# Solved papers for JEE Main & Advanced JEE Main Paper Phase-I (Held on 08-1-2020 Evening)

### done JEE Main Paper Phase-I (Held on 08-1-2020 Evening)

• question_answer1) A uniform sphere of mass 500 g rolls without slipping on a plane horizontal surface with its centre moving at a speed of 5.00 cm/s. Its kinetic energy is [JEE MAIN Held on 08-01-2020 Evening]

A) $6.25\times {{10}^{4}}J$

B) $1.13\times {{10}^{3}}J$

C) $8.75\times {{10}^{\,4}}\text{ }J$

D) $8.75\times {{10}^{3}}\text{ }J$

• question_answer2) In the given circuit, value of Y is [JEE MAIN Held on 08-01-2020 Evening]

A) Toggles between 0 and 1

B) 0

C) 1

D) Will not execute

• question_answer3) Consider a mixture of n moles of helium gas and 2n moles of oxygen gas (molecules taken to be rigid) as an ideal gas. Its ${{C}_{p}}\,/\,{{C}_{v}}$ value will be [JEE MAIN Held on 08-01-2020 Evening]

A) 40/27

B) 19/13

C) 67/45

D) 23/15

• question_answer4) An object is gradually moving away from the focal point of a concave mirror along the axis of the mirror. The graphical representation of the magnitude of linear magnification (m) versus distance of the object from the mirror (x) is correctly given by (Graphs are drawn schematically and are not to scale) [JEE MAIN Held on 08-01-2020 Evening]

A)

B)

C)

D)

• question_answer5) A transverse wave travels on a taut steel wire with a velocity of v when tension in it is$2.06\times \,\,{{10}^{4}}\text{ }N.$ When the tension is changed to T, the velocity changed to v/2. The value of T is close to [JEE MAIN Held on 08-01-2020 Evening]

A) $30.5\times {{10}^{4}}\text{ }N$

B) $2.50\times {{10}^{4}}\text{ }N$

C) $5.15\times {{10}^{3}}\text{ }N$

D) $10.2\times {{10}^{2}}\text{ }N$

• question_answer6) A galvanometer having a coil resistance $100\,\,\Omega$ gives a full scale deflection when a current of 1 mA is passed through it. What is the value of the resistance which can convert this galvanometer into a voltmeter giving full scale deflection for a potential difference of 10 V? [JEE MAIN Held on 08-01-2020 Evening]

A) $10\text{ k}\,\Omega$

B) $9.9\text{ }k\Omega$

C) $8.9\text{ }k\Omega$

D) $7.9\text{ }k\Omega$

• question_answer7) A Carnot engine having an efficiency of $\frac{1}{10}$ is being used as a refrigerator. If the work done on the refrigerator is 10 J, the amount of heat absorbed from the reservoir at lower temperature is: [JEE MAIN Held on 08-01-2020 Evening]

A) 90 J

B) 1 J

C) 99 J

D) 100 J

• question_answer8) As shown in the figure, a battery of emf E is connected to an inductor L and resistance R in series. The switch is closed at$t=0$. The total charge that flows from the battery, between t = 0 and $t={{t}_{C}}$(${{t}_{C}}$ is the time constant of the circuit) is: [JEE MAIN Held on 08-01-2020 Evening]

A) $\frac{EL}{{{R}^{2}}}$

B) $\frac{ER}{e{{L}^{2}}}$

C) $\frac{EL}{{{R}^{2}}}\left( 1-\frac{1}{e} \right)$

D) $\frac{EL}{e{{R}^{2}}}$

• question_answer9) Two liquids of densities ${{\rho }_{1}}$and ${{\rho }_{2}}$$\left( {{\rho }_{2}}\text{=}2{{\rho }_{1}} \right)$are filled up behind a square wall of side 10 m as shown in figure. Each liquid has a height of 5 m. The ratio of the forces due to these liquids exerted on upper part MN to that at the lower part NO is (Assume that the liquids are not mixing) [JEE MAIN Held on 08-01-2020 Evening]

A) 1/4

B) 1/2

C) 2/3

D) 1/3

• question_answer10) A particle of mass m is dropped from a height h above the ground. At the same time another particle of the same mass is thrown vertically upwards from the ground with a speed of$\sqrt{2gh}$. If they collide head-on completely inelastically, the time taken for the combined mass to reach the ground, in units of $\sqrt{\frac{h}{g}}$ is [JEE MAIN Held on 08-01-2020 Evening]

