# Solved papers for JEE Main & Advanced JEE Main Paper Phase-I (Held on 09-1-2020 Morning)

### done JEE Main Paper Phase-I (Held on 09-1-2020 Morning) Total Questions - 75

• question_answer1) A particle moving with kinetic energy E has de Broglie wavelength$\lambda$. If energy $\Delta E$ is added to its energy, the wavelength become$\lambda /2$. Value of $\Delta E$, is:   [JEE MAIN Held on 09-01-2020 Morning]

A)
4E

B)
E

C)
2E

D)
3E

• question_answer2)  Consider a force$\vec{F}=-x\hat{i}+y\hat{j}$. The work done by this force in moving a particle from point A(1, 0) to B(0, 1) along the line segment is: (all quantities are in SI units)
[JEE MAIN Held on 09-01-2020 Morning]

A)
2

B)
1

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

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

• question_answer3) The electric fields of two plane electromagnetic plane waves in vacuum are given by ${{\vec{E}}_{1}}={{E}_{0}}\hat{j}\cos \left( \omega t-kx \right)$ and ${{\vec{E}}_{2}}={{E}_{0}}\hat{k}\cos \left( \omega t-ky \right)$ At t = 0, a particle of chcarge q is at origin with a velocity $\vec{v}=0.8c\hat{j}$ (c is the speed of light in vaccum). The instantaneous force experienced by the particle is:     [JEE MAIN Held on 09-01-2020 Morning]

A)
${{E}_{0}}q\left( 0.4\hat{i}-3\hat{j}+0.8\hat{k} \right)$

B)
${{E}_{0}}q\left( -\,0.8\hat{i}+\hat{j}+\hat{k} \right)$

C)
${{E}_{0}}q\left( 0.8\hat{i}+\hat{j}+0.2\hat{k} \right)$

D)
${{E}_{0}}q\left( 0.8\hat{i}-\hat{j}+0.4\hat{k} \right)$

• question_answer4)  Water flows in a horizontal tube (see figure). The pressure of water changes by 700 $N{{m}^{2}}$ between A and B where the area of cross section are 40 $c{{m}^{2}}$and 20$c{{m}^{2}}$, respectively. Find the rate of flow of water through the tube. (density of water = 1000$kg{{m}^{3}}$)
[JEE MAIN Held on 09-01-2020 Morning]

A)
$3020c{{m}^{3}}/s$

B)
$1810c{{m}^{3}}/s$

C)
$2720c{{m}^{3}}/s$

D)
$2420c{{m}^{3}}/s$

• question_answer5) Two particles of equal mass m have respective initial velocities $u\hat{i}$and u $\left( \frac{\hat{i}+\hat{j}}{2} \right)$.They collide completely inelastically. The energy lost in the process is: [JEE MAIN Held on 09-01-2020 Morning]

A)
$\sqrt{\frac{2}{3}}m{{u}^{2}}$

B)
$\frac{3}{4}m{{u}^{2}}$

C)
$\frac{1}{8}m{{u}^{2}}$

D)
$\frac{1}{3}m{{u}^{2}}$

• question_answer6) Consider two ideal diatomic gases A and B at some temperature T. Molecules of the gas A are rigid, and have a mass m. Molecules of the gas B have an additional vibrational mode, and have a mass$\frac{m}{4}$. The ratio of the specific heats $\left( C_{V}^{A}\,\,\text{and}\,\,C_{V}^{B} \right)$ of gas A and B, respectively is: [JEE MAIN Held on 09-01-2020 Morning]

A)
3 : 5

B)
7 : 9

C)
5 : 7

D)
5 : 9

• question_answer7)  A charged particle of mass 'm' and charge 'q' moving under the influence of uniform electric field $E\hat{i}$ and a uniform magnetic field $B\hat{k}$ follows a trajectory from point P to Q as shown in figure. The velocities at P and Q are respectively, $v\vec{i}$and$-2v\hat{j}$. Then which of the following statements (A, B, C, D) are the correct?           (Trajectory shown is schematic and not to scale) A. $E=\frac{3}{4}\left( \frac{m{{v}^{2}}}{qa} \right)$ B. Rate of work done by the electric field at P is $\frac{3}{4}\left( \frac{m{{v}^{3}}}{a} \right)$ C. Rate of work done by both the fields at Q is zero D. The difference between the magnitude of angular momentum of the particle at P and Q is 2 mav.
[JEE MAIN Held on 09-01-2020 Morning]

