JEE ADV 2016 - Amazon S3s3-ap-southeast-1.amazonaws.com/subscriber.images/jee/... · 2018-12-18 ·...

JEE ADV 2016

Byju’s Classes | 7.5 lakh+ likes on . India’s most liked Education Company | Contact: 9900012400

CODE – 1

PAPER – II Time : 3 Hours Maximum Marks : 186

READ THE INSTRUCTIONS CAREFULLY GENERAL

1. This sealed booklet is your Question Paper. Do not break the seal till you are told to do so. 2. The paper CODE is printed on the right hand top corner of this sheet and the right hand top corner of

the back cover of this booklet. 3. Use the Optical Response Sheet (ORS) provided separately for answering the questions. 4. The paper CODE printed on the left part as well as the right part of the ORS. Ensure that both these

codes are identical and same as that on the question paper booklet. If not, contact the invigilator for change of ORS.

5. Blank spaces are provided within this booklet for rough work. 6. Write your name, roll number and sign in the space provided on the back cover of this booklet. 7. After breaking the seal of the booklet at 9:00 am, verify that the booklet contains 36 pages and that all

the 54 questions along with options are legible. If not, contact the invigilator for replacement of the booklet.

8. You are allowed to take away the Question Paper at the end of the examination. OPTICAL REASPONSE SHEET

9. The ORS (top sheet will) be provided with an attached Candidate’s Sheet (bottom sheet). The Candidate’s Sheet is a carbon – less copy of the ORS.

10. Darken the appropriate bubbles of the ORS by applying sufficient pressure. This will leave an impression at the corresponding place on the Candidate’s Sheet.

11. The ORS will be collected by the invigilator at the end of the examination. 12. You will be allowed to take away the Candidate’s sheet at the end of the examination. 13. Do not atamper with or mutilate the ORS. Don not use the ORS. For rough work. 14. Write your name, roll number and code of the examination centre, and sign with pen in the space

provided for this purpose on the ORS. Do not write any of these details any where else on the ORS. Darken the appropriate bubble under each digit of your roll number.

DARKENING THE BUBBLES ON THE ORS

15. Use a BLACK BALL POINT PEN to darken the bubbles on the ORS. 16. Darken the bubble COMPLETELY. 17. The correct way of darkening a bubble is as: 18. The ORS IS machine – gradable. Ensure that the bubbles are darkened in the correct way. 19. Darken the bubble ONLY IF you are sure of the answer. There is NO WAY to erase or “un – darken” a

darkened bubble.

JEE ADV 2016


PART – I PHYSICS

SECTION – I (Maximum Marks: 18)

1. The electrostatic energy of Z protons uniformly distributed throughout a spherical nucleus of radius R is given by

𝐸 =3

5

𝑍(𝑍−1)𝑒2

4𝜋𝜀0𝑅

The measured masses of the neutron, H11 , N7

15 and O815 are 1.008665 u, 1.007825 u, 15.000109 u and

15.003065 u respectively given that the radii of both the N715 and O8

15 nuclei are same, 1 u = 931.5 MeV/c2 (c is the speed of light) and e2/(4πε0) = 1.44 MeV fm. Assuming that the difference between the binding

energies of N715 and O8

15 is purely due to the electrostatic energy, the radius of either of the nuclei is (1 fm = 10–15 m) (A) 2.85 fm (B) 3.03 fm (C) 3.42 fm (D) 3.80 fm Sol: (C) The reaction at hand is 1H

1 + 7O15 = 8O

15 + 0n1

The Q value for the reaction is, Q = [[15.003065 + 1.008665] – [15.000109 + 1.007825]] × 931.5 = 3.5359 MeV The difference in electrostatic energy is,

E = 3

5𝑅 [8 × 7 – 7 × 6] × 1.44 MeV, where R is in fermi,

= 12.096

𝑅 MeV

We have,

Q = E ⇒ R = 12.096

3.5359 fm

= 3.42 fm Correct Option: (C) 2. An accident in a nuclear laboratory resulted in deposition of a certain amount of radioactive material of half- life 18 days inside the laboratory. Tests revealed that the radiation was 64 times more than the permissible level required for safe operation of the laboratory. What is the minimum number of days after which the laboratory can be considered safe for use? (A) 64 (B) 90 (C) 108 (D) 120

JEE ADV 2016


Sol: (C) The laboratory will be considered safe when the number of atoms decreases by a factor of 64

N = 𝑁0

64

= 𝑁0

26

That is after 6 half lifes

t = 6 × 18 = 108 days Correct option: (C) 3. A gas is enclosed in a cylinder with a movable frictionless piston. Its initial thermodynamic state at pressure 𝑃𝑖 = 105 Pa and volume 𝑉𝑖 = 10–3 m3 changes to a final state at 𝑃𝑓 = (1/32) × 105 Pa and 𝑉𝑓 = 8 × 10–3 m3 in

an adiabatic quasi-static process, such that P3V5 = constant. Consider another thermodynamic process that brings the system from the same initial state to the same final state in two steps: an isobaric expansion at 𝑃𝑖 followed by an isochoric (isovolumetric) process at volume 𝑉𝑓. The amount of heat supplied to the system in

the two-step process is approximately (A) 112 J (B) 294 J (C) 588 J (D) 823 J Sol: (C) Let T1 and T2 be the initial and final temperatures for the adiabatic process.

Using PV= nRT

105 × 10–3 = nRT1 ⇒ T1 = 100

𝑛𝑅

1

32 × 105 × 8 × 10–3 = nRT2 ⇒ T2 =

25

𝑛𝑅

∆T = – 75

𝑛𝑅 ⇒ nR∆T = –75

Also PV5/3 = k ⇒ the gas is monoatomic ⇒ CV = 3

2R

∆U = nCV ∆T = n 3

2 R∆T = –

225

2 J

For the isobaric process, Q1 = ∆U1 + w = ∆U1 + P∆v = ∆U1 + 105[8 – 1] × 10–3 = ∆U1 + 700 For the isochoric process, Q2 = ∆U2 Total heat supplied is Q1 + Q2 = [∆U1 +∆U2] + 700

= – 225

2 + 700

JEE ADV 2016


= 1125

2

= 588 J

Correct Option: (C) 4. The ends Q and R of two thin wires, PQ and RS, are soldered (joined) together. Initially each of the wires has a length of 1 m at 10 °C. Now the end P is maintained at 10 °C, while the end S is heated and maintained at 400 °C. The system is thermally insulated from its surroundings. If the thermal conductivity of wire PQ is twice that of the wire RS and the coefficient of linear thermal expansion of PQ is 1.2 × 10–5 K–1, the change in length of the wire PQ is (A) 0.78 mm (B) 0.90 mm (C) 1.56 mm (D) 2.34 mm Sol: (A)

Let T be the temperature of the junction

2𝑘 𝐴(𝑇−10)

1 = 𝐾𝐴(400−𝑇)

1

⇒ 400 – T = 2T – 20

3T = 420 T = 140°C Consider an element dx of the rod at a distance of x from one end of the rod.

