JEE Main 2020 7th January Evening Shift Question Paper Question Paper with Solutions | JEE Main Mock Test | JeeMate

Q 1PhysicsSection B • Numericalcapacitor

+4 / -0

A 60 pF capacitor is fully charged by a 20 V supply. It is then disconnected from the supply and is conneced to another uncharged 60 pF capacitor in parallel. The electrostatic energy that is lost in this process by the time the charge is redistributed between them is (in nJ) _____

Answer Format:Enter numerical value rounded to the nearest integer

Q 2PhysicsSection B • Numericalrotational motion

+4 / -0

qImage-torque-rotational-motion-mechanics-Jeemate

Consider a uniform cubical box of side a on a rough floor that is to be moved by applying minimum possible force F at a point b above its centre of mass (see figure). If the coefficient of friction is

\mu

= 0.4, the maximum possible value of 100 ×

{b \over a}

for box not to topple before moving is .......

Answer Format:Enter numerical value rounded to the nearest integer

Q 3PhysicsSection A • MCQatoms and nuclei

+4 / -1

The activity of a radioactive sample falls from 700 s^–1 to 500 s^–1 in 30 minutes. Its half life is close to:

A

62 min

B

66 min

C

72 min

D

52 min

Q 4PhysicsSection A • MCQgravitation

+4 / -1

A box weight 196 N on a spring balance at the north pole. Its weight recorded on the same balance if it is shifted to the equator is close to (Take g = 10 ms^–2 at the north pole and the radius of the earth = 6400 km) :

A

194.32 N

B

195.66 N

C

195.32 N

D

194.66 N

Q 5PhysicsSection A • MCQwork power and energy

+4 / -1

An elevator in a building can carry a maximum of 10 persons, with the average mass of each person being 68 kg, The mass of the elevator itself is 920 kg and it moves with a constant speed of 3 m/s. The frictional force opposing the motion is 6000 N. If the elevator is moving up with its full capacity, the power delivered by the motor to the elevator (g = 10 m/s²) must be at least :

A

48000 W

B

62360 W

C

56300 W

D

66000 W

Q 6PhysicsSection A • MCQheat and thermodynamics

+4 / -1

Two ideal Carnot engines operate in cascade (all heat given up by one engine is used by the other engine to produce work) between temperature, T₁ and T₂ . The temperature of the hot reservoir of the first engine is T₁ and the temperature of the cold reservoir of the second engine is T₂ . T is temperature of the sink of first engine which is also the source for the second which is also the source for the second engine. How is T related to T₁ and T₂ . If both engines perform equal amount of work?

A

T = {{2{T_1}{T_2}} \over {{T_1} + {T_2}}}

B

T = \sqrt {{T_1}{T_2}}

C

T = {{{T_1} + {T_2}} \over 2}

D

T = 0

Q 7PhysicsSection A • MCQwaves

+4 / -1

A stationary observer receives sound from two identical tuning forks, one of which approaches and the other one recedes with the same speed (much less than the speed of sound). The observer hears 2 beats/sec. The oscillation frequency of each tuning fork is v₀ = 1400 Hz and the velocity of sound in air is 350 m/s. The speed of each tuning fork is close to :

A

1 m/s

B

{1 \over 8}

m/s

C

{1 \over 4}

m/s

D

{1 \over 2}

m/s

Q 8PhysicsSection A • MCQunits and measurements

+4 / -1

The dimension of

{{{B^2}} \over {2{\mu _0}}}

, where B is magnetic field and

{{\mu _0}}

is the magnetic permeability of vacuum, is :

A

ML²T^–2

B

MLT^–2

C

ML^-1T^–2

D

ML²T^–1

Q 9PhysicsSection B • Numericalcurrent electricity

+4 / -0

The balancing length for a cell is 560 cm in a potentiometer experiment. When an external resistance of 10

\Omega

is connected in parallel to the cell, the balancing length changes by 60 cm. If the internal resistance of the ceil is

{N \over {10}}

\Omega

, where N is an integer then value of N is .............