A)
A, B, C

B)
A, C, D

C)
A, B, C, D

D)
B, C, D

• question_answer8) If the screw on a screw-gauge is given six rotations, it moves by 3 mm on the main scale. If there are 50 divisions on the circular scale the least count of the screw gauge is:              [JEE MAIN Held on 09-01-2020 Morning]

A)
0.01 cm

B)
0.001 mm

C)
0.001 cm

D)
0.02 mm

• question_answer9)  In the given circuit diagram, a wire is joining points B and D. The current in this wire is:
[JEE MAIN Held on 09-01-2020 Morning]

A)
0.4 A

B)
4 A

C)
2 A

D)
zero

• question_answer10) The aperture diameter of a telescope is 5 m. The separation between the moon and the earth is$4\times {{10}^{5}}\text{ }km$. With light of wavelength of 5500$\overset{\text{o}}{\mathop{\text{A}}}\,$, the minimum separation between objects on the surface of moon, so that they are just resolved, is close to:      [JEE MAIN Held on 09-01-2020 Morning]

A)
20 m

B)
200 m

C)
600 m

D)
60 m

• question_answer11)  Three solid spheres each of mass m and diameter d are stuck together such that the lines connecting the centres form an equilateral triangle of side of length d. The ratio ${{I}_{0}}/{{I}_{A}}$ of moment of inertia ${{I}_{0}}$of the system about an axis passing the centroid and about center of any of the spheres ${{I}_{A}}$ and perpendicular to the plane of the triangle is:
[JEE MAIN Held on 09-01-2020 Morning]

A)
$\frac{23}{13}$

B)
$\frac{15}{13}$

C)
$\frac{13}{15}$

D)
$\frac{13}{23}$

• question_answer12) Consider a sphere of radius R which carries a uniform charge density$\rho$. If a sphere of radius $\frac{R}{2}$is carved out of it, as shown, the ratio $\frac{\left| {{{\vec{E}}}_{A}} \right|}{\left| {{{\vec{E}}}_{B}} \right|}$ of magnitude of electric field ${{\vec{E}}_{A}}$ and ${{\vec{E}}_{B}}$respectively, at points A and B due to the remaining portion is: [JEE MAIN Held on 09-01-2020 Morning]

A)
$\frac{18}{34}$

B)
$\frac{18}{54}$

C)
$\frac{21}{34}$

D)
$\frac{9}{17}$

• question_answer13) Radiation, with wavelength 6561$\overset{\text{o}}{\mathop{\text{A}}}\,$ falls on a metal surface to produce photoelectrons. The electrons are made to enter a uniform magnetic field of $3\text{ }\times \text{ }{{10}^{4}}$T. If the radius of the largest circular path followed by the electrons is 10 mm, the work function of the metal is close to       [JEE MAIN Held on 09-01-2020 Morning]

A)
1.6 eV

B)
1.1 eV

C)
0.8 eV

D)
1.8 Ev

• question_answer14) A long straight wire of radius a carries a current distributed uniformly over its cross-section. The ratio of the magnetic fields due to the wire at distance $\frac{a}{3}$ and 2a, respectively from the axis of the wire is [JEE MAIN Held on 09-01-2020 Morning]

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

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

C)
$2$

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

• question_answer15) A body A of mass m is moving in a circular orbit of radius R about a planet. Another body B of mass $\frac{m}{2}$ collides with A with a velocity which is half $\left( \frac{{\vec{v}}}{2} \right)$ the instantaneous velocity $\vec{v}$of A. The collision is completely inelastic. Then the combined body                             [JEE MAIN Held on 09-01-2020 Morning]

A)
Escapes from the Planet's Gravitational field

B)
Continues to move in a circular orbit

C)
Falls vertically downwards towards the planet

D)
Starts moving in an elliptical orbit around the planet

• question_answer16) A vessel of depth 2h is half filled with a liquid of refractive index $2\sqrt{2}$ and the upper half with another liquid of refractive index$\sqrt{2}$. The liquids are immiscible. The apparent depth of the inner surface of the bottom of vessel will be                                       [JEE MAIN Held on 09-01-2020 Morning]

