JEE Main 2020 8th January Morning Shift Question Paper Question Paper with Solutions | JEE Main Mock Test | JeeMate

Q 1PhysicsSection A • MCQray optics

+4 / -1

The critical angle of a medium for a specific wavelength, if the medium has relative permittivity 3 and relative permeability

{4 \over 3}

for this wavelength, will be :

A

45°

B

15°

C

30°

D

60°

Q 2PhysicsSection B • Numericalcenter of mass

+4 / -0

A body A, of mass m = 0.1 kg has an initial velocity of 3

\widehat i

ms^-1 . It collides elastically with another body, B of the same mass which has an initial velocity of 5

\widehat j

ms^-1. After collision, A moves with a velocity

\overrightarrow v = 4\left( {\widehat i + \widehat j} \right)

. The energy of B after collision is written as

{x \over {10}}

. The value of x is ___________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 3PhysicsSection A • MCQcurrent electricity

+4 / -1

The length of a potentiometer wire is 1200 cm and it carries a current of 60 mA. For a cell of emf 5V and internal resistance of 20

\Omega

, the null point on it is found to be a 1000 cm. The resistance of whole wire is :

A

80

\Omega

B

60

\Omega

C

120

\Omega

D

100

\Omega

Q 4PhysicsSection B • Numericalray optics

+4 / -0

A point object in air is in front of the curved surface of a plano-convex lens. The radius of curvature of the curved surface is 30 cm and the refractive index of the lens material is 1.5, then the focal length of the lens (in cm) is ______.

Answer Format:Enter numerical value rounded to the nearest integer

Q 5PhysicsSection A • MCQheat and thermodynamics

+4 / -1

A thermodynamic cycle xyzx is shown on a V-T diagram. qImage-thermodynamics-process-heat-and-thermodynamics-mechanics-Jeemate

The P-V diagram that best describes this cycle is :
(Diagrams are schematic and not to scale)

A

B

C

D

Q 6PhysicsSection A • MCQdual nature of radiation

+4 / -1

When photon of energy 4.0 eV strikes the surface of a metal A, the ejected photoelectrons have maximum kinetic energy T_A eV end de-Broglie wavelength

\lambda _A

. The maximum kinetic energy of photoelectrons liberated from another metal B by photon of energy 4.50 eV is T_B = (T_A – 1.5) eV. If the de-Broglie wavelength of these photoelectrons

\lambda _B

= 2

\lambda _A

, then the work function of metal B is :

A

1.5eV

B

4eV

C

2eV

D

3eV

Q 7PhysicsSection A • MCQelectronic devices

+4 / -1

Boolean relation at the output stage-Y for the following circuit is : qImage-digital-circuits-electronic-devices-modern-physics-Jeemate

A

A.B

B

\overline A + \overline B

C

\overline A .\overline B

D

A + B

Q 8PhysicsSection A • MCQheat and thermodynamics

+4 / -1

The plot that depicts the behavior of the mean free time t (time between two successive collisions) for the molecules of an ideal gas, as a function of temperature (T), qualitatively, is:
(Graphs are schematic and not drawn to scale)

A

B

C

D

Q 9PhysicsSection A • MCQcircular motion

+4 / -1

A particle of mass m is fixed to one end of a light spring having force constant k and unstretched length

\ell

. The other end is fixed. The system is given an angular speed

\omega

about the fixed end of the spring such that it rotates in a circle in gravity free space. Then the stretch in the spring is :

A

{{m\ell {\omega ^2}} \over {k - m{\omega ^2}}}

B

{{m\ell {\omega ^2}} \over {k - m{\omega}}}

C

{{m\ell {\omega ^2}} \over {k + m{\omega ^2}}}

D

{{m\ell {\omega ^2}} \over {k + m{\omega}}}

Q 10PhysicsSection A • MCQcenter of mass

+4 / -1

The coordinates of centre of mass of a uniform flag shaped lamina (thin flat plate) of mass 4kg. (The coordinates of the same are shown in figure) are : qImage-center-of-mass-of-continuous-mass-distribution-center-of-mass-mechanics-Jeemate

qImage-center-of-mass-of-continuous-mass-distribution-center-of-mass-mechanics-Jeemate

A

(0.75m, 1.75m)

B

(1m, 1.75m)

