IIT_PAPER-I_2012

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SRIGAYATRI EUDCATIONAL INSTITUTIONS

Hyderabad Nalgonda Karimnagar Vijayawada Guntur Vizag Kadapa Tirupati


PART I : PHYSICS

SECTION I : Single Correct Answer Type

This section contains 10 multiple choice questions. Each question has fourcho ices (A), (B), (C) and (D) ou t of which ONLY ONE is correct.

1. Three very large plates of same area are kept parallel and close to each other. They are

considered as ideal black surfaces and have very high theermal conductivity. The first and third

plates are maintained at temperatures 2T and 3T respectively. The temperature of the middle(i.e. second) plate under steady state condition is

(A)

1

465

2T

(B)

1

497

4T

(C)

1

497

2T

(D) 1

497 T

Sol. (C)

Under steady state condition, temperature0

T of middle plate remains constant.

Hence heat received by second plate = heat emitted by it

44 4 4 4 4 4 4

0 0 03 27 2 3 2T T T T T

40

972

T T

2. Consider a thin spherical shell of radius r which its centre at the origin, carrying uniform positive surface

charge density. The variation of the magnitude of the electric field E r

and the electric potential V r

with the distance r from the centre, is best represented by which graph?

(A) (B)

(C) (D)

Sol. (D)

Inside a shell of uniform charge, the electric field is zero. Outside it varies as 2

1 E

r .

Inside the shell of charge, potential is constant and outside1

V r

.

Hence (D0 is the correct answer.

3. 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)

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Since D

d

R G B R G B

4. Two large vertical and parallel metal plates having a separation of 1 cm are connected to a DC voltage

source of potential difference X. A proton is released at rest midway between the two plates. It is found

to move at 045 to the vertical JUST after release. Then X is nearly

(A) 51 10 V

(B) 71 10 V

(C) 91 10 V

(D) 101 10 V

Sol. (C) There are two forces acting on the particle

(1) Electrical force =e

F and (2) Gravitational force = mg

As it moves at 045 hence

eF mg

. X

e mgd

27 2

19

1.6 10 10 10

1.6 10

d X mg

e

9

1.0 10 volts.

5. A bi-convex lens is formed with two thin plano-convex lenses as shown in the figure. Refractive index nof 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) 280.0cm (B) 40.0cm (C) 21.5cm (D) 13.3cm

Key B

Sol. Say the focal length of first half lens is1 f and that of second half is

2 f

1

1 1 11.5 1

14 f

1 28 f cm

Similarly2 70 f cm

1 2

1 1 1 1 1

28 70F f f F = 20 cm

1 1 1

V u F

1 1 1

40 20V

V = 40 cm

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6. 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 undeergoing circular motion in the x y plane with centre at O and constant angular speed . If

the angular momentum of the system, calculated about O and P are denoted by0 L

and

P L

respectively,,

then

(A)O L

and

P L

do not vary with time. (B)O L

varies with time while

P L

remains constant

(C)O L

remains constant while

P L

varies with time (D)O L

and

P L

both vary with time

Sol. (C)

The resultant force (of tension and wieght) passes through the point O. Therefore torque about O is zero.

00 0

0d L

L

dt

constant vector

The torque of resultant force about P varies in direction as the mass rotates in a circle.

HenceP L

is not a constant vector

7. 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 rms speeeds

helium

argon

rms

rms

v

v

is

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

2

1

Helium

Argon

rms

rms

V M

V M

3rms

RT V

M

4010 3.16

4

8. A thin uniform rod, pivoted at O, is rotating in the horizontal plane with cosntant angular speed , as

shown in the figure. At time 0t , a small insect starts from O and moves with constant speed v with

respect to the rod towards the other end. It reaches the end of the rod at t T and stops. The angular

speed of the system remains throughout. The magnitude of the torque

on the system about O, as

a function of time is best represented by which plot?

