# Solved papers for JEE Main & Advanced JEE Main Paper Phase-I (Held on 07-1-2020 Evening)

### done JEE Main Paper Phase-I (Held on 07-1-2020 Evening) Total Questions - 75

• question_answer1) 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 ${{v}_{0}}=1400$ Hz and the velocity of sound in air 350 m/s. The speed of each tuning fork is close to:            [JEE MAIN Held on 07-01-2020 Evening]

A)
$\frac{1}{4}\,m/s$

B)
$\frac{1}{2}\,m/s$

C)
$1\,m/s$

D)
$\frac{1}{8}\,m/s$

• question_answer2) The electric field of a plane electromagnetic wave is given by $\vec{E}={{E}_{0}}\frac{\hat{i}+\hat{j}}{\sqrt{2}}\,\cos \,(kz+\omega t)$ At t = 0, a positively charged particle is at the point $(x,y,z)\,=\,\left( 0,0,\,\frac{\pi }{k} \right)$. If its instantaneous velocity at (t = 0) is${{v}_{0}}\,\hat{k},$, the force acting on it due to the wave is: [JEE MAIN Held on 07-01-2020 Evening]

A)
Antiparallel to $\frac{\hat{i}+\hat{j}}{\sqrt{2}}$

B)
Zero

C)
Parallel to $\frac{\hat{i}+\hat{j}}{\sqrt{2}}$

D)
Parallel to $\hat{k}$

• question_answer3) In a Young's double slit experiment, the separation between the slits is 0.15 mm. In the experiment, a source of light of wavelength 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: [JEE MAIN Held on 07-01-2020 Evening]

A)
3.9 mm

B)
6.9 mm

C)
5.9 mm

D)
4.9 mm

• question_answer4) Two ideal Carnot engines operate in cascade (all heat given up by one engine is used by the other engine to produce work) between temperatures, ${{T}_{1}}$and ${{T}_{2}}$. The temperature of the hot reservoir of the first engine is ${{T}_{1}}$ and the temperature of the cold reservoir of the second engine is${{T}_{2}}$. $T$ is temperature of the sink of first engine which is also the source for the second engine. How is $T$ related to ${{T}_{1}}$ and${{T}_{2}}$, if both the engines perform equal amount of work?  [JEE MAIN Held on 07-01-2020 Evening]

A)
$T=\frac{{{T}_{1}}+{{T}_{2}}}{2}$

B)
$T=\sqrt{{{T}_{1}}{{T}_{2}}}$

C)
$T=\frac{2{{T}_{1}}{{T}_{2}}}{{{T}_{1}}+{{T}_{2}}}$

D)
$T=0$

• question_answer5) 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:                                  [JEE MAIN Held on 07-01-2020 Evening]

A)
$\frac{81}{256}$

B)
$\frac{9}{16}$

C)
$\frac{3}{4}$

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

• question_answer6) 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 vacuum)                                                                 [JEE MAIN Held on 07-01-2020 Evening]

A)
$\frac{1}{c}{{\left( \frac{E}{2m} \right)}^{\frac{1}{2}}}$

B)
${{\left( \frac{E}{2m} \right)}^{\frac{1}{2}}}$

C)
$c\,{{(2mE)}^{\frac{1}{2}}}$

D)
$\frac{1}{c}{{\left( \frac{2E}{m} \right)}^{\frac{1}{2}}}$

• question_answer7) Mass per unit area of a circular disc of radius a depends on the distance r from its centre as$\sigma \,(r)=A+Br$. The moment of inertia of the disc about the axis, perpendicular to the plane and passing through its centre is: [JEE MAIN Held on 07-01-2020 Evening]

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

B)
$\pi {{a}^{4}}\,\left( \frac{A}{4}+\frac{aB}{5} \right)$

C)
$2\pi {{a}^{4}}\,\left( \frac{A}{4}+\frac{aB}{5} \right)$

D)
$2\pi {{a}^{4}}\,\left( \frac{aA}{4}+\frac{B}{5} \right)$

• question_answer8) 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 $45{}^\circ$ with the vertical. Then F equals: (Take $g=10\text{ }m{{s}^{2}}$ and the rope to be massless) [JEE MAIN Held on 07-01-2020 Evening]

