everyday-life-organic-chemistry-Jeemate

The correct statement about (A), (B), (C) and (D) is :

A

(A), (B) and (C) are narcotic analgesics

B

(B), (C) and (D) are tranquillizers

C

(A) and (D) are tranquillizers

D

(B) and (C) are tranquillizers

Q 36ChemistrySection A • MCQcompounds containing nitrogen

+4 / -1

Which of the following is not a correct statement for primary aliphatic amines?

A

The intermolecular association in primary amines is less than the intermolecular association in secondary amines.

B

Primary amines on treating with nitrous acid solution from corresponding alcohols except mythyl amine.

C

Primary amines are less basic than the secondary amines.

D

Primary amines can be prepared by the Gabriel phthalimide synthesis.

Q 37ChemistrySection A • MCQsurface chemistry

+4 / -1

Tyndall effect is more effectively shown by :

A

true solution

B

lyophilic colloid

C

lyophobic colloid

D

suspension

Q 38ChemistrySection A • MCQd and f block elements

+4 / -1

The nature of oxides V₂O₃ and CrO is indexed as 'X' and 'Y' type respectively. The correct set of X and Y is :

A

X = basic		Y = amphoteric

B

X = amphoteric		Y = basic

C

X = acidic		Y = acidic

D

X = basic		Y = basic

Q 39ChemistrySection B • Numericald and f block elements

+4 / -1

The number of f electrons in the ground state electronic configuration of Np (Z = 93) is ___________. (Nearest integer)

Answer Format:Enter numerical value rounded to the nearest integer

Q 40ChemistrySection B • Numericalthermodynamics

+4 / -1

200 mL of 0.2 M HCl is mixed with 300 mL of 0.1 M NaOH. The molar heat of neutralization of this reaction is

-

57.1 kJ. The increase in temperature in

^\circ

C of the system on mixing is x

\times

10^{$-$ 2}. The value of x is ___________. (Nearest integer)

[Given : Specific heat of water = 4.18 J g^{$-$ 1} K^{$-$ 1}, Density of water = 1.00 g cm^{$-$ 3}]

[Assume no volume change on mixing)

Answer Format:Enter numerical value rounded to the nearest integer

Q 41ChemistrySection A • MCQs block elements

+4 / -1

The number of water molecules in gypsum, dead burnt plaster and plaster of Paris, respectively are :

A

2, 0 and 1

B

0.5, 0 and 2

C

5, 0 and 0.5

D

2, 0 and 0.5

Official Correct Answer:Option (D)

Step-by-Step Solution:


	Gypsum		$CaS{O_4}.2{H_2}O$
	Plaster of Paris		$CaS{O_4}.{1 \over 2}{H_2}O$
	Dead burnt plaster		$CaS{O_4}$

Q 42ChemistrySection A • MCQp block elements

+4 / -1

In which one of the following molecules strongest back donation of an electron pair from halide to boron is expected?

A

BCl₃

B

BF₃

C

BBr₃

D

BI₃

Q 43ChemistrySection B • Numericalatomic structure

+4 / -1

The kinetic energy of an electron in the second Bohr orbit of a hydrogen atom is equal to

{{{h^2}} \over {xma_0^2}}

. The value of 10x is ___________. (a₀ is radius of Bohr's orbit) (Nearest integer) [Given :

\pi

= 3.14]

Answer Format:Enter numerical value rounded to the nearest integer

Q 44ChemistrySection B • Numericalsome basic concepts of chemistry

+4 / -1

In Carius method for estimation of halogens, 0.2 g of an organic compound gave 0.188 g of AgBr. The percentage of bromine in the compound is ______________. (Nearest integer)

[Atomic mass : Ag = 108, Br = 80]

Answer Format:Enter numerical value rounded to the nearest integer

Q 45ChemistrySection A • MCQp block elements

+4 / -1

In polythionic acid, H₂S_xO₆ (x = 3 to 5) the oxidation state(s) of sulphur is/are :