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

B)
$\frac{h}{2\left( \sqrt{2}+1 \right)}$

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

D)
$\frac{3}{4}h\sqrt{2}$

• question_answer17) An electric dipole of moment $\vec{p}-=\left( -\hat{i}-3\hat{j}+2\hat{k} \right)\times {{10}^{-29}}$C.m is at the origin (0, 0, 0). The electric field due to this dipole at $\vec{r}=+\hat{i}+3\hat{j}+5\hat{k}$(note that $\vec{r}.\vec{p}=0$) is paralled to                     [JEE MAIN Held on 09-01-2020 Morning]

A)
$\left( -\hat{i}-3\hat{j}+2\hat{k} \right)$

B)
$\left( +\hat{i}-3\hat{j}-2\hat{k} \right)$

C)
$\left( -\hat{i}+3\hat{j}-2\hat{k} \right)$

D)
$\left( +\hat{i}+3\hat{j}-2\hat{k} \right)$

• question_answer18) A quantity f is given by $f=\sqrt{\frac{h{{c}^{5}}}{G}}$ where c is speed of light, G universal gravitational constant and h is the Planck's constant. Dimension of f is that of                            [JEE MAIN Held on 09-01-2020 Morning]

A)
Energy

B)
Area

C)
Volume

D)
Momentum

• question_answer19) Three harmonic waves having equal frequency $\nu$and same intensity${{I}_{0}}$, have phase angles $0,\frac{\pi }{4}$ and $-\frac{\pi }{4}$ respectively. When they are superimposed the intensity of the resultant wave is close to [JEE MAIN Held on 09-01-2020 Morning]

A)
$3{{I}_{0}}$

B)
$~5.8{{I}_{0}}$

C)
$0.2{{I}_{0}}$

D)
${{I}_{0}}$

• question_answer20)  Which of the following is an equivalent cyclic process corresponding to the thermodynamic cyclic given in the figure? where, $1\to 2$ is adiabatic. (Graphs are schematic and are not to scale)
[JEE MAIN Held on 09-01-2020 Morning]

A)

B)

C)

D)

• question_answer21) The distance x covered by a particle in one dimensional motion varies with time t as ${{x}^{2}}\text{ }=\text{ }a{{t}^{2}}\text{ }+\text{ }2bt\text{ }+\text{ }c$. If the acceleration of the particle depends on x as${{x}^{n}}$, where n is an integer, the value of n is _____.      [JEE MAIN Held on 09-01-2020 Morning]

• question_answer22) In a fluorescent lamp choke (a small transformer) 100 V of reverse voltage is produced when the choke current changes uniformly from 0.25 A to 0 in a duration of 0.025 ms. The self-inductance of the choke (in mH) is estimated to be _______ [JEE MAIN Held on 09-01-2020 Morning]

• question_answer23) One end of a straight uniform 1 m long bar is pivoted on horizontal table. It is released from rest when it makes an angle $30{}^\circ$ from the horizontal (see figure). Its angular speed when it hits the table is given as $\sqrt{n}$${{s}^{-1}}$, where n is an integer. The value of n is ______                               [JEE MAIN Held on 09-01-2020 Morning]

• question_answer24) Both the diodes used in the circuit shown are assumed to be ideal and have negligible resistance when these are forward biased. Built in potential in each diode is 0.7 V. For the input voltages shown in the figure, the voltage (in Volts) at point A is ___________ [JEE MAIN Held on 09-01-2020 Morning]

• question_answer25) A body of mass m = 10 kg is attached to one end of a wire of length 0.3 m. The maximum angular speed (in rad ${{s}^{-1}}$) with which it can be rotated about its other end in space station in (Breaking stress of wire =$4.8\text{ }\times \text{ }{{10}^{7}}\text{ }N{{m}^{2}}$ and area of cross-section of the wire = ${{10}^{2}}\text{ }c{{m}^{2}}$) is [JEE MAIN Held on 09-01-2020 Morning]

• question_answer26)  ${{\left[ Pd\left( F \right)\left( Cl \right)\left( Br \right)\left( I \right) \right]}^{2}}$ has n number of geometrical isomers. Then, the spin-only magnetic moment and crystal field stabilisation energy [CFSE] of ${{\left[ Fe{{\left( CN \right)}_{6}} \right]}^{n\text{ }\text{ }6}}$, respectively, are [Note: Ignore the pairing energy]
[JEE MAIN Held on 09-01-2020 Morning]

