PHYSICS

[1] IIT JEE – 2012/ Paper - 2

Part I: Physics

Section – I (Single Correct Answer Type)This section contains 8 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D), out ofwhich ONLY ONE is correct.

1. Consider a disc rotating in the horizontal plane with a constant angular speed about its centre O. Thedisc has a shaded region on one side of the diameter and an unshaded region on the other side as shownin the figure.When the disc is in the orientation as shown, two pebbles P and Q are simultaneously pro-jected at an angle towards R. The velocity of projection is in the y-z plane and is same for both pebbles

with respect to the disc. Assume that (i) they land back on the disc before the disc has completed18

rotation, (ii) their range is less than half the disc radius, and (iii) remains constant throughout. Then

(A) P lands in the shaded region and Q in the unshaded region.(B) P lands in the unshaded region and Q in the shaded region.(C) Both P and Q land in the unshaded region.(D) Both P and Q land in the shaded region.

Sol. [C] If the disk were stationary pebble thrown from point P would have landed say at P and from Q atQ .

QQ

P

Now considering rotation of disk, relative velocity of point P w.r.t. P is towards right at time of throw andhence it will now fall to the right of PP i.e. in unshaded region. Similarly relative velocity of Q w.r.t. Q istowards right and this will also fall in unshaded region.

PHYSICS

[2] IIT JEE – 2012/ Paper - 2

2. In the given circuit, a charge of 80 C is given to the upper plate of the 4 F capacitor. Then in thesteady state, the charge on the upper plate of the 3 F capacitor is

(A) 32 C (B) 40 C (C) 48 C (D) 80 C

Sol. [C] The net charge on upper plates both capacitor (2F and 3F ) will be 80C so charge on upperplate of 3C capacitor will be

q C CFHGIKJ

33 2 80 48 c h

3. Two moles of ideal helium gas are in a rubber balloon at 30 0C. The balloon is fully expandable and can beassumed to require no energy in its expansion. The temperature of the gas in the balloon is slowly changedto 350C. The amount of heat required in raising the temperature is nearly (take R = 8.31 J / mol K)(A) 62 J (B) 104 J (C) 124 J (D) 208 J

Sol. [D] The process involved is isobaric so

Q nC T R Jp FHGIKJ ( ) ( )2 5

2 5 208

4. A loop carrying current I lies in the x - y plane as shown in the figure. The unit of vector k is coming outof the plane of the paper. The magnetic moment of the current loop is

(A) a Ik2 (B)21 2FHGIKJa Ik (C) FHG

IKJ

21 2a Ik (D) 2 1 2 b ga Ik

PHYSICS

[3] IIT JEE – 2012/ Paper - 2

Sol. [B] The system is equivalent to the figure shown so

m a a I kFHGIKJ

FHG

IKJ

L

NMM

O

QPP4 2 2

22

FHGIKJ

21 2a I k

5. Two identical discs of same radius R are rotating about their axes in opposite directions with the sameconstant angular speed . The discs are in the same horizontal plane. At time t= 0, the points P and Q arefacing each other as shown in the figure. The relative speed between the two pointsP and Q is vr.. In onetime period (T) of rotation of the discs, vr as a function of time is best represented by

(A) (B)

(C) (D)

PHYSICS

[4] IIT JEE – 2012/ Paper - 2

Sol. [A] From observation we can see

v t T v t T T vr rFHGIKJ FHG

IKJ 0

20

434

2& , &

so correct option is (A).

6. W, closed at both ends, is partially filled with water. It is floating vertically in water in half-submergedstate. If c is the relative density of the material of the shell with respect to water, then the correctstatement is that the shell is(A) more than half-filled if c is less than 0.5. (B) more than half-filled if c is more than 1.0.(C) half-filled if c is more than 0.5. (D) less than half-filled if c is less than 0.5.

Sol. [A] If c 05. , the cylinder will be half filled and if c 05. , then the cylinder will be more than half filled.

7. A student is performing the experiment of Resonance Column. The diameter of the column tube is 4 cm.The frequency of the tuning fork is 512 Hz. The air temperature is 38°C in which the speed of sound is336 m / s. The zero of the meter scale coincides with the top end of the Resonance Column tube. When thefirst resonance occurs, the reading of the water level in the column is(A) 14.0 cm (B) 15.2 cm (C) 16.4cm (D) 17.6 cm

Sol. [B] For first resonance,

1 4 e

or 1 40 3 152

vf

d. . cm

8. An infinitely long hollow conducting cylinder with inner radius R/2 and outer radius R carries a uniformcurrent density along its length. The magnitude of the magnetic field, | |

B as a function of the radial distance

r from the axis is best represented by

(A) (B)

(C) (D)

PHYSICS

[5] IIT JEE – 2012/ Paper - 2

Sol. [D] The cylindrical symmetric current distribution,

Bi

r

0

2enclosed

For, r R B 2 0,

R r R2 , B i

R R

FHG

IKJ

0

22

4

( / )r Rr

2 2 42

and R r , Bir

0

2so correct option is D.

Section–II: (Paragraph Type)This section contains 6 multiple choice questions relating to three paragraphs with two questions on eachparagraph. Each questions has four choices (A), (B), (C) and (D), out of which ONLY ONE is correct.

