Analysis and Solution for Iit Jee 2012

8/2/2019 Analysis and Solution for Iit Jee 2012

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SOLUTIONS

forforforforfor

IIT-JEE 2012

Time : 3 hrs. Max. Marks: 210

Regd. Office: Aakash Tower, Plot No.-4, Sec-11, MLU, Dwarka, New Delhi-110075Ph.: 011-47623456 Fax : 011-47623472

INSTRUCTIONS

1. The question paper consists of 3 parts (Physics, Chemistry and Mathematics). Each part consists

of three sections.

2. Section I contains 10 multiple choice questions. Each question has four choices (A), (B), (C) and

(D) out of which ONLY ONE is correct.3. Section II contains 5 multiple choice questions. Each question has four choices (A), (B), (C) and

(D) out of which ONE or MORE are correct.

4. Section III contains 5 questions. The answer to each question is a single digit integer, ranging

from 0 to 9 (both inclusive).

DATE : 08/04/2012

PAPER - 1 (Code - 0)

PARTI : 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. 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 thermal conductivity. The first and third plates are

maintained at temperatures 2Tand 3Trespectively. 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

Answer (C)


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Hints : At equilibrium,

Heat absorbed = Heat radiated

A(3T)4 +A(2T)4 = 2AT04

1/4

0

97

2

T T

2. 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 and the electric potential V(r) with

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

(A)

| ( )|E r V r( )

0R r

(B)

| ( )|E r V r( )

0R r

(C)

| ( )|E r V r( )

0R r

(D)

| ( )|E r V r( )

0R r

Answer (D)

Hints : Inside, E(r) = 0, outside 2

1( )E r

r

Inside V(r) = constant, outside 1

( )V rr

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,

Rand

B, respectively. Then,

(A) G

> B

> R

(B) B

> G

> R

(C) R

> B

> G

(D) R

> G

> B

Answer (D)

Hints : As R > G > B

R>

G>

Bas

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

of potential differenceX. A proton is released at rest midway between the two plates. It is found to move at

45 to the vertical JUST after release. ThenXis nearly

(A) 1 105 V (B) 1 107 V (C) 1 109 V (D) 1 1010 V

Answer (C)

Hints : Here, qE= mg

19 27

21.6 10 1.6 10 10

(1 10 )

X

X= 109 V


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5. 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

n = 1.5n = 1.2

R = 14 cm

(A) 280.0 cm (B) 40.0 cm (C) 21.5 cm (D) 13.3 cm

Answer (B)

Hints :

1 1 1 1 11.5 1 1.2 1

14 14f

f= 20 cm

As object distance = 40 cm Image distance = 40 cm

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 undergoing 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 andPare denoted by

OL and

PL respectively, then

P

(A)

OL and

PL do not vary with time

(B)

OL varies with time while

PL remains constant

(C)

OL remains constant while

PL varies with time

(D)

OL and

PL both vary with time

Answer (C)

Hints : The torque about O is zero, but aboutPis non-zero.

AsdL

dt

thus PL

must be a variable.

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 speeds

rms

rms

(helium)

(argon)

v

vis

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

Answer (D)

Hints :

He Ar

Ar He

10 3.16M

M


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8. A thin uniform rod, pivoted at O, is rotating in the horizontal plane with constant angular speed , as shownin the figure. At time t = 0, 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 = Tand 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?

z

Ov

(A)

O Tt

(B)

O Tt

(C)

O Tt

(D)

O Tt

Answer (B)

Hints : ( )d

Idt

22

3

d MLmx

dt

2dx

mxdt

Now, x= vt

t

Finally torque becomes zero.

9. In the determination of Youngs modulus

2

4MLgY

ldby using Searles method, a wire of length L= 2 m

and diameter d = 0.5 mm is used. For a load M = 2.5 kg, an extension l = 0.25 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 contributions to

the maximum probable error of the Ymeasurement

(A) Due to the errors in the measurements ofd and l are the same

(B) Due to the error in the measurement ofd is twice that due to the error in the measurement of l

(C) Due to the error in the measurement ofl is twice that due to the error in the measurement of d

(D) Due to the error in the measurement ofd is four times that due to the error in the measurement of l

Answer (A)

Hints :

2y l d

y l d

2l dy y

l d

=

2y l y d

l d

The two quantities are equal.


