2012 IIT Physics Paper01 Soln

7/30/2019 2012 IIT Physics Paper01 Soln

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PHYSICS

[1] IIT JEE 2012

Part I: Physics

Section I (Single Correct Answer Type)

This section contains 10 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D), out of which ONLY ONE is correct.

1. A bi-convex lens is formed with two thin plano-convex lenses as shown in the figure. Refractive index n of the first lens is 1.5 and that of the second lens is 1.2. Both the curved surfaces are of the same radius of curvature R = 14 cm. For this bi-convex lens, for an object distance of 40 cm, the image distance will be

(A) 280 0. cm (B) 40 0. cm (C) 215. cm (D) 13 3. cm

Sol: [B] The focal length of each part

115 1

114

1 1281

1

f F H G

I K J .b g cm

112 1

1 114

1702

1

f F H G

I K J .b g cm

Hence equivalent focal length

1 1 1 1

28

1

70

1

201 2 f f f cm-1

f eq 20 cm

For u cm f v f 40 2 2 40 cm


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[2] IIT JEE 2012

2. A thin uniform rod, pivoted at O, is rotating in the horizontal plane withconstant angular speed , as shown in the figure. At time t = 0, a small

insect starts from O and moves with constant speed v with respect to therod towards the other end. It reaches the end of the rod at t = T andstops. The angular speed of the system remains throughout. The mag-

nitude of the torque | | b gon the system about O, as a function of time is best represented by which plot?

(A) (B)

(C) (D)

Sol: [B] The angular momentum of the system about the rotational axis is

L L Mx L Mvt Rod Rod 2 2c h b g

or L L Mv t Rod 2 2

dLdt

Mv t 2 2

Hence t, till the particle reaches the end and stops, and then becomes zero

3. Three very large plates of same area are kept parallel and close to each other. They are considered asideal black surfaces and have very high thermal conductivity. The first and third plates are maintained attemperatures 2 T and3 T respectively. The temperature of the middle (i.e. second) plate under steady statecondition is

(A)652

14F

H GI K J T (B)

974

14F

H GI K J T (C)

972

14F

H GI K J T (D) 97

14b gT

Sol: [C] The net heat received by the middle plate from the hotter plate is equal to the net heat given to thecooler plate hence

A T T A T T3 24

04

04 4b ge j b ge j


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or 2 81 1604 4 4T T T

or T T04

4

972

T T0

1 4

972

F H G I K J /

4. Consider a thin spherical shell of radius R with its centre at the origin, carrying uniform positive surface

charge density. The variation of the magnitude of the electric field E r bgand the electric potential V (r )with the distance r from the centre, is best represented by which graph ?

(A) (B)

(C) (D)

Sol: [D] For a spherical shell

E 0 r Rb g

E kQr 2

r Rb gwhereas the electric potential

V kQ R

(constant) for r < R

V kQr

for r R

The relevant graph is (D).


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5. In the determination of Youngs modulus Y MLg

d F H G

I K J

42 by using Searle's method, a wire of length L = 2m

and diameter d = 0.5 mm is used. For a load M = 2.5 kg, an extension l 0 25. cm in the length of the wireis observed. Quantities d and l are measured using a screw gauge and a micrometer, respectively. Theyhave the same pitch of 0.5 mm. the number of divisions on their circular scale is 100. The contributions tothe maximum probable error of the Y measurement

(A) due to the errors in the measurements of d and l are the same

(B) due to the error in the measurement of d is twice that due to the error in the measurement of l (C) due to the error in the measurement of l is twice that due to the error in the measurement of d (D) due to the error in the measurement of d is four times that due to the error in the measurement of l

Sol: [A] Y Y

d d max max max

2

Here both screw gauge and micrometer used have same least count and hence maximum probable error inmeasurement of and d will be same i.e.,

max max d x

Thus,c x

max .0 25 mm

and 22

0 50 0 25 d

d x x

max . .mm

and hence contributions due to the errors in the measurement of d and are the same

6. A small block is connected to one end of a massless spring of un -stretched length 4.9 m. The other end of the spring (see the figure) is fixed. The system lies on a horizontal frictionless surface. The block is stretched

by 0.2 m and released from rest at t = 0. It then executes simple harmonic motion with angular frequency

3 rad / s . Simultaneously at t = 0, a small pebble is projected with speed v from point P at an angleof 45 0 as shown in the figure. Point P is at a horizontal distance of 10 m from O. If the pebble hits the block at t = 1s, the value of v is (take g = 10 m / s 2)

(A) 50 m/ s (B) 51 m/ s (C) 52 m/ s (D) 53 m/ s


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Sol: [A] The block starts its oscillation from its positive exrtreme and hence its position xblock from O as afunction of time t can be written as

x t block 4 9 0 2. . cos

x t sblock F H G

I K J 1 4 9 0 2 3 1 50b g . . cos .