A) $\sqrt{\frac{1}{2}}$

B) $\sqrt{\frac{3}{4}}$

C) $\frac{1}{2}$

D) $\sqrt{\frac{3}{2}}$

• question_answer11) A simple pendulum is being used to determine the value of gravitational acceleration g at a certain place. The length of the pendulum is 25.0 cm and a stopwatch with 1 s resolution measures the time taken for 40 oscillation to be 50 s. The accuracy in g is [JEE MAIN Held on 08-01-2020 Evening]

A) 3.40%

B) 2.40%

C) 5.40%

D) 4.40%

• question_answer12) A particle of mass m and charge q is released from rest in a uniform electric field. If there is no other force on the particle, the dependence of its speed v on the distance x travelled by it is correctly given by (graphs are schematic and not drawn to scale) [JEE MAIN Held on 08-01-2020 Evening]

A)

B)

C)

D)

• question_answer13) A capacitor is made of two square plates each of side ?a? making a very small angle $\alpha$ between them, as shown in figure. The capacitance will be close to [JEE MAIN Held on 08-01-2020 Evening]

A) $\frac{{{\varepsilon }_{0}}{{a}^{2}}}{d}\left( 1-\frac{3\alpha a}{2d} \right)$

B) $\frac{{{\varepsilon }_{0}}{{a}^{2}}}{d}\left( 1-\frac{\alpha a}{2d} \right)$

C) $\frac{{{\varepsilon }_{0}}{{a}^{2}}}{d}\left( 1+\frac{\alpha a}{d} \right)$

D) $\frac{{{\varepsilon }_{0}}{{a}^{2}}}{d}\left( 1-\frac{\alpha a}{4d} \right)$

• question_answer14) Consider two charged metallic spheres ${{S}_{1}}$ and ${{S}_{2}}$ of radii ${{R}_{1}}$ and${{R}_{2}}$, respectively. The electric fields ${{E}_{1}}$ (on ${{S}_{1}}$) and ${{E}_{2}}$ (on ${{S}_{2}}$) on their surfaces are such that ${{E}_{1}}$/${{E}_{2}}$ = ${{R}_{1}}$/${{R}_{2}}$. Then the ratio ${{V}_{1}}$(on ${{S}_{1}}$)/${{V}_{2}}$(on ${{S}_{2}}$) of the electrostatic potentials on each sphere is [JEE MAIN Held on 08-01-2020 Evening]

A) ${{\left( {{R}_{1}}\text{/}{{R}_{2}} \right)}^{2}}$

B) $\left( {{R}_{2}}\text{/}{{R}_{1}} \right)$

C) ${{\left( \frac{{{R}_{1}}}{{{R}_{2}}} \right)}^{3}}$

D) ${{R}_{1}}/{{R}_{2}}$

• question_answer15) A plane electromagnetic wave of frequency 25 GHz is propagating in vacuum along the z-direction. At a particular point in space and time, the magnetic field is given by$\vec{B}=5\times {{10}^{-8}}\hat{j}\text{ }T$. The corresponding electric field $\vec{E}$ is (speed of light $c=3\times {{10}^{8}}\text{ }m{{s}^{1}}$) [JEE MAIN Held on 08-01-2020 Evening]

A) $-1.66\times {{10}^{-16}}\hat{i}\text{ }V/m$

B) $1.66\times {{10}^{-16}}\hat{i}\text{ }V/m$

C) $-15\text{ \hat{i} }V/m$

D) $15\,\,\hat{i}\,\,V/m$

• question_answer16) In a double-slit experiment, at a certain point on the screen the path difference between the two interfering waves is $\frac{1}{8}th$ of a wavelength. The ratio of the intensity of light at that point to that at the centre of a bright fringe is [JEE MAIN Held on 08-01-2020 Evening]

A) 0.568

B) 0.760

C) 0.853

D) 0.672

• question_answer17) A very long wire ABDMNDC is shown in figure carrying current I. AB and BC parts are straight, long and at right angle. At D wire forms a circular turm DMND of radius R. AB, BC parts are tangential to circular turn at N and D. Magnetic field at the centre of circle is [JEE MAIN Held on 08-01-2020 Evening]

A) $\frac{{{\mu }_{0}}I}{2\pi R}\left( \pi +1 \right)$

B) $\frac{{{\mu }_{0}}I}{2R}$

C) $\frac{{{\mu }_{0}}I}{2\pi R}\left( \pi +\frac{1}{\sqrt{2}} \right)$

D) $\frac{{{\mu }_{0}}I}{2\pi R}\left( \pi -\frac{1}{\sqrt{2}} \right)$

• question_answer18) As shown in fig. when a spherical cavity (centred at O) of radius 1 is cut out of a uniform sphere of radius R (centred at C), the centre of mass of remaining (shaded) part of sphere is at G, i.e. on the surface of the cavity. R can be determined by the equation [JEE MAIN Held on 08-01-2020 Evening]