A)
75 N

B)
70 N

C)
90 N

D)
100 N

• question_answer9) 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 ${{f}_{l}}$, then the ratio $\frac{{{f}_{l}}}{f}$is closest to the integer [JEE MAIN Held on 07-01-2020 Evening]

A)
17

B)
1

C)
5

D)
9

• question_answer10) A box weighs 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 $g=10\text{ }m{{s}^{2}}$at the north pole and the radius of the earth = 6400 km)                                                                                  [JEE MAIN Held on 07-01-2020 Evening]

A)
194.66 N

B)
195.66 N

C)
195.32 N

D)
194.32 N

• question_answer11) 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 $(g=10\text{ }m/{{s}^{2}})$must be at least  [JEE MAIN Held on 07-01-2020 Evening]

A)
56300 W

B)
66000 W

C)
48000 W

D)
62360 W

• question_answer12) 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 [JEE MAIN Held on 07-01-2020 Evening]

A)
15 A

B)
10 A

C)
20 A

D)
25 A

• question_answer13) 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 $\frac{{{C}_{P}}}{{{C}_{v}}}=\gamma$for this gas then a good estimate for $\frac{{{\tau }_{2}}}{{{\tau }_{1}}}$is given by                                                               [JEE MAIN Held on 07-01-2020 Evening]

A)
${{\left( \frac{1}{2} \right)}^{\frac{\gamma +1}{2}}}$

B)
${{\left( \frac{1}{2} \right)}^{\gamma }}$

C)
$2$

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

• question_answer14) 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 a time $t=\,\,\infty$ and at $t=40$s is close to (Take${{e}^{2}}=7.389$) [JEE MAIN Held on 07-01-2020 Evening]

A)
1.06

B)
1.15

C)
1.46

D)
0.84

• question_answer15)   A particle of mass m and charge q has an initial velocity $\vec{v}={{v}_{0}}\hat{j}$. If an electric field $\vec{E}={{E}_{0}}\hat{i}$ and magnetic field $\vec{B}={{B}_{0}}\hat{i}$act on the particle, its speed will double after a time: [JEE MAIN Held on 07-01-2020 Evening]

A)
$\frac{3\,m{{v}_{0}}}{q{{E}_{0}}}$

B)
$\frac{\sqrt{3\,}m{{v}_{0}}}{q{{E}_{0}}}$

C)
$\frac{2\,m{{v}_{0}}}{q{{E}_{0}}}$

D)
$\frac{\sqrt{2}\,m{{v}_{0}}}{q{{E}_{0}}}$

• question_answer16) In the figure, potential difference between A and B is : [JEE MAIN Held on 07-01-2020 Evening]

A)
Zero

B)
5 V

C)
10 V

D)
15 V

• question_answer17) The activity of a radioactive sample falls from $700\text{ }{{s}^{\,1}}$to $500\text{ }{{s}^{\,1}}$ in 30 minutes. Its half-life is close to [JEE MAIN Held on 07-01-2020 Evening]

A)
66 min

B)
52 min

C)
62 min

D)
72 min

• question_answer18) The figure gives experimentally measured B vs. H variation in a ferromagnetic material. The retentivity, co-ercivity and saturation, respectively, of the material are [JEE MAIN Held on 07-01-2020 Evening]

A)
1.0 T, 50 A/m and 1.5 T

B)
1.5 T, 50 A/m and 1.0 T

C)
1.5 T, 50 A/m and 1.0 T

D)
150 A/m, 1.0 T and 1.5 T

• question_answer19) 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 [JEE MAIN Held on 07-01-2020 Evening]