A

+ 5 only

B

+ 6 only

C

+ 3 and + 5 only

D

0 and + 5 only

Q 46ChemistrySection B • Numericalcoordination compounds

+4 / -1

1 mol of an octahedral metal complex with formula MCl₃ . 2L on reaction with excess of AgNO₃ gives 1 mol of AgCl. The denticity of Ligand L is ____________. (Integer answer)

Answer Format:Enter numerical value rounded to the nearest integer

Q 47ChemistrySection A • MCQenvironmental chemistry

+4 / -1

The gas 'A' is having very low reactivity reaches to stratosphere. It is non-toxic and non-flammable but dissociated by UV-radiations in stratosphere. The intermediates formed initially from the gas 'A' are :

A

Cl\mathop O\limits^ \bullet + \mathop C\limits^ \bullet {F_2}Cl

B

Cl\mathop O\limits^ \bullet + C\mathop {{H_3}}\limits^ \bullet

C

\mathop C\limits^ \bullet {H_3} + C\mathop {{F_2}}\limits^ \bullet Cl

D

\mathop C\limits^ \bullet l + C\mathop {{F_2}}\limits^ \bullet Cl

Q 48ChemistrySection B • Numericalchemical kinetics and nuclear chemistry

+4 / -1

The reaction that occurs in a breath analyser, a device used to determine the alcohol level in a person's blood stream is

2{K_2}C{r_2}{O_7} + 8{H_2}S{O_4} + 3{C_2}{H_6}O \to 2C{r_2}{(S{O_4})_3} + 3{C_2}{H_4}{O_2} + 2{K_2}S{O_4} + 11{H_2}O

If the rate of appearance of Cr₂(SO₄)₃ is 2.67 mol min^{$-$ 1} at a particular time, the rate of disappearance of C₂H₆O at the same time is _____________ mol min^{$-$ 1}. (Nearest integer)

Answer Format:Enter numerical value rounded to the nearest integer

Q 49ChemistrySection A • MCQpractical organic chemistry

+4 / -1

Acidic ferric chloride solution on treatment with excess of potassium ferrocyanide gives a Prussian blue coloured colloidal species. It is :

A

Fe₄[Fe(CN)₆]₃

B

K₅Fe[Fe(CN)₆]₂

C

HFe[Fe(CN)₆]

D

KFe[Fe(CN)₆]

Q 50ChemistrySection B • Numericalchemical equilibrium

+4 / -1

The number of moles of NH₃, that must be added to 2L of 0.80 M AgNO₃ in order to reduce the concentration of Ag⁺ ions to 5.0

\times

10^{$-$ 8} M (K_formation for [Ag(NH₃)₂]⁺ = 1.0

\times

10⁸) is ____________. (Nearest integer)

[Assume no volume change on adding NH₃]

Answer Format:Enter numerical value rounded to the nearest integer

Q 51ChemistrySection A • MCQaldehydes ketones and carboxylic acids

+4 / -1

The structure of the starting compound P used in the reaction given below is :

qImage-preparation-properties-and-uses-of-carboxylic-acids-aldehydes-ketones-and-carboxylic-acids-organic-chemistry-Jeemate

qImage-preparation-properties-and-uses-of-carboxylic-acids-aldehydes-ketones-and-carboxylic-acids-organic-chemistry-Jeemate

A

B

C

D

Q 52ChemistrySection A • MCQbiomolecules

+4 / -1

Out of following isomeric forms of uracil, which one is present in RNA?

A

B

C

D

Q 53ChemistrySection A • MCQchemical bonding and molecular structure

+4 / -1

Match List - I with List - II :

qImage-hybridization-and-vsepr-theory-chemical-bonding-and-molecular-structure-inorganic-chemistry-Jeemate

Choose the most appropriate answer from the options given below :

A

(a)-(iv), (b)-(i), (c)-(ii), (d)-(iii)

B

(a)-(iii), (b)-(iv), (c)-(ii), (d)-(i)

C

(a)-(iii), (b)-(ii), (c)-(iv), (d)-(i)

D

(a)-(iv), (b)-(ii), (c)-(i), (d)-(iii)

Official Correct Answer:Option (D)

Step-by-Step Solution:

Species	(Number of lone pairs of electrons on the central atom)
$Xe{F_2}$
$Xe{O_2}{F_2}$
$Xe{O_3}{F_2}$
$Xe{F_4}$

Q 54ChemistrySection A • MCQhaloalkanes and haloarenes

+4 / -1

In the following sequence of reactions the P is :

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

A

B

C

D

Q 55ChemistrySection B • Numericalsolutions

+4 / -1

1 kg of 0.75 molal aqueous solution of sucrose can be cooled up to

-

4

^\circ

C before freezing. The amount of ice (in g) that will be separated out is __________. (Nearest integer)

[Given : K_f(H₂O) = 1.86 K kg mol^{$-$ 1}]

Answer Format:Enter numerical value rounded to the nearest integer

Q 56ChemistrySection A • MCQgaseous state

+4 / -1

The unit of the van der Waals gas equation parameter 'a' in

\left( {P + {{a{n^2}} \over {{V^2}}}} \right)(V - nb) = nRT

is :

A

kg m s^{$-$ 2}

B

dm³ mol^{$-$ 1}

C

kg m s^{$-$ 1}

D

atm dm⁶ mol^{$-$ 2}

Q 57ChemistrySection B • Numericalpractical organic chemistry

+4 / -1

When 10 mL of an aqueous solution of KMnO₄ was titrated in acidic medium, equal volume of 0.1 M of an aqueous solution of ferrous sulphate was required for complete discharge of colour. The strength of KMnO₄ in grams per litre is __________

\times

10^{$-$ 2}. (Nearest integer)

[Atomic mass of K = 39, Mn = 55, O = 16]

Answer Format:Enter numerical value rounded to the nearest integer

Q 58ChemistrySection A • MCQisolation of elements

+4 / -1

Which refining process is generally used in the purification of low melting metals?

A

Chromatographic method

B

Liquation

C

Electrolysis

D

Zone refining

Q 59ChemistrySection A • MCQchemical bonding and molecular structure

+4 / -1

Match items of List-I with those of List-II :

	List - I (Property)		List - II (Example)
(a)	Diamagnetism	(i)	MnO
(b)	Ferrimagnetism	(ii)	${O_2}$
(c)	Paramagnetism	(iii)	NaCl
(d)	Antiferromagnetism	(iv)	$F{e_3}{O_4}$

Choose the most appropriate answer from the options given below :

A

(a)-(ii), (b)-(i), (c)-(iii), (d)-(iv)

B

(a)-(i), (b)-(iii), (c)-(iv), (d)-(ii)

C

(a)-(iii), (b)-(iv), (c)-(ii), (d)-(i)

D

(a)-(iv), (b)-(ii), (c)-(i), (d)-(iii)

Q 60ChemistrySection B • Numericalpractical organic chemistry

+4 / -1

The number of moles of CuO, that will be utilized in Dumas method for estimation nitrogen in a sample of 57.5 g of N, N-dimethylaminopentane is _____________

\times

10^{$-$ 2}. (Nearest integer)

Answer Format:Enter numerical value rounded to the nearest integer

Q 61MathematicsSection A • MCQparabola

+4 / -1

A tangent and a normal are drawn at the point P(2,

-

4) on the parabola y² = 8x, which meet the directrix of the parabola at the points A and B respectively. If Q(a, b) is a point such that AQBP is a square, then 2a + b is equal to :

A

-

16

B

-

18

C

-

12

D

-

20

Q 62MathematicsSection A • MCQsequences and series

+4 / -1

If for x, y

\in

R, x > 0, y = log₁₀x + log₁₀x^1/3 + log₁₀x^1/9 + ...... upto

\infty

terms

and

{{2 + 4 + 6 + .... + 2y} \over {3 + 6 + 9 + ..... + 3y}} = {4 \over {{{\log }_{10}}x}}

, then the ordered pair (x, y) is equal to :

A

(10⁶, 6)

B

(10⁴, 6)

C

(10², 3)

D

(10⁶, 9)

Q 63MathematicsSection B • Numericalapplication of derivatives

+4 / -1

The number of distinct real roots of the equation 3x⁴ + 4x³

-

12x² + 4 = 0 is _____________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 64MathematicsSection B • Numericalmatrices and determinants

+4 / -1

If the system of linear equations

2x + y

-

z = 3

x

-

y

-

z =

\alpha

3x + 3y +

\beta

z = 3

has infinitely many solution, then

\alpha

+

\beta

-

\alpha

\beta

is equal to _____________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 65MathematicsSection B • Numericalpermutations and combinations

+4 / -1

A number is called a palindrome if it reads the same backward as well as forward. For example 285582 is a six digit palindrome. The number of six digit palindromes, which are divisible by 55, is ____________.