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(A) (B)

(C) (D)

Sol. (B)

J I

secrod in t

I I

2 2

rod

I mv t

22

dJ mv t

dt

t When insect stops, I becomes constant, hence J

0 . Hence B is the correct answer

9. In the determination of Young’s modulus 2

4 MLgY

ld

by using Searle’s method, a wire of length 2 L m

and diameter 0.5d mm is used. For a load M = 2.5 kg, an extension0.25l

mm in the length of the

wire is observed. Quantities d and l are measured using a screw gauge and a micrometer, respectively..

They have the same pitch of 0.5 mm. The number of divisions on their circular scale is 100. The contributionsto the maximum probable error of the Y measurement

(A) Due to the error 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)

2Y l d

Y l d

, , have fixed values M L g

pitch

no. of divisions an circular scalel 30.5

5 10100

l mm

35 10d l mm

Nowl

l

relative error in

35 10

0.25

mml

mm

1

50

l

l

Similarly

35 10 1

6.5 100

d mn

d mn

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10. 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 0t . It then executes simple harmonic motion with angular frequency

/ 3

rad s

. Simultaneously at 0t , a small pebble is projected with speed v from point P at an angle

of 045 as shown in the figure. Point P is at a horizontal distance of 10 m from O. If the pebble hits the

block at 1t s , the value of v is (take 210 / g m s )

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

Sol. (A)

For the block,

2 26

3

T sw

61.5

4 4

T s

Therefore in 1 second it travels a distance less than A.Let cos x A wt for the block

10.2cos 1 0.2 0.1

3 2 x m

Therefore when the pebble hits the block, the block is at a ditance of 4.9 0.2 0.1m m m

5m from 0.

Now time of flight = T = 1 sec.

0cos 45 1V Range = 10 5 5 R m m

5 2 50 / V m s

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SECTION II : Multiple Correct Answer(s) Type

This section contains 5 multiple choice questions. Each question has fourchoices (A), (B), (C) and (D) out of which ONE or M ORE are correct

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

(A) The current through PQ is zero (B)1 3 I A

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

Sol. (A,B,C,D)Due to symmetry in the circuit P and Q are at same potential also S and T are at same potential.

There is no current in PQ. Hence A is correct.

4 AB R

1

123

4 I A

also,2 3

6 12 I I 2 3

2 I I

2 3 1 22 I I I I A

2 2 I A

Q PV V & Kirchhoff’s rule 4P S

V V

4P S

V V

P Q SV V V

12. A person below into open-end of a long pipe. As a result, a high-pressure pulse of air travels down thepipe. 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 travelling 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

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Sol. (B,D)

Since at open end pressure node is formed hence high pressure pulse is relfected as low pressure pulse

B is correct & A is wrong

At closed end pressure antinode is formed.Hence high pressure pulse is relfected as high pressure pulse ‘D’ is correct & C is wrong

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

and magnetic fields0ˆ E E j

and

0ˆ B B j

. At time 0t , 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 0t ?

(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 010 , the charge undergoes helical motion with its pitch increasing with time, along the y-axis

(D) If 090 , the charge undergoes lilnear but accelerated motion along the y-axis.

Sol. (A, C, D)

If 0 , ˆV Vi

, = 0, 0el

F ; 0magF

If 0 ˆ ˆ x yV V i V j , yV is varying due to ˆ E and x

V gives circular path in zx-plane.

helical path with increasing pitchAt 0

90 , 0mag

F ; but 0el

F accelerated linear motion.

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

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

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

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

/ 2 y a .

(C) The net electric flux crossing the entire region is0

q

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

/ 2 x a Sol. (A,C,D)

As q is placed symmetrically w.r.t. origin

flux through y z planes or / 2 x a equal to flux through / 2 x a . Hence A is correct

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As thein

Q q 0 0

Net

q

. Hence C is correct

As the planes at / 2 z a and / 2 z a are symmetric w.r.t. charge distribution.

Flux at planes / 2 Z a Flux at plane / 2 z a . Hence D is correct.