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

3rad/s. Simultaneously at t = 0, a small pebble is projected with speed v from point Pat an angle of

45 as shown in the figure. PointPis at a horizontal distance of 10 m from O. If the pebble hits the blockat t = 1 s, the value of vis (takeg= 10 m/s2)

z

10 m

xP

45O

v

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

Answer (A)

Hints : For the block,

2

T = 6 seconds.

For block, the position in 1 second will be

x= 0.2 cost

At t =1 s, x= 0.1 m

At t = 1 s, block will be at a distance 4.9 m + 0.1 m = 5 m from wall.

Range of pebble is also 5 mNow, for pebble, vcos45 (1 sec) = 5 m

v = 5 2 = 50 m/s

SECTION - II

(Multiple Correct Answer(s) Type)

This section contains 5 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of

which ONE OR MORE are correct.

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

I1 QT

P SI2

2 2

4 4

1 1

2

4

12 V

(A) The current throughPQis zero (B) I1

= 3 A

(C) The potential at Sis less than that at Q (D) I2 = 2 A

Answer (A, B, C, D)


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Hints : The network is an extension of balanced Wheatstone bridge, thus current PQand STis zero.

Req

=

11 1

+6 12

= 4

I1

= 3A

Also, I2

=12

6= 2 A

Now, VS

+ 2I2

+ 2I2

4I3

= VQ

VS

VQ

= 4I3

2I2

2I2

= 4 1 2 2 2 2

= 4 V

12. 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

Answer (B, D)

Hints : At open end, a compression is reflected as a rarefaction

At closed end, a compression is reflected as a compression

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

magnetic fields

0 0 and .E E j B B j 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 = 0, the charge moves in a circular path in the x-z plane

(B) If = 0, the charge undergoes helical motion with constant pitch along they-axis

(C) If = 10, 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 they-axis

Answer (C, D)

Hints : For = 10, path is helical.

Since there is an electric field, pitch is increasing.

For = 90, path is straight line as

B will not exert any force.

14. A cubical region of side a has its centre at the origin. It encloses three fixed point charges, q at 0, , 04

a,

+3q at (0, 0, 0) and q at

0, , 04a

. Choose the correct option(s).

a

z

y3qq

x

q


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(A) The net electric flux crossing the plane x= 2

ais equal to the net electric flux crossing the plane x=

2

a

(B) The net electric flux crossing the planey = 2

ais more than the net electric flux crossing the plane

y =

2

a

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

q

(D) The net electric flux crossing the plane z = 2

ais equal to the net electric flux crossing the plane x=

2

a

Answer (A, C, D)

Hints : Due to symmetry.

15. A small block of mass of 0.1 kg lies on a fixed inclined planePQwhich 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 blockremains stationary if (takeg=10 m/s2)

Q

O

P

1 N

(A) = 45

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

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

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

Answer (A, C)

Hints : If 1 cos = 1 sin

Q= 45, Fnet

= 0

If 1 cos > 1 sin

i.e., < 45, friction must be towards P.

If 1 cos < 1 sin

> 45, friction must be towards Q.

1sin

1cos

P

Q


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

magnitude of the magnetic field at the pointPis given by 0 ,12

NaJ then the value ofNis

aP

2a

O

Answer (5)

Hints : B =B1

B2

Here,B1

is0

2

Ja(field due to complete cylinder)

B2

=

2

0

02

3 122

2

aJ

Ja

a

B = 0512

aJ

17. A proton is fired from very far away towards a nucleus with charge Q= 120 e, where e is the electronic

charge. 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, mp

=

275 10 kg;3

h/e = 4.2 1015 J.s/C;

9 15

0

19 10 . m/F; 1 fm 10 m

4)

Answer (7)

Hints :2 2

15 2

0

12024 (10 10 ) 2

e e p hm m

1422 0

2

4 10

2 120

h

m e

4

04 10

2 120

h

e m

=

1415

9 27

10 34.2 10

9 10 2 120 5 10

= 7 1015 m

= 7fm


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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


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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 inertia of this lamina about axes passing through O and

Pis IO

and IP

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

O

I

Ito

the nearest integer is

2R

O2R P

Answer (3)

Hints : Let the mass density be .