We have T = 10 + (140−10

1) x (Since T varies linearly with x)

= 10 + 130x Change in length of the length dx of PQ is, dx1 = dx ∝ (T – 10) dx1 = dx ∝ (130x)

∫𝑑𝑥1 = 130 ∝ ∫ 𝑥𝑑𝑥1

0

x1 = 130 × 1.2 × 10–5 × 1

2

x1 = 0.78 mm Correct Option: (A)

JEE ADV 2016


5. A small object is placed 50 cm to the left of a thin convex lens of focal length 30 cm. A convex spherical mirror of radius of curvature 100 cm is placed to the right of the lens at a distance of 50 cm. The mirror is tilted such that the axis of the mirror is at an angle θ = 30° to the axis of the lens, as shown in the figure.

If the origin of the coordinate system is taken to be at the centre of the lens, the coordinates (in cm) of the point (x, y) at which the image is formed are

(A) (0, 0) (B) (50 – 25√3, 25) (C) (25, 25√3) (D) (125/3, 25√3) Sol: (C)

For the lens,

1

𝑣−1

𝑢 = 1

𝑅

1

𝑣−

1

(−50) =

1

30

⇒ v = 75 cm ⇒ O’I = 25 cm For the mirror,

u1 = 25 cos 30°= 25√3

2 (virtual object)

h1 = 25 sin 30°= 25

2

JEE ADV 2016


1

𝑣1+

1

𝑢1 = 2

𝑅

1

𝑣1+

2

25√3 =

1

50 ⇒ v1 =

50√3

4−√3 (real image)

Also, m = – 𝑣1

𝑢 =

4

4−√3 (erect image)

nd h2 = 4

4−√3 ×

25

2 = 50√3

4−√3

coordinates of the final image = (x, y), where x = 50 – v1 cos 30° + h2 cos 60°

= 50 – 50√3

4−√3 ×

√3

2 +

50

4−√3 ×

1

2

≈ 25 y = v1 sin 30° + h2 sin 60°

= 50√3

4−√3 ×

1

2 +

50

4−√3 ×

√3

2

≈ 25√3

Thus (x, y) ≡ (25, 25√3) Correct option: (C) 6. There are two Vernier calipers both of which have 1 cm divided into 10 equal divisions on the main scale. The Vernier scale of one of the calipers (C1) has 10 equal divisions that correspond to 9 main scale divisions. The Vernier scale of the other caliper (C2) has 10 equal divisions that correspond to 11 main scale divisions. The readings of the two calipers are shown in the figure. The measured values (in cm) by calipers C1 and C2, respectively, are

(A) 2.85 and 2.82 (B) 2.87 and 2.83 (C) 2.87 and 2.86 (D) 2.87 and 2.87

JEE ADV 2016


Sol: (B) For C1

CVD = 7

LC = 1 mm – 9

10 mm = 0.01 cm

Reading = 2.8 + 7 × 0.01 = 2.87 cm For C2 Distance as measured by maiscale upto CVD is = 2.8 cm + 1 mm (8) = 3.6 cm

Distance as measures by the vernier scale is = [2.8 + x] + 11

10 × 7 = 3.57 + x cm

⇒ 3.6 – 3.57 + x ⇒ x = 0.08 cm Thus the reading is 2.8 + 0.03 = 0.83 cm Correct option: (B)

SECTION – II (Maximum Marks: 32)

7. Two thin circular discs of mass m ad 4m, having radii of a and 2a, respectively, are rigidly fixed by a massless,

rigid rod of length l = √24a through their centers. This assembly is laid on a firm and flat surface, and set rolling without slipping on the surface so that the angular speed about the axis of the rod is ω. The angular

momentum of the entire assembly about the point ‘O’ is �⃗� (see the figure). Which of the following statement(s) is(are) true?

(A) The center of mass of the assembly rotates about the z-axis with an angular speed of ω/5 (B) The magnitude of angular momentum of center of mass of the assembly about the point O is 81 ma2 ω (C) The magnitude of angular momentum of the assembly about its center of mass is 17 ma2 ω/2

(D) The magnitude of the z-component of �⃗� is 55 ma2 ω

JEE ADV 2016


Sol: (A,C)

dcm = n(l)+4m(2l)

5m=9l

5

ω1l = a ω = v (velocity of cm of the first disc)

⇒ ω 1 = aω

a=

ω

√24

ω z = ω 1 cos θ = ω

√24×√24

5=ω

5

L0cm = (5m) (

9l

5× ω1) (

9l

5)

= 81×24a2m ω

5×√24 =

81 ma2 ω√24

5

Labout cm= [ma2

2+4m(2a)2

2]w

= 17 ma2 ω

2

(L⃗ )z= (L0

cm)cosθ − Labout cm sinθ

= 81ma2√24

5×√24

5−17ma2w

2×1

5

= 3803 ma2 ω

50

8. Light of wavelength λph falls on a cathode plate inside a vacuum tube as shown in the figure. The work

function of the cathode surface is Φ and the anode is a wire mesh of conducting material kept at a distance d from the cathode. A potential difference V is maintained between the electrodes. If the minimum de Broglie wavelength of the electrons passing through the anode is 𝜆𝑒, which of the following statement(s) is(are) true?

JEE ADV 2016


(A) 𝜆𝑒 decreases with increase in Φ and 𝜆𝑝ℎ

(B) 𝜆𝑒 is approximately halved, if d is doubled (C) For large potential (V >> Φ/e), 𝜆𝑒 is approximately halved if V is made four times (D) 𝜆𝑒 increases at the same rate as 𝜆𝑝ℎ for 𝜆𝑝ℎ < hc/Φ

Sol: (C)

We have,

hc

λph= ϕ+ k ⇒ k =

hc

λph− ϕ

Also the electrons gain K.E due to the acceleration across a potential V. Therefore, the total K.E is

Ktot = hc

λph− ϕ+ eV

λe =h

√2m ktot=

h

√2m[hc

λph−ϕ+ev]

If λph and ϕ decrease, the denominator decreases. Therefore, λe increases.

λe is independent of d. If eV >> ϕ, then

λe =h

√2n[hc

λph−+eV]

If V increases by a factor of 4 then λe decreases approximately by a factor of 2. Also, h2

2nλe2 =

hc

λps− ϕ+ eV

Differentiating,

JEE ADV 2016


2h2

2mλe3 ×

dλe

dλph= −

hc

λph2 + 0

dλe

dt= (

mcλe3

hλph2 )

dλph

dt

Thus the rates are not the same. Option : (C)

9. In an experiment to determine the acceleration due to gravity g, the formula used for the time period of a

periodic motion is T = 2π √7(𝑅−𝑟)

5𝑔. The values of R and r are measured to be (60 ± 1) mm and (10 ± 1) mm,

respectively. In five successive measurements, the time period is found to be 0.52 s, 0.56 s, 0.57 s, 0.54 s, and 0.59 s. The least count of the watch used for the measurement of time period is 0.01 s. Which of the following statement(s) is(are) true? (A) The error in the measurement of r is 10% (B) The error in the measurement of T is 3.57% (C) The error in the measurement of T is 2% (D) The error in the determined value of g is 11% Sol: (A, B, D) r = 10 ± 1 mm ⇒ r = 10± 10%

T |∆T| 0.52 0.56 0.57 0.54 0.59

0.04 0 0.03 0.02 0.03

T = 0.56 ± 0.02

= 0.56 ± 0.02

0.56× 100

= 0.56 ±3.57%

Tavg = 0.56; ∆Tavg = 0.02

g = r2

5(R−r

T2)

R – r = (50±2)