15. A small block of mass of 0.1 kg lies on a fixed inclined plane PQ which makes an angle with the

horizontal. A horizontal force of 1 N acts on the block through its centre of mass as shown in the figure.

The block remains stationary if 2take g = 10m/s

(A) 0=45

(B) 0>45 and a frictional force acts on the block towards P..

(C) 0>45 and a frictional force acts on the block towards Q.

(D) 0<45 and a frictional force acts on the block towards Q.

Sol. (A,C)

From the figure, cos sin N mg or sin cos N For 0

45 : sin cos Block is stationary..

Hence option (A) is correct

For 045 : sin cos Block will be stationary if frictional force acts towards Q.

Hence (C) is correct

If 045 ,cos sin Block will be stationary if frictional force acts towards P..

Hence option (D) is wrong

SECTION III : Integer Answer Type

This section contains 5 questions. The question to each question is a singledigit integer, ranging from 0 to 9 (both inclusive)

16. A cylindrical cavity of diameter a exists inside a cylinder of diameter 2a as shown in the figure. Both the

cylinder and the cavity are infinitely long. A uniform current density J flows along the length. If the magnitude

of the magnetic field at the point P is given by 012

N aJ , then the value of N is

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Sol. (5)

Magnetic field at P (B) = 1 2 B B

1 B Magnetic field due to whole cylinder without cavity

2 B Magnetic field due to cavity assuming carrying current in opposite direction

01

2

Ja B

and

02

12

Ja B

0

5

12

Ja B

5 N

17. A proton is fired from very far away towards a nucleus with charge 120Q e , 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, 275 / 3 10 ; pm kg 15 / 4.2 10 . / ;h e J s C

9

0

19 10 / ;

4m F

151 10 fm m )

Sol. (7)

KE of proton = Potential energy of system at distance of closest approach

2

0 0

120

2 4

e e p

m r

2

0 0

2 120

4

m e p

r

de-Broglie wavelengthh

p 2

0 0

2 120

4

h

m e

r

Substituting the values 157 10 m

7

18. 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 having two turns is placed with its center at 3 z R along the axis of the circular wire loop,

as shown in figure. The plane of the square loop makes an angle of 045 with respect to the z-axis. If the

mutual inductance between the loops is given by

2

0

/ 22 p

a

R

, then the value of p is

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Sol. (7)2

0

3

2 1

4 8

i r B

r

0

16

i B

20 1. 2

16 2

ia

r

= M.i.

2 2

0 0

7 / 228 2

a a M

r r

Hence 7P 19. An infinitely long solid cylinder of radius R has a uniform volume charge density . It has a spherical

cavity of radius R/2 with its centre on the axis of the cylinder. as shown in the figure. The magnitude of the

electric field at the point P, which is at a distance 2r from the axis of the cylinder, is givenby the expression

0

23

16

R

k

. The value of k is

Sol. (6)

2

0 0

1 1.

2 2 4 4

P

q E

R R

2 R and

3

6

RQ

0

236

96P

R E K

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20. A lamina is made by removing a small disc of diameter 2R from a bigger disc of uniform mass density and

radius 2R, as shown in the figure. The moment of intertia of this lamina about axes passing through O and

P isO

I andP I , respectively. Both these axes are perpendicular to the plane of the lamina. The ratio

P

O

I

I

to the nearest integer is

Sol. (3)

4

0

13

2

R I

4

37

2P I R

0

37 /

13P

I I 3

PART - II : CHEMISTRY

SECTION - I : Single Correct Answer Type

This section contains 10 multiple choice questions. Each question has four choices (A), (B), (C) and

(D) out of which ONLY ONE is correct.