C

(0.75m, 0.75m)

D

(1.25m, 1.50m)

Q 11PhysicsSection A • MCQelectrostatics

+4 / -1

Three charged particle A, B and C with charges –4q, 2q and –2q are present on the circumference of a circle of radius d. the charged particles A, C and centre O of the circle formed an equilateral triangle as shown in figure. Electric field at O along x-direction is : qImage-electric-field-and-electric-field-intensity-electrostatics-electricity-Jeemate

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

A

{3{\sqrt 3 q} \over 4{\pi {\varepsilon _0}{d^2}}}

B

{{\sqrt 3 q} \over 4{\pi {\varepsilon _0}{d^2}}}

C

{{\sqrt 3 q} \over {\pi {\varepsilon _0}{d^2}}}

D

{{2\sqrt 3 q} \over {\pi {\varepsilon _0}{d^2}}}

Q 12PhysicsSection B • Numericalmotion

+4 / -0

A particle is moving along the x-axis with its coordinate with the time 't' given be
x(t) = 10 + 8t – 3t². Another particle is moving the y-axis with its coordinate as a function of time given by y(t) = 5 – 8t³.
At t = 1s, the speed of the second particle as measured in the frame of the first particle is given as

\sqrt v

. Then v (in m/s) is ______.

Answer Format:Enter numerical value rounded to the nearest integer

Q 13PhysicsSection B • Numericalcurrent electricity

+4 / -0

Four resistances of 15

\Omega

, 12

\Omega

, 4

\Omega

and 10

\Omega

respectively in cyclic order to form Wheatstone's network. The resistance that is to be connected in parallel with the resistance of 10

\Omega

to balance the network is _____

\Omega

.

Answer Format:Enter numerical value rounded to the nearest integer

Q 14PhysicsSection A • MCQelectromagnetic induction

+4 / -1

At time t = 0 magnetic field of 1000 Gauss is passing perpendicularly through the area defined by the closed loop shown in the figure. If the magnetic field reduces linearly to 500 Gauss, in the next 5s, then induced EMF in the loop is : qImage-magnetic-flux-faradays-and-lenzs-laws-electromagnetic-induction-electricity-Jeemate

qImage-magnetic-flux-faradays-and-lenzs-laws-electromagnetic-induction-electricity-Jeemate

A

48 μV

B

28 μV

C

56 μV

D

36 μV

Q 15PhysicsSection A • MCQelectrostatics

+4 / -1

In finding the electric field using Gauss Law the formula

\left| {\overrightarrow E } \right| = {{{q_{enc}}} \over {{\varepsilon _0}\left| A \right|}}

is applicable. In the formula

{{\varepsilon _0}}

is permittivity of free space, A is the area of Gaussian surface and q_enc is charge enclosed by the Gaussian surface. The equation can be used in which of the following situation?

A

Only when

\left| {\overrightarrow E } \right|

= constant on the surface.

B

For any choice of Gaussian surface.

C

Only when the Gaussian surface is an equipotential surface.

D

Only when the Gaussian surface is an equipotential surface and

\left| {\overrightarrow E } \right|

is constant on the surface.

Q 16PhysicsSection A • MCQgravitation

+4 / -1

Consider two solid spheres of radii R₁ = 1m, R₂ = 2m and masses M₁ and M₂, respectively. The gravitational field due to sphere (1) and (2) are shown. The value of

{{{M_1}} \over {{M_2}}}

is :

qImage-acceleration-due-to-gravity-and-its-variation-gravitation-mechanics-Jeemate

A

{2 \over 3}

B

{1 \over 6}

C

{1 \over 2}

D

{1 \over 3}

Q 17PhysicsSection A • MCQmagnetics

+4 / -1

Photon with kinetic energy of 1MeV moves from south to north. It gets an acceleration of 10¹² m/s² by an applied magnetic field (west to east). The value of magnetic field : (Rest mass of proton is 1.6 × 10^–27 kg) :

A

0.71mT

B

7.1mT

C

0.071mT

D

71mT

Q 18PhysicsSection A • MCQrotational motion

+4 / -1

Consider a uniform rod of mass M = 4m and length

\ell

pivoted about its centre. A mass m moving with velocity v making angle

\theta = {\pi \over 4}

to the rod's long axis collides with one end of the rod and sticks to it. The angular speed of the rod-mass system just after the collision is :

