JEE Main 9 January 2020 Shift 2 Question Paper

Question 1

If $x = \sum\limits_{n=0}^{\infty} (-1)^{n} 	an^{2n}	heta$ and $y = \sum\limits_{n=0}^{\infty} \cos^{2n}	heta$, for $0 < 	heta < \dfrac{\pi}{4}$, then:

Accepted Answer

(A) $y(1 - x) = 1$

Answer

(B) $y(1 + x) = 1$

Answer

(C) $x(1 + y) = 1$

Answer

(D) $x(1 - y) = 1$

Question 2

If $p \rightarrow (p \wedge \sim q)$ is false, then the truth values of p and q are respectively:

Accepted Answer

(C) T, T

Answer

(A) F, T

Answer

(B) F, F

Answer

(D) T, F

Question 3

Let $a, b \in R, a 
eq 0$ be such that the equation $ax^{2} - 2bx + 5 = 0$ has a repeated root $\alpha$, which is also a root of the equation, $x^{2} - 2bx - 10 = 0$. If $\beta$ is the other root of this equation, then $\alpha^{2} + \beta^{2}$ is equal to:

Accepted Answer

(D) 25

Answer

(A) 28

Answer

(B) 24

Answer

(C) 26

Question 4

Let $[t]$ denote the greatest integer $\leq t$ and $\lim\limits_{x \rightarrow 0} x\left[\dfrac{4}{x}\right] = A$. Then the function, $f(x) = \left[x^{2}\right]\sin(\pi x)$ is discontinuous, when x is equal to:

Accepted Answer

(D) $\sqrt{A + 1}$

Answer

(A) $\sqrt{A + 21}$

Answer

(B) $\sqrt{A + 5}$

Answer

(C) $\sqrt{A}$

Question 5

Let a function $f : [0, 5] \rightarrow R$ be continuous, $f(1) = 3$ and F be defined as: $F(x) = \int\limits_{1}^{x} t^{2} g(t)\,dt$, where $g(t) = \int\limits_{1}^{t} f(u)\,du$. Then for the function F, the point x = 1 is:

Accepted Answer

(A) a point of local minima

Answer

(B) a point of inflection.

Answer

(C) not a critical point

Answer

(D) a point of local maxima

Question 6

A random variable X has the following probability distribution:

Then $P(X > 2)$ is equal to:

Accepted Answer

(C) $\dfrac{23}{36}$

Answer

(A) $\dfrac{1}{36}$

Answer

(B) $\dfrac{7}{12}$

Answer

(D) $\dfrac{1}{6}$

Question 7

If $\int \dfrac{d	heta}{\cos^{2}	heta\,(	an 2	heta + \sec 2	heta)} = \lambda 	an	heta + 2\log_{e}\left|f(	heta)ight| + C$ where C is a constant of integration, then the ordered pair $\left(\lambda,\, f(	heta)ight)$ is equal to:

Accepted Answer

(A) $(-1, 1 + 	an	heta)$

Answer

(B) $(1, 1 - 	an	heta)$

Answer

(C) $(1, 1 + 	an	heta)$

Answer

(D) $(-1, 1 - 	an	heta)$

Question 8

If $\dfrac{dy}{dx} = \dfrac{xy}{x^{2} + y^{2}}$; $y(1) = 1$; then a value of x satisfying $y(x) = e$ is:

Accepted Answer

(C) $\sqrt{3}\,e$

Answer

(A) $\sqrt{2}\,e$

Answer

(B) $\dfrac{1}{2}\sqrt{3}\,e$

Answer

(D) $\dfrac{e}{\sqrt{2}}$

Question 9

Let $a_{n}$ be the $n^{th}$ term of a G.P. of positive terms. If $\sum\limits_{n=1}^{100} a_{2n+1} = 200$ and $\sum\limits_{n=1}^{100} a_{2n} = 100$, then $\sum\limits_{n=1}^{200} a_{n}$ is equal to:

Accepted Answer

(C) 150

Answer

(A) 225

Answer

(B) 300

Answer

(D) 175

Question 10

Accepted Answer

(D) $\sqrt{7}$

Answer

(A) $\sqrt{\dfrac{17}{2}}$

Answer

(B) $\sqrt{10}$

Answer

(C) $\sqrt{8}$

Question 11

Given : $f(x) = \begin{cases} x, & 0 \leq x < \dfrac{1}{2} \[4pt] \dfrac{1}{2}, & x = \dfrac{1}{2} \[4pt] 1 - x, & \dfrac{1}{2} < x \leq 1 \end{cases}$ and $g(x) = \left(x - \dfrac{1}{2}\right)^{2}, x \in R$. Then the area (in sq. units) of the region bounded by the curves, $y = f(x)$ and $y = g(x)$ between the lines $2x = 1$ and $2x = \sqrt{3}$, is:

Accepted Answer

(C) $\dfrac{\sqrt{3}}{4} - \dfrac{1}{3}$

Answer

(A) $\dfrac{1}{3} + \dfrac{\sqrt{3}}{4}$

Answer

(B) $\dfrac{1}{2} + \dfrac{\sqrt{3}}{4}$

Answer

(D) $\dfrac{1}{2} - \dfrac{\sqrt{3}}{4}$

Question 12

If $A = \{x \in R : |x| < 2\}$ and $B = \{x \in R : |x - 2| \ge 3\}$; then:

Accepted Answer

(A) $B - A = R - (-2, 5)$

Answer

(B) $A \cap B = (-2, -1)$

Answer

(C) $A - B = [-1, 2)$

Answer

(D) $A \cup B = R - (2, 5)$

Question 13

Let $a - 2b + c = 1$. If $f(x) = \begin{vmatrix} x + a & x + 2 & x + 1 \ x + b & x + 3 & x + 2 \ x + c & x + 4 & x + 3 \end{vmatrix}$, then:

Accepted Answer

(B) $f(50) = 1$

Answer

(A) $f(-50) = 501$

Answer

(C) $f(50) = -501$

Answer

(D) $f(-50) = -1$

Question 14

Let f and g be differentiable functions on R such that fog is the identity function. If for some $a, b \in R, g'(a) = 5$ and $g(a) = b$, then $f'(b)$ is equal to:

Accepted Answer

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

Answer

(A) $1$

Answer

(B) $5$

Answer

(D) $\frac{2}{5}$

Question 15

If one end of a focal chord AB of the parabola $y^{2} = 8x$ is at $A\left(\frac{1}{2}, -2\right)$, then the equation of the tangent to it at B is:

Accepted Answer

(B) $x - 2y + 8 = 0$

Answer

(A) $2x - y - 24 = 0$

Answer

(C) $2x + y - 24 = 0$

Answer

(D) $x + 2y + 8 = 0$

Question 16

The length of the minor axis (along y − axis) of an ellipse in the standard form is $\frac{4}{\sqrt{3}}$. If this ellipse touches the line, $x + 6y = 8$; then its eccentricity is:

Accepted Answer

(C) $\frac{1}{2}\sqrt{\frac{11}{3}}$

Answer

(A) $\frac{1}{3}\sqrt{\frac{11}{3}}$

Answer

(B) $\frac{1}{2}\sqrt{\frac{5}{3}}$

Answer

(D) $\sqrt{\frac{5}{6}}$

Question 17

If the curves, $x^{2} - 6x + y^{2} + 8 = 0$ and $x^{2} - 8y + y^{2} + 16 - k = 0$, $(k > 0)$ touch each other at a point, then the largest value of k is ______.

Accepted Answer

36

Question 18

If $C_{r} = {}^{25}C_{r}$ and $C_{0} + 5.C_{1} + 9.C_{2} + .......... + (101).C_{25} = 2^{25}.k$, then k is equal to ______

Accepted Answer

51

Question 19

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three vectors such that $|\vec{a}| = \sqrt{3}, |\vec{b}| = 5, \vec{b}.\vec{c} = 10$ and the angle between $\vec{b}$ and $\vec{c}$ is $\frac{\pi}{3}$. If $\vec{a}$ is perpendicular to the vector $\vec{b} 	imes \vec{c}$, then $|\vec{a} 	imes (\vec{b} 	imes \vec{c})|$ is equal to ____.