= 50 ± 4% T2 = 0.31 ±7.14%

∴ R−r

T2=

50

0.31± 11.14%

Options A, B & D

JEE ADV 2016


10. Consider two identical galvanometers and two identical resistors with resistance R. If the internal resistance of the galvanometers RC < R/2, which of the following statement(s) about any one of the galvanometers is(are) true? (A) The maximum voltage range is obtained when all the components are connected in series (B) The maximum voltage range is obtained when the two resistors and one galvanometer are connected in series, and the second galvanometer is connected in parallel to the first galvanometer (C) The maximum current range is obtained when all the components are connected in parallel (D) The maximum current range is obtained when the two galvanometers are connected in series and the combination is connected in parallel with both the resistors Sol: (B,C)

Option (A) Reff = 2r Geff = 2G (R = Re) ⇒ Vmax1 = 2 R ig + 2G ig

Option (B) Reff = 2𝑅

Geff =G

2

Vmax2 = 2R ig +G

2ig

Vmax1 > Vmax2 ∴ Option (A) is correct

Option (C) Reff =R

2

Geff =G

2

Imax1 =2Ig(

G

2+R

2)

R

2

=2Ig(G+R)

R

Option (D) Imax2 =Ig(2G+

R

2)

R

2

=Ig(4G+R)

R

Imax1 > Imax2

∴ Option (D) is correct

JEE ADV 2016


11. In the circuit shown below, the key is pressed at time t = 0. Which of the following statement(s) is(are) true?

(A) The voltmeter displays –5 V as soon as the key is pressed, and displays +5 V after a long time (B) The voltmeter will display 0 V at time t = ln 2 seconds (C) The current in the ammeter becomes 1/e of the initial value after 1 second (D) The current in the ammeter becomes zero after a long time Sol: (A, B, C, D)

Thus Voltmeter reads –5V. I0 = 5×3

50mA =

3

10mA. i1 = 0.1 mA, i2 = 0.2 mA t = ∞ capacitors behave like infinite

resistances. Thus there is no path without an infinite resistance. Therefore i = 0 At all other times between 0 and ∞ voltmeter does not allow any current through it.

i1 = 0.1 e−t

1∆; q1 = 200 [ 1–e–t]

i2 = 0.2 e−t

1; q2 = 100 [1–e–t]

JEE ADV 2016


i = 0.3 e–t mA. After 1s, i = 0.3

e

at t = ∞, i = 0

(V) = VA – VB

= q1

40− 50i

= 5[1 − e−t] − 5e−t

= 5 – 10 e−t At t = ln2

(v) = 5 – 10e−ln (2) = 5 – 5 = 0

Options (A), (B), (C) 12. A block with mass M is connected by a massless spring with stiffness constant k to a rigid wall and moves without friction on a horizontal surface. The block oscillates with small amplitude A about an equilibrium position x0. Consider two cases: (i) When the block is at x0; and (ii) when the block is at x = x0 + A. In both the cases, a particle with mass m(<M) is softly placed on the block after which they stick to each other. Which of the following statement(s) is (are) true about the motion after the mass m is placed on the mass M?

(A) The amplitude of oscillation in the first case changes by a factor of √𝑀

𝑚+𝑀, whereas in the second case it

remains unchanged (B) The final time period of oscillation in both the cases is same (C) The total energy decreases in both the cases (D) The instantaneous speed at x0 of the combined masses decreases in both the cases Sol: (A, B, D) When block is placed at x0

COM : M ω0A0 = (m + n) ωA ⇒ A = [M

M+m] A0 ×

ω0

ω

where ω0 = √K

Mand ω = √

K

M+m

thus, A0 = √M

M+m A

Tf = 2π

ω= 2π√

M+m

K in both cases.

JEE ADV 2016


Since the collision is perfectly inelastic, there is a loss in energy, When block is placed at x0 + A. The system is instantaneously at rest before and after the collision. Options (A), (B) and (D) 13. While conducting the Young’s double slit experiment, a student replaced the two slits with a large opaque plate in the x-y plane containing two small holes that act as two coherent point sources (S1, S2) emitting light of wavelength 600 nm. The student mistakenly placed the screen parallel to the x-z plane (for z > 0) at a distance D = 3 m from the mid-point of S1S2, as shown schematically in the figure. The distance between the sources d = 0.6003 mm. The origin O is at the intersection of the screen and the line joining S1S2. Which of the following is(are) true of the intensity pattern on the screen?

(A) Straight bright and dark bands parallel to the x-axis (B) The region very close to the point O will be dark (C) Hyperbolic bright and dark bands with foci symmetrically placed about O in the x-direction (D) Semicircular bright and dark bands centered at point O Sol: (B, D) The path difference is

∆x0 = 0.6003 mm

= [2(1000)+1

2] × 600 nm

∴ At O there is a dark fringe. Since the sources are spherical, the locus of points on the screen with the same path difference will be semi -circles. Option (B) and (D)

14. A rigid wire loop of square shape having side of length L and resistance R is moving along the x-axis with a constant velocity v0 in the plane of the paper. At t = 0, the right edge of the loop enters a region of length 3L where there is a uniform magnetic field B0 into the plane of the paper, as shown in the figure. For sufficiently large v0, the loop eventually crosses the region. Let x be the location of the right edge of the loop. Let v(x), I(x) and F(x) represent the velocity of the loop, current in the loop, and force on the loop, respectively, as a function of x, Counter- clockwise current is taken as positive.

JEE ADV 2016


Which of the following schematic plot(s) is(are) correct? (Ignore gravity)

Sol: (A, B)

at (1) φ = Blx

and (2) ∈ = Bl 𝑑𝑥

𝑑𝑡

∈ = Blv

i = 𝐵𝑙𝑣

𝑅

JEE ADV 2016


Retarding force on arm AB F = –B i l

F = −𝐵2𝑙2𝑣

𝑅

mv 𝑑𝑣

𝑑𝑥 = –

𝐵2𝑙2

𝑅 v

dv = – 𝐵2𝑙2

𝑚𝑅 dx

v = – 𝐵2𝑙2

𝑅 x; f ∝ x

Thus v decreases linearly with x at (1) and (2)

i = 𝐵𝑙

𝑅 (−

𝐵2𝑙2

𝑚𝑅) x

Thus i also changes linearly with x but at (1) it is clockwise and at (2) it is anticlockwise. F never changes its direction. When the loop is completely inside B, then i and F are zero and v remains constant. Thus, options (A) and (B) are correct.

SECTION – III (Maximum Marks: 12)

PARAGRAPH – I A frame of reference that is accelerated with respect to an inertial frame of reference is called a non-inertial frame of reference. A coordinate system fixed on a circular disc rotating about affixed axis with a constant

angular velocity ω is an example of a non-inertial frame of reference. The relationship between the force 𝐹 𝑟𝑜𝑡 experienced by a particle of mass m moving on the

𝐹 𝑟𝑜𝑡 = 𝐹 𝑖𝑛 + 2m(𝑣 𝑟𝑜𝑡 ×�⃗⃗� ) + m(�⃗⃗� × 𝑟 ) × �⃗⃗� , where 𝑣 𝑟𝑜𝑡 is the velocity of the particle in the rotating frame of reference and 𝑟 is the position vector of the particle with respect to the centre of the disc. Now consider a smooth slot along a diameter of a disc of radius R rotating counter-clockwise with a constant angular speed ω about its vertical axis through its center. We assign a coordinate system with the origin at the center of the disc, the x-axis along the slot, the y-axis perpendicular to the slot and the z-axis along the rotation

axis (�⃗⃗� = ω�⃗� ). A small block of mass m is gently placed in the slot at 𝑟 = (R/2)𝑖 ̂at t = 0 and is constrained to move only along the slot.