Answer Format:Enter numerical value rounded to the nearest integer

Q 10PhysicsSection A • MCQelectromagnetic induction

+4 / -1

A planar loop of wire rotates in a uniform magnetic field. Initially at t = 0, the plane of the loop is perpendicular to the magnetic field. If it rotates with a period of 10 s about an axis in its plane then the magnitude of induced emf will be maximum and minimum, respectively at :

A

2.5 s and 7.5 s

B

5.0 s and 10.0 s

C

5.0 s and 7.5 s

D

2.5 s and 5.0 s

Q 11PhysicsSection A • MCQlaws of motion

+4 / -1

A mass of 10 kg is suspended by a rope of length 4 m, from the ceiling. A force F is applied horizontally at the mid point of the rope such that the top half of the rope makes an angle of 45^o with the vertical. Then F equal : (Take g = 10 ms^–2 and the rope to be massless)

A

100 N

B

75 N

C

90 N

D

70 N

Q 12PhysicsSection A • MCQcurrent electricity

+4 / -1

In a building there are 15 bulbs of 45 W, 15 bulbs of 100 W, 15 small fans of 10 W and 2 heaters of 1 kW. The voltage of electric main is 220 V. The minimum fuse capacity (rated value) of the building will be :

A

15 A

B

20 A

C

25 A

D

10 A

Q 13PhysicsSection A • MCQmagnetics

+4 / -1

A particle of mass m and charge q has an initial velocity

\overrightarrow v = {v_0}\widehat j

. If an electric field

\overrightarrow E = {E_0}\widehat i

and magnetic field

\overrightarrow B = {B_0}\widehat i

act on the particle, its speed will double after a time:

A

{{3m{v_0}} \over {q{E_0}}}

B

{{\sqrt 2 m{v_0}} \over {q{E_0}}}

C

{{\sqrt 3 m{v_0}} \over {q{E_0}}}

D

{{2m{v_0}} \over {q{E_0}}}

Q 14PhysicsSection A • MCQrotational motion

+4 / -1

Mass per unit area of a circular disc of radius

a

depends on the distance r from its centre as

\sigma \left( r \right)

= A + Br . The moment of inertia of the disc about the axis, perpendicular to the plane and assing through its centre is:

A

2\pi {a^4}\left( {{A \over 4} + {{aB} \over 5}} \right)

B

\pi {a^4}\left( {{A \over 4} + {{aB} \over 5}} \right)

C

2\pi {a^4}\left( {{{aA} \over 4} + {B \over 5}} \right)

D

2\pi {a^4}\left( {{A \over 4} + {B \over 5}} \right)

Q 15PhysicsSection A • MCQwave optics

+4 / -1

In a Young's double slit experiment, the separation between the slits is 0.15 mm. in the experiment, a source of light of wavelengh 589 nm is used and the interference pattern is observed on a screen kept 1.5 m away. The separation between the successive bright fringes on the screen is :

A

4.9 mm

B

5.9 mm

C

6.9 mm

D

3.9 mm

Q 16PhysicsSection B • Numericalheat and thermodynamics

+4 / -0

M grams of steam at 100^oC is mixed with 200 g of ice at its melting point in a thermally insulated container. If it produced liquid water at 40^oC [heat of vaporization of water is 540 cal/g and heat of fusion of ice is 80 cal/g] the value of M is ____

Answer Format:Enter numerical value rounded to the nearest integer

Q 17PhysicsSection A • MCQmagnetic properties of matter

+4 / -1

qImage-magnetic-properties-of-matter-magnetic-properties-of-matter-electricity-Jeemate

The figure gives experimentally measured B vs H variation in a ferromagnetic material. The retentivity, coercivity and saturation, respectively, of the material are :

A

1.0 T, 50A/m and 1.5 T

B

1.5 T, 50 A/m and 1.0 T

C

1.5 T, 50 A/m and 2.0 T

D

150 A/m, 1.0 T and 1.5 T

Q 18PhysicsSection A • MCQray optics

+4 / -1

A thin lens made of glass (refractive index = 1.5) of focal length f = 16 cm is immersed in a liquid of refractive index 1.42. If its focal length in liquid is f₁ , then the ratio

{{{f_1}} \over f}

is closest to the integer :

A

17

B

1

C

9

D

5

Q 19PhysicsSection A • MCQheat and thermodynamics

+4 / -1

Under an adiabatic process, the volume of an ideal gas gets doubled. Consequently the mean collision time between the gas molecule changes from

{\tau _1}

to

{\tau _2}

. If

{{{C_p}} \over {{C_v}}} = \gamma

for this gas then a good estimate for

{{{\tau _2}} \over {{\tau _1}}}

is given by :