21. For one mole of a van der Waals gas when b = 0 and T = 300 K, the PV vs1

Vplot is shown below. The

value of the van der Waals constant a (atm. liter2 mol–2) is

A) 1.0 B) 4.5 C) 1.5 D) 3.0

Key: C

Sol. For 1 mole vandewaal’s euqations is

2

aP V b RT

V

b = 0

2

aP V RT

V

aPV RT

V

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1.PV RT aV

Slope =2 1

2 1

1.5 y y

a x x

1.5a

22. The number of optically active products obtained from the complete ozonolysis of the given compound is

A) 0 B) 1 C) 2 D) 4

Key : A

Sol. Ozonolysis products

3

3

|

3 3|

CH H

H CH

CH CHO OCH C CHO OCH C CHO CH CHO

None of the products are asymmetric hence ans = zero23. The colour of light absorbed by an aqueous solution of CuSO

4is

A) orange - red B) blue - green C) yellow D) violet

Key: A

Sol: Orange - Red

24. Which ordering of compounds is according to the decreasing order of the oxidation state of nitrogen ?

A)3 4 2

HNO , NO, NH C , Nl B)3 2 4

HNO , NO, N , NH Cl

C) 3 4 2HNO , NH C , NO, Nl D) 3 4 2NO, HNO , NH C , Nl

Key: B

Sol.

5 2 0 3

3 2 4 HNO NO N N H Cl

25. As per IUPAC nomeclature, the name of the complex 2 3 34 2Co H O NH C l is

A) Tetraaquadiaminecobalt (III) chloride B) Tetraaquadiamminecobalt (III) chloride

C) Diaminetetraaquacobalt (III) chloride D) Diamminetetraaquacobalt (III) chloride

Key : D

Sol. Diamminetetraaquacobalt (III) chloride

26. The carboxyl functional group (-COOH) is present in

A) picric acid B) barbituric acid C) ascorbic acid D) aspirin

Key: D

Sol.

O

COOH

O

C CH3

Barbituric acid O=CNH-C

O

O

NH-C 2CH

Picric acid

2 NO

2 NO

2 NO

OH

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27. The kinetic energy of an electron in the second Bohr orbit of a hydrogen atom is [a0is Bohr radius]

A)

2

2 2

0

h

4 ma B)

2

2 2

0

h

16 ma C)

2

2 2

0

h

32 ma D)

2

2 2

0

h

64 ma

Key: C

Sol.21

.2

K E mv and also2

nhmvr

2nhmv

r ,

2 22 2

2 24n hm v

r ,

2 22

2 24n hmv

mr

2 2 2 2 2

22 2 2 2 2 220 00

1 1 4

2 4 2 32 324 4

n h h hKE

mr ma mam a

28. In allene (C

3H

4), the type(s) of hybridisation of the carbon atoms is (are)

A) sp and sp3 B) sp and sp2 C) only sp2 D) sp2 and sp3

Key : B

Sol. CH2

= C= CH2

sp2 sp sp2

29. A compound MpXq has cubic close packing (ccp) arrangement of X. Its unit cell structure is shownbelow. The empirical formula of the compound is

A) MX B) MX2

C) M2X D) M

5X

14

Key: BSol. M is at M edge centers = 4 × ¼ = M

M is at body center = 1×1 = M Total 2M

X at 8 corners and 6 face centers =1 1

8 6 1 3 48 2

X

Formuale = M2X

4= MX

2

30. The number of aldol reaction(s) that occurs in the given transformation is

A) 1 B) 2 C) 3 D) 4

Key : C

3 Aldol condensations as there are 3– hydrogens

31. Choose the correct reason(s) for the stability of the lyophobic colloidal particles.

A) preferential adsorption of ions on their surface from the solution

B) preferential adsorption of solvent on their surface from the solution

C) attraction between different particles having opposite charges on their surface

D) potential diffrence between the fixed layer and the diffused layer of ooposite charges around the

colloidal particles.

Key : A & D [Conceptual]

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32. For an ideal gas, consider of P - V work in going from an initial state X to the final state Z. The final state

Z can be reached by either of the two paths shown in the figure. Which of the following choice(s) is (are)

correct ? [Take S as change in entropy and w as work done]

A) x z x y y zS S S B) x z x y y z

w w w

C) x y z x yw w D) x y z x y

S S

Key : A & C

Entropy changes from x y and y z x y y zs s s

Work done from y z 0 x y z x yw w

33. Which of the following hydrogen halides react(s) with AgNO3(aq) to give a precipitate that dissolves in

Na2S2O3(aq) ?A) HCl B) HF C) HBr D) HI

Key : AgF is soluble in H2O hence A,C,D

34. Identify the binary mixture(s) that can be separated into individual compounds, by differential extraction,

as shown in the given scheme.