JEE ADV 2016


15. The distance r of the block at time t is

(A) 𝑅

4(𝑒𝜔𝑡 + 𝑒−𝜔𝑡) (B)

𝑅

4 cos ωt

(C) 𝑅

4(𝑒2𝜔𝑡 + 𝑒−2𝜔𝑡) (D)

𝑅

4 cos 2ωt

Sol: (A)

ma = mω2r

v 𝑑𝑣

𝑑𝑟 = ω2r

∫ 𝑣𝑣

0 dv = ω2 ∫ 𝑟

𝑅𝑅

2

dr

v = ω√𝑟2 −𝑅2

4

Then, ∫𝑑𝑟

√𝑟2−𝑅2

4

𝑅

20

= ω∫ 𝑑𝑡𝑡

0

ln [𝑟+√𝑟2−

𝑅2

4𝑅

2

] = ωt

on simplifying, r = 𝑅

4[𝑒𝜔𝑡 + 𝑒−𝜔𝑡]

JEE ADV 2016


v = 𝜔𝑅

4[𝑒𝜔𝑡 − 𝑒−𝜔𝑡]

Option (A) 16. The net reaction of the disc on the block is

(A) 1

2𝑚𝜔2𝑅(𝑒2𝜔𝑡 − 𝑒−2𝜔𝑡)𝑗̂ + 𝑚𝑔�̂� (B)

1

2𝑚𝜔2𝑅(𝑒𝜔𝑡 − 𝑒−𝜔𝑡)𝑗̂ + 𝑚𝑔�̂�

(C) −𝑚𝜔2𝑅 cos𝜔𝑡 𝑗̂ − 𝑚𝑔�̂� (D) 𝑚𝜔2𝑅 sin𝜔𝑡 𝑗̂ − 𝑚𝑔�̂�

Sol: (B) 𝐹 𝑟𝑜𝑡 = 𝐹 𝑖𝑛 + 2m (𝑣 𝑟𝑜𝑡 × �⃗⃗� ) + m(�⃗⃗� × 𝑟 ) × �⃗⃗�

mω2r 𝑖 ̂= 𝐹 𝑖𝑛 + 2m(v𝑖 ̂× ω�̂�) + mω2r (�̂� × 𝑖̂) × �̂�

⇒ 𝐹 𝑖𝑛 = 2mωv𝑗̂ [Provided by the normal reaction by the wall of the slot]

�⃗⃗� = 2mω × 𝜔𝑅

4[𝑒𝜔𝑡 + 𝑒−𝜔𝑡] 𝑗̂ =

1

2 mω2R [𝑒𝜔𝑡 + 𝑒−𝜔𝑡] 𝑗

Normal force from the floor of the slot is N1 = mg�̂�

Thus, 𝑁𝑛𝑒𝑡 = 1

2 mω2R[𝑒𝜔𝑡 + 𝑒−𝜔𝑡]𝑗̂ + mg�̂�

Option (B)

PARAGRAPH – II Consider an evacuated cylindrical chamber of height h having rigid conducting plates at the ends and an insulating curved surface as shown in the figure. A number of spherical balls made of a light weight and soft material and coated with a conducting material are placed on the bottom plate. The balls have a radius r << h. Now a high voltage source (HV) is connected across the conducting plates such that the bottom plate is at +V0 and the top plate at –V0. Due to their conducting surface, the balls will get charged, will become equipotential with the plate and are repelled by it. The balls will eventually collide with the top plate, where the coefficient of restitution can be taken to be zero due to the soft nature of the material of the balls. The electric field in the chamber can be considered to the that of a parallel plate capacitor. Assume that field in the chamber can be considered to be that of a parallel plate capacitor. Assume that there are no collisions between the balls and the interaction between them is negligible (Ignore gravity)

JEE ADV 2016


17. Which one of the following statements is correct? (A) The balls will stick to the top plate and remain there (B) The balls will bounce back to the bottom plate carrying the same charge they went up with (C) The balls will bounce back to the bottom plate carrying the opposite charge they went up with (D) The balls will execute simple harmonic motion between the two plates Sol: (C). Each sphere will gain a potential Vo from the lower plate and a charge q. Then they are repelled by the lower plate and accelerate towards the upper plate. They undergo plastic collision there and get charged to a potential -V0 and a charge -q, Thus, a charge 2q is supplied by each sphere to the upper plate. Then they are repelled towards the lower plate and the same process continues.

Where V0 =

𝑞

4𝜋𝜀0𝑟 ⇒ q ∝ V0

Option (C) 18. The average current in the steady state registered by the ammeter in the circuit will be (A) zero (B) proportional to the potential V0

(C) proportional to 𝑉01/2

(D) proportional to 𝑉02

Sol: (D) Time taken by each sphere to travel a distance h is

h = 1

2

𝑞𝐸

𝑚𝑡2

⇒ t2 = 2𝑚ℎ

𝑞×2𝑣0ℎ

= 𝑚ℎ2

𝑞V0 ⇒ t ∝ 1 / V0

A charge 2q is transferred by each sphere every ‘t’ seconds

∴ i ∝ 2𝑞

𝑡 ⇒ i ∝ 𝑉0

2

Option (D)

JEE ADV 2016


PART – II CHEMISTRY

SECTION – I (Maximum Marks : 18)

This section contains SIX questions.

Each question has FOUR options (A), (B),(C) and (D). ONLY ONE of these four options is correct.

For each question, darken the bubble corresponding to the correct option in the ORS.

For each question, marks will be awarded in one of the following categories; Full Marks : +3 If only the bubble corresponding to the correct option is darkened. Zero Marks : 0 If none of the bubbles in darkened. Negative Marks : - 1 In all other cases.

19. For the following electrochemical cell at 298 K, Pt(s)|𝐻2(𝑔, 1 𝑏𝑎𝑟)| 𝐻

+(𝑎𝑞, 1 𝑀)||𝑀4+(𝑎𝑞),𝑀2+(𝑎𝑞)|𝑃𝑡(𝑠)

𝐸𝑐𝑒𝑙𝑙 = 0.092 𝑉 𝑤ℎ𝑒𝑛 [𝑀2+ (𝑎𝑞)]

[𝑀4+ (𝑎𝑞)]= 10𝑥

Given: 𝐸𝑀4+/𝑀

2+0 = 0.151 V; 2.303

𝑅𝑇

𝐹= 0.59 𝑉

The value of x is (A) – 2 (B) – 1 (C) 1 (D) 2 Ans: (D) Solution: Anode H2⟶ 2𝐻+ + �̅�

Cathode 𝑀+4 + 2�̅�⟶ 𝑀+2

H2 + 𝑀+4⟶ 2𝐻++ 𝑀+2

Qc=[𝐻+]2[𝑀+2]

𝑃𝐻2[𝑀+4]

= 1 × [𝑀+2]

1× [𝑀+4] = 10𝑥

∴𝐸𝑐𝑒𝑙𝑙 = 𝐸𝑐𝑒𝑙𝑙𝑜 −

0.059

𝑛log𝑄𝐶

Substituting them, we get x =2

JEE ADV 2016


20. The qualitative sketches I, II and III given below show the variation of surface tension with molar concentration of three different aqueous solutions of KCI, 𝐶𝐻3 𝑂𝐻 𝑎𝑛𝑑 𝐶𝐻3(𝐶𝐻2)11𝑂𝑆𝑂3