Accepted Answer

30

Question 20

The number of terms common to the two A.P.'s 3, 7, 11,…………..407 and 2, 9, 16, …..709 is ______

Accepted Answer

14

Question 21

A small spherical droplet of density d is floating exactly half immersed in a liquid of density ρ and surface tension T. The radius of the droplet is (take note that the surface tension applies an upward force on the droplet):

Accepted Answer

(A) $r = \sqrt{\frac{3T}{(2d - \rho)g}}$

Answer

(B) $r = \sqrt{\frac{T}{(d - \rho)g}}$

Answer

(C) $r = \sqrt{\frac{2T}{3(d + \rho)g}}$

Answer

(D) $r = \sqrt{\frac{T}{(d + \rho)g}}$

Question 22

There is a small source of light at some depth below the surface of water (refractive index = $\frac{4}{3}$) in a tank of large cross sectional surface area. Neglecting any reflection from the bottom and absorption by water, percentage of light that emerges out of surface is (nearly):
[Use the fact that surface area of spherical cap of height h and radius of curvature r is 2πrh]

Accepted Answer

(B) 17%

Answer

(A) 21%

Answer

(C) 50%

Answer

(D) 34%

Question 23

A plane electromagnetic wave is propagating along the direction $\frac{\hat{i} + \hat{j}}{\sqrt{2}}$, with its polarization along the direction $\hat{k}$. The correct form of the magnetic field of the wave would be (here $B_{0}$ is an appropriate constant)

Accepted Answer

(C) $B_{0}\frac{\hat{i} - \hat{j}}{\sqrt{2}}\cos\left(\omega t - k\frac{\hat{i} + \hat{j}}{\sqrt{2}}\right)$

Answer

(A) $B_{0}\frac{\hat{j} - \hat{i}}{\sqrt{2}}\cos\left(\omega t + k\frac{\hat{i} + \hat{j}}{\sqrt{2}}\right)$

Answer

(B) $B_{0}\,\hat{k}\cos\left(\omega t - k\frac{\hat{i} + \hat{j}}{\sqrt{2}}\right)$

Answer

(D) $B_{0}\frac{\hat{i} + \hat{j}}{\sqrt{2}}\cos\left(\omega t - k\frac{\hat{i} + \hat{j}}{\sqrt{2}}\right)$

Question 24

A wire of length L and mass per unit length $6.0 	imes 10^{-3}$ kg $\mathrm{m}^{-1}$ is put under tension of 540 N. Two consecutive frequencies that it resonates at are: 420 Hz and 490 Hz. Then L in meters is

Accepted Answer

(C) 2.1 m

Answer

(A) 1.1 m

Answer

(B) 5.1 m

Answer

(D) 8.1 m

Question 25

A particle of mass m is projected with a speed u form the ground at an angle $	heta = \frac{\pi}{3}$ w.r.t. horizontal (x-axis). When it has reached its maximum height, it collides completely inelastically with another particle of the same mass and velocity $u\hat{i}$. The horizontal distance covered by the combined mass before reaching the ground is:

Accepted Answer

(C) $\frac{3\sqrt{3}}{8}\frac{u^{2}}{g}$

Answer

(A) $\frac{5}{8}\frac{u^{2}}{g}$

Answer

(B) $\frac{3\sqrt{2}}{4}\frac{u^{2}}{g}$

Answer

(D) $2\sqrt{2}\frac{u^{2}}{g}$

Question 26

In LC circuit the inductance L = 40 mH and capacitance C = 100 μF. If a voltage V(t) = 10 sin (314 t) is applied to the circuit, the current in the circuit is given as

Accepted Answer

(C) 0.52 cos 314 t

Answer

(A) 5.2 cos 314 t

Answer

(B) 0.52 sin 314 t

Answer

(D) 10 cos 314 t

Question 27

The current i in the network is:

Accepted Answer

(D) 0.3 A

Answer

(A) 0.6 A

Answer

(B) 0 A

Answer

(C) 0.2 A

Question 28

A rod of length L has non-uniform linear mass density given by $\rho(x) = a + b\left(\frac{x}{L}\right)^{2}$, where a and b are constants and $0 \le x \le L$. The value of x for the centre of mass of the rod is at:

Accepted Answer

(C) $\frac{3}{4}\left(\frac{2a + b}{3a + b}\right)L$

Answer

(A) $\frac{3}{2}\left(\frac{a + b}{2a + b}\right)L$

Answer

(B) $\frac{4}{3}\left(\frac{a + b}{2a + 3b}\right)L$

Answer

(D) $\frac{3}{2}\left(\frac{2a + b}{3a + b}\right)L$

Question 29

Two steel wires having same length are suspended from a ceiling under the same load. If the ratio of their energy stored per unit volume is 1 : 4, the ratio of their diameters is

Accepted Answer

(C) $\sqrt{2} : 1$

Answer

(A) $1 : \sqrt{2}$

Answer

(B) $2 : 1$

Answer

(D) $1 : 2$

Question 30

Planet A has mass M and radius R. Plant B has half the mass and half the radius of Planet A. If the escape velocities from the planets A and B are $v_{A}$ and $v_{B}$, respectively, then $\frac{v_{A}}{v_{B}} = \frac{n}{4}$. The value of ‘n’ is:

Accepted Answer

(C) 4

Answer

(A) 2

Answer

(B) 1

Answer

(D) 3

Question 31

The energy required to ionize a hydrogen like ion in its ground state is 9 Rydbergs. What is the wavelength of the radiation emitted when the electron in this ion jumps from the second excited state to the ground state?

Accepted Answer

(A) 11.4 nm

Answer

(B) 35.8 nm

Answer

(C) 24.2 nm

Answer

(D) 8.6 nm

Question 32

An electron of mass m and magnitude of charge $|e|$ initially at rest gets accelerated by a constant electric field E. The rate of change of de-Broglie wavelength of this electron at time t ignoring relativistic effects is

Accepted Answer

(D) $-\frac{h}{|e|Et^{2}}$

Answer

(A) $\frac{|e|Et}{h}$

Answer

(B) $-\frac{h}{|e|Et}$

Answer

(C) $-\frac{h}{|e|E\sqrt{t}}$

Question 33

A particle starts from the origin at t = 0 with an initial velocity of 3.0 $\hat{i}$ m/s and moves in the x-y plane with a constant acceleration $(6.0\hat{i}+4.0\hat{j})$ m/s$^{2}$. The x-coordinate of the particle at the instant when its y-coordinate is 32 m is D meters. The value of D is

Accepted Answer

(B) 60

Answer

(A) 40

Answer

(C) 32

Answer

(D) 50

Question 34

A small circular loop of conducting wire has radius a and carries current I. It is placed in a uniform magnetic field B perpendicular to its plane such that when rotated slightly about its diameter and released, it starts performing simple harmonic motion of time period T. If the mass of the loop is m then:

Accepted Answer

(C) $T=\sqrt{\frac{2\pi m}{IB}}$

Answer

(A) $T=\sqrt{\frac{\pi m}{IB}}$

Answer

(B) $T=\sqrt{\frac{\pi m}{2IB}}$

Answer

(D) $T=\sqrt{\frac{2m}{IB}}$

Question 35

An electron gun is placed inside a long solenoid of radius R on its axis. The solenoid has n turns/length and carries a current I. The electron gun shoots an electron along the radius of the solenoid with speed v. If the electron does not hit the surface of the solenoid, maximum possible value of v is (all symbols have their standard meaning):