A

{{3\sqrt 2 } \over 7}{v \over \ell }

B

{3 \over 7}{v \over \ell }

C

{3 \over {7\sqrt 2 }}{v \over \ell }

D

{4 \over 7}{v \over \ell }

Q 19PhysicsSection A • MCQproperties of matter

+4 / -1

Consider a solid sphere of radius R and mass density

\rho \left( r \right) = {\rho _0}\left( {1 - {{{r^2}} \over {{R^2}}}} \right)

,

0 < r \le R

The minimum density of a liquid in which it will float is :

A

{{2{\rho _0}} \over 3}

B

{{2{\rho _0}} \over 5}

C

{{{\rho _0}} \over 5}

D

{{{\rho _0}} \over 3}

Q 20PhysicsSection A • MCQcapacitor

+4 / -1

Effective capacitance of parallel combination of two capacitors C₁ and C₂ is 10 μF. When these capacitors are individually connected to a voltage source of 1V, the energy stored in the capacitor C₂ is 4 times that of C₁. If these capacitors are connected in series, their effective capacitance will be :

A

4.2 μF

B

8.4 μF

C

1.6 μF

D

3.2 μF

Q 21PhysicsSection A • MCQproperties of matter

+4 / -1

A leak proof cylinder of length 1m, made of a metal which has very low coefficient of expansion is floating vertically in water at 0°C such that its height above the water surface is 20 cm. When the temperature of water is increased to 4°C, the height of the cylinder above the water surface becomes 21 cm. The density of water at T = 4°C, relative to the density at T = 0°C is close to :

A

1.04

B

1.26

C

1.01

D

1.03

Q 22PhysicsSection A • MCQray optics

+4 / -1

The magnifying power of a telescope with tube 60 cm is 5. What is the focal length of its eye piece ?

A

40 cm

B

10 cm

C

30 cm

D

20 cm

Q 23PhysicsSection B • Numericalwaves

+4 / -0

A one metre long (both ends open) organ pipe is kept in a gas that has double the density of air at STP. Assuming the speed of sound in air at STP is 300 m/s, the frequency difference between the fundamental and second harmonic of this pipe is ________ Hz.

Answer Format:Enter numerical value rounded to the nearest integer

Q 24PhysicsSection A • MCQunits and measurements

+4 / -1

The dimension of stopping potential V₀ in photoelectric effect in units of Planck's constant 'h', speed of light 'c' and Gravitational constant 'G' and ampere A is :

A

h^1/3 G^2/3 c^1/3 A^–1

B

h⁰ c⁵ G^-1 A^-1

C

h^2/3 c^5/3 G^1/3 A^–1

D

h² G^3/2 c^1/3 A^–1

Q 25PhysicsSection A • MCQatoms and nuclei

+4 / -1

The graph which depicts the results of Rutherford gold foil experiment with

\alpha

-particales is :

\theta

: Scattering angle
Y : Number of scattered

\alpha

-particles detected (Plots are schematic and not to scale)

A

B

C

D

Q 26ChemistrySection A • MCQalcohols phenols and ethers

+4 / -1

The major product of the following reaction is : qImage-properties-preparation-and-uses-of-phenols-alcohols-phenols-and-ethers-organic-chemistry-Jeemate

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

A

B

C

D

Q 27ChemistrySection B • Numericalchemistry in everyday life

+4 / -0

The number of chiral centres in penicillin is _________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 28ChemistrySection B • Numericalsome basic concepts of chemistry

+4 / -0

The volume (in mL) of 0.125 M AgNO₃ required to quantitatively precipitate chloride ions in 0.3 g of [Co(NH₃)₆]Cl₃ is ________.
M_{[Co(NH₃)₆Cl₃]} = 267.46 g/mol
M_AgNO₃ = 169.87 g/mol

Answer Format:Enter numerical value rounded to the nearest integer

Q 29ChemistrySection A • MCQsolutions

+4 / -1

A graph of vapour pressure and temperature for three different liquids X, Y, and Z is shown below : qImage-henrys-law-solutions-physical-chemistry-Jeemate

qImage-henrys-law-solutions-physical-chemistry-Jeemate

The following inferences are made :
(A) X has higher intermolecular interactions compared to Y.
(B) X has lower intermolecular interactions compared to Y.
(C) Z has lower intermolecular interactions compared to Y.
The correct inference (s) is / are :