A

{\left( 2 \right)^{{{1 + \gamma } \over 2}}}

B

2

C

{\left( {{1 \over 2}} \right)^{{{1 + \gamma } \over 2}}}

D

{\left( {{1 \over 2}} \right)^\gamma }

Q 20PhysicsSection A • MCQalternating current

+4 / -1

An emf of 20 V is applied at time t = 0 to a circuit containing in series 10 mH inductor and 5

\Omega

resistor. The ratio of the currents at time t =

\infty

and at t = 40 s is close to : (Take e² = 7.389)

A

1.06

B

0.84

C

1.15

D

1.46

Q 21PhysicsSection B • Numericalvector algebra

+4 / -0

The sum of two forces

\overrightarrow P

and

\overrightarrow Q

is

\overrightarrow R

such that

\left| {\overrightarrow R } \right| = \left| {\overrightarrow P } \right|

. The angle

\theta

(in degrees) that the resultant of 2

{\overrightarrow P }

and

{\overrightarrow Q }

will make with

{\overrightarrow Q }

is , ..............

Answer Format:Enter numerical value rounded to the nearest integer

Q 22PhysicsSection A • MCQdual nature of radiation

+4 / -1

An electron (of mass m) and a photon have the same energy E in the range of a few eV. The ratio of the de-Broglie wavelength associated with the electron and the wavelength of the photon is (c = speed of light in vaccuum)

A

{1 \over c}{\left( {{{2E} \over m}} \right)^{{1 \over 2}}}

B

{1 \over c}{\left( {{E \over {2m}}} \right)^{{1 \over 2}}}

C

{\left( {{E \over {2m}}} \right)^{{1 \over 2}}}

D

c{\left( {2mE} \right)^{{1 \over 2}}}

Q 23PhysicsSection A • MCQelectromagnetic waves

+4 / -1

The electric field of a plane electromagnetic wave is given by

\overrightarrow E = {E_0}{{\widehat i + \widehat j} \over {\sqrt 2 }}\cos \left( {kz + \omega t} \right)

At t = 0, a positively charged particle is at the point (x, y, z) =

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

.
If its instantaneous velocity at (t = 0) is

{v_0}\widehat k

, the force acting on it due to the wave is :

A

parallel to

\widehat k

B

parallel to

{{\widehat i + \widehat j} \over {\sqrt 2 }}

C

antiparallel to

{{\widehat i + \widehat j} \over {\sqrt 2 }}

D

zero

Q 24PhysicsSection A • MCQelectronic devices

+4 / -1

In the figure, potential difference between A and B is : qImage-semiconductor-and-p-n-junction-diode-electronic-devices-modern-physics-Jeemate

A

10 V

B

5 V

C

zero

D

15 V

Q 25PhysicsSection A • MCQproperties of matter

+4 / -1

An ideal fluid flows (laminar flow) through a pipe of non-uniform diameter. The maximum and minimum diameters of the pipes are 6.4 cm and 4.8 cm, respectively. The ratio of the minimum and the maximum velocities of fluid in this pipe is :

A

{3 \over 4}

B

{9 \over {16}}

C

{{\sqrt 3 } \over 2}

D

{{81} \over {256}}

Q 26ChemistrySection A • MCQisolation of elements

+4 / -1

The refining method used when the metal and the impurities have low and high melting temperatures, respectively, is :

A

liquation

B

vapour phase refining

C

distillation

D

zone refining

Q 27ChemistrySection A • MCQhaloalkanes and haloarenes

+4 / -1

For the following reactions

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

K_s and K_e, are respectively, the rate constants for substitution and elimination and

\mu = {{{k_s}} \over {{k_e}}}

the correct options is

A

{\mu _B} > {\mu _A}

and K_e(A) > K_e(B)

B

{\mu _A} > {\mu _B}

and K_e(B) > K_e(A)

C

{\mu _B} > {\mu _A}

and K_e(B) > K_e(A)

D

{\mu _A} > {\mu _B}

and K_e(A) > K_e(B)

Q 28ChemistrySection B • Numericald and f block elements

+4 / -0

Consider the following reactions :

NaCl{\rm{ }} + {\rm{ }}{K_2}C{r_2}{O_7} + \mathop {{H_2}S{O_4}}\limits_{(conc.)}

\to

(A) + side products

(A) + NaOH

\to

(B) + Side products

\left( B \right){\rm{ }} + \mathop {{H_2}S{O_4}}\limits_{(dilute)} + {\rm{ }}{H_2}{O_2}

\to

(C) + Side products

The sum of the total number of atoms in one molecule each of (A) and (B) and (C) is

Answer Format:Enter numerical value rounded to the nearest integer

Q 29ChemistrySection A • MCQhaloalkanes and haloarenes

+4 / -1

Consider the following reactions : qImage-haloarenes-haloalkanes-and-haloarenes-organic-chemistry-Jeemate

Which of these reactions are possible ?

