solved paper 2014

Instructions

1. This test consists of 90 questions.

2. There are three parts in the question paper A,B,C consisting of Physics, Chemistry and Mathematics

having 30 questions in each part of equal weightage. Each question is allotted 4 marks for correct

response.

3. Candidates will be awarded marks as stated above in instruction 2 for correct response of each

question. 1/4 mark will be deducted for indicating incorrect response for an item in the answer sheet.

4. There is only one correct response for each question. Filling up more than one response in any

question will be treated as wrong response and marks for wrong response will be deducted

accordingly as per instructions.

Physics1. The current voltage relation of diode is

given by I e T= ( )/1000 1V mA, where the

applied voltage V is in volt and thetemperature T is in kelvin. If a studentmakes an error measuring 0 01. V whilemeasuring the current of 5 mA at 300K,what will be the error in the value ofcurrentinmA?

(a) 0.2mA (b) 0.02mA

(c) 0.5mA (d) 0.05mA

2. From a tower of height H, a particle isthrown vertically upwards with a speed u.The time taken by the particle to hit theground, is n times that taken by it to reachthe highest point of its path. The relationbetween H, u and n is

(a) 2 2 2gH n u= (b) gH n u= ( )2 2 2

(c) 2 22gH nu n= ( ) (d) gH n u= ( )2 2 2

3. A mass m supported by amassless string woundaround a uniform hollowcylinder of mass m andradius R. If the string doesnot slip on the cylinder,with what acceleration willthemassfallonrelease?

(a)23g

(b)g2

(c)56g

(d) g

4. A block of mass m is placed on a surface

with a vertical cross-section given by

y x= 3 6/ . If the coefficient of friction is

0.5, the maximum height above the

ground at which the block can be placed

withoutslippingis

(a)16

m (b)23

m (c)13

m (d)12

m

Solved Paper 2014

Joint Entrance ExaminationJoint Entrance Examination

Time : 3 hrs MM : 360

R

m

m

5. When a rubber band is stretched by adistance x, it exerts a restoring force ofmagnitude F ax bx= + 2 , where a and b are

constants. The work done in stretching theunstretched rubber-bandby L is

(a) aL bL2 3+ (b) 12

2 3( )aL bL+

(c)aL bL2 3

2 3+ (d) 1

2 2 3

2 3aL bL+

6. A bob of mass m attached to aninextensible string of length l is suspendedfrom a vertical support. The bob rotates in ahorizontal circle with an angular speed rad/s about the vertical support. About thepointofsuspension

(a) angularmomentumisconserved

(b) angular momentum changes in magnitude butnotindirection

(c) angular momentum changes in direction but notinmagnitude

(d) angular momentum changes both in directionandmagnitude

7. Four particles, each of mass M and

equidistant from each other, move along a

circle of radius R under the action of their

mutual gravitational attraction, the speed

ofeachparticleis

(a)GMR

(b) 2 2GMR

(c)GMR

( )1 2 2+

(d)12

1 2 2GMR

( )+

8. The pressure that has to be applied to the

ends of a steel wire of length 10 cm to keep

its length constant when its temperature is

raised by 100C is (For steel, Youngs

modulus is 2 1011 Nm2 and coefficient of

thermal expansion is1.1 10 5 1K )

(a) 2 2 108. Pa (b) 2 2 109. Pa

(c) 2 2 107. Pa (d) 2 2 106. Pa

9. There is a circulartube in a verticalplane. Two liquidswhich do not mix andof densities d1 and d2are filled in the tube.Each liquid subtends90 angle at centre. Radius joining theirinterface makes an angle with vertical.Ratio d d1 2/ is

(a)11

+

sinsin

(b)11

+

coscos

(c)11

+

tantan

(d)11

+

sincos

10. On heating water, bubbles beingsformed at the bottom of the vessel detachand rise. Take the bubbles to be spheresof radius R and making a circular contactof radius r with the bottomof the vessel. If r

12. One mole of diatomicideal gas undergoes acyclic process ABC asshown in figure. Theprocess BC is adiabatic.The temperatures at A,B and C are 400 K, 800 K and 600 K,respectively. Choose the correctstatement.