A

(B)

B

(A)

C

(C)

D

(A) and (C)

Q 30ChemistrySection A • MCQchemical kinetics and nuclear chemistry

+4 / -1

The rate of a certain biochemical reaction at physiological temperature (T) occurs 10⁶ times faster with enzyme than without. The change in the activation energy upon adding enzyme is :

A

+ 6RT

B

– 6 (2.303)RT

C

– 6RT

D

+ 6(2.303)RT

Q 31ChemistrySection A • MCQp block elements

+4 / -1

The number of bonds between sulphur and oxygen atoms in S₂O₈^2- and the number of bonds between sulphur and sulphur atoms in rhombic sulphur, respectively, are :

A

4 and 8

B

8 and 6

C

4 and 6

D

8 and 8

Q 32ChemistrySection A • MCQperiodic classification of elements

+4 / -1

The first ionization energy (in kJ/mol) of Na, Mg, Al and Si respectively, are :

A

496, 577, 786, 737

B

496, 737, 577, 786

C

786, 737, 577, 496

D

496, 577, 737, 786

Q 33ChemistrySection A • MCQs block elements

+4 / -1

When gypsum is heated to 393 K, it forms :

A

CaSO₄ . 0.5H₂O

B

CaSO₄ . 5H₂O

C

Dead burnt plaster

D

Anhydrous CaSO₄

Q 34ChemistrySection A • MCQhaloalkanes and haloarenes

+4 / -1

The decreasing order of reactivity towards dehydrohalogenation (E₁) reaction of the following compounds is : qImage-haloalkanes-haloalkanes-and-haloarenes-organic-chemistry-Jeemate

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

A

B

>

D

>

A

>

C

B

D

>

B

>

C

>

A

C

B

>

A

>

D

>

C

D

B

>

D

>

C

>

A

Q 35ChemistrySection B • Numericalelectrochemistry

+4 / -0

What would be the electrode potential for the given half cell reaction at pH = 5? ______.

2H₂O

\to

O₂ + 4H^$\oplus$ + 4e^– ;

E_{red}^0

= 1.23 V

(R = 8.314 J mol^–1 K^–1 ; Temp = 298 k;

oxygen under std. atm. pressure of 1 bar)

Answer Format:Enter numerical value rounded to the nearest integer

Q 36ChemistrySection A • MCQenvironmental chemistry

+4 / -1

Among the gases (a) – (e), the gases that cause greenhouse effect are :
(a) CO₂
(b) H₂O
(c) CFCs
(d) O₂
(e) O₃

A

(a), (b), (c) and (d)

B

(a) and (d)

C

(a), (b), (c) and (e)

D

(a), (c), (d) and (e)

Q 37ChemistrySection A • MCQaldehydes ketones and carboxylic acids

+4 / -1

The predominant intermolecular forces present in ethyl acetate, a liquid, are:

A

Dipole-dipole and hydrogen bonding

B

hydrogen bonding and London dispersion

C

London dispersion, dipole-dipole and hydrogen bonding

D

London dispersion and dipole-dipole

Q 38ChemistrySection A • MCQalcohols phenols and ethers

+4 / -1

Arrange the following compounds in increasing order of C–OH bond length :
methanol, phenol, p-ethoxyphenol

A

phenol < methanol < p-ethoxyphenol

B

methanol < p-ethoxyphenol < phenol

C

methanol < phenol < p-ethoxyphenol

D

phenol < p-ethoxyphenol < methanol

Q 39ChemistrySection B • Numericalsome basic concepts of chemistry

+4 / -0

Ferrous sulphate heptahydrate is used to fortify foods with iron. The amount (in grams) of the salt required to achieve 10 ppm of iron in 100 kg of wheat is _______.
Atomic weight : Fe = 55.85; S = 32.00; O = 16.00

Answer Format:Enter numerical value rounded to the nearest integer

Q 40ChemistrySection A • MCQsurface chemistry

+4 / -1

As per Hardy-Schulze formulation, the flocculation values of the following for ferric hydroxide sol are in the order :