Paragraph for Questions 9 and 10

Most materials have the refractive index, n > 1. So, when a light ray from air enters a naturally occurring

material, then by Snell's law,sinsin

1

2

2

1

nn it is understood that the refracted ray bends towards the normal. But

it never emerges on the same side of the normal as the incident ray. According to electromagnetism, the refractive

index of the medium is given by the relation ncv r r FHGIKJ , where c is the speed of electromagnetic waves

in vacuum, v its speed in the medium, r and r the relative permittivity and permeability of the mediumrespectively.In normal materials both r and r are positive, implying positiven for the medium. When both rand r are negative, one must choose the negative root of n. Such negative refractive index materials can nowbe artificially prepared and are called meta-materials. They exhibit significantly different optical behavior, withoutviolating any physical laws. Since n is negative, it results in a change in the direction of propagation of therefracted light. However, similar to normal materials, the frequency of light remains unchanged upon refractioneven in meta-materials.

PHYSICS

[6] IIT JEE – 2012/ Paper - 2

9. For light incident from air on a meta- material, the appropriate ray diagram is

(A) (B)

(C) (D)

Sol. [C] sinsin

21 1

2

nn

as n2 0 so the ray will be refracted on the same side of the normal on entering the material. Thus, correctoption is (C)

10. Choose the correct statement.(A) The speed of light in the meta- material is v c n | |

(B) The speed of light in the meta-material is vcn

| |

(C) The speed of light in the meta-material is v c(D) The wavelength of the light in the meta-material mb g is given by m air n | |, where air is thewavelength of the light in air

Sol. [B] Form the given expression, ncv

so v cn

PHYSICS

[7] IIT JEE – 2012/ Paper - 2

Paragraph for Questions 11 and 12The general motion of a rigid body can be considered to be a combination of (i) a motion of its centre of massabout an axis, and (ii) its motion about an instantaneous axis passing through the centre of mass. These axes neednot be stationary. Consider, for example, a thin uniform disc welded (rigidly fixed) horizontally at its rim to amassless stick, as shown in the figure. When the disc-stick system is rotated about the origin on a horizontalfriction plane with angular speed the motion at any instant can be taken as a combination of (i) a rotation of thecentre of mass of the disc about the z-axis and (ii) a rotation of the disc through an instantaneous vertical axispassing through its centre of mass (as is seen from the changed orientation of points P and Q). Both thesemotions have the same angular speed in this case.

Now consider two similar systems as shown in the figure: Case (a) the disc with its face vertical and parallel tox-z plane; Case (b) the disc with its face making an angle of 45° with x-y plane and its horizontal diameter parallelto x-axis. In both the cases, the disc is welded at point P, and the systems are rotated with constant angularspeed about the z-axis.

11. Which of the following statements regarding the angular speed about the instantaneous axis(passing throughthe centre of mass) is correct?

(A) It is 2 for both the cases (B) It is for case (a): and2 for case (b)

(C) It is for case (a): and 2 for case (b) (D) It is for both cases.

PHYSICS

[8] IIT JEE – 2012/ Paper - 2

12. Which of the following statements about the instantaneous axis (passing through the centre of mass) iscorrect?(A) It is vertical for both the cases (a) and (b)(B) It is vertical for case (a); and is at 450 to the x-z plane and lies in the plane of the disc for case (b)(C) It is vertical for case (a); and is at 450 to the x-z plane and is normal to the plane of the disc for case (b)(D) It is vertical for case (a); and is at 450 to the x-z plane and is normal to the plane of the disc for case (b)

Sol. [D, A]

BA

P

QB

A

From figure, for case (a) when the complete system turns by 180o, the disk also turns by 180o about itsvertical axis.

B

A

P

Q

BA

P

Q

From figure, for case (b)when the complete system turns by 180o, the disk also turns by 180o, about vertical axis. Hence angularspeed is for both cases.

Paragraph for Questions 13 and 14

The decay process, discovered around 1900, is basically the decay of a neutron (n). In the laboratory, a

proton (p) and an electron ec h are observed as the decay products of the neutron. Therefore, considering thedecay of a neutron as a two-body decay process, it was predicted theoretically that the kinetic energy of theelectron should be a constant. But experimentally, it was observed that the electron kinetic energy has a continuousspectrum. Considering a three-body decay process, i.e. n p e ve , around 1930, Pauli explained the

observed electron energy spectrum. Assuming the anti-neutrino veb g to be massless and possessing negligibleenergy, and the neutron to be at rest, momentum and energy conservation principles are applied. From thiscalculation, the maximum kinetic energy of the electron is 08 106. eV. The kinetic energy carried by the protonis only the recoil energy.

13. If the anti-neutrino had a mass of 3 eV/c2 (where c is the speed of light) instead of zero mass, what shouldbe the range of the kinetic energy, K, of the electron?