(a) The change in internal energy in whole cyclicprocess is 250 R

(b) The change in internal energy in the process CAis700 R

(c) The change in internal energy in the process ABis 350 R

(d) The change in internal energy in the process BCis 500 R

13. An open glass tube is immersed in mercuryin such a way that a length of 8cm extendsabove the mercury level. The open end ofthe tube is then closed and sealed and thetube is raised vertically up by additional 46cm. What will be length of the air columnabove mercury in the tube now?(Atmospheric pressure = 76 cmofHg)

(a) 16cm (b) 22cm (c) 38cm (d) 6cm

14. A particle moves with simple harmonicmotion in a straight line. In first sec, afterstarting from rest it travels a distance a andin next sec, it travels 2a, in samedirection,then

(a) amplitudeofmotionis3a

(b) timeperiodofoscillationsis 8pi

(c) amplitudeofmotionis4a

(d) timeperiodofoscillationsis 6pi

15. A pipe of length 85 cm is closed from oneend. Find the number of possible naturaloscillations of air column in the pipe whosefrequencies lie below 1250 Hz. Thevelocityofsoundinairis340m/s.

(a) 12 (b) 8 (c) 6 (d) 4

16. Assume that an electric field E i= 30 2x $

exists in space. Then, the potentialdifference V VA O , where VO is thepotential at the origin and VA the potentialat x = 2 m,is

(a) 120J (b) 120 J (c) 80 J (d) 80J

17. A parallel plate capacitor is made of twocircular plates separated by a distance of5mm and with a dielectric of dielectricconstant 2.2 between them. When theelectric field in the dielectric is3 104 V/m, the charge density of the

positiveplatewillbecloseto

(a) 6 10 7 C/m2 (b) 3 10 7 C/m2

(c) 3 104 C/m2 (d) 6 104 C/m2

18. In a large building, there are 15 bulbs of40W, 5bulbs of 100 W, 5 fans of 80 Wand 1heater of 1kW. The voltage of theelectric mains is 220 V. The minimumcapacity of the main fuse of the buildingwillbe

(a) 8A (b) 10A (c) 12A (d) 14A

19. A conductor lies along the z-axis at

21. In the circuit shown here, the point Cis kept connected to point A till thecurrent flowing through the circuitbecomes constant. Afterward,suddenly point C is disconnectedfrom point A and connected to pointB at time t = 0. Ratio of the voltageacross resistance and the inductor att L R= / willbeequalto

(a)e

e1 (b) 1

(c) 1 (d)1 e

e

22. During the propagation ofelectromagnetic waves in a medium,

(a) electric energy density is double of themagneticenergydensity

(b) electric energy density is half of themagneticenergydensity

(c) electric energy density is equal to themagneticenergydensity

(d) Both electric and magnetic energydensitiesarezero

23. A thin convex lens made from crown

glass = 32 has focal length f.When

it is measured in two different liquids

having refractive indices43

and53

. It

has the focal lengths f1 and f2 ,respectively. The correct relationbetweenthefocallengthis

(a) f f f1 2= and f2 becomesnegative

(c) f f2 > and f1 becomesnegative

(d) f1 and f2 bothbecomenegative

24. A green light is incident from the water to theair-water interface at the critical angle ().Selectthecorrectstatement.

(a) The entire spectrum of visible light will come out ofthewateratanangleof90tothenormal

(b) The spectrum of visible light whose frequency is lessthan that of green light will come out of the airmedium.

(c) The spectrum of visible light whose frequency ismore than that of green light will come out to the airmedium.

(d) The entire spectrum of visible light will come out ofthewateratvariousanglestothenormal.