IO

=

2 2

2 2(2 ) (2 ) 3 ( ) ( )2 2

R RR R

= 413

2R

IP

=

2 22 2 2 23 ( )(2 ) (2 ) ( 5 )

2 2

R RR R R R

= 437

2R

RatioP

O

I

I=

373

13

PARTII : CHEMISTRY

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.

21. For one mole of a van der Waals gas when b = 0 and T = 300 K, the PV vs. 1/V plot is shown below. The

value of the van der Waals constant 'a' (atm. litre2 mol2) is

24.6

23.1

21.6

20.1

2.0 3.00

1

V (mol litre )

1

PV(litre-a

tmm

ol)

1[Graph not to scale]

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


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Answer (C)

Hints :

2

aP

VV = RT

PV +a

V = RT

PV = RT a

V

So, Plot of (PV) vs

1

Vhas slope a.

slope =

20.1 24.6

1.53

a = 1.5 atm L2 mol2

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

CH CH = CH C CH = CH C CH = CH CH33

H CH3

CH3 H

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

Answer (A)

Hints :

Ozonolysis produces CH3CHO and CH3CH(CHO)2

23. The colour of light absorbed by an aqueous solution of CuSO4

is

(A) Orange-red (B) Blue-green (C) Yellow (D) Violet

Answer (A)

Hints :

Complementary colour of blue is (Orange-red)

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

(A) HNO3, NO, NH

4Cl, N

2(B) HNO

3, NO, N

2, NH

4Cl

(C) HNO3, NH4Cl, NO, N2 (D) NO, HNO3, NH4Cl, N2

Answer (B)

Hints :

3 2 4

( 5) ( 2) (0) ( 3)

H NO , NO , N , NH Cl

25. As per IUPAC nomenclature, the name of the complex [Co(H2O)

4(NH

3)2]Cl

3is

(A) Tetraaquadiaminecobalt (III) chloride

(B) Tetraaquadiamminecobalt (III) chloride

(C) Diaminetetraaquacobalt(III) chloride

(D) Diamminetetraaquacobalt (III) chloride

Answer (D)


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

(A) Picric acid (B) Barbituric acid (C) Ascorbic acid (D) Aspirin

Answer (D)

Hints :

(Picric acid)

(Barbituric acid)

OH

NO2O2N

NO2

,

NH

O

HN

OO

(Ascorbic acid) (Aspirin)

COOH,

OH

O

HO OH

HOO

H

OCCH3

O

27. The kinetic energy of an electron in the second Bohr orbit of a hydrogen atom is [a0

is 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

Answer (C)

Hints :

Centripital force = Coulombic force of attraction

2mV

r=

2

2

KZe

r

V2 =2KZe

mr

V =

nh

2 mr

V2 =

2 2

2 2 2

n h

4 m r

22 2 2 2 20

2 2 2 2 2

a nKZe n h h nr

mr Z Z4 m r 4 mKe

Vn

= 0

hZ

2 ma n

K.E. =

2 22

n 2 2 2

0

1 h Zmv

2 8 ma n; K.E. (for x = 2 and Z = 1) =

2

2 2

0

h

32 ma


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28. In allene (C3H

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

Answer (B)

Hints :

C = C = CH

( )sp

( )sp2

(Allene)

H

H

H

( )sp2

29. A compound MpX

qhas cubic close packing (ccp) arrangement of X. Its unit cell structure is shown below. The

empirical formula of the compound is

M =

X =

(A) MX (B) MX2

(C) M2X (D) M

5X

14

Answer (B)