A)
0 BM and -2.4 ${{\Delta }_{0}}$

B)
5.92 BM and 0

C)
1.73 BM and -2.0 ${{\Delta }_{0}}$

D)
2.84 BM and -1.6 ${{\Delta }_{0}}$

• question_answer27) If enthalpy of atomisation for $B{{r}_{2\left( I \right)}}$ is x kJ/mol and bond enthalpy for $B{{r}_{2}}$is y kJ/mol, the relation between them [JEE MAIN Held on 09-01-2020 Morning]

A)
is x > y

B)
does not exist

C)
is x = y

D)
is x < y

• question_answer28)  The major product Z obtained in the following reaction scheme is
[JEE MAIN Held on 09-01-2020 Morning]

A)

B)

C)

D)

• question_answer29) Complex X of composition $Cr{{\left( {{H}_{2}}O \right)}_{6}}C{{l}_{n}}$ has a spin only magnetic moment of 3.83 BM. It reacts with $AgN{{O}_{3}}$ and shows geometrical isomerism. The IUPAC nomenclature of X is [JEE MAIN Held on 09-01-2020 Morning]

A)

B)
Hexaaqua chromium (III) chloride

C)

D)
Dichloridotetraaqua chromium (IV) chloride dehydrate

• question_answer30) The electronic configurations of bivalent europium and trivalent cerium are (atomic number: Xe = 54, Ce = 58, Eu = 63)  [JEE MAIN Held on 09-01-2020 Morning]

A)
[Xe] $4{{f}^{7}}$and [Xe] $4{{f}^{1}}$

B)
[Xe] $4{{f}^{7}}$$6{{s}^{2}}$ and [Xe] $4{{f}^{2}}\text{ }6{{s}^{2}}$

C)
[Xe] $4{{f}^{2}}$and [Xe] $4{{f}^{7}}$

D)
[Xe] $4{{f}^{4}}$and [Xe] $4{{f}^{9}}$

• question_answer31) The de Broglie wavelength of an electron in the ${{4}^{th}}$ Bohr orbit is [JEE MAIN Held on 09-01-2020 Morning]

A)
$4\pi {{a}_{0}}$

B)
$6\pi {{a}_{0}}$

C)
$8\pi {{a}_{0}}$

D)
$2\pi {{a}_{0}}$

• question_answer32)  Identify in the following reaction sequence.
[JEE MAIN Held on 09-01-2020 Morning]

A)

B)

C)

D)

• question_answer33)  The major product (Y) in the following reaction is
[JEE MAIN Held on 09-01-2020 Morning]

A)
$C{{H}_{3}}-\overset{C{{H}_{3}}}{\mathop{\overset{|}{\mathop{CH}}\,}}\,-\underset{C{{H}_{3}}}{\mathop{\underset{|}{\mathop{C}}\,}}\,=CH-C{{H}_{3}}$

B)
$C{{H}_{3}}-\overset{C{{H}_{3}}}{\mathop{\overset{|}{\mathop{C}}\,}}\,=\underset{C{{H}_{2}}C{{H}_{3}}}{\mathop{\underset{|}{\mathop{C}}\,}}\,-C{{H}_{3}}$

C)
${{H}_{3}}C-\overset{C{{H}_{2}}}{\mathop{\overset{||}{\mathop{C}}\,}}\,-\underset{{{C}_{2}}{{H}_{5}}}{\mathop{\underset{|}{\mathop{CH}}\,}}\,-C{{H}_{3}}$

D)
$C{{H}_{3}}-\overset{C{{H}_{3}}}{\mathop{\overset{|}{\mathop{CH}}\,}}\,-\underset{C{{H}_{2}}C{{H}_{3}}}{\mathop{\underset{|}{\mathop{C}}\,}}\,=C{{H}_{2}}$

• question_answer34)  A chemist has 4 samples of artificial sweetener A, B, C and D. To identify these samples, he performed certain experiments and noted the following observations : (i) A and D both form blue-violet colour with ninhydrin. (ii) Lassaigne extract of C gives positive $AgN{{O}_{3}}$  test and negative $F{{e}_{4}}{{\left[ Fe{{\left( CN \right)}_{6}} \right]}_{3}}$test. (iii) Lassaigne extract of B and D gives positive sodium nitroprusside test. Based on these observations which option is correct?
[JEE MAIN Held on 09-01-2020 Morning]