-Na+at room temperature. The correct assignment of the sketches is

(A) I : KCI II : CH3 OH III : CH3(CH2)11OSO3–Na+

(B) I : CH3(CH2)11OSO3–Na+ II : CH3 OH III : KCI

(C) I : KCI II : CH3(CH2)11OSO3–Na+ III : CH3 OH

(D) I : CH3 OH II : KCI III : CH3(CH2)11OSO3–Na+

Ans: (D) Solution: Conceptual 21. In the following reaction sequence in aqueous solution, the species X, Y and Z, respectively, are

(A) [Ag (S2O3)2]

3−, Ag2S2O3,Ag2S (B) [Ag (S2O3)2]5−, Ag2SO3, Ag2S

(C) [Ag (𝑆2O3)2]

3−, Ag2S2O3,Ag (D) [Ag (𝑆2O3)3]3−, Ag2SO4, Ag

Ans: (A)

Solution:S2O3−2

Ag+

→ [Ag(S2O3)2]−3

Ag+

→ Ag2S2O3↓ ThiosulphateArgentothiosulphate silver thiosulphate ↓withtime Ag2 S ↓ (black) 22. The geometries of the ammonia complexes of 𝑁𝑖2+, 𝑃𝑡2+ 𝑎𝑛𝑑 𝑍𝑛2+, respectively, are (A) octahedral, square planar and tetrahedral (B) square planar, octahedral and tetrahedral

(C) tetrahedral, square planar and octahedral (D) octahedral, tetrahedral and square planar

JEE ADV 2016


Ans: (A) Solution: [Ni(NH3)6]+2 Octahedral

[Pt(NH3)4]+2 Square Plannar

[Zn(NH3)4]+2 Tetrahedral

23. The correct order of acidity for the following compounds is

(A) I > II > III > IV (B) III > I > II > IV (C) III > IV > II > I (D) I > III > IV > II Ans: (A) Solution:

24. The major product of the following reaction sequence is

JEE ADV 2016


Ans: (A) Solution:

SECTION – II (Maximum Marks : 32)

This section contains EIGHT questions.

Each question has FOUR options (A), (B),(C) and (D) ONE OR MORE THAN ONE of these four option(s) is

(are) correct.

For each question, darken the bubble(s) corresponding to all the correct option(s)in the ORS.

For each question, marks will be awarded in one of the following categories:

Full Marks : +4 If only the bubble(s) corresponding to all the correct option(s) in (are) darkened.

Partial Marks : + 1 For darkening a bubble corresponding to each correct option, provided NO

incorrect option id darkened.

Zero Marks : 0 If none of the bubbles is darkened.

Negative Marks: - 2 If none of the bubbles is darkened.

For example, if(A), (C) and (D) are all the correct options for a question, darkening all these three will

result in +4 marks; darkening only (A) and (D) will result in +2 marks; and darkening (A) and (B) will

result in – 2 marks, as a wrong option is also darkened.

25. According to Molecular Orbital Theory,

(A) C22−is expected to be diamagnetic (B) O2

2+ is expected to have a longer bond length than 𝑂2 (C) N2

+and𝑁2− have the same bond order (D) He2

+ has the same energy as two isolated He atoms Ans: (A, C) Solution:

i) 𝐶2−2 is diamagnetic

ii) 𝑂2+2bond order is 3.0 and O2 bond order is 2.0

iii) 𝑁2

+&𝑁2−have same bond order of 2.5

JEE ADV 2016


iv) 𝐻𝑒2+2has less energy as compare to He atoms

26. Mixture(s) showing positive deviation from Raoult’s law at 35° C is (are) (A) carbon tetrachloride + methanol (B) carbon dissulphide + acetone (C) benzene + toluene (D) Phenol + aniline Ans: (A, B) Solution: Conceptual 27. The CORRECT statement(s) for cubic close packed (ccp) three dimensional structure is (are) (A) The number of the nearest neighbours of an atom present in the topmost layer is 12 (B) The efficiency of atom packing is 74% (C) The number of octahedral and tetrahedral voids per atom are 1 and 2, respectively

(D) The unit cell edge length is 2√2 times the radius of the atom. Ans: (B, C, D) Solution:

i) Co. No of an atom in top most layer is 9 ii) Packing Fraction in CCP = 0.74

iii) Octahedral voids & Tetrahedral voids = 4 & 8 iv) edge length a = 4𝑟

√2⇒ r =

𝑎

2√2

28. Extraction of copper from copper pyrite (CuFeS2) involves

(A) crushing followed by concentration of the ore by froth-flotation (B) removal of iron as slag (C) self-reduction step to produce blister copper following evolution of SO2 (D) refining of ‘blister copper’ by carbon reduction

Ans: (A, B, C) Solution: Usually blister copper is refined either by poling (or) electrolytic refining methods( D is also possible ) 29. The nitrogen containing compound produced in the reaction of HNO3 with P4O10

(A) can also be prepared by reaction of P4 and HNO3 (B) is diamagnetic (C) contains one N-N bond (D) reacts with Na metal producing a brown gas

Ans: (B, D) Solution:

2HNO3

𝑃2𝑂5→ N2O5 + 2HPO3

i) P4 + 20HNO3⟶4H3PO4 + 20NO2 + 4H2O ii) N2O5 has no unpaired electron iii) It contains N – O – N linkage iv) Na + N2O5⟶ NaNO3 + NO2↑

JEE ADV 2016


30. For ‘invert sugar’, the correct statement(s) is (are) (Given: specific rotations of (+) –sucrose, (+)-maltose, L-(–)-glucose and L-(+)- fructose in aqueous solution are +660, +1400, –520 and +920, respectively)

(A) ‘invert sugar’ is prepared by acid catalyzed hydrolysis of maltose (B) ‘invert sugar’ is an equimolar mixture of D-(+)-glucose and D-(–)-fructose (C) specific rotation of ‘invert sugar’ is -200 (D) on reaction with Br2 water, ‘invert sugar’ forms saccharic acid as one of the products

Ans: (B, C) Solution:

i) Invert sugar is prepared by hydrolysis of sucrose

ii) specific rotation of mixture = +52𝑜+ (−92𝑜)

2= −20𝑜 on bromination

D – Glucose +Br2 water⟶Gluconic acid but not Saccharic acid D – Fructose + Br2 water⟶ No reaction 31. Reagent(s) which can be used to bring about the following transformation is (are)

(A) LiA1H in (C2H5)2O (B) BH3 in THF (C) NaBH4 in C2H5OH (D) Raney Ni/H2 in THF

Ans: (C) Solution:

NaBH4 is mild reducing agent [selective reductant] It only reduces – CHO ⟶ –CH2OH (1°) but not ester, – COOH - - - etc groups

32. Among the allowing, reaction(s) which gives (give) tert-butyl benzene as the major product is (are)

JEE ADV 2016


Ans: (B, C, D) Solution:

i)

ii) Friedal craft’s alkylation, ter – butyl, Carbocation attacks on benzene

iii)

iv)

SECTION 3 (Maximum Marks: 12)

This section contains TWO paragraphs.

Based on each paragraph, there are TWO questions.