25. Two beams, A and B, of plane polarised lightwith mutually perpendicular planes ofpolarisation are seen through a polaroid. Fromthe position when the beam A has maximumintensity (and beam B has zero intensity), arotation of polaroid through 30 makes the twobeams appear equally bright. If the initialintensities of the two beams are IA and IBrespectively,then I IA B/ equals

(a) 3 (b)32

(c) 1 (d)13

26. The radiation corresponding to 3 2 transitionof hydrogen atom falls on a metal surface toproduce photoelectrons. These electrons aremade to enter a magnetic field of 3 10 4 T. If

the radius of the largest circular path followedby these electrons is 10.0 mm, the workfunctionofthemetaliscloseto

(a) 1.8eV (b) 1.1eV (c) 0.8eV (d) 1.6eV

27. Hydrogen( )11H , deuterium( )1

2H , singly ionised

helium ( He )24 + and doubly ionised lithium

( Li )38 + + all have one electron around the

nucleus. Consider an electron transition fromn = 2 to n = 1. If the wavelengths of emittedradiation are 31 2, , and 4 , respectively forfour elements, then approximately which oneofthefollowingiscorrect?

(a) 4 1 = = =2 22 3 4(b) 1 = = =2 22 3 4(c) 1 = = =2 3 44 9(d) 1 = = =2 3 42 3 4

4 JEE Main Solved Papers

A C R

B

L

28. Theforwardbiaseddiodeconnectionis

(a)

(b)

(c)

(d)

29. A student measured the length of a rod andwrote it as 3.50 cm. Which instrument didheusetomeasureit?

(a) Ameterscale

(b) A vernier calliper where the 10 divisions in vernierscale matches with 9 divisions in main scale andmainscalehas10divisionsin1cm

(c) A screw gauge having 100 divisions in thecircularscaleandpitchas1mm

(d) A screw gauge having 50 divisions in the circularscaleandpitchas1mm

30. Match List I (Electromagnetic wavetype) with List II (Itsassociation/application) and select thecorrect option from the choices givenbelowthelists.

ListI ListII

A. Infrared waves 1. Totreatmuscularstrain

B. Radiowaves 2. Forbroadcasting

C. X-rays 3. Todetectfractureofbones

D. Ultraviolet 4. Absorbed by the ozone layeroftheatmosphere

Codes

A B C D

(a) 4 3 2 1

(b) 1 2 4 3

(c) 3 2 1 4

(d) 1 2 3 4

JEE Main Solved Paper 2014 5

Chemistry31. The correct set of four quantum numbers

for the valence electrons of rubidium atom( )Z = 37 is

(a) 5 0 012

, , , + (b) 5 1 0 12

, , , +

(c) 5 1 112

, , , , + (d) 5 0 1 12

, , , +

32. If Z is a compressibility factor, van derWaals equation at low pressure can bewrittenas

(a) ZRTpb

= +1 (b) Z aVRT

= 1

(c) ZpbRT

= 1 (d) ZpbRT

= +1

33. CsCl crystallises in body centred cubiclattice. If a its edge length, then which ofthefollowingexpressionsiscorrect?

(a) r r aCs Cl+

+ = 3

(b) r ra

Cs Cl+

+ = 32

(c) r r aCs Cl+

+ = 32

(d) r r aCs Cl+

+ = 3

34. For the estimation of nitrogen, 1.4 g of anorganic compound was digested byKjeldahl's method and the evolved ammonia

was absorbed in 60 mL ofM

10sulphuric acid.

The unreacted acid required 20 mL of M /10sodium hydroxide for completeneutralisation. The percentage of nitrogenin the compound is

(a) 6% (b) 10%

(c) 3% (d) 5%

35. Resistance of 0.2 M solution of an

electrolyte is 50 . The specificconductance of the solution of 0.5 M

solution of same electrolyte is 1.4 S m 1

and resistance of same solution of the

same electrolyte is 280 . The molarconductivity of 0.5M solution of the

electrolytein Sm mol2 1 is

(a) 5 10 4 (b) 5 10 3

(c) 5 103 (d) 5 102

2V +2V

2V 4V

3V 3V

+2V 2V

36. For the complete combustion of ethanol,

C H OH( )+3O ( ) 2CO ( )+3H O( ),2 5 2 2 2l g g lthe amount of heat produced as measured in

bomb calorimeter, is 1364.47 kJ mol 1 at

25C. Assuming ideality the enthalpy of

combustion, CH, for the reaction will be( = 8.314 J mol )1 1R K

(a) 1366 95. kJ mol 1

(b) 1361 95. kJ mol 1

(c) 1460 50. kJ mol 1

(d) 1350 50. kJ mol 1

37. The equivalent conductance of NaCl atconcentration C and at infinite dilution are C and , respectively. The correctrelationship between C and is given as(where,theconstant B ispositive)

(a) C (B)C= +(b) C B C= ( )(c) C B C= ( )(d) C C= + ( )B

38. Consider separate solution of 0.500M

C H OH2 5 (aq), 0.100 M Mg3(PO )4 2(aq),

0.250M KBr(aq) and 0.125 M Na PO3 4 (aq)

at 25C. Which statement is true about these

solution, assuming all salts to be strong

electrolytes?