A) $\left( {{R}^{2}}\text{+}R+1 \right)\left( 2R \right)=1$

B) $\left( {{R}^{2}}R+1 \right)\left( 2-R \right)=1$

C) $\left( {{R}^{2}}R1 \right)\left( 2R \right)\,\,\text{=}\,1$

D) $\left( {{R}^{2}}+R1 \right)\left( 2R \right)=1$

• question_answer19) A particle moves such that its position vector $\vec{r}\left( t \right)=cos\omega t\,\hat{i}\text{+}sin\omega t\,\hat{j}$ where $\omega$ is a constant and t is time. Then which of the following statements is true for the velocity $\vec{v}(t)$ and acceleration $\vec{a}(t)$ of the particle [JEE MAIN Held on 08-01-2020 Evening]

A) $\overrightarrow{v}$and $\overrightarrow{a}$both are perpendicular to $\overrightarrow{r}$

B) $\overrightarrow{v}$ is perpendicular to $\overrightarrow{r}$ and $\overrightarrow{a}$ is directed towards the origin

C) $\overrightarrow{v}$ and $\overrightarrow{a}$ both are parallel to $\overrightarrow{r}$

D) $\overrightarrow{v}$ is perpendicular to $\overrightarrow{r}$and $\overrightarrow{a}$ is directed away from the origin

• question_answer20) An electron (mass m) with initial velocity $\vec{v}={{v}_{0}}\,\hat{i}+{{v}_{0}}\,\hat{j}$ is in an electric field $\vec{E}=-{{E}_{0}}\hat{k}$. If ${{\lambda }_{0}}$ is initial de-Broglie wavelength of electron, its de-Broglie wavelength at time t is given by  [JEE MAIN Held on 08-01-2020 Evening]

A) $\frac{{{\lambda }_{0}}}{\sqrt{1+\frac{{{e}^{2}}E_{0}^{2}{{t}^{2}}}{{{m}^{2}}v_{0}^{2}}}}$

B) $\frac{{{\lambda }_{0}}\sqrt{2}}{\sqrt{1+\frac{{{e}^{2}}{{E}^{2}}{{t}^{2}}}{{{m}^{2}}v_{0}^{2}}}}$

C) $\frac{{{\lambda }_{0}}}{\sqrt{1+\frac{{{e}^{2}}{{E}^{2}}{{t}^{2}}}{2{{m}^{2}}v_{0}^{2}}}}$

D) $\frac{{{\lambda }_{0}}}{\sqrt{2+\frac{{{e}^{2}}{{E}^{2}}{{t}^{2}}}{{{m}^{2}}v_{0}^{2}}}}$

• question_answer21) Three containers ${{C}_{1}}$, ${{C}_{2}}$ and ${{C}_{3}}$ have water at different temperatures. The table below shows the final temperature T when different amounts of water (given in liters) are taken from each container and mixed (assume no loss of heat during the process)

 ${{C}_{1}}$ ${{C}_{2}}$ ${{C}_{3}}$ T $1\ell$ $2\ell$ - $60{}^\circ C$ - $1\ell$ $2\ell$ $30{}^\circ C$ $2\ell$ - $1\ell$ $60{}^\circ C$ $1\ell$ $1\ell$ $1\ell$ $\theta$
The value of $\theta$ (in ${}^\circ C$ to the nearest integer) is _______.               [JEE MAIN Held on 08-01-2020 Evening]

• question_answer22) An asteroid is moving directly towards the centre of the earth. When at a distance of 10 R (R is the radius of the earth) from the earths centre, it has a speed of 12 km/s. Neglecting the effect of earths atmosphere, what will be the speed of the asteroid when it hits the surface of the earth (escape velocity from the earth is 11.2 km/s)? Give your answer to the nearest integer in kilometer/s _______.                       [JEE MAIN Held on 08-01-2020 Evening]

• question_answer23) The series combination of two batteries, both of the same emf 10 V, but different internal resistance of $20\text{ }\Omega$ and$\text{5 }\Omega$, is connected to the parallel combination of two resistors $\text{30 }\Omega$ and $\text{R }\Omega$. The voltage difference across the battery of internal resistance $20\text{ }\Omega$ is zero, the value of R$\left( \text{in }\Omega \right)$ is _______. [JEE MAIN Held on 08-01-2020 Evening]