m

Hence range of the pebble R = 10 5 = 5 m

Sov

g v

2 25 50

sin m / s

7. Young's double slit experiment is carried out by using green, red and blue light, one color at a time. The

fringe widths recorded are G R, and B , respectively. Then,(A) G B R (B) B G R (C) R B G (D) R G B

Sol: [D] Fringe width, Dd

and hence

Since red green blue

So red green blue 8. A small mass m is attached to a massless string whose other end is fixed at P as shown in the figure. The

mass is undergoing circular motion in the x - y plane with centre at O and constant angular speed . If theangular momentum of the system, calculated about O and P are denoted by LO and L P respectively, then

(A) LO and L P do not vary with time

(B) LO varies with time while L P remains constant

(C) LO remains constant while L P varies with time

(D) LO and L P both vary with time


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Sol: [C] The net force acting on the mass m is m r 2 acting towards point O and hence net torque about point O is zero but about point P is non-zero. Hence L0 remains constant while L P varies with time.

9. A mixture of 2 moles of helium gas (atomic mass = 4 amu) and 1 mole of argon gas (atomic mass = 40

amu) is kept at 300 K in a container. The ratio of the r m s speedv heliumv on

rms

rms

b gb garg

F H G

I K J is

(A) 0.32 (B) 0.45 (C) 2.24 (D) 3.16

Sol: [D] Both gases are at same temperature and thus v M rms1

where M is molar mass of the gas

v

v

M

M rms Helium

rms Argon

Argon

Helium

b g

b g

404

316.

10. Two large vertical and parallel metal plates having a separation of 1 cm are connected to a DC voltagesource of potential difference X. A Proton is released at rest midway between the two plates. It is foundto move at 45 0 to the vertical JUST after release. Then X is nearly

(A) 1 10 5 V (B) 1 10 7 V (C) 1 10 9 V (D) 1 10 10 V

Sol: [C] As the net force is at 45 0 to the vertical so

1 cm

qE

Mg

= 45

qE mg

or eX d

mg

or X mgd

e

16 10 10 1016 10

27 2

19

.. 1 10

9 V


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SectionII (Multiple Correct Answer Type)

This section contains 5 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D), out of which ONE or MORE THAN ONE is / are correct.

11. A cubical region of side a has its centre at the origin. It encloses three fixed point charges, q at

0 4 0 3, / , , a qb g at (0, 0, 0) and q at 0 4, / ,a Ob g. Choose the correct option(s)

(A) The net electric flux crossing the plane x = + a/ 2 is equal to the net electric flux crossing the plane x = a /2.

(B) The net electric flux crossing the plane y = +a /2 is more than the net electric flux crossing the plane y = a /2.

(C) The net electric flux crossing the entire region isq 0

(D) The net electric flux crossing the plane z = +a/2 is equal to the net electric flux crossing the plane

x = + a/2.

Sol: [A, C, D] The given distribution is symmetric about planes xa

2

and z a

2

. Further the distribution

is separately symmetric about planes ya

2

. The total charge enclosed is q so accordingly all options

are correct.


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12. For the resistance network shown in the figure, choose the correct option (s)

(A) The current through PQ is zero (B) I A1 3

(C) The potential at S is less than that at Q (D) I A2 2

Sol: [A, B, C, D] Due to the symmtry potentials V V P Q and V V S T

So current through PQ is zero.

I 2 I 3

6

12

Further I V

1

126 12

3 ||b g A and I

V 2

126

2 A

Across each resistor potential decreases by 4V in the directin of current so V V S Q

P

Q

S

T


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13. A small block of mass of 0.1 kg lies on a fixed inclined plane PQ which makes an angle with thehorizontal. A horizontal force of 1 N acts on the block through its center of mass as shown in the figure.

The block remains stationary if (take g = 10 m/s2

)

(A) 45(B) 45 and a fricitional force acts on the block towards P.

(C) 45 and a frictional force acts on the block towards Q.

(D) 45

and a fricitional force acts on the block towards Q.

Sol: [A, C]

1N

1N

The resultant of weight (1 N dowanrd) and applied force (1N) is

F net 1bgb gsin cos

For 450 , F net is down the incline so friction acts towards Q where as for 450 , F net is up the incline

so friction acts towards P.

For 450 , the block remains at rest without friction


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14. Consider the motion of a positive point charge in a region where there are simultaneous uniform electric

and magnetic fields E E j0 and B B j0 . At time t = 0, this charge has velocity v in the x - y plane,making an angle with the x axis. Which of the following option(s) is(are) correct for time t > 0.(A) If 00 , the charge moves in a circular path in the x-z plane(B) If 00 , the charge undergoes helical motion with constant pitch along the y- axis.(C) If 100 , the charge undergoes helical motion with its pitch increasing with time, along the y-axis.(D) If 90 , the charge undergoes linear but accelerated motion along the y-axis .