Accepted Answer

(D) $\frac{e\mu_{0}nIR}{2m}$

Answer

(A) $\frac{2e\mu_{0}nIR}{m}$

Answer

(B) $\frac{e\mu_{0}nIR}{m}$

Answer

(C) $\frac{e\mu_{0}nIR}{4m}$

Question 36

A spring mass system (mass m, spring constant k and natural length $\ell$) rests in equilibrium on a horizontal disc. The free end of the spring is fixed at the centre of the disc. If the disc together with spring mass system, rotates about it's axis with an angular velocity $\omega$, $(k >> m\omega^{2})$ the relative change in the length of the spring is best given by the option:

Accepted Answer

(B) $\frac{m\omega^{2}}{k}$

Answer

(A) $\frac{m\omega^{2}}{3k}$

Answer

(C) $\frac{2m\omega^{2}}{3k}$

Answer

(D) $\sqrt{\frac{2}{3}}\left(\frac{m\omega^{2}}{k}\right)$

Question 37

In a meter bridge experiment S is a standard resistance. R is a resistance wire. It is found that balancing length is $\ell$ = 25 cm. If R is replaced by a wire of half length and half diameter that of R of same material, then the balancing distance $\ell'$ (in cm) will now be __________.

Accepted Answer

40

Question 38

An electric field $\vec{E}=4x\hat{i}-(y^{2}+1)\hat{j}$ N/C passes through the box shown in figure. The flux of the electric field through surfaces ABCD and BCGF are marked as $\phi_{I}$ and $\phi_{II}$ respectively. The difference between ($\phi_{I}-\phi_{II}$) is (in N m$^{2}$/C) __________.

Accepted Answer

-48

Question 39

In a Young's double's slit experiment 15 fringes are observed on a small portion of the screen when light of wavelength 500 nm is used. Ten fringes are observed on the same section of the screen when another light source of wavelength $\lambda$ is used. Then the value of $\lambda$ is (in nm) __________.

Accepted Answer

750

Question 40

Amongst the following, the form of water with the lowest ionic conductance at 298 K is :

Accepted Answer

(B) distilled water

Answer

(A) sea water

Answer

(C) saline water used for intravenous injection

Answer

(D) water from a well

Question 41

Which polymer has 'chiral' monomer(s)?

Accepted Answer

(C) PHBV

Answer

(A) Nylon 6, 6

Answer

(B) Neoprene

Answer

(D) Buna-N

Question 42

The first and second ionisation enthalpies of a metal are 496 and 4560 kJ mol$^{-1}$, respectively. How many moles of HCl and H$_2$SO$_4$, respectively, will be needed to react completely with 1 mole of the metal hydroxide ?

Accepted Answer

(D) 1 and 0.5

Answer

(A) 1 and 1

Answer

(B) 1 and 2

Answer

(C) 2 and 0.5

Question 43

Among the statements (a) − (d), the correct ones are:
(a) Lithium has the highest hydration enthalpy among the alkali metals.
(b) Lithium chloride is insoluble in pyridine.
(c) Lithium cannot form ethynide upon its reaction with ethyne
(d) Both lithium and magnesium react slowly with H$_2$O

Accepted Answer

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

Answer

(A) (a), (b) and (d) only

Answer

(B) (a) and (d) only

Answer

(C) (b) and (c) only

Question 44

The solubility product of Cr(OH)$_3$ at 298 K is $6.0 	imes 10^{-31}$. The concentration of hydroxide ions in a saturated solution of Cr(OH)$_3$ will be :

Accepted Answer

(A) $(18 	imes 10^{-31})^{1/4}$

Answer

(B) $(4.86 	imes 10^{-29})^{1/4}$

Answer

(C) $(18 	imes 10^{-31})^{1/2}$

Answer

(D) $(2.22 	imes 10^{-31})^{1/4}$

Question 45

5 g of zinc is treated separately with an excess of
(a) dilute hydrochloric acid and
(b) aqueous sodium hydroxide.
The ratio of the volumes of H$_2$ evolved in these two reactions is : -

Accepted Answer

(B) 1 : 1

Answer

(A) 1 : 4

Answer

(C) 1 : 2

Answer

(D) 2 : 1

Question 46

A mixture of gases O$_2$, H$_2$ and CO are taken in a closed vessel containing charcoal. The graph that represents the correct behaviour of pressure with time is:

Accepted Answer

(D) (D)

Answer

(A) (A)

Answer

(B) (B)

Answer

(C) (C)

Question 47

The number of sp$^2$ hybrid orbitals in a molecule of benzene is

Accepted Answer

(A) 18

Answer

(B) 6

Answer

(C) 12

Answer

(D) 24

Question 48

Biochemical Oxygen Demand (BOD) is the amount of oxygen required (in ppm):

Accepted Answer

(C) by bacteria to break-down organic waste in a certain volume of a water sample.