• question_answer24) The first member of the Balmer series of hydrogen atom has a wavelength of 6561$\overset{\text{o}}{\mathop{\text{A}}}\,$. The wavelength of the second member of the Balmer series (in nm) is _______.                                                          [JEE MAIN Held on 08-01-2020 Evening]

• question_answer25) A ball is dropped from the to of a 100 m high tower on a planet. In the last $\frac{1}{2}$s before hitting the ground, it covers a distance of 19 m. Acceleration due to gravity (in $m{{s}^{-2}}$) near the surface on that planet is _______. [JEE MAIN Held on 08-01-2020 Evening]

• question_answer26) Among the compounds A and B with molecular formula${{C}_{9}}{{H}_{18}}{{O}_{3}}$, A is having higher boiling point the B. The possible structures of A and B are [JEE MAIN Held on 08-01-2020 Evening]

A)

B)

C)

D)

• question_answer27) Which of the following compounds is likely to show both Frenkel and Schottky defects in its crystalline form? [JEE MAIN Held on 08-01-2020 Evening]

A) ZnS

B) CsCl

C) AgBr

D) KBr

 Among the reactions (A) - (D), the reaction(s) that does/do not occur in the blast furnace during the extraction of iron is/are (A) $CaO+Si{{O}_{2}}\to CaSi{{O}_{3}}$ (B) $3F{{e}_{2}}{{O}_{3}}+CO\to 2F{{e}_{3}}{{O}_{4}}+C{{O}_{2}}$ (C) $FeO+Si{{O}_{2}}\to FeSi{{O}_{3}}$ (D) $\text{FeO}\to \text{ Fe+}\frac{1}{2}{{O}_{2}}$
[JEE MAIN Held on 08-01-2020 Evening]

A) (C) and (D)

B) (D)

C) (A)

D) (A) and (D)

• question_answer29) The increasing order of the atomic radii of the following elements is [JEE MAIN Held on 08-01-2020 Evening]

 (A) C (B) O (C) F (D) Cl (E) Br

A) (D) < (C) < (B) < (A) < (E)

B) (B) < (C) < (D) < (A) < (E)

C) (C) < (B) < (A) < (D) < (E)

D) (A) < (B) < (C) < (D) < (E)

• question_answer30) The radius of the second Bohr orbit, in terms of the Bohr radius, ${{a}_{0}}$, in $L{{i}^{2+}}$ is [JEE MAIN Held on 08-01-2020 Evening]

A) $\frac{4{{a}_{0}}}{3}$

B) $\frac{4{{a}_{0}}}{9}$

C) $\frac{2{{a}_{0}}}{3}$

D) $\frac{2{{a}_{0}}}{9}$

 The correct order of the calculated spin-only magnetic moments of complexes (A) to (D) is (A) $Ni{{\left( CO \right)}_{4}}$ (B) $\left[ Ni{{\left( {{H}_{2}}O \right)}_{6}} \right]C{{l}_{2}}$ (C) $N{{a}_{2}}\left[ Ni{{\left( CN \right)}_{4}} \right]$ (D) $PdC{{l}_{2}}{{\left( PP{{h}_{3}} \right)}_{2}}$
[JEE MAIN Held on 08-01-2020 Evening]

A) $\left( A \right)\approx \left( C \right)\approx \left( D \right)<\left( B \right)$

B) $\left( C \right)\approx \left( D \right)<\left( B \right)<\left( A \right)$

C) $\left( A \right)\approx \left( C \right)<\left( B \right)\approx \left( D \right)$

D) $\left( C \right)<\left( D \right)<\left( B \right)<\left( A \right)$

• question_answer32) The major product [B] in the following sequence of reactions is [JEE MAIN Held on 08-01-2020 Evening]

A)

B)

C)

D)

• question_answer33) Hydrogen has three isotopes (A), (B) and (C). If the number of neutron(s) in (A), (B) and (C) respectively, are (x), (y) and (z), the sum of (x), (y) and (z) is [JEE MAIN Held on 08-01-2020 Evening]

A) 4

B) 2

C) 3

D) 1

• question_answer34) The major product in the following reaction is [JEE MAIN Held on 08-01-2020 Evening]

A)

B)

C)

D)

 For the following Assertion and Reason, the correct option is Assertion: The pH of water increases with increase in temperature. Reason: The dissociation of water into ${{H}^{+}}$ and $O{{H}^{-}}$ is an exothermic reaction.
[JEE MAIN Held on 08-01-2020 Evening]