Hints :

Number of X atoms/ ions per unit cell =1 1

8 6 48 2

Number of M atoms/ions per unit cell =1

1 4 24

Empirical formula of the compound is MX2

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

CH3CHO + 4HCHOconc. aq. NaOH

OH OH

HO OH

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

Answer (C)

Hints :

OH + HCHO/OH3 2 2Aldol Aldol

CH CHO + H CHO HOCH CH CHO HOCH CH CHO2

CH OH

|2

+ HCHO/OHAldol

HOCH C CHO2

CH OH

|

|

CH OH

2

2

+ HCHO/OHCannizzaro

HOCH C CH OH2 2

CH OH

|

|

CH OH

2

2

Number of aldol reactions = 3


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SECTION - II




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 absorption of solvent on their surface from the solution

(C) Attraction between different particles having opposite charges on their surface

(D) Potential difference between the fixed layer and the diffused layer of opposite charges around the colloidal

particles

Answer (A, D)

Hints :

Lyophobic colloids carry either +ve or ve charge. In order to acquire stability the colloidal particles attract

oppositely charged ions forming a diffused layer. A potential is developed between fixed layer and diffused layer

called zeta potential. Higher the value of zeta potential higher will be the stability of lyophobic colloid.

32. For an ideal gas, consider only 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]

P(atmosphere)

V(liter)

YX

Z

(A) S S Sx z x y y z+ (B) w w wx z x y y z+

(C) w wx y z x y (D) S Sx y z x y

Answer (A, C)

Hints :

Entropy is a state function. In this diagram initial state is X and final state is Z.

Therefore, SX Z = SX Y + SY Z

Work is a path function. Therefore WX Y Z

= WX Y

33. Which of the following hydrogen halides react(s) with AgNO3

(aq) to give a precipitate that dissolves in

Na2S

2O

3(aq)?

(A) HCl (B) HF

(C) HBr (D) HI

Answer (A, C, D)

Hints :

Na2S2O3 + AgX Ag2S2O32 2 32Na S O

Na2[Ag(S2O3)3][X = Cl and Br]


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

shown in the given scheme.

Binary mixture containing

Compound 1 and Compound 2

NaOH(aq)

NaHCO (aq)3

Compund 1

Compund 1

+ Compund 2

Compund 2+

(A) C6H

5OH and C

6H

5COOH (B) C

6H

5COOH and C

6H

5CH

2OH

(C) C6H

5CH

2OH and C

6H

5OH (D) C

6H

5CH

2OH and C

6H

5CH

2COOH

Answer (B, D)

Hints :

C6H

5COOH + NaOH C

6H

5COONa + H

2O ; C

6H

5COOH + NaHCO

3 C

6H

5COONa + CO

2+ H

2O

C6H

5CH

2 OH + NaOH No reaction ; C

6H

5CH

2OH + NaHCO

3 No reaction

C6H5CH2COOH + NaOH C6H5CH2 COONa + H2O ; C6H5CH2COOH + NaHCO3 C6H5CH2COONa

+ CO2

+ H2O

C6H

5CH

2OH + NaOH No reaction ; C

6H

5CH

2OH + NaHCO

3 No reaction

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

(A) (B)

(C)

O

(D)

O

Answer (B, C)

Hints :

The compound is antiaromatic and hence unstable at room temperature. The other compound

O O

+

is also unstable at room temperature due to partial positive charge at carbonyl C-atom

SECTION - III(Integer Answer Type)



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

nuclear reaction yielding elementXas shown below. To which group, element Xbelongs in the periodic table?

63 1 1 129 1 0 1

Cu H 6 n 2 H X

Answer (8)

Hints :

63 1 1 4 1 5229 1 0 2 1 26Cu H 6 n ( He) 2 H X( Fe)

Fe belongs to group-8.