A)
2.5 s and 7.5 s

B)
5.0 s and 7.5 s

C)
2.5 s and 5.0 s

D)
5.0 s and 10.0 s

• question_answer20) The dimension of $\frac{{{B}^{2}}}{2{{\mu }_{0}}},$ where B is magnetic field and ${{\mu }_{0}}$ is the magnetic permeability of vaccum, is [JEE MAIN Held on 07-01-2020 Evening]

A)
$M{{L}^{1}}\text{ }{{T}^{\,2}}$

B)
$M{{L}^{2}}\text{ }{{T}^{\,\,2}}$

C)
$M{{L}^{2}}\text{ }{{T}^{\,\,1}}$

D)
$ML{{T}^{\,2}}$

• question_answer21) 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 $100\times \frac{b}{a}$ for box not to topple before moving is _________. [JEE MAIN Held on 07-01-2020 Evening]

• question_answer22) A 60 pF capacitor is fully charged by a 20 V supply. It is then disconnected from the supply and is connected 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) _________. [JEE MAIN Held on 07-01-2020 Evening]

• question_answer23) 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 cell is $\frac{N}{10}\,\Omega$, where N is an integer then value of N is _________. [JEE MAIN Held on 07-01-2020 Evening]

• question_answer24) The sum of two forces $\vec{P}$ and $\vec{Q}$ is $\vec{R}$ such that$\left| {\vec{R}} \right|=\left| {\vec{P}} \right|$. The angle $\theta$ (in degrees) that the resultant of $2\vec{P}$and  $\vec{Q}$ will make with $\vec{Q}$is_________. [JEE MAIN Held on 07-01-2020 Evening]

• question_answer25) M grams of steam at $100{}^\circ C$ is mixed with 200 g of ice at its melting point in a thermally insulated container. If it produces liquid water at $40{}^\circ C$ [heat of vaporization of water is 540 cal/g and heat of fusion of ice is 80 cal/g], the value of M is _________.  [JEE MAIN Held on 07-01-2020 Evening]

• question_answer26)  In the following reactions, products and (B), respectively, are $\underset{(hot\,\,and\,\,conc.)}{\mathop{NaOH}}\,+C{{l}_{2}}\to \,(A)+\,side\,products$ $\underset{(dry)}{\mathop{Ca(OH)}}\,+C{{l}_{2}}\to \,(B)+\,side\,products$ [JEE MAIN Held on 07-01-2020 Evening]

A)
$NaOCl$ and $Ca{{(OCl)}_{2}}$

B)
$NaCl{{O}_{3}}$and $Ca{{(Cl{{O}_{3}})}_{2}}$

C)
$NaOCl$ and $Ca{{(Cl{{O}_{3}})}_{2}}$

D)
$NaCl{{O}_{3}}$and $Ca{{(OCl)}_{2}}$

• question_answer27)  For the reaction $2{{H}_{2}}(g)\,+2NO(g)\,\to \,{{N}_{2}}(g)+2{{H}_{2}}O(g)$ the observed rate expression is, rate = $~{{k}_{f}}{{[NO]}^{2}}\,[{{H}_{2}}]$. The rate expression for the reverse reaction is:
[JEE MAIN Held on 07-01-2020 Evening]

A)
${{k}_{b}}[{{N}_{2}}]\,{{[{{H}_{2}}O]}^{2}}/[NO]$

B)
$~{{k}_{b}}\,[{{N}_{2}}]\,[{{H}_{2}}O]$

C)
$~{{k}_{b}}\,[{{N}_{2}}]\,{{[{{H}_{2}}O]}^{2}}$

D)
${{k}_{b}}[{{N}_{2}}]{{[{{H}_{2}}O]}_{2}}/[{{H}_{2}}]$

• question_answer28)    Among the statements (a)-(d), the incorrect ones are Octahedral Co(III) complexes with strong field ligands have very high magnetic moments When${{\Delta }_{0}} [JEE MAIN Held on 07-01-2020 Evening] A) (c) and (d) only B) (a) and (d) only C) (a) and (b) only D) (b) and (c) only View Answer play_arrow • question_answer29)  Consider the following reactions [JEE MAIN Held on 07-01-2020 Evening] Which of these reactions are possible? A) [b] and [d] B) [a] and [d] C) [a] and [b] D) [b], [c] and [d] View Answer play_arrow • question_answer30) For the following reactions where, \[{{k}_{s}}$ and ${{k}_{e}}$, are, respectively, the rate constants for substitution and elimination, and $\mu =\frac{{{k}_{s}}}{{{k}_{e}}}$the correct option is _________ .  [JEE MAIN Held on 07-01-2020 Evening]