A

AICl₃ > K₃[Fe(CN)₆] > K₂CrO₄ > KBr = KNO₃

B

K₃[Fe(CN)₆] < K₂CrO₄ < KBr = KNO₃ = AlCl₃

C

K₃[Fe(CN)₆] < K₂CrO₄ < AlCl₃ < KBr < KNO₃

D

K₃[Fe(CN)₆] > AlCl₃ > K₂CrO₄ > KBr > KNO₃

Q 41ChemistrySection A • MCQionic equilibrium

+4 / -1

The stoichiometry and solubility product of a salt with the solubility curve given below is, respectively : qImage-solubility-product-and-common-ion-effect-ionic-equilibrium-physical-chemistry-Jeemate

qImage-solubility-product-and-common-ion-effect-ionic-equilibrium-physical-chemistry-Jeemate

A

XY, 2 × 10^–6 M³

B

XY₂, 1 × 10^–9 M³

C

XY₂, 4 × 10^–9 M³

D

X₂Y, 2 × 10^–9 M³

Q 42ChemistrySection A • MCQcompounds containing nitrogen

+4 / -1

The major products A and B in the following reactions are :

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

A

B

C

D

Q 43ChemistrySection A • MCQatomic structure

+4 / -1

For the Balmer series in the spectrum of H atom,

\overline \nu = {R_H}\left\{ {{1 \over {n_1^2}} - {1 \over {n_2^2}}} \right\}

, the correct statements among (I) to (IV) are :
(I) As wavelength decreases, the lines in the series converge
(II) The integer n₁ is equal to 2
(III) The lines of longest wavelength corresponds to n₂ = 3
(IV) The ionization energy of hydrogen can be calculated from wave number of these lines

A

(II), (III), (IV)

B

(I), (II), (III)

C

(I), (III), (IV)

D

(I), (II), (IV)

Q 44ChemistrySection A • MCQbasics of organic chemistry

+4 / -1

A flask contains a mixture of isohexane and 3-methylpentane. One of the liquids boils at 63^oC while the other boils at 60^oC What is the best way to separate the two liquids and which one will be distilled out first ?

A

fractional distillation, 3-methylpentane

B

simple distillation, 3-methylpentane

C

fractional distillation, isohexane

D

simple distillation, isohexane

Q 45ChemistrySection A • MCQionic equilibrium

+4 / -1

The strength of an aqueous NaOH solution is most accurately determined by titrating :
(Note : consider that an appropriate indicator is used)

A

Aq. NaOH in a pipette and aqueous oxalic acid in a burette

B

Aq. NaOH in a burette and aqueous oxalic acid in a conical flask

C

Aq. NaOH in a volumetric flask and concentrated H₂SO₄ in a conical flask

D

Aq. NaOH in a burette and concentrated H₂SO₄ in a conical flask

Q 46ChemistrySection B • Numericalthermodynamics

+4 / -0

The magnitude of work done by a gas that undergoes a reversible expansion along the path ABC shown in the figure is _______. qImage-first-law-of-thermodynamics-thermodynamics-physical-chemistry-Jeemate

qImage-first-law-of-thermodynamics-thermodynamics-physical-chemistry-Jeemate

Answer Format:Enter numerical value rounded to the nearest integer

Q 47ChemistrySection A • MCQaldehydes ketones and carboxylic acids

+4 / -1

The most suitable reagent for the given conversion 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

LiAlH₄

B

B₂H₆

C

NaBH₄

D

H₂/Pd

Q 48ChemistrySection A • MCQperiodic classification of elements

+4 / -1

The third ionization enthalpy is minimum for :

A

Ni

B

Co

C

Mn

D

Fe

Q 49ChemistrySection A • MCQcoordination compounds

+4 / -1

The complex that can show fac- and mer-isomers is :

A

[CoCl₂(en)₂]

B

[Pt(NH₃)₂Cl₂]

C

[Co(NH₃)₃(NO₂)₃]

D

[Co(NH₃)₄Cl₂]⁺

Q 50ChemistrySection A • MCQbiomolecules

+4 / -1

Which of the following statement is not true for glucose?