Answer Format:Enter numerical value rounded to the nearest integer

Official Correct Answer:100

Step-by-Step Solution:

5	a	b	b	a	5

For divisible by 55 it shall be divisible by 11 and 5 both, for divisibility by 5 unit digit shall be 0 or 5 but as the number is six digit palindrome unit digit is 5.

A number is divisible by 11 if the difference between sum of the digits in the odd places and the sum of the digits in the even places is a multiple of 11 or zero.

Sum of the digits in the even place = a + b + 5

Sum of the digits in the odd places = a + b + 5

Difference between the two sums = (a + b + 5 ) - (a + b + 5) = 0

0 is divisible by 11.

Hence, 5abba5 is divisible by 11.

So, required number = 10

\times

10 = 100

Q 66MathematicsSection A • MCQinverse trigonometric functions

+4 / -1

If

{({\sin ^{ - 1}}x)^2} - {({\cos ^{ - 1}}x)^2} = a

; 0 < x < 1, a

\ne

0, then the value of 2x²

-

1 is :

A

\cos \left( {{{4a} \over \pi }} \right)

B

\sin \left( {{{2a} \over \pi }} \right)

C

\cos \left( {{{2a} \over \pi }} \right)

D

\sin \left( {{{4a} \over \pi }} \right)

Q 67MathematicsSection A • MCQdifferential equations

+4 / -1

Let us consider a curve, y = f(x) passing through the point (

-

2, 2) and the slope of the tangent to the curve at any point (x, f(x)) is given by f(x) + xf'(x) = x². Then :

A

{x^2} + 2xf(x) - 12 = 0

B

{x^3} + xf(x) + 12 = 0

C

{x^3} - 3xf(x) - 4 = 0

D

{x^2} + 2xf(x) + 4 = 0

Q 68MathematicsSection A • MCQmathematical reasoning

+4 / -1

The statement (p

\wedge

(p

\to

q)

\wedge

(q

\to

r))

\to

r is :

A

a tautology

B

equivalent to p

\to

\sim

r

C

a fallacy

D

equivalent to q

\to

\sim

r

Q 69MathematicsSection A • MCQbinomial theorem

+4 / -1

\sum\limits_{k = 0}^{20} {{{\left( {{}^{20}{C_k}} \right)}^2}}

is equal to :

A

{}^{40}{C_{21}}

B

{}^{40}{C_{19}}

C

{}^{40}{C_{20}}

D

{}^{41}{C_{20}}

Q 70MathematicsSection A • MCQellipse

+4 / -1

If x² + 9y²

-

4x + 3 = 0, x, y

\in

R, then x and y respectively lie in the intervals :

A

\left[ { - {1 \over 3},{1 \over 3}} \right]

and

\left[ { - {1 \over 3},{1 \over 3}} \right]

B

\left[ { - {1 \over 3},{1 \over 3}} \right]

and [1, 3]

C

[1, 3] and [1, 3]

D

[1, 3] and

\left[ { - {1 \over 3},{1 \over 3}} \right]

Q 71MathematicsSection A • MCQdefinite integration

+4 / -1

\int\limits_6^{16} {{{{{\log }_e}{x^2}} \over {{{\log }_e}{x^2} + {{\log }_e}({x^2} - 44x + 484)}}dx}

is equal to :

A

6

B

8

C

5

D

10

Q 72MathematicsSection A • MCQ3d geometry

+4 / -1

The distance of the point (1,

-

2, 3) from the plane x

-

y + z = 5 measured parallel to a line, whose direction ratios are 2, 3,

-

6 is :