A)
${{\mu }_{B}}>{{\mu }_{A}}$ and ${{k}_{e}}(A)>{{k}_{e}}(B)$

B)
${{\mu }_{A}}>{{\mu }_{B}}$ and ${{k}_{e}}(A)>{{k}_{e}}(B)$

C)
${{\mu }_{B}}>{{\mu }_{A}}$ and ${{k}_{e}}(A)>{{k}_{e}}(B)$

D)
${{\mu }_{A}}>{{\mu }_{B}}$ and ${{k}_{e}}(A)>{{k}_{e}}(B)$

• question_answer31) 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 : ethyl acetate (20 : 80), the sequence of obtained compounds is                     [JEE MAIN Held on 07-01-2020 Evening]

A)
(A), (B) and (C)

B)
(C), (A) and (B)

C)
(B), (C) and (A)

D)
(B), (A) and (C)

• question_answer32) The number of possible optical isomers for the complexes $M{{A}_{2}}{{B}_{2}}$ with $s{{p}^{3}}$ and $ds{{p}^{2}}$hybridized metal atom, respectively, is Note: A and B are unidentate neutral and unidentate monoanionic ligands, respectively. [JEE MAIN Held on 07-01-2020 Evening]

A)
2 and 2

B)
0 and 2

C)
0 and 1

D)
0 and 0

• question_answer33) Which of the following statements is correct?                  [JEE MAIN Held on 07-01-2020 Evening]

A)
Gluconic acid is a dicarboxylic acid

B)
Gluconic acid can form cyclic (acetal/ hemiacetal) structure

C)
Gluconic acid is a partial oxidation product of glucose

D)
Gluconic acid is obtained by oxidation of glucose with $HN{{O}_{3}}$

• question_answer34) The bond order and the magnetic characteristics of CN- are [JEE MAIN Held on 07-01-2020 Evening]

A)
$2\frac{1}{2}$, paramagnetic

B)
3, diamagnetic

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

D)
3, paramagnetic

• question_answer35)  Among statements (A)-(D) the correct ones are: (A) Decomposition of hydrogen peroxide gives dioxygen. (B) Like hydrogen peroxide, compounds, such as $KCl{{O}_{3}},Pb\,{{(N{{O}_{3}})}_{2}}$ and $NaN{{O}_{3}}$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.
[JEE MAIN Held on 07-01-2020 Evening]

A)
(A) and (C) only

B)
(A), (B) and (C) only

C)
(A), (B), (C) and (D)

D)
(A), (C) and (D) only

• question_answer36) Two open beakers one containing a solvent and the other containing a mixture of that solvent with a nonvolatile solute are together sealed in a container. Over time: [JEE MAIN Held on 07-01-2020 Evening]

A)
The volume of the solution and the solvent does not change

B)
The volume of the solution increases and the volume of the solvent decreases

C)
The volume of the solution does not change and the volume of the solvent decreases

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

• question_answer37) The redox reaction among the following is:                  [JEE MAIN Held on 07-01-2020 Evening]

A)
Reaction of $[Co{{({{H}_{2}}O)}_{6}}]C{{l}_{3}}$with $AgN{{O}_{3}}$

B)
Formation of ozone from atmospheric oxygen in the presence of sunlight.