Sol: [C, D] y

x

v

E B

For 0 900 0 , charge particle, undergoes helical motion with increasing pitch. For 900 , chargeundergoes linear but accelerated motion along the y-axis.

15. A person blows into open-end of a long pipe. As a result, a high-pressure pulse of air travels down the pipe. When this pulse reaches the other end of the pipe

(A) a high-pressure pulse starts traveling up the pipe, if the other end of the pipe is open.(B) a low-pressure pulse starts traveling up the pipe, if the other end of the pipe is open(C) a low-pressure pulse starts traveling up the pipe, if the other end of the pipe is closed(D) a high-pressure pulse starts traveling up the pipe, if the other end of the pipe is closed.

Sol: [B, D] Pressure wave undergoes a phase charge of rad on refletion from open end and no phase changeoccurs on reflection from closed end. Accordingly B, D are correct.


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Section III (Integer Answer Type)

This section contains 5 questions . The answer to each of the questions is a single-digit integer , ranging from0 to 9. (both enclusive)

16. A ninfinitely long solid cylinder of radius R has a uniform volume charge density . It has a spherical cavityof radius R/2 with its centre on the axis of the cylinder, as shown in the figure. The magnitude of the electricfield at the point P, which is at a distance 2R from the axis of the cylinder, is given by the expression

2316 0

Rk . The value of k is

Sol. [6] Let us assume that the spherical cavity is filled with same charge density then

E E E complete cylinder spherical cavity

E Q R

Q R

10

2

022 2 4 2 b g b g

R R

R

R

2

0

3

022 2

43 8

4 4b g

LNMOQP

R R R k 0 0 0

14

196

2396

2316

k 6


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17. A cylindrical cavity of diameter a exists inside a cylinder of diameter 2 a as shown in the figure. Both thecylinder and the cavity are infinitely long. A uniform current density J flows along the length. If the magni-

tude of the magnetic field at the point P is given by N aJ 12 0

, then the value of N is

Sol: [5] Let us assume that cylindrical cavity is also carrying current with same current density

B B Bnet complete cylinder cavity

or Ba Ja

aJ

a

L

NMM

O

QPPF

H GI K J

0

2 0

2

2

4

232

( )

LNM

OQP0

12

112

Ja 512

0Ja


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18. A lamina is made by removing a small disc of diameter 2R from a bigger disc of uniform mass density andradius 2R, as shown in the figure. The moment of inertia of this lamina about axes passing through O and

P is I 0 and I P , respectively. Both these axes are perpendicular to the plane of the lamina. The ratioII

P

0

to the nearest integer is

Sol: [2] Let us assume that the smaller disk is also filled with same surface mass density. Then as mass is propotional to area so if

m mbO then mm

S D4

I m R m R

R0

2 222

2 4 2

LNM

OQP

b g F H G

I K J mR

mR22

238

138

I mR

Rm R

R P ' LNMM

OQPP

LNM

OQP

22

24 2

52

22

2b g b g F H GI K J mR

2 6118

378

2mR

II

P

O

37 813 8

2 8 3//

. ~


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19. A circular wire loop of radius R is placed in the x-y plane centered at the origin O. A square loop of side

a a R

b ghaving two turns is placed with its center at z R 3 along the axis of the circular wire loop,

as shown in figure. The plane of the square loop makes an angle of 45 with respect to the z-axis. If the

mutual inductance between the loops is given by 02

22a

R p/, then the value of p is

Sol: [7] The mutual inductance

M N B A

iS C S

e

cos 45b g

Here N s = no of turns in square coil and B C be the magnetic field of circular coil at the position of squarecoil.

m

22 3

12

0 2

2 2 3 22

bg c h i R

R aa

i

C

C

/

~ ~/

02

02

7 28 2 2

aR

a

R b g p = 7


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20. A proton is fired from very far away towards a nucleus with charge Q e120 , where e is the electroniccharge. It makes a closest approach of 10 fm to the nucleus. The de Broglie wavelength (in units of fm) of

the proton at its start is (take the proton mass, m kg p 5 3 1027/b g ; h e J s C/ . . / 4 2 10 15 ;

14

9 100

9 m F/ ; 1 fm m10 15 )

Sol: [7] ve

Q

At closest approach U K e

kQer

pm

h

mmin

2

22 2

h r mkQe

he

r mk

2

2 2 120min min

4 2 10

10 10 3

2 5 10 9 10 120

1515

27 9.

4 2 10 1015. 7 fm

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2012 IIT Physics Paper01 Soln