A

3

B

5

C

2

D

1

Q 73MathematicsSection B • Numericalcircle

+4 / -1

Let the equation x² + y² + px + (1

-

p)y + 5 = 0 represent circles of varying radius r

\in

(0, 5]. Then the number of elements in the set S = {q : q = p² and q is an integer} is __________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 74MathematicsSection A • MCQstraight lines and pair of straight lines

+4 / -1

Let A be a fixed point (0, 6) and B be a moving point (2t, 0). Let M be the mid-point of AB and the perpendicular bisector of AB meets the y-axis at C. The locus of the mid-point P of MC is :

A

3x²

-

2y

-

6 = 0

B

3x² + 2y

-

6 = 0

C

2x² + 3y

-

9 = 0

D

2x²

-

3y + 9 = 0

Q 75MathematicsSection A • MCQlimits continuity and differentiability

+4 / -1

If

\alpha

,

\beta

are the distinct roots of x² + bx + c = 0, then

\mathop {\lim }\limits_{x \to \beta } {{{e^{2({x^2} + bx + c)}} - 1 - 2({x^2} + bx + c)} \over {{{(x - \beta )}^2}}}

is equal to :

A

b² + 4c

B

2(b² + 4c)

C

2(b²

-

4c)

D

b²

-

4c

Q 76MathematicsSection A • MCQsequences and series

+4 / -1

If 0 < x < 1, then

{3 \over 2}{x^2} + {5 \over 3}{x^3} + {7 \over 4}{x^4} + .....

, is equal to :

A

x\left( {{{1 + x} \over {1 - x}}} \right) + {\log _e}(1 - x)

B

x\left( {{{1 - x} \over {1 + x}}} \right) + {\log _e}(1 - x)

C

{{1 - x} \over {1 + x}} + {\log _e}(1 - x)

D

{{1 + x} \over {1 - x}} + {\log _e}(1 - x)

Q 77MathematicsSection B • Numericalstatistics

+4 / -1

Let n be an odd natural number such that the variance of 1, 2, 3, 4, ......, n is 14. Then n is equal to _____________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 78MathematicsSection A • MCQprobability

+4 / -1

When a certain biased die is rolled, a particular face occurs with probability

{1 \over 6} - x

and its opposite face occurs with probability

{1 \over 6} + x

. All other faces occur with probability

{1 \over 6}

. Note that opposite faces sum to 7 in any die. If 0 < x <

{1 \over 6}

, and the probability of obtaining total sum = 7, when such a die is rolled twice, is

{13 \over 96}

, then the value of x is :

A

{1 \over 16}

B

{1 \over 8}

C

{1 \over 9}

D

{1 \over 12}

Q 79MathematicsSection A • MCQdifferential equations

+4 / -1

Let y = y(x) be the solution of the differential equation

{{dy} \over {dx}} = 2(y + 2\sin x - 5)x - 2\cos x

such that y(0) = 7. Then y(

\pi

) is equal to :

A

2{e^{{\pi ^2}}} + 5

B

{e^{{\pi ^2}}} + 5

C

3{e^{{\pi ^2}}} + 5

D

7{e^{{\pi ^2}}} + 5

Q 80MathematicsSection A • MCQcomplex numbers

+4 / -1

If

S = \left\{ {z \in C:{{z - i} \over {z + 2i}} \in R} \right\}

, then :

A

S contains exactly two elements

B

S contains only one element

C

S is a circle in the complex plane

D

S is a straight line in the complex plane

Q 81MathematicsSection A • MCQdefinite integration

+4 / -1

If

{U_n} = \left( {1 + {1 \over {{n^2}}}} \right)\left( {1 + {{{2^2}} \over {{n^2}}}} \right)^2.....\left( {1 + {{{n^2}} \over {{n^2}}}} \right)^n

, then

\mathop {\lim }\limits_{n \to \infty } {({U_n})^{{{ - 4} \over {{n^2}}}}}

is equal to :

A

{{{e^2}} \over {16}}

B

{4 \over e}

C

{{16} \over {{e^2}}}

D

{4 \over {{e^2}}}

Q 82MathematicsSection A • MCQproperties of triangles

+4 / -1

Let

{{\sin A} \over {\sin B}} = {{\sin (A - C)} \over {\sin (C - B)}}

, where A, B, C are angles of triangle ABC. If the lengths of the sides opposite these angles are a, b, c respectively, then :

A

b²

-

a² = a² + c²

B

b², c², a² are in A.P.