(A) 0 08 106 K . eV (B) 30 08 106. .eV eV K

(C) 30 08 106. .eV eV K (D) 0 08 106 K . eV

PHYSICS

[9] IIT JEE – 2012/ Paper - 2

14. What is the maximum energy of the anti-neutrino?(A) Zero(B) Much less than 08 106. eV(C) Nearly 08 106. eV(D) Much larger than 08 106. eV

Sol. [D, C] Treating neutrino as mass particle, from momentum conservation, p p pp e ve 0

It is possible that pe 0 & still momentum is conserved. Thus minimum kinetic energy of electron is zero.Total energy (Q - value) released will be lesser than 08 106. eV due to finite mass of neutrino. Hence,maximum KE of electron will be less than 08 106. eV.Maximum energy of antinutrino will be nearly08 106. eV

Section–III (Multiple Correct Answer(s) Type)This section contains 6 multiple choice questions. Each questions has four choices (A), (B), (C) and (D) outof which ONE or MORE are correct.

15. A current carrying infinitely long wire is kept along the diameter of a circular wire loop, without touching it.The correct statement (s) is / (are)

(A) The emf induced in the loop is zero if the current is constant

(B) The emf induced in the loop is finite if the current is constant(C) The emf induced in the loop is zero if the current decreases at a steady rate(D) The emf induced in the loop is finite if the current decreases at a steady rate

Sol. [A, C] The net flux passing through the loop due to wire is zero and hence no emf is induced in loopirrespective of the current being constant or varying.If zero is considered as finite also the options (A, B, C, D) can also be a possible answer.

PHYSICS

[10] IIT JEE – 2012/ Paper - 2

16. In the given circuit, AC source has 100 rad / s. Considering the inductor and capacitor to be ideal, thecorrect choice (s) is (are)

(A) The current through the circuit, I is 0.3 A

(B) The current through the circuit, I is 03 2. A

(C) The voltage across 100 resistor 10 2 V

(D) The voltage across 50 resistor = 10 V

Sol. [C] In the upper branch, X RC 100 100 , and than Z1 100 2 and current I120

100 2 . A

which leads by 450 w.r.t. voltage. While in lower branch X L 50, R 50 and thus Z2 50 2

and current I22050 2

A which lags by 450 w.r.t. voltage.

Thus total current I is given by summation of phasors I1 and I2 which differ by 900 in phase and hence

I I I A 12

22 1

10

Also voltage across100 resistor I R1 20

100 2100 10 2V

Similarly across 50 resistor I R22050 2

50 V =10 2V

PHYSICS

[11] IIT JEE – 2012/ Paper - 2

17. Six point charges are kept at the vertices of a regular hexagon of side L and centre O, as shown in the

figure. Given that KqL

14 0

2, which of the following statgement(s) is (are) correct

(A) The electric field at O is 6K along CD

(B) The potential at O is zero(C) The potential at all points on the line PR is same(D) The potential at all points on the line ST is same

Sol. [A, B, C] From the symmetry of distribution, potential at O is zero and is same at all points on the line PR.Also electric field at O is 6 K along CD.

18. Two spherical planets P and Q have the same uniform density , masses MP and MQ , and surface areas

A and 4A, respectively. A spherical plane R also has uniform density and its mass is M MP Qd i . The

escape velocities from the planets P, Q and R, areV VP Q, andVR , respectively. Then

(A) V V VQ R P (B) V V VR Q P

(C)V VR P/ 3 (D) V VP Q/ 12

PHYSICS

[12] IIT JEE – 2012/ Paper - 2

Sol. [B, D] For spherical planets having same density area and mass, M R 3 and as A R 2

Thus,RRQ

p

41

21 and

MM

Q

P

81

so M M M MR P Q P 9

and hence R RR P 3

Now, v GMR Resc

2

V V VR Q P andVV

P

Q

12

19. The figure shows a system consisting of (i) a ring of outer radius 3R rolling clockwise without slipping on ahorizontal surface with angular speed and (ii) an inner disc of radius 2R rotating anti-clockwise withangular speed / 2. The ring and disc are separated by frictionless ball bearings. The system is in thex - z plane. The point P on the inner disc is at a distance R from the origin, where OP makes an angle of300 with the horizontal. Then with respect to the horizontal surface.

(A) the point O has a linear velocity 3R i

(B) the point P has a linear velocity114

34

R i R k

(C) the point P has a linear velocity134

34

R i R k

(D) the point P has a linear velocity 3 34

14

FHG

IKJ R i R k

PHYSICS

[13] IIT JEE – 2012/ Paper - 2

Sol. [A, B] Since outer ring is rolling without slipping

V V R icm o 3

Also,

V V VP cm P cm /

FHGIKJ 3

230 30R i R k i cos sin e j

114

34R i R k

20. Two solids cylinders P and Q of same mass and same radius start rolling down a fixed inclined plane fromthe same height at the same time. Cylinder P has most of its mass concentrated near its surface, while Qhas most of its mass concentrated near the axis. Which statement (s) is (are) correct?

(A) Both cylinders P and Q reach the grough at the same time

(B) Cylinder P has larger linear acceleration than cylinder Q(C) Both cylinders reach the ground with same translational kinetic energy(D) Cylinder Q reaches the ground with larger angular speed

Sol. [D] From the given information radius of gyration of P is greater than that Q i.e., k kP Q . Now forrolling cylinder down a fixed inclined plane

a gk Rcm sin

/

1 2 2 and thusQ QP Q and hence Q reaches earlier. Also Q reaches the ground with larger

angular speed.