A)6 5

C H OH and6 5

C H COOH B)6 5

C H COOH and6 5 2

C H CH OH

C) 6 5 2 6 5C H CH OH and C H OH D) 6 5 2 6 5 2C H CH OH and C H CH COOH

Key : B,D

35. Which of the following molecules, in pure form, is (are) unstable at room temperature ?

A) B) C) D)

Key : B & C are anti aromatic and

36. The periodic table consists of 18 groups. An isotope of copper, on bombardment with protons, undergoes

a nuclear reaction yielding element X as shown below. To which group, element X belongs in the periodic

table ?

63 1 1 129 1 0 1Cu H 6 n 2 H X

Key : 63 1 1 4 129 1 0 2 1Cu H 6 n He 2 H 52

26Fe = 8

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37. An organic compound undergoes first - order decomposition. The time taken for its decomposition to 1/

8 and 1/10 of its initial concentration are t1/8

and t1/10

respectively. What is the value of 1/ 8

1/10

[t ]10?

[t ]

(take log10

2 = 0.3)

Key :

31/8 1/8

1/10 1/10

t tlog 8 log(2)10 10 9

t log10 t 1

38. 29.2% (w/w) HCl stock solution has a density of 1.25 g mL –1. The molecular weight of HCl is 36.5 g

mol–1. The volume (mL) of stock solution required to prepare a 200 mL solution of

0.4 M HCl is

Key :10dx 10 1.25 29.2

M 10GMW 36.5

V1M

1= V

2M

2

1V 10 200 0.4

V1

= 8ml

39. The substituents R1

and R2

for nine peptides are listed in the table given below. How many of these

peptides are positively charged at pH = 7.0 ?

3

| | | |

1 2

H N CH CO NH CH CO NH CH CO NH CH CO O

H R R H

Key : 4 [IV, VI, VIII, IX]

40. When the following aldohexose exists in its D-configuration, the total number of stereoisomers in its

pyranose form is

|

2

|

|

|

2

CHO

CH

CHOH

|

CHOH

CHOH

CH OH

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Key : 8

O

OH

OH

OH

H

H H

H

H

*

* *

*

H

HOH C2

Total no.of stereoisomers = 24 = 16No.of stereoisomers of D - Pyranose = 8

PART - II : MATHEMATICS

SINGLE ANSWER TYPE QUESTIONS :

41. Let 2 cos , 0

, ,

0, 0

x x f x x x

x

then f is(A) differentiable both at 0 x and at 2 x (B) differentiable at 0 x but not differentiable at 2 x (C) not differentiable at 0 x but differentiable at 2 x (D) differentiable neither at 0 x nor at 2 x

Sol :

0

0' 0 lim

0 x

f x f f

x

2

0

cos

lim 0 x

x x

finite x

2

0

cos

lim 0 x

x x

finite x

0.

Similarly at 2 x .

42. The function : 0,3 1, 29 , f defined by 3 22 15 36 1, f x x x x is

(A) one-one and onto (B) onto but not one-one

(C) one-one but not onto (D) neither one-one nor onto

Sol : 3 22 15 36 1 f x x x x

2' 6 30 36 f x x x 6 2 3 x x

neither increasing nor decreasing in 0,3

f is not one - one

Range codomain onto.

43. The ellipse

2 2

1 : 19 4

x y E is inscribed in a rectangle R whose sides are parallel to the coordinate axes.

Another ellipse2

E passing through the point 0,4 circumscribes the rectangle R. The eccentricity of the

ellipse2

E is.