C)
Combination of dinitrogen with dioxygen at 2000 K

D)
Reaction of ${{H}_{2}}S{{O}_{4}}$with NaOH

• question_answer38) In the following reaction sequence:  the major product B is:     [JEE MAIN Held on 07-01-2020 Evening]

A)

B)

C)

D)

• question_answer39) Within each pair of elements F & Cl, S & Se, and Li & Na, respectively, the elements that release more energy upon an electron gain are:                                   [JEE MAIN Held on 07-01-2020 Evening]

A)
F, S and Li

B)
F, Se and Na

C)
Cl, S and Li

D)
Cl, Se and Na

• question_answer40) Identify the correct labels of A, B and C in the following graph from the options given below: Root mean square speed$({{V}_{rms}})$; most probable speed$({{V}_{mp}})$; Average speed $({{V}_{av}})$ [JEE MAIN Held on 07-01-2020 Evening]

A)
$A{{V}_{av}};\text{ }B{{V}_{rms}};\text{ }C{{V}_{mp}}$

B)
$A{{V}_{rms}};\text{ }B{{V}_{mp}};C{{V}_{av}}$

C)
$A{{V}_{mp}};\text{ }B{{V}_{rms}};\text{ }C{{V}_{av}}$

D)
$A{{V}_{mp}};\,B{{V}_{av}};\,C{{V}_{rms}}$

• question_answer41) The correct order of stability for the following alkoxides is                     [JEE MAIN Held on 07-01-2020 Evening]

A)
(B) > (A) > (C)

B)
(C) > (B) > (A)

C)
(B) > (C) > (A)

D)
(C) > (A) > (B)

• question_answer42) The equation that is incorrect is                                     [JEE MAIN Held on 07-01-2020 Evening]

A)
${{(\Lambda _{m}^{{}^\circ })}_{NaBr}}-(\Lambda _{m}^{{}^\circ }){{\,}_{NaCl}}={{(\Lambda _{m}^{{}^\circ })}_{kBr}}-{{(\Lambda _{m}^{{}^\circ })}_{KCl}}$

B)
${{(\Lambda _{m}^{{}^\circ })}_{{{H}_{2}}O}}={{(\Lambda _{m}^{{}^\circ })}_{\,HCl}}+{{(\Lambda _{m}^{{}^\circ })}_{NaO{{H}^{-}}}}{{(\Lambda _{m}^{{}^\circ })}_{NaCl}}$

C)
${{(\Lambda _{m}^{{}^\circ })}_{NaBr}}-(\Lambda _{m}^{{}^\circ }){{\,}_{NaI}}={{(\Lambda _{m}^{{}^\circ })}_{kBr}}-{{(\Lambda _{m}^{{}^\circ })}_{NaBr}}$

D)
${{(\Lambda _{m}^{{}^\circ })}_{KCl}}-(\Lambda _{m}^{{}^\circ }){{\,}_{NaCl}}={{(\Lambda _{m}^{{}^\circ })}_{kBr}}-{{(\Lambda _{m}^{{}^\circ })}_{NaBr}}$

• question_answer43) In the following reaction sequence, structures of A and B, respectively will be  [JEE MAIN Held on 07-01-2020 Evening]

A)

B)

C)

D)

• question_answer44) The refining method used when the metal and the impurities have low and high melting temperatures, respectively, is                                                                      [JEE MAIN Held on 07-01-2020 Evening]

A)
Zone refining

B)
Vapour phase refining

C)
Liquation

D)
Distillation

• question_answer45) The ammonia $(N{{H}_{3}})$released on quantitative reaction of 0.6 g urea $(N{{H}_{2}}CON{{H}_{2}})$with sodium hydroxide (NaOH) can be neutralized by [JEE MAIN Held on 07-01-2020 Evening]