A

(b), (c) and (d)

B

(a) and (b)

C

(a) and (d)

D

(b) and (d)

Q 30ChemistrySection A • MCQsolutions

+4 / -1

Two open beakers one containing a solvent and the other containing a mixture of that solvent with a non voltatile solute are together selated in a container. Over time :

A

The volume of the solution increases and the volume of the solvent decreases.

B

The volume of solution does not change and the volume of the solvent decreases.

C

The volumer of the solution and the solvent does not change.

D

The volume of the solution decreases and the volume of the solvent increases.

Q 31ChemistrySection A • MCQcoordination compounds

+4 / -1

Among the statements(a)-(d) the incorrect ones are :
(a) Octahedral CO(III) complexes with strong fields ligands have very high magnetic moments.
(b) When

\Delta

₀ < P, the d-electron configuration of Co(III) in an octahedral complex is

t_{eg}^4e_g^2

(c) Wavelength of light absorbed by [Co(en)₃]³⁺ is lower than that of [CoF₆]^3-
(d) If the

\Delta

₀ for an octahedral complex of CO(III) is 18,000 cm^-1, the

\Delta

_t for its tetrahedral complex with the same ligand be 16,000 cm^-1

A

(a) and (b) only

B

(b) and (c) only

C

(c) and (d) only

D

(a) and (d) only

Q 32ChemistrySection A • MCQcoordination compounds

+4 / -1

The number of possible optical isomers for the complexes MA₂B₂ with sp³ and dsp² hydridized metal atom. respectively, is :
Note : A and B are unidentate netural and unidentate monoanionic ligands, respectively.

A

0 and 2

B

0 and 0

C

0 and 1

D

2 and 2

Q 33ChemistrySection A • MCQcompounds containing nitrogen

+4 / -1

In the following reaction sequence,

qImage-diazonium-salts-and-other-nitrogen-containing-compounds-compounds-containing-nitrogen-organic-chemistry-Jeemate

qImage-diazonium-salts-and-other-nitrogen-containing-compounds-compounds-containing-nitrogen-organic-chemistry-Jeemate

The major product B is :

A

B

C

D

Q 34ChemistrySection A • MCQhydrocarbons

+4 / -1

In the following reactions sequence, structure of A and B respectively will be :
qImage-reaction-of-alkenes-hydrocarbons-organic-chemistry-Jeemate

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

A

B

C

D

Q 35ChemistrySection A • MCQbiomolecules

+4 / -1

Which of the following statements is correct ?

A

Gluconic acid can from cyclic (acteal/hemiacetal) structure

B

Gluconic acid is a partial oxidation product of glucose

C

Gluconic acid is obtained by oxidation of glucose with (HNO₃)

D

Gluconic acid is a dicarboxylic acid.

Q 36ChemistrySection A • MCQelectrochemistry

+4 / -1

The equation that is incorrect is :

A

{\left( {\Lambda _m^0} \right)_{KCl}} - {\left( {\Lambda _m^0} \right)_{NaCl}} = {\left( {\Lambda _m^0} \right)_{KBr}} - {\left( {\Lambda _m^0} \right)_{NaBr}}

B

{\left( {\Lambda _m^0} \right)_{NaBr}} - {\left( {\Lambda _m^0} \right)_{NaI}} = {\left( {\Lambda _m^0} \right)_{KBr}} - {\left( {\Lambda _m^0} \right)_{NaBr}}

C

{\left( {\Lambda _m^0} \right)_{NaBr}} - {\left( {\Lambda _m^0} \right)_{NaCl}} = {\left( {\Lambda _m^0} \right)_{KBr}} - {\left( {\Lambda _m^0} \right)_{KCl}}

D

{\left( {\Lambda _m^0} \right)_{{H_2}O}} = {\left( {\Lambda _m^0} \right)_{HCl}} + {\left( {\Lambda _m^0} \right)_{NaOH}} - {\left( {\Lambda _m^0} \right)_{NaCl}}