A

Glucose gives Schiff's test for aldehyde

B

Glucosse reacts with hydroxylamine to form oxime

C

The pentaacetate of glucose does not react with hydroxylamine to give oxime

D

Glucose exists in two crystalline forms

\alpha

and

\beta

Q 51MathematicsSection A • MCQfunctions

+4 / -1

The inverse function of

f(x) =

{{{8^{2x}} - {8^{ - 2x}}} \over {{8^{2x}} + {8^{ - 2x}}}}

, x

\in

(-1, 1), is :

A

{1 \over 4}{\log _e}\left( {{{1 - x} \over {1 + x}}} \right)

B

{1 \over 4}\left( {{{\log }_8}e} \right){\log _e}\left( {{{1 - x} \over {1 + x}}} \right)

C

{1 \over 4}\left( {{{\log }_8}e} \right){\log _e}\left( {{{1 + x} \over {1 - x}}} \right)

D

{1 \over 4}{\log _e}\left( {{{1 + x} \over {1 - x}}} \right)

Q 52MathematicsSection A • MCQprobability

+4 / -1

Let A and B be two independent events such that
P(A) =

{1 \over 3}

and P(B) =

{1 \over 6}

.
Then, which of the following is TRUE?

A

P\left( {{A \over {A \cup B}}} \right) = {1 \over 4}

B

P\left( {{A \over B}} \right) = {2 \over 3}

C

P\left( {{{A'} \over {B'}}} \right) = {1 \over 3}

D

P\left( {{A \over {B'}}} \right) = {1 \over 3}

Q 53MathematicsSection A • MCQstraight lines and pair of straight lines

+4 / -1

Let two points be A(1, –1) and B(0, 2). If a point P(x', y') be such that the area of

\Delta

PAB = 5 sq. units and it lies on the line, 3x + y – 4

\lambda

= 0, then a value of

\lambda

is :

A

4

B

1

C

-3

D

3

Q 54MathematicsSection B • Numericalsequences and series

+4 / -0

The sum

\sum\limits_{k = 1}^{20} {\left( {1 + 2 + 3 + ... + k} \right)}

is :

Answer Format:Enter numerical value rounded to the nearest integer

Q 55MathematicsSection A • MCQ3d geometry

+4 / -1

The shortest distance between the lines

{{x - 3} \over 3} = {{y - 8} \over { - 1}} = {{z - 3} \over 1}

and

{{x + 3} \over { - 3}} = {{y + 7} \over 2} = {{z - 6} \over 4}

is :

A

3

B

{7 \over 2}\sqrt {30}

C

3\sqrt {30}

D

2\sqrt {30}

Q 56MathematicsSection A • MCQlimits continuity and differentiability

+4 / -1

\mathop {\lim }\limits_{x \to 0} {\left( {{{3{x^2} + 2} \over {7{x^2} + 2}}} \right)^{{1 \over {{x^2}}}}}

is equal to

A

e

B

e²

C

{1 \over {{e^2}}}

D

{1 \over e}

Q 57MathematicsSection A • MCQdifferential equations

+4 / -1

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

\sqrt {1 - {x^2}} {{dy} \over {dx}} + \sqrt {1 - {y^2}} = 0

, |x| < 1.

If

y\left( {{1 \over 2}} \right) = {{\sqrt 3 } \over 2}

, then

y\left( { - {1 \over {\sqrt 2 }}} \right)

is equal to :

A

- {{\sqrt 3 } \over 2}

B

None of those

C

{{1 \over {\sqrt 2 }}}

D

-{{1 \over {\sqrt 2 }}}

Q 58MathematicsSection A • MCQdifferentiation

+4 / -1

Let ƒ(x) = (sin(tan^–1x) + sin(cot^–1x))² – 1, |x| > 1.
If

{{dy} \over {dx}} = {1 \over 2}{d \over {dx}}\left( {{{\sin }^{ - 1}}\left( {f\left( x \right)} \right)} \right)

and

y\left( {\sqrt 3 } \right) = {\pi \over 6}

, then y(

{ - \sqrt 3 }

) is equal to :

A

{{5\pi } \over 6}

B

- {\pi \over 6}

C

{\pi \over 3}

D

{{2\pi } \over 3}

Q 59MathematicsSection A • MCQmatrices and determinants

+4 / -1

For which of the following ordered pairs (

\mu

,

\delta

), the system of linear equations
x + 2y + 3z = 1
3x + 4y + 5z =

\mu

4x + 4y + 4z =

\delta

is inconsistent ?