A)
200 ml of 0.4 N HCl

B)
100 ml of 0.1 N HCl

C)
200 ml of 0.2 N HCl

D)
100 ml of 0.2 N HCl

• question_answer46)  Consider the following reactions: $NaCl+{{K}_{2}}C{{r}_{2}}{{O}_{7}}+\underset{(Conc.)}{\mathop{{{H}_{2}}S{{O}_{4}}}}\,\to (A)+side\,products$ $(A)+NaOH\to (B)+Side\,products$ $(B)+\underset{(dilute)}{\mathop{{{H}_{2}}S{{O}_{4}}}}\,+{{H}_{2}}{{O}_{2}}\to \text{(}C\text{)}+Side\text{ }products$ The sum of the total number of atoms in one molecule each of (A), and is _____.
[JEE MAIN Held on 07-01-2020 Evening]

• question_answer47) The flocculation value of HCl for arsenic sulphide sol. is$30\text{ }m\text{ }mol\text{ }{{L}^{1}}$. If ${{H}_{2}}S{{O}_{4}}$is used for the flocculation of arsenic sulphide, the amount, in grams, of ${{H}_{2}}S{{O}_{4}}$ in 250 ml required for the above purpose is ______. (molecular mass of ${{H}_{2}}S{{O}_{4}}$ = 98 g/mol) [JEE MAIN Held on 07-01-2020 Evening]

• question_answer48) The number of $s{{p}^{2}}$ hybridised carbons present in "Aspartame" is ______. [JEE MAIN Held on 07-01-2020 Evening]

• question_answer49)  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 solution $\frac{1}{2}\,mL$of 5 M NaOH is added. The pH of the solution is_______. [Given: $p{{K}_{a}}$ of acetic acid = 4.75, molar mass of acetic acid = 60 g/mol, log3 = 0.4771] Neglect any changes in volume.
[JEE MAIN Held on 07-01-2020 Evening]

• question_answer50) The standard heat of formation $({{\Delta }_{f}}H_{298}^{0})$ ethane (in kJ/mol), if the heat of combustion of ethane, hydrogen and graphite are -1560, -393.5 and -286 kJ/mol, respectively is_______.         [JEE MAIN Held on 07-01-2020 Evening]

• question_answer51) The value of c in the Lagrange's mean value theorem for the function $f(x)={{x}^{3}}4{{x}^{2}}+8x+11,$ when $x\in [0,1]$  is [JEE MAIN Held on 07-01-2020 Evening]

A)
$\frac{4-\sqrt{7}}{2}$

B)
$\frac{4-\sqrt{5}}{3}$

C)
$\frac{2}{3}$

D)
$\frac{\sqrt{7}-2}{3}$

• question_answer52) Let y= y(x) be the solution curve of the differential equation, $({{y}^{2}}-x)\frac{dx}{dx}=1,$ satisfying y(0) = 1. This curve intersects the x-axis at a point whose abscissa is [JEE MAIN Held on 07-01-2020 Evening]

A)
2 - e

B)
2 + e

C)
- e

D)
2

• question_answer53) 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 $y\left( \frac{1}{2} \right)=-\frac{1}{4}.$Then $\frac{dy}{dx}$at $x=\frac{1}{2}$, is equal to [JEE MAIN Held on 07-01-2020 Evening]

A)
$-\frac{\sqrt{5}}{2}$

B)
$\frac{2}{\sqrt{5}}$

C)
$\frac{\sqrt{5}}{2}$

D)
$-\frac{\sqrt{5}}{4}$

• question_answer54) Let $A=[{{a}_{ij}}]$ and $B=[{{b}_{ij}}]$ be two $3\times 3$ real matrices such that ${{b}_{ij}}={{(3)}^{(i+j2)}}\,{{a}_{ji}},$where i, j = 1, 2, 3. If the determinant of B is 81, then the determinant of A is         [JEE MAIN Held on 07-01-2020 Evening]

A)
1/9

B)
1/81

C)
3

D)
1/3

• question_answer55) Let $\alpha$ and $\beta$ be the roots of the equation ${{x}^{2}}-x-1=0$.  If ${{p}_{k}}={{(\alpha )}^{k}}+{{(\beta )}^{k}}$, $k\ge 1,$then which one of the following statements is not true? [JEE MAIN Held on 07-01-2020 Evening]