(A)2

2(B)

3

2(C)

1

2(D)

3

4

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Sol:

2 2

2 2: 1

16

x y E

a .

It passes 3,4

2

2

9 41 12

16a

a

2 2 2

2 2: 1 112 16

x y b E e

a

12 11 .

16 2e

44. The point P is the intersection of the straight line joining the points 2,3,5Q and 1, 1, 4 R with the

plane 5 4 1. x y z If S is the foot of the perpendicular drawn from the point 2,1,4T to ,QR then

the length of the line segment PS is

(A)1

2(B) 2 (C) 2 (D) 2 2

Sol: Any point on QR is 2 ,3 4 ,5t t t

it lies on plane2

3t .

it is perpendicular to line joining T to this9 1 9 1

, ,2 2 2 2 2 2

4 1 13 3 9, , ,1,

3 3 3 2 2P S

1

2PS

45. The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line

4 5 20 x y to the circle 2 2 9 x y is

(A) 2 220 36 45 0 x y x y (B) 2 220 36 45 0 x y x y

(C) 2 236 20 45 0 x y x y (D) 2 236 20 45 0 x y x y

Sol: Let ,

Then 4 5 20 1

Chord of contact of w.r.t circle 9 2 x y

Equation of chord in terms of midpoints is

2 2

1 11 1 13 xx yy x y

2 & 3 same

1

2 2

1 1

9 x

x y

1

2 2

1 1

9 y

x y

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substitute in 1 we get locus as

2 2

1 1 1 120 36 45 0 x y x y

46. Let ijP a be a 3 3 matrix and let ,ijbQ where 2i j

ij ijb a

for 1 , 3.i j If the determi-

nant of P is 2, then the determinant of the matrix Q is.

(A) 102 (B) 112 (C) 122 (D) 132

Sol: 2P

2 3 4

11 12 13

3 4 5

21 22 23

4 5 6

31 32 33

2 2 2

2 2 2

2 2 2

a a a

Q a a a

a a a

2 3 4 132 .2 .2 .2.4. 2Q P

47. The integral

92

2sec

sec tan

xdx

x x equals (for some arbitrary constant K)

(A)

2

11 2

1 1 1sec tan

11 7sec tan x x K

x x

(B)

2

11 2

1 1 1sec tan

11 7sec tan x x K

x x

(C)

2

11 2

1 1 1sec tan

11 7sec tan x x K

x x

(D)

2

11 2

1 1 1sec tan

11 7sec tan x x K

x x

Sol: sec tan x x t

1sec tan x x

t

1 1sec

2 x t

t

sec

dt

x dx t 48. The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that

each person gets at least one ball is

(A) 75 (B) 150 (C) 210 (D) 243

Sol: Number of onto functions from set having 5 elements to set having 3 elements

5 3 5 3

1 23 2 243 96 3 150c c

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49. If

2 14,

1lim

x

x xax b

x

then

(A) 1, 4a b (B) 1, 4a b (C) 2, 3a b (D) 2, 3a b

Sol:

2 21lim 4

1 x

x x ax ax bx b

x

Equating the coeff of

2

0 x and coeff 4 x 1 0 1a a

1 4 4.a b b

50. Let z be a complex number such that the imaginary part of z is non zero and 21 za z is real. Then

a cannot take the value

(A) 1 (B)1

3(C)

1

2(D)

3

4

Sol: If Z x iy then 2 2 1 2 .a x y x i xy y real

12 0 0.2

xy y x y

23 33 4

4 4a y y a a and 0 y

MULTIPLE TYPE QUESTIONS :

51. Let S be the area of the region enclosed by2 , 0, 0,

x y e y x

and 1. x Then

(A)1

Se

(B)1

1Se

(C)1 1

14

Se

(D)

1 1 11

2 2S

e

Sol: tan 2 secdy

y x x xdx

. .G S is 2cos y x x c

0 0 0 y c

2

cos

x y

x

2

4 8 2

y A

2

2

cos .2 sin'

cos

x x x x y

x

24 2

'3 3 3 3

y D

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52. If y x satisfies the differential equation y y tan 2 sec x x x and 0 0, y then

(A)

2

4 8 2 y

(B)

2

4 18 y

(C)

2

3 9 y

( D )

24 2

3 3 3 3 y

Sol: (A)

1 2 3

1

21 2 3 1 3 1 2 3 1 2 3

C P E E E

P X X P E E E P E E E P E E E P E E E

132 1

88

32

.