A

(1, 0)

B

(4, 3)

C

(4, 6)

D

(3, 4)

Q 60MathematicsSection A • MCQapplication of derivatives

+4 / -1

Let ƒ(x) = xcos^–1(–sin|x|),

x \in \left[ { - {\pi \over 2},{\pi \over 2}} \right]

, then which of the following is true?

A

ƒ' is decreasing in

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

and increasing in

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

B

ƒ '(0) =

{ - {\pi \over 2}}

C

ƒ is not differentiable at x = 0

D

ƒ' is increasing in

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

and decreasing in

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

Q 61MathematicsSection A • MCQindefinite integration

+4 / -1

If

\int {{{\cos xdx} \over {{{\sin }^3}x{{\left( {1 + {{\sin }^6}x} \right)}^{2/3}}}}} = f\left( x \right){\left( {1 + {{\sin }^6}x} \right)^{1/\lambda }} + c

where c is a constant of integration, then

\lambda f\left( {{\pi \over 3}} \right)

is equal to

A

{9 \over 8}

B

2

C

-2

D

-{9 \over 8}

Q 62MathematicsSection B • Numericalapplication of derivatives

+4 / -0

Let the normal at a point P on the curve
y² – 3x² + y + 10 = 0 intersect the y-axis at

\left( {0,{3 \over 2}} \right)

.
If m is the slope of the tangent at P to the curve, then |m| is equal to

Answer Format:Enter numerical value rounded to the nearest integer

Q 63MathematicsSection A • MCQstatistics

+4 / -1

The mean and the standard deviation (s.d.) of 10 observations are 20 and 2 resepectively. Each of these 10 observations is multiplied by p and then reduced by q, where p

\ne

0 and q

\ne

0. If the new mean and new s.d. become half of their original values, then q is equal to

A

10

B

-20

C

-10

D

-5

Q 64MathematicsSection B • Numericalmatrices and determinants

+4 / -0

The number of all 3 × 3 matrices A, with enteries from the set {–1, 0, 1} such that the sum of the diagonal elements of AA^T is 3, is

Answer Format:Enter numerical value rounded to the nearest integer

Q 65MathematicsSection A • MCQcomplex numbers

+4 / -1

If the equation, x² + bx + 45 = 0 (b

\in

R) has conjugate complex roots and they satisfy |z +1| = 2

\sqrt {10}

, then :

A

b² – b = 42

B

b² + b = 12

C

b² + b = 72

D

b² – b = 30

Q 66MathematicsSection B • Numericalpermutations and combinations

+4 / -0

An urn contains 5 red marbles, 4 black marbles and 3 white marbles. Then the number of ways in which 4 marbles can be drawn so that at the most three of them are red is ___________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 67MathematicsSection A • MCQarea under the curves

+4 / -1

For a > 0, let the curves C₁ : y² = ax and C₂ : x² = ay intersect at origin O and a point P. Let the line x = b (0 < b < a) intersect the chord OP and the x-axis at points Q and R, respectively. If the line x = b bisects the area bounded by the curves, C₁ and C₂, and the area of

\Delta

OQR =

{1 \over 2}

, then 'a' satisfies the equation :

A

x⁶ – 12x³ + 4 = 0

B

x⁶ – 12x³ – 4 = 0

C

x⁶ + 6x³ – 4 = 0

D

x⁶ – 6x³ + 4 = 0

Q 68MathematicsSection A • MCQapplication of derivatives

+4 / -1

If c is a point at which Rolle's theorem holds for the function,
f(x) =

{\log _e}\left( {{{{x^2} + \alpha } \over {7x}}} \right)

in the interval [3, 4], where a

\in

R, then ƒ''(c) is equal to

A

{1 \over {12}}

B

{{\sqrt 3 } \over 7}

C

-{1 \over {12}}

D

-{1 \over {24}}

Q 69MathematicsSection A • MCQmathematical reasoning

+4 / -1

Which one of the following is a tautology?