Each question has FOUR options (A), (B), (C) and (D). ONLY ONE of these four options is correct.

For each question, darken the bubble corresponding to the correct option in the ORS.

For each question, marks will be awarded in one of the following categories: Full Marks : +3 If only the bubble corresponding to the correct option is darkened. Zero Marks : 0 In all other cases

PARAGRAPH – I

Thermal decomposition of gaseous X2 to gaseous X at 298 K takes place according to the following equation:

X2(g) ⇌ 2X(g) The standard reaction Gibbs energy, ∆,G0, of this reaction is positive. At the start of the reaction, there is one mole of X2 and no X. As the reaction proceeds, the number of moles of X formed is given by β. Thus, βequilibrium is the number of moles of X formed at equilibrium. The reaction is carried out at a constant total pressure of 2 bar. Consider the gases to behave ideally. (Given: R = 0.083 L bar K–1mol–1) 33. The equilibrium constant KP for this reaction at 298 K, in terms of βequilibrium, IS

(A) 8β𝑒𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚2

2−β𝑒𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚 (B)

8β𝑒𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚2

4−β𝑒𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚2 (C)


2−β𝑒𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚 (D)


4−β𝑒𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚2

Ans: (B) Solution:𝑋2(𝑔) ⇌ 2𝑋(𝑔)

1 – 𝛽𝑒𝑞

2 𝛽𝑒𝑞

JEE ADV 2016


KP = 𝑃𝑋2

𝑃𝑋2 =

{

𝛽𝑒𝑞

1+𝛽𝑒𝑞2

𝑃𝑇

1−𝛽𝑒𝑞2

1+𝛽𝑒𝑞2

.𝑃𝑇

}

2

= 𝛽𝑒𝑞2

1−𝛽𝑒𝑞2

4

.PT = 2𝛽𝑒𝑞

2

1−𝛽𝑒𝑞2

4

KP = 8𝛽𝑒𝑞2

4−𝛽𝑒𝑞2

34. The INCORRECT statement among the following, for this reaction, is

(A) Decrease in the total pressure will result in formation of more moles of gaseous X (B) At the start of the reaction, dissociation of gaseous X2 takes place spontaneously (C) β𝑒𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚 = 0.7

(D) KC< 1 Ans: (C) Solution: If 𝛽𝑒𝑞 = 0.7

𝐾𝑃 =8 × 0.49

4 − 0.49=3.92

3.51> 1

∵ ∆ G° >O thenKp< 1

PARAGRAPH – II

Treatment of compound O with KMnO4/H+ gave p, which on heating with ammonia gave Q. The compound Q on treatment with Br2/NaOH produced R. On strong heating, Q gave S, which on further treatment with ethyl 2-bromopropanoate in the presence of KOH followed by acidification, gave a compound T.

35. The compound R is Ans: (A)

JEE ADV 2016


36. The compound T is (A) glycine (B) alanine (C) valine (D) serine Ans: (B) Solution: Solution: For 35 & 36 Question

JEE ADV 2016


PART III: MATHEMATICS

SECTION – I (Maximum Marks: 18)

37. Let P = [1 0 04 1 016 4 1

] and I be the identity matrix of order 3. If Q = [𝑞𝑖𝑗] is a matrix such that

P50 – Q = I, then 𝑞31+𝑞32

𝑞21 equals

(A) 52 (B) 103 (C) 201 (D) 205 Ans: B Solution:

P = [1 0 04 1 016 4 1

] p2 = [1 0 08 1 048 8 1

] P3 = [1 0 012 1 096 12 1

]

∴ 𝑃𝑛= [1 0 04𝑛 1 0

8(𝑛2 + 𝑛) 4𝑛 1] P50 = [

1 0 0200 1 020400 200 1

]

Q = P50 – I = [0 0 0200 0 020400 200 0

]

𝑞31+𝑞32

𝑞21 =

20400+200

200 = 103

38. Let 𝑏𝑖 > 1 for I = 1, 2, …, 101. Suppose 𝑙𝑜𝑔𝑒 b1, 𝑙𝑜𝑔𝑒 b2, …, 𝑙𝑜𝑔𝑒 b101 are in Arithmetic Progression (A.P.) with the common difference 𝑙𝑜𝑔𝑒 2. Suppose a1, a2, …, a101 are in A.P. such that a1 = b1 and a51 = b51. If t = b1 + b2 + … + b51 and s = a1 + a2 + … + a51, then (A) s > t and a101 > b101 (B) s > t and a101 < b101 (C) s < t and a101 > b101 (D) s < t and a101 < b101 Ans: B Solution: log b1, log b2, ……log b101 are in A.P. will C.d. log 2

b1, b2 ……b101 are in .G.P. with C.r. 2

T = b1 (251 – 1), S =

51

2 (a1 + a51) =

51

2 (b1 + b51) =

51.b1

2 (1 + 250)

S – t = b1 (53

2+ 249. 47) > 0 ⇒ S > t

b101 = 2100 b1

a101 = a1 + 100d = 2a51 – a1 = 2b51 – b1 = (251 – 1)b1

∴ b101 > a101

JEE ADV 2016


39. The value of ∑1

sin(𝜋

4+(𝑘−1)𝜋

6) sin(

𝜋

4+𝑘𝜋

6)

13𝑘=1 is equal to

(A) 3 – √3 (B) 2(3 – √3) (C) 2(√3 – 1) (D) 2(2 – √3) Ans: C Solution:

G.E. = 2 ∑sin[(

𝜋

4+𝑘𝜋

6)−(

𝜋

4+(𝑘−1)𝜋

6)]

sin(𝜋

4+(𝑘−1)𝜋

6). sin(

𝜋

4+𝑘𝜋

6)

13𝑘=1

= 2 ∑ [cot (𝜋

4+ (𝑘 − 1)

𝜋

6) − cot (

𝜋

4+𝑘𝜋

6)]13

𝑘=1

= 2 [cot𝜋

4− cot (

𝜋

4+13𝜋

6)]

= 2 [1 – (2 – √3)]

= 2(√3 – 1)

40. The value of ∫𝑥2 cos𝑥

1+𝑒𝑥

𝜋

2

−𝜋

2

dx is equal to

(A) 𝜋2

4 – 2 (B)

𝜋2

4 + 2 (C) 𝜋2 − 𝑒

𝜋

2 (D) 𝜋2 + 𝑒𝜋

2

Ans: A

Solution: 𝐼 = ∫𝑥2.𝑐𝑜𝑠 𝑥

1+𝑒𝑥𝜋/2

−𝜋/2 dx . . . . . . (1)

Use ∫ 𝑓(𝑥)𝑏

𝑎 dx = ∫ 𝑓(𝑎 + 𝑏 − 𝑥)

𝑏

𝑎 dx

I = ∫𝑥2.cos𝑥.𝑒𝑥

𝑒𝑥+1

𝜋/2

−𝜋/2 dx . . . . . . (2)

(1) + (2) ⇒ 2I = ∫ 𝑥2𝜋/2

−𝜋/2. cos x dx = 2 ∫ 𝑥2

𝜋/2

0. cos x dx

⇒ I = 𝜋2

4 – 2 (Apply by Parts)

41. Area of the region {(x, y) ∈ ℝ2: y ≥ √|𝑥 + 3|, 5y ≤ x + 9 ≤ 15} is equal to

(A) 1

6 (B)

4

3 (C)