C

c², a², b² are in A.P.

D

a², b², c² are in A.P.

Q 83MathematicsSection B • Numericalvector algebra

+4 / -1

Let

\overrightarrow a = \widehat i + 5\widehat j + \alpha \widehat k

,

\overrightarrow b = \widehat i + 3\widehat j + \beta \widehat k

and

\overrightarrow c = - \widehat i + 2\widehat j - 3\widehat k

be three vectors such that,

\left| {\overrightarrow b \times \overrightarrow c } \right| = 5\sqrt 3

and

{\overrightarrow a }

is perpendicular to

{\overrightarrow b }

. Then the greatest amongst the values of

{\left| {\overrightarrow a } \right|^2}

is _____________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 84MathematicsSection A • MCQ3d geometry

+4 / -1

Equation of a plane at a distance

\sqrt {{2 \over {21}}}

from the origin, which contains the line of intersection of the planes x

-

y

-

z

-

1 = 0 and 2x + y

-

3z + 4 = 0, is :

A

3x - y - 5z + 2 = 0

B

3x - 4z + 3 = 0

C

- x + 2y + 2z - 3 = 0

D

4x - y - 5z + 2 = 0

Q 85MathematicsSection B • Numericalellipse

+4 / -1

If the minimum area of the triangle formed by a tangent to the ellipse

{{{x^2}} \over {{b^2}}} + {{{y^2}} \over {4{a^2}}} = 1

and the co-ordinate axis is kab, then k is equal to _______________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 86MathematicsSection B • Numericalindefinite integration

+4 / -1

If

\int {{{dx} \over {{{({x^2} + x + 1)}^2}}} = a{{\tan }^{ - 1}}\left( {{{2x + 1} \over {\sqrt 3 }}} \right) + b\left( {{{2x + 1} \over {{x^2} + x + 1}}} \right) + C}

, x > 0 where C is the constant of integration, then the value of

9\left( {\sqrt 3 a + b} \right)

is equal to _____________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 87MathematicsSection A • MCQmatrices and determinants

+4 / -1

If the matrix

A = \left( {\matrix{ 0 & 2 \cr K & { - 1} \cr } } \right)

satisfies

A({A^3} + 3I) = 2I

, then the value of K is :

A

{1 \over 2}

B

-

{1 \over 2}

C

-

1

D

1

Q 88MathematicsSection A • MCQapplication of derivatives

+4 / -1

A wire of length 20 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a regular hexagon. Then the length of the side (in meters) of the hexagon, so that the combined area of the square and the hexagon is minimum, is :

A

{5 \over {2 + \sqrt 3 }}

B

{{10} \over {2 + 3\sqrt 3 }}

C

{5 \over {3 + \sqrt 3 }}

D

{{10} \over {3 + 2\sqrt 3 }}

Q 89MathematicsSection B • Numericalsets and relations

+4 / -1

If A = {x

\in

R : |x

-

2| > 1},
B = {x

\in

R :

\sqrt {{x^2} - 3}

> 1},
C = {x

\in

R : |x

-

4|

\ge

2} and Z is the set of all integers, then the number of subsets of the
set (A

\cap

B

\cap

C)^c

\cap

Z is ________________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 90MathematicsSection B • Numericaldifferential equations

+4 / -1

If

{y^{1/4}} + {y^{ - 1/4}} = 2x

, and

({x^2} - 1){{{d^2}y} \over {d{x^2}}} + \alpha x{{dy} \over {dx}} + \beta y = 0

, then |

\alpha

-

\beta

| is equal to __________.

Answer Format:Enter numerical value rounded to the nearest integer

JEE Main 2021 27th August Morning Shift Question Paper with Solutions