P300

300

O

VP/cm

CHEMISTRY

[1] IIT JEE – 2012

Part IISection–I

Single Correct Answer TypeThis section contains 8 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D), out ofwhich ONLY ONE is correct.

21. The compound that undergoes decarboxylation must readily under mild condition is

(A) (B)

(C) (D)

Sol: [B] It is a -keto acid

22. The shape of XeO F2 2 molecule is(A) trigonal bipyramidal (B) square planar(C) tetrahedral (D) see-saw

Sol: [D]

F

Xe

F

|:

|

O

O See saw structure

23. For a dilute solution containing 2.5 g of a non-volatile non-electrolyte solute in 100 g of water, the eleva-tion in boiling point at 1 atm pressure is 2C. Assuming concentration of solute is much lower than the

concentration of solvent, the vapour pressure (mm of Hg) of the solution is (take K K kg molb 0 76 1. )

(A) 724 (B) 740 (C) 736 (D) 718Sol: [A] mass = 2.5 g

mol of solute =2 5.M

so, moladity of solute =25M

T k mb b

CHEMISTRY

[2] IIT JEE – 2012

2 0 76 25 . M M

0 76 252

.M 9 5.

PP Xsolute

PP

0 26555.. as dilute solution

P mmsolution 724 of Hg

24. NiCl P C H C H2 2 5 2 6 5 2b g b g exhibits temperature dependent magnetic behaviour (paramagnetic/ diamag-

netic). The coodination geometries of Ni2 in the paramagnetic and diamagnetic states are respectively(A) tetrahedral and tetrahedral (B) square planar and square planar(C) tetrahedral and square planar (D) square planar and tetrahedral

Sol: [C] sp3 and dsp2 hybridisation

25. The major product H of the given reaction sequence is

CH CH CO CH G HCN H SOHeat3 2 3

95% 2 4 –

(A) CH CH C COOH

CH3

3

|

(B) CH CH C CN

CH3

3

|

(C)OH

CH CH C COOH

CH

|

|3 2

3

(D) CH CH C CO NH

CH3 2

3

|

Sol: [A] CH CH C CH

O3 2 3

||CN

CN

CH CH C CH

OHa

|

|

( )

3 2 3 95% 2 4H SO

OH

CH CH C COOH

CH

|

|3 2

3

CH CH C COOH

CH3 2

3

|

CHEMISTRY

[3] IIT JEE – 2012

26. The reaction of white phosphorus with equeous NaOH gives phosphine along with another phosphoruscontaining compound. The reaction type; the oxidation states of phosphorus in phosphine and the otherproduct are respectively(A) redox reaction; –3 and –5 (B) redox reaciton + 3 and + 5(C) disproportionation reaction; –3 and +5 (D) disproportionation reaction; –3 and +3

Sol: [D]

27. Using the data provided, calcualte the multiple bond energy (kJ mol–1) of a C C bond in C H2 2 . That

energy is (take the bond energy of C H bond as 350 1kJ mol )

2 2 2 2C s H g C H g( ) ( ) ( ) H kJ mol 225 1

2 2C s C g( ) ( ) H kJ mol 1410 1

H g H g2 2( ) ( ) H kJ mol 330 1

(A) 1165 (B) 837(C) 865 (D) 815

Sol: [D]

2 C (g)

2 2 2 2C s H g C H g( ) ( ) ( )

1410

2 H (g)

2 × 330– 2 DH–C– DCC H kJ mol 225 /

1410 330 2 350 225 ( )DC C

1410 330 700 225 DC C

DC C 1410 330 925 = 81528. In the cyanide process of silver from argentite ore, the oxidizing and reducing agents used are

(A) O2 and CO respectively (B) O2 and Zn dust respectively

(C) HNO3 and Zn dust respectively (D) HNO3 and CO dust respectivelySol: [B]

CHEMISTRY

[4] IIT JEE – 2012

SECTION II :

Paragraph TypeThis section contains 6 multiple choice questions relating to three paragraphs with two questions on each para-graph. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct

Paragraph for Questions 29 and 30

Bleaching powder and bleach solution are produced on a large scale and used in several house hold products.The effectiveness of bleach solution is often measured by iodometry.

29. Bleaching powder contains a salt of an oxoacid as one of its components. The anhydride of that oxoacidis(A) Cl O2 (B) Cl O2 7 (C) ClO2 (D) Cl O2 6

30. 25 mL of household bleach solution was mixed with 30 mL of 0.50 MKI and 10 mL of 4 N acetic acid.In the titration of the liberated iodine, 48 mL of 0.25 N Na S O2 2 3 was used to reach the end point. Themolarity of the household bleach solutions is(A) 0 48. M (B) 0 96. M (C) 0 24. M (D) 0 024. M

Sol: [A,C] Bleaching powder Ca OCl Clb g , so oxoacid is HClO.