A)
 A : Aspartame; B : Alitame; C : Saccharin; D : Sucralose

B)
 A : Saccharin; B : Alitame; C : Sucralose; D : Aspartame

C)
 A : Alitame; B : Saccharin; C : Aspartame; D : Sucralose

D)
 A : Aspartame; B : Saccharin; C : Sucralose; D : Alitame

• question_answer35) If the magnetic moment of a dioxygen species is 1.73 B.M, it may be   [JEE MAIN Held on 09-01-2020 Morning]

A)
$O_{2}^{-}$ or $O_{2}^{+}$

B)
${{O}_{2}},O_{2}^{-}$ or $O_{2}^{+}$

C)
${{O}_{2}}$ or $O_{2}^{+}$

D)
${{O}_{2}}$ or $O_{2}^{-}$

• question_answer36)  The increasing order of basicity for the following intermediates is (from weak to strong)
[JEE MAIN Held on 09-01-2020 Morning]

A)
(v) < (i) < (iv) < (ii) < (iii)

B)
(iii) < (iv) < (ii) < (i) < (v)

C)
(v) < (iii) < (ii) < (iv) < (i)

D)
(iii) < (i) < (ii) < (iv) < (v)

• question_answer37) According to the following diagram, A reduces $B{{O}_{2}}$when the temperature is [JEE MAIN Held on 09-01-2020 Morning]

A)
$>\text{ }1400{}^\circ C$

B)
$>\text{ }1200{}^\circ C$ but $<\text{ }1400{}^\circ C$

C)
$<\text{ }1200{}^\circ C$

D)
$<\text{ }1400{}^\circ C$

• question_answer38) The acidic, basic and amphoteric oxides, respectively, are [JEE MAIN Held on 09-01-2020 Morning]

A)
$N{{a}_{2}}O,\text{ }S{{O}_{3}},\text{ }A{{l}_{2}}{{O}_{3}}$

B)
$C{{l}_{2}}O,\text{ }CaO,\text{ }{{P}_{4}}{{O}_{10}}$

C)
$MgO,\text{ }C{{l}_{2}}O,\text{ }A{{l}_{2}}{{O}_{3}}$

D)
${{N}_{2}}{{O}_{3}},\text{ }L{{i}_{2}}O,\text{ }A{{l}_{2}}{{O}_{3}}$

• question_answer39) Which of these will produce the highest yield in Friedel Crafts reaction?

A)

B)

C)

D)

• question_answer40) The ${{K}_{sp}}$ for the following dissociation is $1.6\text{ }\times \text{ }{{10}^{5}}$ $PbC{{l}_{2\left( s \right)}}\rightleftharpoons Pb_{\left( aq \right)}^{2+}+2Cl_{\left( aq \right)}^{-}$ Which of the following choices is correct for a mixture of 300 mL 0.134 M $Pb{{\left( N{{O}_{3}} \right)}_{2}}$and 100 mL 0.4 M NaCl? [JEE MAIN Held on 09-01-2020 Morning]

A)
$Q\text{ }<\text{ }{{K}_{sp}}$

B)
$Q\text{ }=\text{ }{{K}_{sp}}$

C)
Not enough data provided

D)
$Q\text{ }>\text{ }{{K}_{sp}}$

• question_answer41) The compound that cannot act both as oxidising and reducing agent is  [JEE MAIN Held on 09-01-2020 Morning]

A)
${{H}_{3}}P{{O}_{4}}$

B)
${{H}_{2}}S{{O}_{3}}$

C)
${{H}_{2}}{{O}_{2}}$

D)
$HN{{O}_{2}}$

• question_answer42) 'X' melts at low temperature and is a bad conductor of electricity in both liquid and solid state. X is [JEE MAIN Held on 09-01-2020 Morning]

A)
Zinc sulphide

B)
Carbon tetrachloride

C)
Mercury

D)
Silicon carbide

• question_answer43)  B has a smaller first ionization enthalpy than Be. Consider the following statements. (I) It is easier to remove 2p electron than 2s electron (II) 2p electron of B is more shielded from the nucleus by the inner core of electrons than the 2s electrons of Be (III) 2s electron has more penetration power than 2p electron (IV) Atomic radius of B is more than Be (atomic number B = 5, Be = 4) The correct statements are
[JEE MAIN Held on 09-01-2020 Morning]

A)
(I), (II) and (IV)

B)
(I), (III) and (IV)

C)
(I), (II) and (III)

D)
(II), (III) and (IV)