A) Both assertion and reason are false

B) Assertion is not true, but reason is true

C) Both assertion and reason are true, and the reason is the correct explanation for the assertion

D) Both assertion and reason are true, but the reason is not the correct explanation for the assertion

• question_answer36) A metal (A) on heating in nitrogen gas gives compound B. B on treatment with ${{H}_{2}}O$ gives a colourless gas which when passed through $CuS{{O}_{4}}$ solution gives a dark blue-violet coloured solution. A and B respectively, are [JEE MAIN Held on 08-01-2020 Evening]

A) Mg and $M{{g}_{3}}{{N}_{2}}$

B) Na and $N{{a}_{3}}N$

C) Mg and $Mg{{\left( N{{O}_{3}} \right)}_{2}}$

D) Na and $NaN{{O}_{3}}$

• question_answer37) Preparation of Bakelite proceeds via reactions [JEE MAIN Held on 08-01-2020 Evening]

A) Electrophilic substitution and dehydration

D) Condensation and elimination

• question_answer38) Consider the following plots of rate constant versus $\frac{1}{T}$ for four different reactions. Which of the following orders is correct for the activation energies of these reactions? [JEE MAIN Held on 08-01-2020 Evening]

A) ${{E}_{b}}>{{E}_{a}}>{{E}_{d}}>{{E}_{c}}$

B) ${{E}_{c}}>{{E}_{a}}>{{E}_{d}}>{{E}_{b}}$

C) ${{E}_{a}}>{{E}_{c}}>{{E}_{d}}>{{E}_{b}}$

D) ${{E}_{b}}>{{E}_{d}}>{{E}_{c}}>{{E}_{a}}$

 For the following Assertion and Reason, the correct option is Assertion: For hydrogenation reactions, the catalytic activity increases from Group 5 to Group 11 metals with maximum activity shown by Group 7-9 elements. Reason: The reactants are most strongly adsorbed on group 7-9 elements. [JEE MAIN Held on 08-01-2020 Evening]

A) Both assertion and reason are true and the reason is the correct explanation for the assertion.

B) Both assertion and reason are false.

C) The assertion is true, but the reason is false.

D) Both assertion and reason are true but the reason is not the correct explanation for the assertion.

• question_answer40) Arrange the following bonds according to their average bond energies in descending order $C-Cl,C-Br,C-F,C-l$ [JEE MAIN Held on 08-01-2020 Evening]

A) $C-Cl>C-Br>C-l>C-F$

B) $C-Br>C-l>C-Cl>C-F$

C) $C-F>C-Cl>C-Br>C-l$

D) $C-l>C-Br>C-Cl>C-F$

• question_answer41) White phosphorus on reaction with concentrated $NaOH$solution in an inert atmosphere of $C{{O}_{2}}$gives phosphine and compound (X). (X) on acidification with $HCl$ gives compound (Y). The basicity of compound (Y) is [JEE MAIN Held on 08-01-2020 Evening]

A) 3

B) 2

C) 4

D) 1

• question_answer42) Two monomers in maltose are [JEE MAIN Held on 08-01-2020 Evening]

A) $\alpha$-D-glucose and $\alpha$-D-glucose

B) $\alpha$-D-glucose and $\beta$-D-glucose

C) $\alpha$-D-glucose and $\alpha$-D-galactose

D) $\alpha$-D-glucose and $\alpha$-D-Fructose

• question_answer43) Kjeldahl's method cannot be used to estimate nitrogen for which of the following compounds? [JEE MAIN Held on 08-01-2020 Evening]

A) $C{{H}_{3}}C{{H}_{2}}C\equiv N$

B) $N{{H}_{2}}\overset{\overset{O}{\mathop{\parallel }}\,}{\mathop{C}}\,N{{H}_{2}}$

C) ${{C}_{6}}{{H}_{5}}N{{O}_{2}}$

D) ${{C}_{6}}{{H}_{5}}N{{H}_{2}}$

 Among (A) - (D), the complexes that can display geometrical isomerism are [JEE MAIN Held on 08-01-2020 Evening] (A) ${{\left[ Pt{{\left( N{{H}_{3}} \right)}_{3}}Cl \right]}^{+}}$ (B) ${{\left[ Pt\left( N{{H}_{3}} \right)C{{l}_{5}} \right]}^{}}$ (C) $\left[ Pt{{\left( N{{H}_{3}} \right)}_{2}}Cl\left( N{{O}_{2}} \right) \right]$ (D) ${{\left[ Pt{{\left( N{{H}_{3}} \right)}_{4}}ClBr \right]}^{2+}}$

A) (C) and (D)

B) (A) and (B)