3

2 (D)

5

3

Ans: C

Solution: y ≥ √|𝑥 + 3|

JEE ADV 2016


⇒ y2 ≥ {𝑥 + 3, 𝑥 ≥ −3−𝑥 − 3, 𝑥 < −3

5y ≤ x + 9 ≤ 15 ⇒ 5y ≤ x + 9, x ≤ 6

A = ∫ (𝑥+9

5)

−3

−4 – √−𝑥 − 3 dx + ∫ (

𝑥+9

5)

1

−3 – √𝑥 + 3 dx

= 3

2

42. Let P be the image of the point (3, 1, 7) with respect to the plane x – y + z = 3. Then the equation

of the plane passing through P and containing the straight line 𝑥

1 = 𝑦

2 = 𝑧

1 is

(A) x + y – 3z = 0 (B) 3x + z = 0 (C) x – 4y + 7z = 0 (D) 2x – y = 0 Ans. C Solution: Let (α, β, γ) be the image of (3, 1, 7) with respect to x – y + z = 3

𝛼−3

1 = 𝛽−1

−1 = 𝛾−7

1 = −2(3−1+7−3)

3

⇒ α = –1, β = 5, γ = 3

Required plane is P.T. (–1, 5, 3), (0, 0, 0) and containing the line 𝑥

1 =

𝑦

2 = 𝑧

1

Equation of the plane is |𝑥 𝑦 𝑧−1 5 31 2 1

| = 0

⇒ x – 4y + 7z = 0

JEE ADV 2016


SECTION – II (Maximum Marks: 32)

43. Let f(x) = 𝐿𝑖𝑚𝑛 → ∞

(𝑛𝑛(𝑥+𝑛)(𝑥+

𝑛

2)…..(𝑥+

𝑛

𝑛)

𝑛!(𝑥2+𝑛2)(𝑥2+𝑛2

4)…..(𝑥2+

𝑛2

𝑛2))

𝑥

𝑛

, for all x > 0. Then

(A) 𝑓 (1

2) ≥ 𝑓(1) (B) 𝑓 (

1

3) ≤ 𝑓 (

2

3)

(C) 𝑓′(2) ≤ 0 (D) 𝑓′(3)

𝑓(3)≥𝑓′(2)

𝑓(2)

Ans: B, C

Solution: log f(x) = 𝐿𝑖𝑚𝑛 → ∞

𝑥

𝑛{∑ log (1 +

𝑟𝑥

𝑛)𝑛

𝑟=1 − ∑ log (1 +𝑟2𝑥2

𝑛2)𝑛

𝑟=1 }

𝑥 ∫ log(1 + 𝑥𝑦) 𝑑𝑦 − 𝑥 ∫ log (1 + 𝑥2𝑦2)𝑑𝑦1

0

1

0

Put xy = t ⇒ xdy = dt

log f(x) = ∫ log(1 + 𝑡)𝑑𝑡 − ∫ log(1 + 𝑡2)𝑑𝑟𝑥

0

𝑥

0

⇒ 𝑓′(𝑥)

𝑓(𝑥) = log (

1+𝑥

1+ 𝑥2) ⇒

𝑓′(2)

𝑓(2) ≥

𝑓′(3)

𝑓(3) and f’(2) < 0

𝑓′(𝑥)

𝑓(𝑥) > 0 in (0, 1) and

𝑓′(𝑥)

𝑓(𝑥) < 0 in (1, ∞)

⇒ f(1) ≥ f(1

2) and f(

1

3) ≤ f(

2

3)

44. Let a, b ∈ ℝ and f:ℝ ⟶ ℝ be defined by f(x) = a cos (|x3 – x|) + b|x| sin (|x3 + x|). Then f is (A) differentiable at x = 0 if a = 0 and b = 1 (B) differentiable at x = 1 if a = 1 and b = 0 (C) NOT differentiable at x = 0 if a = 1 and b = 0 (D) NOT differentiable at x = 1 if a = 1 and b = 1 Ans: A, B Solution: f(x) = a cos (|x3 – x|) + b|x| sin (|x3 + x|) (I) a = 0, b = 1 f(x) = |x| in (|x3 + x|) = x sin (x3 + x), ∀ x ∈ R

f(x) is differentiable at x = 0 (II) a = 1, b = 0 f(x) = cos (|x3 – x|) = cos (x3 – x), ∀ x ∈ R ∴ f(x) is differentiable at x = 1 , x = 0 (III) a = 1, b = 1 f(x) = cos (x3 – x) + x sin (x3 + x)

f(x) is differentiable at x = 1

JEE ADV 2016


45. Let f: ℝ → (0, ∞) and g: ℝ ⟶ ℝ be twice differentiable functions such that f’’ and g’’ are continuous functions on ℝ. Suppose f’(2) = g(2) = 0, f’’(2) ≠ 0 and g’(2) ≠ 0.

If 𝐿𝑖𝑚𝑥 → 2

𝑓(𝑥)𝑔(𝑥)

𝑓′(𝑥)𝑔′(𝑥) = 1, then

(A) f has a local minimum at x = 2 (B) f has a local maximum at x = 2 (C) f’’(2) > f(2) (D) f(x) – f’’(x) = 0 for at least one x ∈ ℝ Ans A, D

Solution: 𝐿𝑖𝑚𝑥 → 2

𝑓(𝑥).𝑔(𝑥)

𝑓′(𝑥).𝑔′(𝑥) = 1 ⇒

𝐿𝑖𝑚𝑥 → 2

𝑓′(𝑥).𝑔(𝑥)+𝑔′(𝑥)𝑓(𝑥)

𝑓′′(𝑥).𝑔′(𝑥)+𝑔′′(𝑥)𝑓′(𝑥) = 1

⇒ f’’(x) = f(2)

f’’(x) = f(x) for atleast one x ∈ 𝑅 We have f’’(2) = f(2)>0

f has local minimum at x = 2

46. Let 𝑓: [−1

2, 2] → ℝ and 𝑔: [−

1

2, 2] ⟶ ℝ be functions defined by f(x) = [x2 – 3] and g(x) = |x|f(x) +

|4x – 7|f(x), where [y] denotes the greatest integer less than or equal to y for y ∈ ℝ. Then

(A) f is discontinuous exactly at three points in [−1

2, 2]

(B) f is discontinuous exactly at four points in [−1

2, 2]

(C) g is NOT differentiable exactly at four points in (−1

2, 2)

(D) g is NOT differentiable exactly at five points in (−1

2, 2)

Ans: B, C Solution: F(x) = [x2 – 3] = [x2] – 3

∴f(x) discontinuous at x = 1, √2, √3, 2 → 4 points g(x) = (|x| + |4x – 7|)([x2] – 3) = 15x – 21, x < 0 = 9x – 21, 0 ≤ x < 1

= 6x – 14, 1 ≤ x < √2

= 3x – 7, √2 ≤ x < √3

= 0, √3 ≤ x < 2 = 3, x = 2

∴ g(x) is not differentiable at x = 0, 1, √2, √3 ⟶ 4 points.

JEE ADV 2016


47. Let a, b ∈ ℝ and a2 + b2 ≠ 0. Suppose S = {𝑧 ∈ ℂ: 𝑧 =1

𝑎+𝑖𝑏𝑡, 𝑡 ∈ ℝ, 𝑡 ≠ 0}, Where 𝑖 = √−1.