Anhydride of HClO is Cl O2

OCl I H Cl I H O 2 2 2 2

I S O I S O3 2 42

4 622 3

for S O2 42 its normality is equal to its molarity

12mmol of S Omol I

mol of S Omol of OCl

mol of I mm of OCl2 32 2

2 32

2

12

11 6

So its Mmm of

mL 625 0 24.

Paragraph for Questions 31 and 32

In the following reaction sequence, the compound J is an intermediate

J ( )C H O9 8 2 gives effervescence on treatment with NaHCO3 and positive Baeyer's test

CHEMISTRY

[5] IIT JEE – 2012

31. The compound K is

(A) (B) (C) (D)

Sol: [C]

O

C H|| CH CO O

CH COONa3 2

3

b g CH CH C OH

O

|| H Pd c2 , /

OHO

SOCl 2

O

AlCl3 OCl

(K)

32. The compound I is

(A) (B) (C) (D)

Sol: [A]Paragraph for Questions 33 and 34

The electrochemical cell sown below is a concentration cell.

M M| 2 (saturated solution of a sparingly soluble salt,) MX M mol dm M22 30 001 .c h

The emf of the cell depends on the difference in concentrations of M2 ions at the two electrodes. The emf ofthe cell at 298 K is 0.059 V.

33. The value of G kJ mol1c h for the given cell is (take 1 96500 1F C mol )

(A) – .57 (B) 5.7 (C) 11.4 (D) – .114

CHEMISTRY

[6] IIT JEE – 2012

34. The solubility product K mol dmsp3 9d i of MX2 at 298 K based on the information available for the

given concentration cell is (take 2 303 298 0 059. / . R F V )

(A) 1 10 15 (B) 4 10 15 (C) 1 10 12 (D) 4 10 12

Sol: [D, B] For the given cell

M M| 2 (saturated solution of a sparingly soluble salt,) MX M mol dm M22 30 001 .c h

E Vcell 0 059.

G nFE 2 96500 0 059. 11 40. kJ

E

Ksp

cell cell

FHGIKJ

0 059

24

0 001

1 3

. log .

/

0 059 0 0592

40 001

1 3

. . log .

/

FHGIKJ

Ksp

10 10 43 2

1 3

FHGIKJ

Ksp/

4 10 15 Ksp

CHEMISTRY

[7] IIT JEE – 2012

Section–III

Section III : Muliple Corect Answer(s) TypeThis section contains 6 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out ofwhich ONE or MORE are correct.

35. The given graphs / data I, II, III and IV represent general trends observed for different physisorption andchemisorption processes under mild conditions of temperature and pressure. Which of the following choice(s) about I, II, III and IV is (are) correct?

(A) I is physisorption and II is chemisorption (B) I is physisorption and III is chemisorption(C) IV is chemisorption and II is chemisorption (D) IV is chemisorption and III is chemisorption

Sol: [A, C]

CHEMISTRY

[8] IIT JEE – 2012

36. The reversible expansion of an ideal gas under adiabatic and isothermal conditions is shown in the figure.Which of the following statements(s) is (are) correct ?

(A) T T1 2 (B) T T3 1

(C) w wisothermal adiabatic (D) U Uisothermal adiabaticSol: [A, C, D]

37. For the given aqueous reactions, which of the statements(s) is (are) true?

excess KI K Fe CN dilute H SO 3 62 4b g brownish - yellow solution

ZnSO 4

white precipitate + brownish yellow filtrate

Na S O2 2 3

colourless solution(A) The first reaction is a redox reaction

(B) White precipitate is Zn Fe CN2 6 2b g

(C) Addition of filtrate to starch solution gives blue colour.(D) White precipitate is soluble in NaOH solution

Sol: [A, C, D]

CHEMISTRY

[9] IIT JEE – 2012

38. With reference to the scheme given, which of the given statement(s) about T, U, V and W is (are) correct?

(A) T is soluble in ot aqueous NaOH(B) U is optically active(C) Molecular formula of W is C H O10 18 4

(D) V gives effervescence on treatment with aqueous NaHCO3

Sol: [A, C, D]

39. Which of the given statement(s) about N, O, P and Q with respect to M is (are) correct?

(A) M and N are non-mirror image stereoisomers(B) M and O are identical(C) M and P are enantiomers(D) M and Q are identical

Sol: [A, B, C]40. With respec to graphite and diamond, wich of the statement(s) given below is (are) correct?

(A) Graphite is harder than diamond.(B) Graphite has higher electrical conductivity than diamon.(C) Graphite has higher thermal conductivity than diamond.(D) Graphite has higher C-C bond order than diamond.

Sol: [B, D]

MATHEMATICS

[17] IIT JEE – 2012

Part III

Section–I : Straight Correct Answer TypeThis section contains 8 multiple choice questions. Each question has fcour choices (A), (B), (C) and (D), outof which ONLY ONE is correct.