Q 37ChemistrySection A • MCQperiodic classification of elements

+4 / -1

Within each pair of elements F & Cl, S & Se, and Li & Na, respectively, the elements that release more energy upon and electron gain are :

A

F, S and Li

B

Cl, Se and Na

C

Cl, S and Li

D

F, Se and Na

Q 38ChemistrySection B • Numericalionic equilibrium

+4 / -0

3 g of acetic acid is added to 250 mL of 0.1 M HCL and the solution made up to 500 mL. To 20 mL of this solutions

{1 \over 2}

mL of 5 M NaOH is added. The pH of the solution is __________.

[Given : pKa of acetic acid = 4.75, molar mass of acetic of acid = 60 g/mol, log 3 = 0.4771] Neglect any changes in volume.

Answer Format:Enter numerical value rounded to the nearest integer

Official Correct Answer:5.22TO5.24

Step-by-Step Solution:

milimole of acetic acid in 20 ml
=

{3 \over {60}} \times 1000 \times {{20} \over {500}}

= 2

milimole of HCl in 20 ml = 25

\times

{{20} \over {500}}

= 1

milimole of NaOH 20 ml =

{1 \over 2} \times 5

= 2.5

NaOH	+	CH₃COOH	$\to$	CH3COONa	+	H₂O
	1.5	2		0		0
	0	0.5		1.5

pH = pKa + log

{{1.5} \over {0.5}}

= 4.74 + log 3 = 5.22

Q 39ChemistrySection A • MCQhydrogen

+4 / -1

Among the statements (a) - (d), the correct ones are :
(a) Decomposition of hydrogen peroxide gives dioxygen.
(b) Like hydrogen peroxide, compounds, such as KCIO₃, Pb(NO₃)₂ and NaNO₃ when heated liberate dioxygen.
(c) 2-Ethylanthraquinone is useful for the industrial preparation of hydrogen peroxide.
(d) Hydrogen peroxide is used for the manufacture of sodium perborate.

A

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

B

(a) and (c) only

C

(a), (b) and (c) only

D

(a), (c) and (d) only

Q 40ChemistrySection A • MCQredox reactions

+4 / -1

The redox reaction among the following is :

A

reaction of H₂SO₄ with NaOH.

B

formation of ozone form atmosphere oxygen in the presence of sunlight.

C

combination of dinitrogen with dioxygen at 2000 K

D

reaction of [Co(H₂O)₆]Cl₃ With AgNO₃

Q 41ChemistrySection A • MCQchemical bonding and molecular structure

+4 / -1

The bond order and the magnetic characteristics of CN^- are :

A

3, paramagnetic

B

2{1 \over 2}

, paramagnetic

C

3, diamagnetic

D

2{1 \over 2}

, diamagnetic

Q 42ChemistrySection A • MCQp block elements

+4 / -1

In the following reaction, products (A) and (B) respectively, are :

NaOH + Cl₂

\to

(A) + side products
(hot and conc.)

Ca(OH)₂ + Cl₂

\to

(B) + side products (dry)

A

NaOCl and Ca(OCl)₂

B

NaClO₃ and Ca(ClO₃)₂

C

NaOCl and Ca(ClO₃)₂

D

NaClO₃ and Ca(OCl)₂

Q 43ChemistrySection A • MCQbasics of organic chemistry

+4 / -1

A chromatography column, packed with silica gel as stationary phase, was used to separate a mixture of compounds consisting of (A) benzanilide (B) aniline and (C) acetophenone. When the column is eluted with a mixture of solvents, hexane : ethylacelate (20:80), the sequence of obtained compounds is :

A

(A), (B) and (C)

B

(B), (A) and (C)

C

(C), (A) and (B)

D

(B), (C) and (A)

Q 44ChemistrySection B • Numericalsome basic concepts of chemistry

+4 / -0

The flocculation value of HCl for arsenic sulphide sol.
is 30 m mol L^-1 If H₂SO₄ is used for the flocculatiopn of arsenic sulphide, the amount in grams, of H₂SO₄ in 250 ml required for the above purposed is ______.
(molecular mass of H₂SO₄ = 98 g/mol)