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37. An organic compound undergoes first-order decomposition. The time taken for its decomposition to1

8and

1

10

of its initial concentration are 18

tand 1

10

trespectively. What is the value of

1

8

1

10

t

10 ?

t

(take log10

2 = 0.3)

Answer (9)

Hints :

1/8 102.303 1

t log ...(i)k 1 8

1/10 102.303 1

t log ...(ii)k 1 10

1/8

1/10

[t ] log810 10 3 0.3 10 9

[t ] log10

38. 29.2% (w/w) HCl stock solution has a density of 1.25 g mL1. The molecular weight of HCl is 36.5 g mol1.

The volume (mL) of stock solution required to prepare a 200 mL solution of 0.4 M HCl is

Answer (8)

Hints :

Molarity of HCl =1.25 29.2 1000

10(M)100 1 36.5

V 10 = 200 0.4

V = 8 mL

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?

H N3 CH CO NH CH NH CH CO NH CH COOCO

HR2

R1

H

Peptide R1 R2

I H H

II H CH3

III CH2COOH HIV CH2CONH2 (CH2)4NH2

V CH2CONH2 CH2CONH2

VI (CH2)4NH2 (CH2)4NH2

VII CH2COOH CH2CONH2

VIII CH2OH (CH2)4NH2

IX (CH2)4NH2 CH3

Answer (4)

Hints :

When any group in R1

and R2

is basic group then amino acid is positively charged at pH = 7.0. So, answers

are peptide IV, VI, VIII and IX.


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

form is

CHO

CH2

CHOH

CHOH

CHOH

CH OH2

Answer (8)

Hints :

Given

CHO

CH2

CHOH

CHOH

CHOH

CH OH2

has D configuration

H

CH OH2

O

HH

HO

HO

OH*

*

*C* are chiral carbon atoms. Hence total stereoisomers are 8.

PARTIII : MATHEMATICS

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 whichONLY ONE is correct.

41. Letf(x)

2 cos , 0,

0, 0

x xxx

x

thenfis

(A) Differentiable both at x= 0 and at x= 2

(B) Differentiable at x= 0 but not differentiable at x= 2

(C) Not differentiable at x= 0 but differentiable at x= 2

(D) Differentiable neither at x= 0 nor at x= 2Answer (B)


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Hints :

2 cos , 0( )

0 , 0

x xf x xx

x

Letp(x) = x2 and ( ) cosg xx

Nowg(x) is a continuous non-differentiable at x= 0, and 2 but sincep(0) = 0, so,p(x).g(x) is non-differentiable

at x= 2.

42. The functionf: [0, 3] [1, 29], defined byf(x) = 2x3 15x2 + 36x+1, is

(A) One-one and onto

(B) Onto but not one-one

(C) One-one but not onto

(D) Neither one-one nor onto

Answer (B)Hints :

3 2( ) 2 15 36 1f x x x x

f(x) = 6x2 30x+ 36

= 6(x2 5x+ 6)

= 6(x 2) (x 3)

Clearly the derivative changes sign in [0, 3] so, fis NOT one-one.

Now the function is increasing in [0, 2] and decreasing in [2, 3]

Also, f(0) = 1

f(2) = 29

f(3) = 8

Hence the range is [1, 29] and so, the function is onto.

43. The ellipse E1

: 2 2

19 4

x yis inscribed in a rectangle R whose sides are parallel to the coordinate axes.

Another ellipse E2

passing through the point (0, 4) circumscribes the rectangle R. The eccentricity of the ellipse

E2

is

(A)2

2(B)

3

2(C)

1

2(D)

3

4

Answer (C)Hints :

Let the equation ofE2

be

2 2

2 21

x y

a b

Clearly the pointPhas coordinates (3, 2)

(0,4)

P

E2

E1

So, (3, 2) and (0, 4) satisfy E2

22

161 16b

b

and 2 29 4

1a b


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22

9 312

4a

a

12 1

116 2

e

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

plane 5x 4y z = 1. IfSis the foot of the perpendicular drwan from the point T(2, 1, 4) to QR, then the

length of the line segmentPSis

(A)1

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

Answer (A)