C) (B) and (C)

D) (D) and (A)

• question_answer45) An unsaturated hydrocarbon X absorbs two hydrogen molecules on catalytic hydrogenation, and also gives following reaction B (3-oxo-hexanedicarboxylic acid) X will be [JEE MAIN Held on 08-01-2020 Evening]

A)

B)

C)

D)

• question_answer46) At constant volume, 4 mol of an ideal gas when heated from 300 K to 500 K changes its internal energy by 5000 J. The molar heat capacity at constant volume is __________.   [JEE MAIN Held on 08-01-2020 Evening]

 For an electrochemical cell $Sn\left( s \right)\left| S{{n}^{2+}}\left( aq,1M \right) \right|\left| P{{b}^{2+}}\left( aq,1M \right) \right|Pb\left( s \right)$the ratio $\frac{\left[ S{{n}^{2+}} \right]}{\left[ P{{b}^{2+}} \right]}$ when this cell attains equilibrium is _______ . Given: $E_{S{{n}^{2+}}|Sn}^{0}=-0.14V,$ $E_{P{{b}^{2+}}|Pb}^{0}=-0.\left. 13V,\frac{2.303RT}{F}=0.06 \right)$
[JEE MAIN Held on 08-01-2020 Evening]

 $NaCl{{O}_{3}}$ is used, even in spacecrafts, to produce${{O}_{2}}$. The daily consumption of pure ${{O}_{2}}$by a person is 492 L at 1 atm, 300 K. How much amount of$NaCl{{O}_{3}}$, in grams, is required to produce ${{O}_{2}}$for the daily consumption of a person at 1 atm, 300 K? __________. $NaCl{{O}_{3}}\left( s \right)+Fe\left( s \right)\to {{O}_{2}}\left( g \right)+NaCl\left( s \right)+FeO\left( s \right)$$R=0.082\text{ }L\text{ }atm\text{ }mo{{l}^{1}}\text{ }{{K}^{1}}$
[JEE MAIN Held on 08-01-2020 Evening]

• question_answer49) In the following sequence of reactions the maximum number of atoms present in molecule "C" in one plane is __________.  (A is a lowest molecular weight alkyne)     [JEE MAIN Held on 08-01-2020 Evening]

 Complexes $(\text{M}\,{{\text{L}}_{\text{5}}})$ of metals Ni and Fe have ideal square pyramidal and trigonal bipyramidal geometries, respectively. The sum of the $90{}^\circ$, $120{}^\circ$ and $180{}^\circ$ L-M-L angles in the two complexes is __________.
[JEE MAIN Held on 08-01-2020 Evening]

• question_answer51) If and then $10{{A}^{-1}}$ is equal to [JEE MAIN Held on 08-01-2020 Evening]

A) $6l-A$

B) $4l-A$

C) $A-4l$

D) $A-6l$

• question_answer52) The differential equation of the family of curves, ${{x}^{2}}=4b\left( y+b \right),b\in R,$ is [JEE MAIN Held on 08-01-2020 Evening]

A) $x{{\left( y' \right)}^{2\text{ }}}=x-2yy'$

B) $x{{\left( y' \right)}^{2}}=2yy'-x$

C) $x{{\left( y' \right)}^{2}}=x+2yy'$

D) $xy''=y'$

• question_answer53) The area (in sq. units) of the region $\left\{ \left( x,\text{ }y \right)\in {{R}^{2}}:{{x}^{2}}\le y\le 3-2\text{ }x \right\},$is[JEE MAIN Held on 08-01-2020 Evening]

A) $\frac{31}{3}$

B) $\frac{29}{3}$

C) $\frac{34}{3}$

D) $\frac{32}{3}$

• question_answer54) Let $\vec{a}=\hat{i}-2\hat{j}+\hat{k}$ and $\vec{b}=\hat{i}-\hat{j}+\hat{k}$ be two vectors. If $\vec{c}$is a vector such that $\vec{b}\times \vec{c}=\vec{b}\times \vec{a}$ and $\overrightarrow{c}\cdot \overrightarrow{a}=0$ , then $\vec{c}\cdot \vec{b}$ is equal to  [JEE MAIN Held on 08-01-2020 Evening]

A) $-\frac{1}{2}$

B) $-\frac{3}{2}$

C) $-1$

D) $\frac{1}{2}$

• question_answer55) Let S be the set of all functions $f:\left[ 0,1 \right]\to R,$which are continuous on [0, 1] and differentiable on (0, 1). Then for every f in S, there exists a c $\in$(0, 1), depending on f, such that [JEE MAIN Held on 08-01-2020 Evening]