(B)Exactly twoengines of theship are functioning

P

X

3 2 11 2 1 3 2 3

3 2 11 2 1 3 2 3 1 2 3

P E E E P E E E P E E E

P E E E P E E E P E E E P E E E

7

8

(C)

11 2 3 2 3 1 2 3

2

1 2 3 1 2 3 1 2 3 1 2 3

P E E E P E E E P E E E P X X


5

8

(D)

1 2 3 1 2 3 1 2 3

1

31 2 3 1 2 3 1 2 3 1 2

P E E E P E E E P E E E P X X


7 32 7

1616 32

53. Tangents are drawn to the hyperbola2 2

1,9 4

x y parallel to the straight line 2 1. x y The

points of contact of the tangents on the hyperbola are

(A )9 1

.2 2 2

(B)9 1

.2 2 2

(C) 3 3, 2 2 (D) 3 3,2 2

Sol: 2cos 1 sin 2sin cos 1

2 co s 1 2 s in

1

c o s s i n2

31 sin

2 , tan 0

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53

2 3

x

1cos .

2o

1 1cos 1

2 2

1 sin 12

52 2 , 0,1, 2

6 6n n n

54. Let , 0, 2 be such that 22cos 1 sin sin tan cot cos 1,2 2

tan 2 0 and3

1 sin .2

Then cannot satisfy..

(A) 02

(B)

4

2 3

(C)

4 3

3 2

(D)

32

2

Sol:21

0. xS e dx

2 x

f x e is decreasing function

1 A S b a me

0 1 B x 2 2 x x x x

2 x x

e e

1

011 xS e dxe

2

11 1 14

2 2

x D f x e x x

1 1 11

2 2S

e

55. A ship is fitted with three engines 1 2, E E and 3. E The engines function independently of each

other with respective probabilities

1 1

,2 4 and

1

.4 For the ship to be operational at least two of

its engines must function. Let X denote the event that the ship is operational and let1 2, X X

and3

X denote respectively the events that the engines1 2, E E and

3 E are functioning. Which

of the following is (are) true ?

(A) 1

3

16

cP X X (B) P [Exactly two engines of the ship are functioning7

8 X

(C) 2

5

16P X X (D) 1

7

16P X X

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Sol: Any line Parallel to 2 1 x y is 2 0 x y k

4 2k

2 4 2 0 x y

Point of contact9 1 9 1

, ,2 2 2 2 2 2

INTEGER TYPE QUESTIONS

56. The value of 3

2

1 1 1 16 log 4 4 4 ....

3 2 3 2 3 2 3 2

is

Sol: 2 2 22 4 0, 8 x y x y y x

solving

4 22 4 0

64 4

y y y y

3

0, 48 256 0 y y y 0,0 2,4 2,0P Q S

Area1

8 4 . .2

sq u

57. Let S be the focus of the parabola 2 8 y x and let PQ be the common chord of the circle

2 2 2 4 0 x y x y and the given parabola. The area of the triangle PQS is

Sol: Let 1 21 3 9 3P x k x x k x x

Then 3

22 33

xP x K x x C

1

1 2 3 63

P K C

4

63

K C

3 2 0 2P K C

4

4 33

K

K

1 3 1 3 0 3 3 9.P x x x P

58. If ,a b and c

are unit vectors satisfying

2 2 29,a b b c c a

then 2 5 5a b c

is

Sol:

2

2

2

2

1 1

1 1 0

1 0 1

1 1

x x if x

x x x f x

x x x

x x x

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IIT_PAPER-I_2012

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