Answer Format:Enter numerical value rounded to the nearest integer

Q 45ChemistrySection B • Numericalthermodynamics

+4 / -0

The standard heat of formation

\left( {{\Delta _f}H_{298}^0} \right)

of ethane (in kj/mol), if the heat of combustion of ethane, hydrogen and graphite are - 1560, -393.5 and -286 Kj/mol, respectively is :

Answer Format:Enter numerical value rounded to the nearest integer

Q 46ChemistrySection A • MCQsome basic concepts of chemistry

+4 / -1

The ammonia (NH₃) released on quantitative reaction of 0.6 g urea (NH₂CONH₂) with sodium hydroxide (NaOH) can be neutralized by :

A

200 ml of 0.02 N HCl

B

100 ml of 0.2 N HCl

C

100 ml of 0.1 HCl

D

200 ml of 0.4 N HCl

Q 47ChemistrySection B • Numericalchemistry in everyday life

+4 / -0

The number of sp² hybridised carbons present in "Aspartame" is

Answer Format:Enter numerical value rounded to the nearest integer

Q 48ChemistrySection A • MCQgaseous state

+4 / -1

Identify the correct labels of A,B and C in the following graph from the options given below :

qImage-kinetic-theory-of-gases-gaseous-state-physical-chemistry-Jeemate

qImage-kinetic-theory-of-gases-gaseous-state-physical-chemistry-Jeemate

Root mean square speed (V_rms); most probable speed (V_mp); Average speed (V_av)

A

A - V_av ; B - V_rms ; C - V_mp

B

A - V_rms ; B - V_mp ; C - V_av

C

A - V_mp; B- V_av; C - V_rms

D

A - V_mp ; B - V_rms ; C - V_av

Q 49ChemistrySection A • MCQalcohols phenols and ethers

+4 / -1

The correct order of stability for the following alkoxides is :

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

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

A

(C) > (B) > (A)

B

(C) > (A) > (B)

C

(B) > (C) > (A)

D

(B) > (A) > (C)

Q 50ChemistrySection A • MCQchemical kinetics and nuclear chemistry

+4 / -1

For the reaction
2H₂(g) + 2NO(g)

\to

N₂(g) + 2H₂O(g)
the observed rate expression is, rate = K_f[NO]²[H₂]. The rate expression for the reverse reaction is :

A

K_b[N₂][H₂O]

B

K_b[N₂][H₂O]²/[H₂]

C

K_b[N₂][H₂O]²/[NO]

D

K_b[N₂][H₂O]²

Q 51MathematicsSection B • Numericalstatistics

+4 / -0

If the mean and variance of eight numbers 3, 7, 9, 12, 13, 20, x and y be 10 and 25 respectively, then x.y is equal to _______.

Answer Format:Enter numerical value rounded to the nearest integer

Q 52MathematicsSection A • MCQcomplex numbers

+4 / -1

If

{{3 + i\sin \theta } \over {4 - i\cos \theta }}

,

\theta

\in

[0, 2

\theta

], is a real number, then an argument of
sin

\theta

+ icos

\theta

is :

A

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

B

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

C

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

D

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

Q 53MathematicsSection A • MCQsequences and series

+4 / -1

Let

{a_1}

,

{a_2}

,

{a_3}

,....... be a G.P. such that

{a_1}

< 0,

{a_1}

+

{a_2}

= 4 and

{a_3}

+

{a_4}

= 16.
If

\sum\limits_{i = 1}^9 {{a_i}} = 4\lambda

, then

\lambda

is equal to:

A

171

B

-171

C

-513

D

{{511} \over 3}

Q 54MathematicsSection A • MCQapplication of derivatives

+4 / -1

Let ƒ(x) be a polynomial of degree 5 such that x = ±1 are its critical points.

If

\mathop {\lim }\limits_{x \to 0} \left( {2 + {{f\left( x \right)} \over {{x^3}}}} \right) = 4

, then which one of the following is not true?

A

ƒ(1) - 4ƒ(-1) = 4.

B

x = 1 is a point of minima and x = -1 is a point of maxima of ƒ.

C

x = 1 is a point of maxima and x = -1 is a point of minimum of ƒ.

D

ƒ is an odd function.