Answer

(A) for the photochemical breakdown of waste present in 1 m$^3$ volume of a water body.

Answer

(B) for sustaining life in a water body.

Answer

(D) by anaerobic bacteria to breakdown inorganic waste present in a water body.

Question 49

A, B and C are three biomolecules. The results of the tests performed on them are given below:
A, B and C are respectively:

Accepted Answer

(B) A = Lactose, B = Glucose, C = Albumin

Answer

(A) A = Lactose, B = Glucose, C = Alanine

Answer

(C) A = Glucose, B = Fructose, C = Albumin

Answer

(D) A = Lactose, B = Fructose, C = Alanine

Question 50

Consider the following reactions:

The compound[P] is

Accepted Answer

(B) (B)

Answer

(A) (A)

Answer

(C) (C)

Answer

(D) (D)

Question 51

Which of the following reactions will not produce a racemic product?

Accepted Answer

(D) (D)

Answer

(A) (A)

Answer

(B) (B)

Answer

(C) (C)

Question 52

Which of the following has the shortest C–Cl bond?

Accepted Answer

(A) Cl – CH = CH – NO$_2$

Answer

(B) Cl – CH = CH – OCH$_3$

Answer

(C) Cl – CH = CH – CH$_3$

Answer

(D) Cl – CH = CH$_2$

Question 53

The decreasing order of basicity of the following amines is

Accepted Answer

(D) (III) > (II) > (I) > (IV)

Answer

(A) (III) > (I) > (II) > (IV)

Answer

(B) (II) > (III) > (IV) > (I)

Answer

(C) (I) > (III) > (IV) > (II)

Question 54

The reaction of H$_3$N$_3$B$_3$Cl$_3$(A) with LiBH$_4$ in tetrahydrofuran gives inorganic benzene (B). Further, the reaction of (A) with (C) leads to H$_3$N$_3$B$_3$(Me)$_3$. Compounds (B) and (C) respectively, are:

Accepted Answer

(D) Borazine and MeMgBr

Answer

(A) Borazine and MeBr

Answer

(B) Boron nitride and MeBr

Answer

(C) Diborane and MeMgBr

Question 55

In the following reaction A is :

Accepted Answer

(B) (B)

Answer

(A) (A)

Answer

(C) (C)

Answer

(D) (D)

Question 56

The isomer (s) of [Co(NH$_3$)$_4$Cl$_2$] that has/have a Cl – Co – Cl angle of 90º, is/are :

Accepted Answer

(B) cis only

Answer

(A) meridional and trans

Answer

(C) trans only

Answer

(D) cis and trans

Question 57

The sum of the total number of bonds between chromium and oxygen atoms in chromate and dichromate  ions is ________________

Accepted Answer

12

Question 58

Consider the following reactions

The mass percentage of carbon in A is____.

Accepted Answer

66.67

Question 59

10.30 mg of O$_2$ is dissolved into a liter of sea water of density 1.03 g/mL. The concentration of O$_2$ in ppm  is________________.

Accepted Answer

10

	Molisch's Test	Barfoed Test	Biuret Test
(A)	Positive	Negative	Negative
(B)	Positive	Positive	Negative
(C)	Negative	Negative	Positive

JEE Main 9 January 2020 Shift 2 Question Paper with Answers

Physics — JEE Main 9 January 2020 Shift 2

Chemistry — JEE Main 9 January 2020 Shift 2

Mathematics — JEE Main 9 January 2020 Shift 2

Chapters tested in this paper

Attempt this paper under exam timing.