• question_answer44)  For following reactions $A\xrightarrow{700K}$ Product $A\xrightarrow[catalyst]{500K}$ Product it was found that the ${{E}_{a}}$is decreased by 30 kJ/ mol in the presence of catalyst. If the rate remains unchanged, the activation energy for catalysed reaction is (Assume pre exponential factor is same)
[JEE MAIN Held on 09-01-2020 Morning]

A)
75 kJ/mol

B)
198 kJ/mol

C)
105 kJ/mol

D)
135 kJ/mol

• question_answer45) The correct order of heat of combustion for following alkadienes is [JEE MAIN Held on 09-01-2020 Morning]

A)
(c) < (b) < (a)

B)
(a) < (c) < (b)

C)
(b) < (c) < (a)

D)
(a) < (b) < (c)

• question_answer46) How much amount of NaCl should be added to 600 g of water $\left( \rho =1.00\text{ }g/mL \right)$ to decrease the freezing point of water to $\,0.2{}^\circ C$? ________. (The freezing point depression constant for water$=\text{ }2\text{ }K\text{ }kg\text{ }mo{{l}^{1}}$) [JEE MAIN Held on 09-01-2020 Morning]

• question_answer47) The molarity of $HN{{O}_{3}}$in a sample which has density 1.4 g/mL and mass percentage of 63% is _____. (Molecular Weight of $HN{{O}_{3}}$= 63)                                                [JEE MAIN Held on 09-01-2020 Morning]

• question_answer48) The hardness of a water sample containing ${{10}^{-3}}M\,\,\,MgS{{O}_{4}}$ expressed as $CaC{{O}_{3}}$equivalents (in ppm) is _____. (molar mass of $MgS{{O}_{4}}$ is$120.37\text{ }g/mol$)                       [JEE MAIN Held on 09-01-2020 Morning]

• question_answer49) $108\text{ }g$of silver (molar mass $108\text{ }g\text{ }mo{{l}^{1}}$) is deposited at cathode from $AgN{{O}_{3}}(aq)$ solution by a certain quantity of electricity. The volume (in L) of oxygen gas produced at 273 K and 1 bar pressure from water by the same quantity of electricity is ______.         [JEE MAIN Held on 09-01-2020 Morning]

• question_answer50) The mass percentage of nitrogen in histamine is _____.                    [JEE MAIN Held on 09-01-2020 Morning]

• question_answer51) Let z be a complex number such that $\left| \frac{z-i}{z+2i} \right|=1$ and $|z|=\frac{5}{2}.$Then the value of $|z+3i|$ is [JEE MAIN Held on 09-01-2020 Morning]

A)
$2\sqrt{3}$

B)
$\frac{7}{2}$

C)
$\sqrt{10}$

D)
$\frac{15}{4}$

• question_answer52)  If for some $\alpha$ and $\beta$ in R, the intersection of the following three planes $x+4y-2z=1$ $x+7y-5z=\beta$ $x+5y+\alpha z=5$ is a line in ${{R}^{3}},$ then $\alpha +\beta$ is equal to
[JEE MAIN Held on 09-01-2020 Morning]

A)
$-10$

B)
$0$

C)
$2$

D)
10

• question_answer53) A circle touches the y-axis at the point $(0,4)$ and passes through the point. Which of the following lines is not a tangent to this circle?                                                    [JEE MAIN Held on 09-01-2020 Morning]

A)
$3x-4y-24=0$

B)
$4x-3y+17=0$

C)
$4x+3y-8=0$

D)
$3x+4y-6=0$

• question_answer54) In a box, there are 20 cards, out of which 10 are labelled as A and the remaining 10 are labelled as B. Cards are drawn at random, one after the other and with replacement, till a second A-card is obtained. The probability that the second A-card appears before the third B-card is                          [JEE MAIN Held on 09-01-2020 Morning]

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

B)
$\frac{13}{16}$

C)
$\frac{11}{16}$

D)
$\frac{15}{16}$

• question_answer55) Let f be any function continuous on $[a,b]$and twice differentiable on $(a,b)$. If for all $x\in (a,b),$ $f'(x)>0$ and $f''(x)<0,$ then for any $c\in (a,b),\frac{f(c)-f(a)}{f(b)-f(c)}$is greater than                   [JEE MAIN Held on 09-01-2020 Morning]