A

P

\wedge

(P

\vee

Q)

B

P

\vee

(P

\wedge

Q)

C

Q

\to

(P

\wedge

(P

\to

Q))

D

(P

\wedge

(P

\to

Q))

\to

Q

Official Correct Answer:Option (D)

Step-by-Step Solution:

Option A :

P	Q	P $\vee$ Q	P $\wedge$ (P $\vee$ Q)
T	T	T	T
T	F	T	T
F	T	T	F
F	F	F	F

Option B :

P	Q	P $\wedge$ Q	P $\vee$ (P $\wedge$ Q)
T	T	T	T
T	F	F	T
F	T	F	F
F	F	F	F

Option C :

Q

\to

(P

\wedge

(P

\to

Q))

\equiv

~ Q

\vee

(P

\wedge

(P

\to

Q))

\equiv

~ Q

\vee

(P

\wedge

(~P

\vee

Q))

\equiv

~ Q

\vee

(P

\wedge

~P)

\vee

(P

\wedge

Q))

\equiv

~ Q

\vee

(ƒ

\vee

(P

\wedge

Q))

\equiv

~ Q

\vee

(P

\wedge

Q)

\equiv

(~Q

\vee

P)

\wedge

(~Q

\vee

Q)

\equiv

(~ Q

\vee

P)

\wedge

t

\equiv

~ Q

\vee

P

Option D :

(P

\wedge

(P

\to

Q))

\to

Q

\equiv

(P

\wedge

(~P

\vee

Q))

\to

Q

\equiv

((P

\wedge

~P)

\vee

(P

\wedge

Q))

\to

Q

\equiv

(ƒ

\vee

(P

\wedge

Q))

\to

Q

\equiv

P

\wedge

Q

\to

Q

\equiv

~ (P

\wedge

Q)

\vee

Q

\equiv

(~ P)

\vee

(~Q)

\vee

Q

\equiv

(~ P)

\vee

t

\equiv

t

Q 70MathematicsSection A • MCQvector algebra

+4 / -1

Let the volume of a parallelopiped whose coterminous edges are given by

\overrightarrow u = \widehat i + \widehat j + \lambda \widehat k

,

\overrightarrow v = \widehat i + \widehat j + 3\widehat k

and

\overrightarrow w = 2\widehat i + \widehat j + \widehat k

be 1 cu. unit. If

\theta

be the angle between the edges

\overrightarrow u

and

\overrightarrow w

, then cos

\theta

can be :

A

{7 \over {6\sqrt 3 }}

B

{7 \over {6\sqrt 6 }}

C

{5 \over 7}

D

{5 \over {3\sqrt 3 }}

Q 71MathematicsSection A • MCQellipse

+4 / -1

Let the line y = mx and the ellipse 2x² + y² = 1 intersect at a ponit P in the first quadrant. If the normal to this ellipse at P meets the co-ordinate axes at

\left( { - {1 \over {3\sqrt 2 }},0} \right)

and (0,

\beta

), then

\beta

is equal to :

A

{{\sqrt 2 } \over 3}

B

{2 \over 3}

C

{{2\sqrt 2 } \over 3}

D

{2 \over {\sqrt 3 }}

Q 72MathematicsSection B • Numericalquadratic equation and inequalities

+4 / -0

The least positive value of 'a' for which the equation

2x² + (a – 10)x +

{{33} \over 2}

= 2a has real roots is

Answer Format:Enter numerical value rounded to the nearest integer

Q 73MathematicsSection A • MCQparabola

+4 / -1

The locus of a point which divides the line segment joining the point (0, –1) and a point on the parabola, x² = 4y, internally in the ratio 1 : 2, is :

A

9x² – 3y = 2

B

4x² – 3y = 2

C

x² – 3y = 2

D

9x² – 12y = 8

Q 74MathematicsSection A • MCQpermutations and combinations

+4 / -1

If a, b and c are the greatest value of ¹⁹C_p, ²⁰C_q and ²¹C_r respectively, then :

A

{a \over {11}} = {b \over {22}} = {c \over {21}}

B

{a \over {10}} = {b \over {22}} = {c \over {21}}

C

{a \over {10}} = {b \over {11}} = {c \over {42}}

D

{a \over {11}} = {b \over {22}} = {c \over {42}}

Q 75MathematicsSection A • MCQsequences and series

+4 / -1

Let ƒ : R

\to

R be such that for all x

\in

R
(2^1+x + 2^1–x), ƒ(x) and (3^x + 3^–x) are in A.P.,
then the minimum value of ƒ(x) is

A

2

B

0

C

3

D

4

JEE Main 2020 8th January Morning Shift Question Paper with Solutions