Hints :

The equation of the line QR is

2 3 51 4 1

x y z k

so any arbitrary point is ( 2, 4 3, 5)k k k

Now this will satisfy the equation of the plane

5( 2) 4( 4 3) ( 5) 1k k k

2

3k

So, the point is4 1 13

, ,3 3 3

Now, let point Sbe (+2, 4+ 3, + 5)

So, D.R's ofSTis (, 4+ 2, + 1)

() (1) 4(4+ 2) 1(+ 1) = 0

1

2

The point is3 9

,1,2 2

So,PSis

1

2

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

4x 5y = 20 to the circle x2 + y2 = 9 is

(A) 20(x2 +y2) 36x+ 45y = 0

(B) 20(x2 +y2) + 36x 45y = 0

(C) 36(x2 +y2) 20x+ 45y = 0

(D) 36(x2 +y2) + 20x 45y = 0

Answer (A)

Hints :Let,P(, ) be a point on the line 4x 5y = 20.


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i.e., 4 5 = 20 ... (1)

If (h, k) be the mid point of chord of contact, then T= S1, gives

xh +yk 9 = h2 + k2 9

xh + yk = h2 + k2. ... (2)

Also, equation of chord of contact w.r.t.P(, ) isx+ y = 9 ... (3)

as equations (2) & (3) are identical,

(0, 0)P( , )

( , )h k

So,

2 29

h k h k

2 2 2 2

9 9,

h k

h k h k

Using equation (1), we get

2 2 2 2

9 94 5 20

h k

h k h k

i.e., locus of (h, k) is given by

20(x2 +y2) 36x+ 45y = 0

46. LetP= [aij] be a 3 3 matrix and let Q= [b

ij], where b

ij= 2i+ja

ijfor 1 i,j 3. If the determinant ofPis

2, then the determinant of the matrix Qis

(A) 210 (B) 211 (C) 212 (D) 213

Answer (D)

Hints :

Given,

11 12 13

3 3 21 22 33

31 32 33

[ ]ij

a a a

P a a a a

a a a

Now, 2 [ ]i j ij ijQ a b

11 12 13

21 22 23

31 32 33

b b b

Q b b b

b b b

=

2 3 411 12 13

3 4 521 22 23

4 5 631 32 33

2 2 2

2 2 2

2 2 2

a a a

a a a

a a a

So,

11 12 13

1221 22 23

31 32 33

2

a a a

Q a a a

a a a

= 213.

(as |P| = 2)


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47. The integral

2

92

sec

sec tan

xdx

x xequals (for some arbitrary constantK)

(A)

2

112

1 1 1 sec tan

11 7sec tanx x K

x x

(B)

2

112

1 1 1 sec tan

11 7sec tanx x K

x x

(C)

2

112

1 1 1 sec tan

11 7sec tanx x K

x x

(D)

2

112

1 1 1sec tan

11 7sec tanx x K

x x

Answer (C)

Hints :Let secx+ tanx= t

secx(secx+ tanx) dx= dt

dx= 22

1dt

t

Now, I =

2

2

9/2

12

2 1

tt dt

t

t

=

9 13 2 21

2t t dt

=

7 11

2 21 2 2

2 7 11

t t K

=

11

221 1

7 11

t t K

=

2

11

2

1 1 1 sec tan

11 7

sec tan

x x K

x x

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

Answer (B)

Hints :

x y z

3

2

1

2

1

1

3 = 605 2 1C C C3 1 1

3 = 90

5 3 1

C C C2 2 1

Total = 150


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49. If 2 1

41

limx

x xax b

x, then

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

Answer (B)

Hints :

Here,

2 1 1 1lim 4

1x

x x ax x b x

x

2 1 1 1 lim 4

1x

x a x a b b

x

This is possible iff

1 a = 0 a = 1

and 1 a b = 4 b = 4

50. Let z be a complex number such that the imaginary part of z is nonzero and a = z2 + z +1 is real. Then a

cannot take the value

(A) 1 (B)1

3(C)