A)
$1$

B)
$\frac{b+a}{b-a}$

C)
$\frac{c-a}{b-c}$

D)
$\frac{b-c}{c-a}$

• question_answer56) The number of real roots of the equation, ${{e}^{4x}}+{{e}^{3x}}-4{{e}^{2x}}+{{e}^{x}}+1=0$ is           [JEE MAIN Held on 09-01-2020 Morning]

A)
4

B)
2

C)
3

D)
1

• question_answer57) If ${{e}_{1}}$ and ${{e}_{2}}$ are the eccentricities of the ellipse, $\frac{{{x}^{2}}}{18}+\frac{{{y}^{2}}}{4}=1$and the hyperbola, $\frac{{{x}^{2}}}{9}-\frac{{{y}^{2}}}{4}=1$ respectively and $({{e}_{1}},{{e}_{2}})$is a point on the ellipse, $15{{x}^{2}}+3{{y}^{2}}=k,$then k is equal to                     [JEE MAIN Held on 09-01-2020 Morning]

A)
14

B)
15

C)
17

D)
16

• question_answer58) Let C be the centroid of the triangle with vertices $(3,-1),\,\,(1,3)$ and $(2,4)$. Let P be the point of intersection of the lines $x+3y-1=0$ and$3x-y+1=0$. Then the line passing through the points C and P also passes through the point [JEE MAIN Held on 09-01-2020 Morning]

A)
$(-9,-6)$

B)
$(-9,-7)$

C)
$(9,7)$

D)
$(7,6)$

• question_answer59) The integral $\int{\frac{dx}{{{(x+4)}^{\frac{8}{7}}}{{(x-3)}^{\frac{6}{7}}}}}$ is equal to (where C is a constant of integration) [JEE MAIN Held on 09-01-2020 Morning]

A)
$\frac{1}{2}\,\,{{\left( \frac{x-3}{x+4} \right)}^{3/7}}+C$

B)
$-\frac{1}{13}{{\left( \frac{x-3}{x+4} \right)}^{-13/7}}+C$

C)
${{\left( \frac{x-3}{x+4} \right)}^{1/7}}+C$

D)
$-{{\left( \frac{x-3}{x+4} \right)}^{-1/7}}+C$

• question_answer60) If the matrices  B = adj A and $C=3A,$then $\frac{|adj\,B|}{|C|}$   is equal to [JEE MAIN Held on 09-01-2020 Morning]

A)
16

B)
2

C)
72

D)
8

• question_answer61) The value of $\int\limits_{0}^{2\pi }{\frac{x{{\sin }^{8}}x}{{{\sin }^{8}}x+{{\cos }^{8}}x}}\,\,\,dx$ is equal to [JEE MAIN Held on 09-01-2020 Morning]

A)
${{\pi }^{2}}$

B)
$2\pi$

C)
$2{{\pi }^{2}}$

D)
$4\pi$

• question_answer62) The product ${{2}^{\frac{1}{4}}}.\,{{4}^{\frac{1}{16}}}.\,{{8}^{\frac{1}{48}}}.\,{{16}^{\frac{1}{128}}}.....$ to $\infty$ is equal to                                                         [JEE MAIN Held on 09-01-2020 Morning]

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

B)
${{2}^{\frac{1}{4}}}$

C)
$2$

D)
$1$

• question_answer63) Negation of the statement: ?$\sqrt{5}$ is an integer or 5 is irrational' is          [JEE MAIN Held on 09-01-2020 Morning]

A)
$\sqrt{5}$ is not an integer and 5 is not irrational

B)
$\sqrt{5}$ is an integer and 5 is irrational

C)
$\sqrt{5}$ is not an integer or 5 is not irrational

D)
$\sqrt{5}$ is irrational or 5 is an integer

• question_answer64) Let the observations ${{x}_{i}}(1\le i\le 10)$ satisfy the equations, $\sum\limits_{i=1}^{10}{({{x}_{i}}-5)=10}$  and $\sum\limits_{i=1}^{10}{{{({{x}_{i}}-5)}^{2}}=40}$ If $\mu$ and $\lambda$ are the mean and the variance of the observations, ${{x}_{1}}-3,$ ${{x}_{2}}-3,.....,{{x}_{10}}-3,$ then the ordered pair $(\mu ,\lambda )$is equal to [JEE MAIN Held on 09-01-2020 Morning]