Q 55MathematicsSection A • MCQvector algebra

+4 / -1

Let

\overrightarrow a

,

\overrightarrow b

and

\overrightarrow c

be three unit vectors such that

\overrightarrow a + \vec b + \overrightarrow c = \overrightarrow 0

. If

\lambda = \overrightarrow a .\vec b + \vec b.\overrightarrow c + \overrightarrow c .\overrightarrow a

and

\overrightarrow d = \overrightarrow a \times \vec b + \vec b \times \overrightarrow c + \overrightarrow c \times \overrightarrow a

, then the ordered pair,

\left( {\lambda ,\overrightarrow d } \right)

is equal to :

A

\left( {{3 \over 2},3\overrightarrow a \times \overrightarrow c } \right)

B

\left( { - {3 \over 2},3\overrightarrow c \times \overrightarrow b } \right)

C

\left( { - {3 \over 2},3\overrightarrow a \times \overrightarrow b } \right)

D

\left( {{3 \over 2},3\overrightarrow b \times \overrightarrow c } \right)

Q 56MathematicsSection A • MCQcircle

+4 / -1

Let the tangents drawn from the origin to the circle,
x² + y² - 8x - 4y + 16 = 0 touch it at the points A and B. The (AB)² is equal to :

A

{{56} \over 5}

B

{{32} \over 5}

C

{{52} \over 5}

D

{{64} \over 5}

Q 57MathematicsSection B • Numericalmatrices and determinants

+4 / -0

If the system of linear equations,
x + y + z = 6
x + 2y + 3z = 10
3x + 2y +

\lambda

z =

\mu

has more than two solutions, then

\mu

-

\lambda

² is equal to ______.

Answer Format:Enter numerical value rounded to the nearest integer

Q 58MathematicsSection A • MCQprobability

+4 / -1

In a workshop, there are five machines and the probability of any one of them to be out of service on a day is

{{1 \over 4}}

. If the probability that at most two machines will be out of service on the same day is

{\left( {{3 \over 4}} \right)^3}k

, then k is equal to :

A

{{{17} \over 4}}

B

{{{17} \over 2}}

C

{{{17} \over 8}}

D

4

Q 59MathematicsSection B • Numericallimits continuity and differentiability

+4 / -0

If the function ƒ defined on

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

by

f(x) =

\left\{ {\matrix{ {{1 \over x}{{\log }_e}\left( {{{1 + 3x} \over {1 - 2x}}} \right),} & {when\,x \ne 0} \cr {k,} & {when\,x = 0} \cr } } \right.

is continuous, then k is equal to_______.

Answer Format:Enter numerical value rounded to the nearest integer

Q 60MathematicsSection A • MCQapplication of derivatives

+4 / -1

The value of c in the Lagrange's mean value theorem for the function
ƒ(x) = x³ - 4x² + 8x + 11, when x

\in

[0, 1] is:

A

{2 \over 3}

B

{{\sqrt 7 - 2} \over 3}

C

{{4 - \sqrt 5 } \over 3}

D

{{4 - \sqrt 7 } \over 3}

Q 61MathematicsSection A • MCQstraight lines and pair of straight lines

+4 / -1

The locus of the mid-points of the perpendiculars drawn from points on the line, x = 2y to the line x = y is :

A

3x - 2y = 0

B

7x - 5y = 0

C

2x - 3y = 0

D

5x - 7y = 0

Q 62MathematicsSection A • MCQsequences and series

+4 / -1

If the sum of the first 40 terms of the series,
3 + 4 + 8 + 9 + 13 + 14 + 18 + 19 + ..... is (102)m, then m is equal to :

A

20

B

5

C

10

D

25

Q 63MathematicsSection A • MCQdefinite integration

+4 / -1

The value of

\alpha

for which

4\alpha \int\limits_{ - 1}^2 {{e^{ - \alpha \left| x \right|}}dx} = 5

, is:

A

{\log _e}2

B

{\log _e}\sqrt 2

C

{\log _e}\left( {{4 \over 3}} \right)

D

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

Q 64MathematicsSection A • MCQarea under the curves

+4 / -1

The area (in sq. units) of the region
{(x, y)

\in

R² | 4x²

\le

y

\le

8x + 12} is :

A

{{125} \over 3}

B

{{128} \over 3}

C

{{127} \over 3}

D

{{124} \over 3}

Q 65MathematicsSection A • MCQbinomial theorem

+4 / -1

The coefficient of x⁷ in the expression
(1 + x)¹⁰ + x(1 + x)⁹ + x²(1 + x)⁸ + ......+ x¹⁰ is:

A

120

B

330

C

420

D

210

Q 66MathematicsSection A • MCQmathematical reasoning

+4 / -1

Let A, B, C and D be four non-empty sets. The contrapositive statement of "If A

\subseteq

B and B

\subseteq

D, then A

\subseteq

C" is :

A

If A ⊈ C, then A ⊈ B or B ⊈ D

B

If A ⊈ C, then A ⊈ B and B

\subseteq

D

C

If A

\subseteq

C, then B

\subset

A or D

\subset

B

D

If A ⊈ C, then A

\subseteq

B and B

\subseteq

D

Q 67MathematicsSection A • MCQdifferential equations

+4 / -1

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

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

, satisfying y(0) = 1. This curve intersects the x-axis at a point whose abscissa is :

A

2 + e

B

-e

C

2

D

2 - e

Q 68MathematicsSection A • MCQmatrices and determinants

+4 / -1

Let A = [a_ij] and B = [b_ij] be two 3 × 3 real matrices such that b_ij = (3)^(i+j-2)a_ji, where i, j = 1, 2, 3. If the determinant of B is 81, then the determinant of A is:

A

3

B

{1 \over 3}

C

{1 \over 9}

D

{1 \over {81}}

Q 69MathematicsSection B • Numericalsets and relations

+4 / -0

Let X = {n

\in

N : 1

\le

n

\le

50}. If
A = {n

\in

X: n is a multiple of 2} and
B = {n

\in

X: n is a multiple of 7}, then the number of elements in the smallest subset of X containing both A and B is ________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 70MathematicsSection A • MCQellipse

+4 / -1

If 3x + 4y = 12

\sqrt 2

is a tangent to the ellipse

{{{x^2}} \over {{a^2}}} + {{{y^2}} \over 9} = 1

for some

a

\in

R, then the distance between the foci of the ellipse is :

A

2\sqrt 5

B

2\sqrt 7

C

4

D

2\sqrt 2

Q 71MathematicsSection A • MCQquadratic equation and inequalities

+4 / -1

Let

\alpha

and

\beta

be the roots of the equation x² - x - 1 = 0.
If p_k =

{\left( \alpha \right)^k} + {\left( \beta \right)^k}

, k

\ge

1, then which one of the following statements is not true?

A

(p₁ + p₂ + p₃ + p₄ + p₅) = 26

B

p₅ = 11

C

p₃ = p₅ – p₄

D

p₅ = p₂ · p₃

Q 72MathematicsSection A • MCQdefinite integration

+4 / -1

If

\theta

₁ and

\theta

₂ be respectively the smallest and the largest values of

\theta

in (0, 2

\pi

) - {

\pi

} which satisfy the equation,
2cot²

\theta

-

{5 \over {\sin \theta }}

+ 4 = 0, then

\int\limits_{{\theta _1}}^{{\theta _2}} {{{\cos }^2}3\theta d\theta }

is equal to :

A

{\pi \over 9}

B

{{2\pi } \over 3}

C

{{\pi } \over 3}

D

{\pi \over 3} + {1 \over 6}

Q 73MathematicsSection A • MCQpermutations and combinations

+4 / -1

The number of ordered pairs (r, k) for which
6.³⁵C_r = (k² - 3). ³⁶C_{r + 1}, where k is an integer, is :

A

6

B

3

C

2

D

4

Q 74MathematicsSection B • Numerical3d geometry

+4 / -0

If the foot of the perpendicular drawn from the point (1, 0, 3) on a line passing through (

\alpha

, 7, 1) is

\left( {{5 \over 3},{7 \over 3},{{17} \over 3}} \right)

, then

\alpha

is equal to______.

Answer Format:Enter numerical value rounded to the nearest integer

Q 75MathematicsSection A • MCQdifferentiation

+4 / -1

Let y = y(x) be a function of x satisfying

y\sqrt {1 - {x^2}} = k - x\sqrt {1 - {y^2}}

where k is a constant and

y\left( {{1 \over 2}} \right) = - {1 \over 4}

. Then

{{dy} \over {dx}}

at x =

{1 \over 2}

, is equal to :

A

{2 \over {\sqrt 5 }}

B

- {{\sqrt 5 } \over 2}

C

{{\sqrt 5 } \over 2}

D

- {{\sqrt 5 } \over 4}

JEE Main 2020 7th January Evening Shift Question Paper with Solutions