41. Four fair dice D D D1 2 3, , and D4 , each having six faces numbered 1, 2, 3, 4, 5 and 6, are rolledsimultaneously. The probability that D4 shows a number appearing on one of D D1 2, and D3 is

(A)91

216 (B)108216 (C)

125216 (D)

127216

Ans. [A]

Sol. E1 : number of D4 is shown by exactly one of D D D1 2 3, ,E2 : number of D4 is shown by exactly two of D D D1 2 3, ,E3 : number of D4 is shown by exactly all of D D D1 2 3, ,E : D4 shows a number appearing on one of D D D1 2 3, ,

P E P E E P E E P E Eb g b g b g b g 1 2 3

FHGIKJFHGIKJ

FHGIKJFHGIKJ FHGIKJ

31

23

2

2 316

56

16

56

16

91216

C Cc h c h

42. If P is a 3 × 3 matrix such that P P IT 2 , where PT is the transpose of P and I is the 3 × 3 identity

matrix, then there exists a column matrix Xxyz

L

NMMM

O

QPPPL

NMMM

O

QPPP

000

such that

(A) PX L

NMMM

O

QPPP

000

(B) PX X (C) PX X 2 (D) PX X

Ans. [D]

Sol. Let Pa a aa a aa a a

L

NMMM

O

QPPP

11 12 13

21 22 23

31 32 33

P P IT 2

MATHEMATICS

[18] IIT JEE – 2012

a a aa a aa a a

a a aa a aa a a

11 21 31

12 22 32

13 23 33

11 12 13

21 22 23

31 32 33

2 2 22 2 22 2 2

1 0 00 1 00 0 1

L

NMMM

O

QPPPL

NMMM

O

QPPPL

NMMM

O

QPPP

a a a11 22 33 1

and a a a a a a12 21 31 13 23 32 0

so P

L

NMMM

O

QPPP

1 0 00 1 00 0 1

Hence PXxyz

xyz

X

L

NMMM

O

QPPP

L

NMMM

O

QPPP

L

NMMM

O

QPPP

1 0 00 1 00 0 1

43. Let ab g and ab g be the roots of the equation

1 1 1 1 1 1 03 2 6 a x a x ad i d i d i where a 1

Then lima

a 0

b g and lima

a 0

b g are

(A) 52

and 1 (B) 12

and –1 (C) 72

and 2 (D) 92

and 3

Ans. [B]

Sol.

FHG

IKJ

F

HGGG

I

KJJJ

lim lima a

aa

a

a0 3 0

12

23

1 11 1

12

1

13

1

32

b gb g

lim lima a

a

a

a

a0

16

13

0

56

23

1 1

1 1

16

1

13

1

12

b gb g

b gb g

32

12

,

12 , 1

MATHEMATICS

[19] IIT JEE – 2012

44. The equation of a plane passing through the line of intersection of the planes x y z 2 3 2 and

x y z 3 and at a distance23

from the point (3, 1, –1) is

(A) 5 11 17x y z (B) 2 3 2 1x y (C) x y z 3 (D) x y 2 1 2

Ans. [A]

Sol. Let equation of plane through the line of intersectionof the planesx y z 2 3 2 and x y z 3 is

x y z x y z 2 3 2 3 0b g b g . . . (i)

1 2 3 2 3 0 b g b g b gx z

perpendicular distance from (3, 1, –1)23

3 1 2 3 2 3

1 2 3

232 2 2

b g b g b gb g b g b g

2

3 4 14232

3 3 4 142

3 3 4 14 72

2 2

Required plane

1 72

2 72

3 72

2 3 72

0FHGIKJ FHG

IKJ FHG

IKJ FHG

IKJ x y z

5 11 17 0x y z 5 11 17 0x y z

45. Let a a a1 2 3, , ,... be in harmonic progression with a1 5 and a20 25 . The least positive integer n forwhich a n 0 is(A) 22 (B) 23 (C) 24 (D) 25

Ans. [D]

Sol. In corresponding AP, T115

, T20125

T d2015

19 125

(Let common difference of AP is d)

d 40

25 19

MATHEMATICS

[20] IIT JEE – 2012

Now T nn

0 15

14

25 190b g b g

n n 954

1 24 34

lest value of x = 25

46. If a andb are vectors such that

a b 29 and a i j k i j k b 2 3 4 2 3 4 e j e j , then a pos-

sible value of a b i j k d i e j7 2 3 is

(A) 0 (B) 3 (C) 4 (D) 8

Ans. [C]

Sol. a i j k i j k b 2 3 4 2 3 4 e j e j

a i j k i j k b 2 3 4 2 3 4 0 e j e j

a b i j k d i e j2 3 4 0 but

a b 0

a b i j k d i e j 2 3 4 (say)

a b a a a 2 2 2 1 1

a b i j k 2 3 4 e j

Now a b i j k i j k i j k d i e j e j e j7 2 3 2 3 4 7 2 3

14 6 12 4b g

47. The value of the integral

x xx

x dx2

2

2

FHG

IKJ

z ln cos

is

(A) 0 (B)2

24 (C)

2

24 (D)

2

2

Ans. [B]

MATHEMATICS

[21] IIT JEE – 2012

Sol. x xx

x dx2

2

2

FHG

IKJ

z ln cos

FHGIKJ

z zx x dx

even function

xx

x dx

odd function

2

2

2

2

2

cos ln cos

b g b g

z2 02

0

2

x x dxcos

2 2 22

02x x x x xsin cos sinc h b g b g

FHGIKJ

LNM

OQP 2

42 0

44

2 2 b g

48. Let PQR be a triangle of area with a b 2 72

, and c 52

, where a, b and c are the lengths of the

sides of the triangle copposite to the angles at P, Q and R respectively. Then 2 22 2

sin sinsin sin

P PP P

equals

(A)3

4 (B)454 (C)