1

2(D)

3

4

Answer (D)

Hints :

As, a is real,

So a a gives

z2 + z + 1 = 2 1z z

( )( 1) 0z z z zAs, z z

So, 1z z

1

{where }2

x z x iy

Now,

a = z2 + z + 1

21 1

12 2

iy iy

23

4 y

As,y 0 so, 3

4a

SECTION - II




51. Let Sbe the area of the region enclosed by2 xy e , y = 0, x= 0, and x= 1. Then

(A)1

Se

(B)1

1 Se

(C)1 1

14

S

e(D)

1 1 11

2 2

S

e


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Answer (A, B, D)

Hints :

x [0, 1] x2x x2 xex2ex

21 1

10

0 0

1[ ] 1x x xS e dx e dx e

e

i.e.,1

1Se

Option (B) is correct.

(0, 1)

1,1e

x= 0 x= 1

y = 0

1

A

Q

1e

R

y-axis

x-axis0

0,1e

S

1P(1, 0)

y e=x2

Sis the area under the curve y = ex2

Hence, S area of the rectangle OPQS

i.e., S 1 1 1

e e

Hence, option (A) is correct.

B

A

y-axis

(0, 1)

0

x= 1x= 0 x=1

2

1

21

e,

1,1e

x-axis

21

0

x dxS e

2 2

1

12

10

2

x xe dx e dx

AreaA + AreaB

Now,

1

2

0

1Area 1

2A = dx

1

1

2

1 1 1Area 1

2B = dx

e e

Hence,1 1 1

12 2

Se

Hence, option (D) is correct

Now, consider the integral3/2

1

12 1

edx

x e

Substitute t = lnx

1/2

3/21 0

etdx e dt

x

Now, for 0 < x< 1

2 /2x xe e


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2

1

0

12 1xe dx

e

Now,1 1 1 7 9

2 1 14 4 4e e e

7 9 0

4

e

e

So, option (C) is not correct.

52. If y(x) satisfies the differential equationy y tan x= 2xsec xand y(0) = 0, 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

Answer (A, D)

Hints :

tan 2 secdy

y x x xdx

cos (sin ) 2dy

x x y x dx

d(y cosx) = d(x2)

ycosx= x2 + c

y(0) = c = 0

y = x2secx

also,y = 2xsecx+ x2secxtanx

53. Tangents are drawn to the hyperbola2 2

19 4

x y

, parallel to the straight line 2xy = 1. The point 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( , )

Answer (A, B)

Hints :

As, points of contact

2 2

2 2 2 2 2 2,

a m b

a m b a m b

Here, a2

= 9, b2

= 4, m = 2

so, required points are

9 1,

2 2 2

54. Let , [0, 2] be such that

22cos (1 sin ) sin tan cot cos 1

2 2,

tan(2 ) 0 and 3

1 sin2

.

Then cannot satisfy

(A)

02

(B)

4

2 3(C)

4 3

3 2(D)

32

2


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Answer (A, C, D)

Hints :

2 12cos 1 sin sin cos 1

sin cos2 2

2cos 1 sin 2sin cos 1

2cos + 1 = 2sin( + )(i)

Now,

3

tan 0 and 1 sin2

gives

3 5,

2 3

Using (i) ;

1

sin cos2

1 sin( ) 12

i.e.

5 13 17

or,6 6 6 6

13 17

6 6

2 7

3 6So, options (A), (C) and (D) are correct.

55. A ship is fitted with three engines E1, E

2and E

3. The engines function independently of each other with

respective probabilities 12

, 14

and 14

. For the ship to be operational at least two of its engines must function.

LetXdenote the event that the ship is operational and letX1,X

2andX

3denote respectively the events that

the engines E1, E

2and E

3are functioning. Which of the following is (are) true?