A) $\left| f\left( c \right)-f\left( 1 \right)< \right|f'\left( c \right)|$

B) $\frac{f(1)-f(c)}{1-c}f'\left( c \right)$

C) $\left| f\left( c \right)-f\left( 1 \right)<\left( 1-c \right) \right|f'\left( c \right)|$

D) $\left| f\left( c \right)+f\left( 1 \right)<\left( 1+c \right) \right|f'\left( c \right)|$

• question_answer56) The mirror image of the point (1, 2, 3) in a plane is $\left( -\frac{7}{3},-\frac{4}{3},-\frac{1}{3} \right)$. Which of the following points lies on this plane? [JEE MAIN Held on 08-01-2020 Evening]

A) $\left( -1,-1,-1 \right)$

B) $(1,1,1)$

C) $\left( -1,-1,\,\,1 \right)$

D) $\left( 1,-1,\,\,1 \right)$

• question_answer57) Which of the following statements is a tautology? [JEE MAIN Held on 08-01-2020 Evening]

A) $\tilde{\ }\left( p\wedge \tilde{\ }q \right)\to p\vee q$

B) $p\vee (~\sim q)\to p\wedge q$

C) $\sim \left( p\vee \tilde{\ }q \right)\to p\vee q$

D) $~\sim (p\vee \sim ~q)\to p\wedge q$

 The system of linear equations $\lambda x+2y+2z=5$ $2\lambda x+3y+5z=8$ $4x+\lambda y+6z=10$ has
[JEE MAIN Held on 08-01-2020 Evening]

A) Infinitely many solutions when $\lambda =2$

B) No solution when $\lambda =8$

C) A unique solution when $\lambda =-\,8$

D) No solution when $\lambda =2$

• question_answer59) The length of the perpendicular from the origin, on the normal to the curve, ${{x}^{2}}+2xy-3{{y}^{2}}=0$ at the point (2, 2) is [JEE MAIN Held on 08-01-2020 Evening]

A) $2\sqrt{2}$

B) $\sqrt{2}$

C) $4\sqrt{2}$

D) $2$

• question_answer60) If $I=\int\limits_{1}^{2}{\frac{dx}{\sqrt{2{{x}^{3}}-9{{x}^{2}}+12x+4}}}~$, then [JEE MAIN Held on 08-01-2020 Evening]

A) $\frac{1}{9}<{{I}^{2}}<\frac{1}{8}$

B) $\frac{1}{8}<{{I}^{2}}<\frac{1}{4}$

C) $\frac{1}{6}<{{I}^{2}}<\frac{1}{2}$

D) $\frac{1}{16}<{{I}^{2}}<\frac{1}{9}$

• question_answer61) If a hyperbola passes through the point P (10, 16) and it has vertices at $\left( \pm \,6,\text{ }0 \right)$ then the equation of the normal to it at P is [JEE MAIN Held on 08-01-2020 Evening]

A) $x+2y=42$

B) $2x+5y=100$

C) $x+3y=58$

D) $3x+4y=94$

• question_answer62) $\underset{x\to 0}{\mathop{lim}}\,\frac{\int\limits_{0}^{x}{t\sin (10t)dt}}{x}$is equal to [JEE MAIN Held on 08-01-2020 Evening]

A) $0$

B) $\frac{1}{10}$

C) $-\frac{1}{5}$

D) $-\frac{1}{10}$

• question_answer63) If the 10th term of an A.P. is $\frac{1}{20}$ and its 20th term is $\frac{1}{10}$, then the sum of its first 200 terms is [JEE MAIN Held on 08-01-2020 Evening]

A) ${{50}^{\frac{1}{4}}}$

B) $50$

C) $100$

D) ${{100}^{\frac{1}{2}}}$

• question_answer64) Let S be the set of all real roots of the equation, ${{3}^{X}}\left( {{3}^{X\text{ }}}-1 \right)+2=\left| {{3}^{x}}-1\text{ } \right|+\left| {{3}^{x}}-2\text{ } \right|.$ Then S [JEE MAIN Held on 08-01-2020 Evening]

A) Contains at least four elements

B) Is a singleton

C) Contains exactly two elements

D) Is an empty set

• question_answer65) The mean and variance of 20 observations are found to be 10 and 4, respectively. On rechecking, it was found that an observation 9 was incorrect and the correct observation was 11. Then the correct variance is [JEE MAIN Held on 08-01-2020 Evening]