A)
${{p}_{3}}={{p}_{5}}{{p}_{4}}$

B)
$({{p}_{1}}+{{p}_{2}}+{{p}_{3}}+{{p}_{4}}+{{p}_{5}})=26$

C)
${{p}_{5}}=11$

D)
${{p}_{5}}={{p}_{2}}\cdot {{p}_{3}}$

• question_answer56) 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                                                   [JEE MAIN Held on 07-01-2020 Evening]

A)
If $AC$, then $A\subseteq B$ and $B\subseteq D$

B)
If $AC$, then $AB$ and $B\subseteq D$

C)
If $AC$, then $AB$ or $BD$

D)
If $A\subseteq C$, then $B\subset A$ or $D\subset B$

• question_answer57) The area (in sq. units) of the region $\{(x,\text{ }y)\in {{R}^{2}}|4{{x}^{2}}\le y\le 8x+12\}$ is [JEE MAIN Held on 07-01-2020 Evening]

A)
$\frac{128}{3}$

B)
$\frac{125}{3}$

C)
$\frac{127}{3}$

D)
$\frac{124}{3}$

• question_answer58) Let the tangents drawn from the origin to the circle, ${{x}^{2}}+{{y}^{2}}8x4y+16=0$touch it at the points A and B. The ${{(AB)}^{2}}$is equal to    [JEE MAIN Held on 07-01-2020 Evening]

A)
$\frac{64}{5}$

B)
$\frac{52}{5}$

C)
$\frac{56}{5}$

D)
$\frac{32}{5}$

• question_answer59) Let ${{a}_{1}},\,\,{{a}_{2}},\,\,{{a}_{3}}\text{ }...$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 [JEE MAIN Held on 07-01-2020 Evening]

A)
$-\,513$

B)
$-\,171$

C)
$\frac{511}{3}$

D)
$171$

• question_answer60) In a workshop, there are five machines and the probability of any one of them to be out of service on a day is $\frac{1}{4}$. If the probability that at most two machines will be out of service on the same day is ${{\left( \frac{3}{4} \right)}^{3}}$k, then k is equal to                     [JEE MAIN Held on 07-01-2020 Evening]

A)
$4$

B)
$\frac{17}{4}$

C)
$\frac{17}{8}$

D)
$\frac{17}{2}$

• question_answer61) The number of ordered pairs (r, k) for which $6\cdot {{\,}^{35}}{{C}_{r}}=({{k}^{2}}3)\,\cdot {{\,}^{36}}{{C}_{r}}_{+1},$where k is an integer, is [JEE MAIN Held on 07-01-2020 Evening]

A)
3

B)
6

C)
2

D)
4

• question_answer62) The locus of the mid-points of the perpendiculars drawn from points on the line, $x=2y$to the line $x=y$ is [JEE MAIN Held on 07-01-2020 Evening]

A)
$5x7y=0$

B)
$2x3y=0$

C)
$3x2y=0$

D)
$7x5y=0$

• question_answer63) Let f(x) be a polynomial of degree 5 such that $x=\pm 1$ are its critical points. If $\underset{x\to 0}{\mathop{\lim }}\,\left( 2+\frac{f(x)}{{{x}^{3}}} \right)=4$, then which one of the following is not true? [JEE MAIN Held on 07-01-2020 Evening]

A)
f is an odd function

B)
x =1 is a point of minima and $x=1$ is a point of maxima of f.

C)
$f\left( 1 \right)4f\left( 1 \right)=4$

D)
x =1 is a point of maxima and $x=1$ is a point of minimum of f.