(A) 13

[ | ]16

cP X X

(B) P [Exactly two engines of the ship are functioning |X] =7

8

(C) 25

[ | ]16

P X X

(D) 17

[ | ]16

P X X

Answer (B, D)

Hints :

(A) 1cP X X

1

P X X

P X

1 1 1

2 4 4

1 1 1 1 1 1 1 1 1 1 3

2 4 4 2 4 4 2 4 4 2 4 4

18


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(B) Required probability =

3 3 1

2 4 4 2 4 4 2 4 4

1 1 3 1 3 1 1 1 1 1 1 1

2 4 4 2 4 4 2 4 4 2 4 4

7

8

(C) 2P X X

22

( )P X X

P X

1 1 1 1 3 1 1

4 2 4 2 4 2 4

1

4

1 5

54 81 8

4

(D)

1

1

P X X

P X

1 1 1 3 1 1 3

2 4 4 4 4 4 4

1

2

7

16

SECTION - III

(Integer Answer Type)



56. The value of

3

2

1 1 1 16 log 4 4 4 ...

3 2 3 2 3 2 3 2is

Answer (4)

Hints :

As, 1

43 2

y y

4

9y (as y > 0)

so,

3

2

46 log 4

9

57. Let Sbe the focus of the parabola y2 = 8xand letPQbe the common chord of the circle x2 +y2 2x 4y =

0 and the given parabola. The area of the triangle PQSis

Answer (4)


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Hints :

ClearlyP(2t2, 4t) satisfies the equation of circle,

x2 +y2 2x 4y = 0

4t4 + 16t2 4t216t = 0

t(t3 + 3t 4) = 0

t(t 1) (t2 + t + 4) = 0

Y

P t t(2 ,4 )2

S(2, 0)QX t = 0,1

i.e.,Pbe (2, 4)

So, area ofPQS= 1

2 4 42

units

58. If

,a b andc are unit vectors satisfying

2 2 2| | | | | | 9a b b c c a , then

|2 5 5 |a b c is

Answer (3)

Hints :

2 2 2| | | | | | 9a b b c c a

2 2 22(| | | | | | . ) 9a b c a b b c c a (i)

2(3 ) 9a b b c c a

3

2a b b c c a angle between each pair is 120.

Now, 2 2 2 2|2 5 5 | 4| | 25| | 25| | 20 20 50a b c a b c a b a c a c

=

20 20 504 25 25

2 2 2

= 54 10 10 25

= 9

|2 5 5 | 3a b c

59. Letf: be defined asf(x) = |x| + |x2 1|. The total number of points at which fattains either alocal maximum or a local minimum is

Answer (5)

Hints :

f(x) = |x| + |x2 1|

f(x) = |x| + |(x 1)(x+ 1)| 1 10

2

2

2

2

1 , 1

1 , 1 0( )

1 , 0 1

1 , 1

x x x

x x xf x

x x x

x x x

2 1 , 1

2 1 , 1 0( )

2 1 , 0 1

2 1 , 1

x x

x xf x

x x

x x


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f(x) is not differentiable at x= 0, 1 andf(x) = 0 at 1 1

,2 2

x

Also sign scheme off(x)

x= 1, 0, 1 are point of minima 1 1/2 0 1/2 1 + + +

1 1,2 2

x are point of maxima

total number of point of maxima

and minima = 5

60. Letp(x) be a real polynomial of least degree which has a local maximum at x= 1 and a local minimum at

x= 3. Ifp(1) = 6 andp(3) = 2, thenp(0) is

Answer (9)

Hints :

p(x) will be of degree 31 3

+

p(x) = k(x 1) (x 3), k > 0 as 1 is point of maxima and 3 is point of minima

= k(x2 4x+ 3)

Now

32( ) 2 3

3

xp x k x x c

given,p(1) = 6

4k + 3c = 18 (i)

andp(3) = 2

c = 2

so, k = 3

p(x) = 3(x 1) (x 3)

p(0) = 3( 1) ( 3) = 9

Analysis and Solution for Iit Jee 2012

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Transcript of Analysis and Solution for Iit Jee 2012