A)
$(6,\,3)$

B)
$(3,\,6)$

C)
$(3,\,3)$

D)
$(6,\,6)$

• question_answer65) If for all real triplets (a, b, c), $f(x)=a+bx+c{{x}^{2}};$then $\int_{0}^{1}{f(x)\,dx}$ is equal to [JEE MAIN Held on 09-01-2020 Morning]

A)
$2\left\{ 3f(1)+2f\left( \frac{1}{2} \right) \right\}$

B)
$\frac{1}{2}\left\{ f(1)+3f\left( \frac{1}{2} \right) \right\}$

C)
$\frac{1}{6}\left\{ f(0)+f(1)+4f\left( \frac{1}{2} \right) \right\}$

D)
$\frac{1}{3}\left\{ f(0)+f\left( \frac{1}{2} \right) \right\}$

• question_answer66) If $f'(x)={{\tan }^{-1}}(\sec x+\tan x),$ $-\frac{\pi }{2}<x<\frac{\pi }{2},$and $f(0)=0,$ then $f(1)$ is equal to [JEE MAIN Held on 09-01-2020 Morning]

A)
$\frac{\pi +1}{4}$

B)
$\frac{1}{4}$

C)
$\frac{\pi +2}{4}$

D)
$\frac{\pi -1}{4}$

• question_answer67)   The value of ${{\cos }^{3}}\left( \frac{\pi }{8} \right).\cos \left( \frac{3\pi }{8} \right)+{{\sin }^{3}}\left( \frac{\pi }{8} \right).\sin \left( \frac{3\pi }{8} \right)$ is   [JEE MAIN Held on 09-01-2020 Morning]

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

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

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

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

• question_answer68) If the number of five digit numbers with distinct digits and 2 at the 10th place is 336 k, then k is equal to [JEE MAIN Held on 09-01-2020 Morning]

A)
8

B)
6

C)
7

D)
4

• question_answer69) If  is continuous at $x=0,$then $a+2b$ is equal to [JEE MAIN Held on 09-01-2020 Morning]

A)
$-1$

B)
$1$

C)
$0$

D)
$-2$

• question_answer70) A spherical iron ball of 10 cm radius is coated with a layer of ice of uniform thickness that melts at a rate of $50\,\,c{{m}^{3}}/\min .$When the thickness of ice is 5 cm, then the rate (in cm/min.) at which of the thickness of ice decreases, is                                                                               [JEE MAIN Held on 09-01-2020 Morning]

A)
$\frac{1}{36\pi }$

B)
$\frac{1}{18\pi }$

C)
$\frac{1}{54\pi }$

D)
$\frac{5}{6\pi }$

• question_answer71) The projection of the line segment joining the points $(1,-1,3)$ and $(2,-4,11)$ on the line joining the points $(-1,2,3)$ and $(3,-2,10)$ is_______.     [JEE MAIN Held on 09-01-2020 Morning]

• question_answer72) The coefficient of ${{x}^{4}}$ in the expansion of ${{(1+x+{{x}^{2}})}^{10}}$is _______. [JEE MAIN Held on 09-01-2020 Morning]

• question_answer73) If   the   vectors,  $\overset{\to }{\mathop{p}}\,=(a+1)\hat{i}+a\hat{j}+a\hat{k},$ $\overset{\to }{\mathop{q}}\,=a\,\,\hat{i}+(a+1)\hat{j}+a\hat{k},$ and $\overset{\to }{\mathop{r}}\,=a\,\,\hat{i}+a\hat{j}+(a+1)\hat{k}\,\,(a\in R)$are coplanar and $3{{(\overset{\to }{\mathop{p}}\,.\overset{\to }{\mathop{q}}\,)}^{2}}-\lambda {{\left| \overset{\to }{\mathop{r}}\,\times \overset{\to }{\mathop{q}}\, \right|}^{2}}=0,$then the value of $\lambda$ is           [JEE MAIN Held on 09-01-2020 Morning]

• question_answer74) The number of distinct solutions of the equation, ${{\log }_{\frac{1}{2}}}|\sin x|=2-{{\log }_{\frac{1}{2}}}|\cos x|$ in the interval $[0,\,\,2\pi ],$ is _______. [JEE MAIN Held on 09-01-2020 Morning]

• question_answer75) If for $x\ge 0,$ $y=y(x)$ is the solution of the differential equation, $(x+1)dy=({{(x+1)}^{2}}+y-3)dx,\,\,y(2)=0,$. then $y(3)$ is equal to _______                                  [JEE MAIN Held on 09-01-2020 Morning]