• question_answer64) 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, $2\,{{\cot }^{2}}\theta -\frac{5}{\sin \theta }+4=0$, then $\int\limits_{{{\theta }_{1}}}^{{{\theta }_{2}}}{{{\cos }^{2}}\,3\theta \,d\theta },$ is equal to [JEE MAIN Held on 07-01-2020 Evening]

A)
$\frac{\pi }{3}+\frac{1}{6}$

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

C)
$\frac{2\pi }{3}$

D)
$\frac{\pi }{9}$

• question_answer65) Let $\vec{a},\text{ }\vec{b}$ and $\vec{c}$ be three unit vectors such that $\vec{a}+\vec{b}+\vec{c}=\vec{0}.$ If $\lambda \,=\vec{a}.\vec{b}+\vec{b}.\vec{c}+\vec{c}.\vec{a}.$ and $\vec{d}=\vec{a}\times \vec{b}+\vec{b}\times \vec{c}+\vec{c}\times \vec{a}$ then the ordered pair, $\left( \lambda ,\vec{d} \right)$ is equal to                                 [JEE MAIN Held on 07-01-2020 Evening]

A)
$\left( \frac{3}{2},3\vec{a}\times \vec{c} \right)$

B)
$\left( -\frac{3}{2},3\vec{c}\times \vec{b} \right)$

C)
$\left( -\frac{3}{2},3\vec{a}\times \vec{b} \right)$

D)
$\left( -\frac{3}{2},3\vec{b}\times \vec{c} \right)$

• question_answer66) If $3x+4y=12\sqrt{2}$is a tangent to the ellipse $\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{9}=1$for some $a\in R$, then the distance between the foci of the ellipse is                                                             [JEE MAIN Held on 07-01-2020 Evening]

A)
$2\sqrt{5}$

B)
$~2\sqrt{7}$

C)
$4$

D)
$2\sqrt{2}$

• question_answer67) 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                                                                              [JEE MAIN Held on 07-01-2020 Evening]

A)
5

B)
20

C)
25

D)
10

• question_answer68) If $\frac{3+i\sin \theta }{4-i\cos \theta },\,\theta \in [0,\,2\pi ]$, is a real number, then an argument of $\sin \,\theta +\,i\cos \theta$ [JEE MAIN Held on 07-01-2020 Evening]

A)
$\pi -{{\tan }^{-1}}\left( \frac{3}{4} \right)$

B)
$\pi -{{\tan }^{-1}}\left( \frac{4}{3} \right)$

C)
$-{{\tan }^{-1}}\left( \frac{3}{4} \right)$

D)
${{\tan }^{-1}}\left( \frac{4}{3} \right)$

• question_answer69) The coefficient of ${{x}^{7}}$ in the expression ${{(1+x)}^{10}}+x{{(1+x)}^{9}}+{{x}^{2}}{{(1+x)}^{8}}+\text{ }\ldots \text{ }+\text{ }{{x}^{10}}$ is [JEE MAIN Held on 07-01-2020 Evening]

A)
120

B)
330

C)
420

D)
210

• question_answer70) The value of $\alpha$ for which $4\alpha \int\limits_{-1}^{2}{{{e}^{-\,\alpha \left| x \right|}}}$ dx=5, is [JEE MAIN Held on 07-01-2020 Evening]

A)
${{\log }_{e}}\left( \frac{4}{3} \right)$

B)
${{\log }_{e}}\left( \frac{3}{2} \right)$

C)
${{\log }_{e}}2$

D)
${{\log }_{e}}\sqrt{2}$

• question_answer71)  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 ____________.
[JEE MAIN Held on 07-01-2020 Evening]

• question_answer72) If the foot of the perpendicular drawn from the point (1, 0, 3) on a line passing through $(\alpha ,\text{ }7,\text{ }1)$is $\left( \frac{5}{3},\frac{7}{3},\frac{17}{3} \right),$then $\alpha$ is equal to ____________. [JEE MAIN Held on 07-01-2020 Evening]

• question_answer73) 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 __________.                                               [JEE MAIN Held on 07-01-2020 Evening]

• question_answer74) If the function f defined on $\left( -\frac{1}{3},\frac{1}{3} \right)$ by   is continuous, then k is equal to _________.                  [JEE MAIN Held on 07-01-2020 Evening]

• question_answer75) 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 ________. [JEE MAIN Held on 07-01-2020 Evening]