34

2

FHGIKJ (D)

454

2

FHGIKJ

Ans. [C]

Sol. a b c 2 72

52

, , , semiparimeter s

2 7

252

24

2 22 2

2 12 1

sin sinsin sin

sin cossin cos

P PP P

P PP P

b gb g

11 2

2coscos

tanPP

P

LNM

OQP

FHGIKJ FHGIKJ

L

N

MMMM

O

Q

PPPP FHGIKJ

s b s cb gb g

2

2

24 72

4 52 3

4

MATHEMATICS

[22] IIT JEE – 2012

Section–II : Paragrah TypeThis section contains 6 multiple choice questions relating to three paragraphs with two questions on eachparagraph. Each question has four choices (A), (B), (C) and (D), out of which ONE ONE is correct.

Paragraph for Questions 49 and 50

Let a n denote the number of alln-digit positive integers formed by the digits 0, 1 or both such that no consecu-tive digits in them are 0. Let bn the number of suchn-digit integers ending with digit 1 and cn the number ofsuchn-digit integers ending with digit 0.

49. The value of b6 is(A) 7 (B) 8 (C) 9 (D) 11

50. Which of the following is correct?(A) a a a17 16 15 (B) c c c17 16 15 (C) b b c17 16 16 (D) a c b17 17 16

Ans. [B, A]Sol. required number b6 are

1 1 1 1 1 1 1 0 1 1 1 1

1 1 0 1 1 1 1 0 1 0 1 1

1 1 1 0 1 1 1 0 1 1 0 1

1 1 1 1 0 1 1 1 0 1 0 1

a C C C C1716

115

214

39

81 ...

also a C C C1615

114

28

81 ...

a C C C1514

113

28

71 ...

a a C C C C C C16 1515

114

28

814

113

28

71 1 ... ...c h c h 2 15

115

214

39

8 17C C C C a...

MATHEMATICS

[23] IIT JEE – 2012

Paragraph for Questions 51 and 52

A tangent PT is drawn to the circle x y2 2 4 at the point P 3 1,d i . A straight line L, perpendicular to PT is

a tangent to the circle x y 3 12 2b g .

51. A common tagnet of the two circles is

(A) x 4 (B) y 2 (C) x y 3 4 (D) x y 2 2 6

52. A possible equation of L is

(A) x y 3 1 (B) x y 3 1 (C) x y 3 1 (D) x y 3 5

Ans. [D, A]

Sol. T divides A(0, 0), B(3, 0) in 2 : 1

T 2 3 02 1

0 02 1

6 0

FHG

IKJ , ,b g

Line through (6, 0) y m x 0 6b g mx y m 6 0 . . . (i)

it touches x y2 2 4

0 0 6

12

2

m

m 9 1 12 2

2 2m m m,

(i) 1

2 26

2 20x y x y 2 2 6 0

and 2

2 26

2 20x y x y2 2 6 0 x y 2 2 6

Tangent at P 3 1,d i to the circle

x y2 2 4 . . . (i)

is x y 3 1 4 . . . (ii)

equation of line perpendicular to (i) which touches circle x y 3 12 2b g is

L y x: 13

3 1 13

1b g

3 3 2y x

x y 3 5 or x y 3 1

y

A TB2(3, 0)

MATHEMATICS

[24] IIT JEE – 2012

Paragraph for Questions 53 and 54

Let f x x x xb g b g 1 2 2 2sin for all x IR , and let g xt

tt f t dt

x

b g b g b g

FHG

IKJz 2 1

11

ln for all x 1,b g

53. Consider the statements:

P : There exists some x IR such that f x x xb g c h 2 2 1 2

Q : There exists some x IR such that 2 1 2 1f x x xb g b g Then(A) both P and Q are true (B) P is true and Q is false(C) P is false and Q is true (D) both P and Q are false

54. Which of the following is true?(A) g is increasing on 1, b g(B) g is decreasing on 1, b g(C) g is increasing on (1, 2) and decreasing on 2, b g(D) g is decreasing on (1, 2) and increasing on 2, b g

Ans. [C, B]

Sol. f x x xb g c h 2 2 1 2

1 2 2 12 2 2 2 x x x x xb g c hsin

1 2 22 2 2 x x x xb g sin

1 1 12 2 2 x x xb g b gsin . . . (i)

for x 1 sin221 1

1x

x

b g which is false

for x = 1 equation (i) is NOT satisfied P is FALSELet h x f x x xb g b g b g 2 1 2 1