A) 3.98

B) 4.02

C) 3.99

D) 4.01

• question_answer66) If $\alpha$ and $\beta$ be the coefficients of ${{x}^{4}}$ and ${{x}^{2}}$respectively in the expansion of ${{\left( x+\sqrt{{{x}^{2}}-1} \right)}^{6}}+{{\left( x-\sqrt{{{x}^{2}}-1} \right)}^{6}}$, then [JEE MAIN Held on 08-01-2020 Evening]

A) $\alpha -\beta =60$

B) $\alpha +\beta =60$

C) $\alpha -\beta =-132$

D) $\alpha +\beta =-30$

• question_answer67) If a line, $y=mx+c$ is a tangent to the circle, ${{\left( x-3 \right)}^{2}}+{{y}^{2}}=1$ and it is perpendicular to a line ${{L}_{1}}$, where ${{L}_{1}}$ is the tangent to the circle, ${{x}^{2}}+{{y}^{2}}=1$ at the point $\left( \frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}} \right)$; then [JEE MAIN Held on 08-01-2020 Evening]

A) ${{c}^{2}}+6c+7=0$

B) ${{c}^{2}}-7c+6=0$

C) ${{c}^{2}}+7c+6=0$

D) ${{c}^{2}}-6c+7=0$

• question_answer68) Let$\alpha =\frac{-1+i\sqrt{3}}{2}$. If $a=(1+\alpha )\underset{k=0}{\overset{100}{\mathop{\sum }}}\,{{\alpha }^{2k}}$ and $b=\underset{k=0}{\overset{100}{\mathop{\sum }}}\,{{\alpha }^{3k}}$, then a and b are the roots of the quadratic equation [JEE MAIN Held on 08-01-2020 Evening]

A) ${{x}^{2}}-101x+100=0$

B) ${{x}^{2}}-102x+101=0$

C) ${{x}^{2}}+101x+100=0$

D) ${{x}^{2}}+102x+101=0$

• question_answer69) Let A and B be two events such that the probability that exactly one of them occurs is $\frac{2}{5}$and the probability that A or B occurs is $\frac{1}{2}$ , then the probability of both of them occur together is [JEE MAIN Held on 08-01-2020 Evening]

A) 0.01

B) 0.20

C) 0.02

D) 0.10

• question_answer70) Let $f:\left( 1,\text{ }3 \right)\to R$ be a function defined by$f\left( x \right)=\frac{x\left[ x \right]}{1+{{x}^{2}}}$, where [x] denotes the greatest integer$\le x$. Then the range of f is [JEE MAIN Held on 08-01-2020 Evening]

A) $\left( \frac{2}{5},\left. \frac{3}{5} \right] \right.\cup \left( \frac{3}{4},\frac{4}{5} \right)$

B) $\left( \frac{2}{5},\frac{1}{2} \right)\cup \left( \frac{3}{5},\frac{4}{5} \right]$

C) $\left( \frac{2}{5},\frac{4}{5} \right]$

D) $\left( \frac{3}{5},\frac{4}{5} \right)$

• question_answer71) If $\frac{\sqrt{2}\sin \alpha }{\sqrt{1+\cos 2\alpha }}=\frac{1}{7}$ and $\sqrt{\frac{1-\cos 2\beta }{2}}=\frac{1}{\sqrt{10}}$, $\alpha ,\beta \in \left( 0,\frac{\pi }{2} \right)$, then $tan\left( \alpha +2\beta \right)$ is equal to [JEE MAIN Held on 08-01-2020 Evening]

• question_answer72) The number of 4 letter words (with or without meaning) that can be formed from the eleven letters of the word 'EXAMINATION' is _____.                                               [JEE MAIN Held on 08-01-2020 Evening]

• question_answer73) Let f(x) be a polynomial of degree 3 such that $f\left( -1 \right)=10,f\left( 1 \right)=-6,$ f(x) has a critical point at $x=-1$, and f'(x) has a critical point x = 1. Then f(x) has a local minima at x = _____. [JEE MAIN Held on 08-01-2020 Evening]

• question_answer74) Let a line $y=mx\left( m>0 \right)$ intersect the parabola, ${{y}^{2}}=x$at a point P, other than the origin. Let the tangent to it at P meet the x-axis at the point Q. If area $\left( \Delta OPQ \right)=4\,\,sq.$sq. units, then m is equal to_____. [JEE MAIN Held on 08-01-2020 Evening]

• question_answer75) The sum, $\sum\limits_{n\,=\,1}^{7}{\frac{n\left( n+1 \right)\left( 2n+1 \right)}{4}}$ is equal to [JEE MAIN Held on 08-01-2020 Evening]