If z = x + iy and z ∈ S, then (x, y) lies on

(A) the circle with radius 1

2𝑎 and centre (

1

2𝑎, 0) for a > 0, b ≠ 0

(B) the circle with radius – 1

2𝑎 and centre (−

1

2𝑎, 0) for a < 0, b ≠ 0

(C) the x-axis for a ≠ 0, b = 0 (D) the y-axis for a = 0, b ≠0 Ans: A, C, D

Solution: Z = 1

𝑎+𝑖𝑏𝑡 =

𝑎−𝑖𝑏𝑡

𝑎2+𝑏2𝑡2

⇒ x = 𝑎

𝑎2+𝑏2𝑡2, y =

−𝑏𝑡

𝑎2+𝑏2𝑡2

Eliminating ‘t’, we get x2 + y2 = 𝑥

𝑎 ⇒ (𝑥 −

1

2𝑎)2

+ y2 = (1

2𝑎)2

⇒ Z lies on the circle with radius 1

2𝑎 and centre (

1

2𝑎, 0)

⟶ a ≠ 0, b = 0 ⇒ x = 1

𝑎, y = 0 ⇒ Z lies on x-axis

⟶ a = 0, b ≠ 0 ⇒ x = 0, y = −1

𝑏𝑡 ⇒ Z lies on y-axis

48. Let P be the point on the parabola y2 = 4x which is at the shortest distance from the center S of the circle x2 + y2 – 4x – 16y + 64 = 0. Let Q be the point on the circle dividing the line segment SP internally. Then

(A) SP = 2√5

(B) SQ:QP = (√5 + 1):2 (C) the x-intercept of the normal to the parabola at P is 6

(D) the slope of the tangent to the circle at Q is 1

2

Ans: A, C, D Solution: Centre, S = (2, 8), radius, r = 2

Equation of normal: y = mx – 2m – m3 which P.T. (2, 8)

⟹ m = − 2 , equation of normal is y = -2x + 12

P = (am2, –2am) = (4, 4)

SP = 2√5

SQ = 2, QP = 2√5 – 2

JEE ADV 2016


SQ : QP = 2:(2√5 – 2) = 1:(√5 – 1)

X-int of normal at P is 6

Slope of tangent at Q is 1

2

49. Let a, λ, μ ∈ ℝ. Consider the system of linear equations ax + 2y = λ 3x – 2y = μ Which of the following statement(s) is(are) correct? (A) If a = –3, then the system has infinitely many solutions for all values of λ and μ (B) If a ≠ –3, then the system has a unique solution for all values of λ and μ (C) If λ + μ = 0, then the system has infinitely many solutions for a = –3 (D) If λ + μ ≠ 0, then the system has no solution for a = –3 Ans: B, C, D Solution: ax + 2y = λ 3x – 2y = μ

∆ = |𝑎 23 −2

| = –2a – 6

∆1 = |𝜆 2𝜇 −2

| = –2(𝜆 + μ)

∆2 = |𝑎 𝜆3 𝜇

| = aμ – 3λ

⟶ If a ≠ –3, ∆ ≠ 0 ⇒ unique sol ⟶ If a = –3, λ + μ = 0 ⇒ Infinite sol ⟶ If a = –3, λ + μ ≠ 0 ⇒ No sol

50. Let �̂� = u1 𝑖 ̂+ u2 𝑗̂ + u3 �̂� be a unit vector in ℝ3 and �̂� = 1

√6 (𝑖 ̂+ 𝑗̂ + 2�̂�). Given that there exists a

vector 𝑣 in ℝ3 such that |�̂� × 𝑣 | = 1 and �̂�.( �̂� × 𝑣 ) = 1. Which of the following statement(s) is(are) correct? (A) There is exactly one choice for such 𝑣 (B) There are infinitely many choices for such 𝑣 (C) If �̂� lies in the xy-plane then |u1| = |u2| (D) If �̂� lies in the xz-plane then 2|u1| = |u3| Ans: B, C Solution: |�̂� × 𝑣 | = 1 ⇒ |�̂�||𝑣 | sin α = 1 ⇒ |𝑣 | sin α = 1 ⇒ There are infinitely many choices for 𝑣 �̂�.( �̂� × 𝑣 ) = 1 ⇒ |�̂�||�̂� × 𝑣 | cos θ = 1 ⇒ θ = 0 ⇒ �̂� is perpendicular to �̂� and 𝑣 ⇒ �̂�.�̂� = 0 ⇒ u1 + u2 + 2u3 = 0 If �̂� lies in xy-plane, u1 + u2 = 0 ⇒ |u1| = |u2| If �̂� lies in xz-plane, u1 + 2u3 = 0 ⇒ |u1| = 2|u3|

JEE ADV 2016


SECTION – III (Maximum M arks: 12)

PARAGRAPH – I Football teams T1 and T2 have to play two games against each other. It is assumed that the outcomes of the

two games are independent. The probabilities of T1 winning, drawing and losing a game against T2 are 1

2, 1

6 and

1

3, respectively. Each team gets 3 points for a win, 1 point for a draw and 0 point for a loss in a game. Let X and

Y denote the total points scored by teams T1 and T2, respectively, after two games. 51. P(X > Y) is

(A) 1

4 (B)

5

12 (C)

1

2 (D)

7

12

Ans: B Solution: P(X > Y) = P(W, W) + P(W, D) + P(D, W)

= 1

2.1

2 + 1

2.1

6 + 1

6.1

2 =

5

12

52. P(X = Y) is

(A) 11

36 (B)

1

3 (C)

13

36 (D)

1

2

Ans: C Solution: P(X = Y) = P(W, L) + P(L, W) + P(D, D)

= 1

2.1

3 + 1

3.1

2 + 1

6.1

6 = 13

36

PARAGRAPH – II

Let F1(x1, 0) and F2(x2, 0), for x1 < 0 and x2 > 0, be the foci of the ellipse 𝑥2

9 + 𝑦2

8 = 1. Suppose a parabola having

vertex at the origin and focus at F2 intersects the ellipse at point M in the first quadrant and at point N in the fourth quadrant. 53. The orthocentre of the triangle F1MN is

(A) (−9

10, 0) (B) (

2

3, 0) (C) (

9

10, 0) (D) (

2

3, √6)

Ans: A

Solution: a2 = 9, b2 = 8 ⇒ e = 1

3

F1 = (–1, 0), F2 = (1, 0) ∴ Equation of parabola is y2 = 4x

JEE ADV 2016


∴ M = (3

2, √6), N = (

3

2, −√6)

Orthocentre of ∆F1MN = (−9

10, 0)

54. If the tangents to the ellipse at M and N meet at R and the normal to the parabola at M meets the x-axis at Q, then the ratio of area of the triangle MQR to area of the quadrilateral MF1NF2 is (A) 3:4 (B) 4:5 (C) 5:8 (D) 2:3 Ans: C

Solution: R = (6, 0), Q = (7

2, 0)

Area of ∆MQR = 5

4√6

Area of quadrilateral MF1 NF2 = 2√6

Ratio = 5

8 = 5:8

JEE ADV 2016 - Amazon S3s3-ap-southeast-1.amazonaws.com/subscriber.images/jee/... · 2018-12-18 ·...

Documents

Transcript of JEE ADV 2016 - Amazon S3s3-ap-southeast-1.amazonaws.com/subscriber.images/jee/... · 2018-12-18 ·...