h x x x xb g b g 2 1 2 12 2sin

h 0 1b g h 1 1b g

MATHEMATICS

[25] IIT JEE – 2012

From IVT, at least one h 0 1,b gh xb g 0

Q is TRUE

g xt

tt f t dt

x

b g b g b g

FHG

IKJz 2 1

11

ln

Differentiating: g xx

xx f x

x

' lnb g b g b g

LNM

OQP z 2 1

10

1

g xx

xx x x x' ln sinb g b gb g b ge j

LNM

OQP

2 1

11 2 2 2

Now Let xx

xx

xxb g b gb g

2 1

12 4

1ln ln

' xx x

x xx x

b g b gb gb g

4

11 4 1

12

2

2

' xx

x xb g b g

b g

11

2

2

clearly ' ,x xb g 0 1 is decreasing function

x xb g b g 1 1

x xb g 0 1,

g x x x x x x' sinb g b g b ge j 1 0 12 2 2

g xb g is a decreasing function in 1, b g

+ ––1 0

+ –1

MATHEMATICS

[26] IIT JEE – 2012

Section–III : Multiple Correct Answer(s) TypeThis section contains 6 multiple choice questions. Each question has four choices (A), (B), (C) and (D), out ofwhich ONE or MORE are correct.

55. If f x e t t dttx

b g b gb g z 2

0

2 3 for all x 0,b g , then

(A) f has a local maximum at x 2(B) f is decreasing on (2, 3)(C) there exists some c 0,b g such that f c"b g 0

(D) f has a local minimum at x 3

Ans. [A, B, C, D]

Sol. f x e t t dt xtx

b g b gb g b g z 2

2 3 00

, ,

f x e x xx'b g b gb g 2

2 3

f x'b g 0 have two roots

f x"b g 0 will have one root.

56. For every integer n, let a n and bn be real numbers. Let function f IR IR: be given by

f xa x for x n nb x for x n n

n

nb g b g

RSTsin , ,cos , ,

2 2 12 1 2

, for all integers n.

If f is continuoues, then which of the following hold(s) for all n?(A) a bn n 1 1 0 (B) a bn n 1 (C) a bn n 1 1 (D) a bn n 1 1

Ans. [B, D]

Sol. f x

a x x n nb x x n na x x n nb x x n n

n

n

n

n

b g

RS||

T||

1

1

2 2 2 12 3 2 22 2 12 1 2

sin ,cos ,

sin ,cos ,

continuity at x n 2 2

a n b nn n 1 12 2 2 2sin cosb g b g

a b n nn n 1 1 2 2 2 2 1 0 1cos sinb g b g

+ –f (x)’2

+3

f(x)

MATHEMATICS

[27] IIT JEE – 2012

a bn n 1continuity at x n 2 1

a n b nn n 1 2 1 2 1sin cosb g b g a bn n 1 1

also a bn n 1 1

57. If the straight lines x yk

z

12

12

and x y zk

15

12

are coplanar, then the the plane(s) contain-

ing these two lines is(are)(A) y z 2 1 (B) y z 1 (C) y z 1 (D) y z 2 1

Ans. [B, C]

Sol. x yk

z

12

12

and x y zk

15

12

are coplanar

1 1 1 1 0 02 25 2

0

b g b g

kk

2 0 02 25 2

0kk k2 4 k 2

If k 2 then required plane

x y z

1 1 02 2 25 2 2

0b g b g

x y z 1 4 4 1 4 10 4 10 0b gb g b gb g b g 6 6 6 0y z y z 1if k 2 then plane

x y z

1 12 2 25 2 2

0

x y z 1 4 4 1 4 10 4 10 0b gb g b gb g b g 14 1 14 0y z b g y z 1 0 y z 1

MATHEMATICS

[28] IIT JEE – 2012

58. If the adjoint of a 3 × 3 matrix P is1 4 42 1 71 1 3

L

NMMM

O

QPPP

, then the possible value(s) of the determinant of P is(are)

(A) –2 (B) –1 (C) 1 (D) 2

Ans. [A, D]

Sol. adj P 1 4 42 1 71 1 3

4 P 2 4 P 2

59. Let f IR: , 1 1b g be such that f cossec

4 22 2

b g

for

FHGIKJFHGIKJ0

4 4 2, , . Then the value(s)

of f 13FHGIKJ is(are)

(A) 1 32

(B) 1 32

(C) 1 23

(D) 1 23

Ans. [A, B]

Sol. f cossec

4 22 2

b g

2

2 11 2

2

2

2

coscos

coscos

Now cos4 13

2 2 13

12cos

cos2 2 46

cos2 23

2 02 2

FHGIKJ FHGIKJ, ,

f cos coscos cos

4 1 22

12

1

b g

f 13

1 32

FHGIKJ

MATHEMATICS

[29] IIT JEE – 2012

60. Let X and Y be two events such that P X Y|b g 12

, P Y X|b g 13

and P X Yb g 16

. Which of the

following is(are) correct?

(A) P X Yb g 23

(B) X and Y are independent

(C) X and Y are not independent (D) P X Yc c h 13

Ans. [A, B]

Sol. P XYFHGIKJ

12

, P YXFHGIKJ

13

, P X Y b g 16

P XY

P X YP Y

FHGIKJ

b gb g

12 P Yb g 1

3

P YX

P Y XP X

FHGIKJ

b gb g

13 P Xb g 1

2

Now P X Y P X P Y b g b g b g16

X and Y are independentAlso P X Y P X P Y P X Y b g b g b g b g

12

13

16

23

P X Y P Y P X Yc c h b g b g

13

16

16