(a) Theyallhavethesameosmoticpressure

(b) 0.100 M Mg (PO )3 4 2(aq) has the highest osmoticpressure

(c) 0.125 M Na PO3 4(aq) has the highest osmoticpressure

(d) 0.500 M C H OH2 5 (aq) has the highest osmoticpressure

39. Forthereaction,

SO O SO2 2 3( ) ( ) ( )g g g+12

q

if K K RTp Cx

= ( )

where, the symbols have usual meaning,thenthevalueof x is(assuming ideality)

(a) 1 (b) 12

(c)12

(d) 1

40. For the non-stoichiometric reaction2 A B C D+ + , the following kineticdata were obtained in three separateexperiments,allat298K.

Initial

concentration

( )A

Initial

concentration

( )B

Initialrateof

formationof C

(molL1S1)

(i) 0.1M 0.1M 1 2 10 3.

(ii) 0.1M 0.2M 1 2 10 3.

(iii) 0.2M 0.1M 2 4 10 3.

Theratelawfortheformationof C is

(a)dCdt

= k A[ [B]] (b)dCdt

= k A B[ ] [ ]2

(c)dCdt

= k A B[ ][ ]2 (d)dCdt

= k A[ ]

41. Among the following oxoacids, thecorrect decreasing order of acidstrengthis

(a) HOCl > HClO > HClO > HClO2 3 4(b) HClO > HOCl > HClO > HClO4 2 3(c) HClO > HClO > HClO > HOCl4 3 2(d) HClO > HClO > HClO > HOCl2 4 3

42. The metal that cannot be obtained by

electrolysis of an aqueous solution of its

saltsis

(a) Ag (b) Ca(c) Cu (d) Cr

43. The octahedral complex of a metal ion

M3+ with four monodentate ligands L1,

L2 , L3 and L4 absorb wavelengths in the

region of red, green, yellow and blue,

respectively. The increasing order of

ligandstrengthofthefour ligands is

(a) L < L , L < L4 3 2 1(b) L < L < L < L1 3 2 4(c) L < L < L < L3 2 4 1(d) L < L < L < L1 2 4 3

44. Which of the following properties is notshownbyNO?

(a) Itis diamagnetic ingaseousstate

(b) Itisaneutraloxide

(c) It combines with oxygen to form nitrogendioxide

(d) Itsbondorderis2.5


45. In which of the following reactions H O2 2actsasareducingagent?

I. H O H H O2 2 22 2 2+ + + e

II. H O O H2 2 22 ++ 2e

III. H O OH2 2 2 2+ e

IV. H O OH O H O2 2 2 22 2 2+ +

e

(a) IandII (b) IIIandIV

(c) IandIII (d) IIandIV

46. The correct statement for the molecule,CsI3 is

(a) itisacovalentmolecule

(b) itcontains Cs+ and I3

(c) itcontains Cs3+ and I ions

(d) itcontains Cs+ , I andlattice I2 molecule

47. The ratio of masses of oxygen and nitrogenof a particular gaseous mixture is 1:4.Theratioofnumberoftheirmoleculeis

(a) 1:4 (b) 7:32 (c) 1:8 (d) 3:16

48. Givenbelowarethehalf-cellreactions

Mn Mn2 2+ + e ; E = 1.18 eV

2 2( );Mn Mn3+ ++ e E = + 1.51eV

The E for 3Mn Mn + 2Mn2 3++ willbe(a) 2 69. V;thereactionwillnotoccur(b) 2 69. V;thereactionwilloccur

(c) 0 33. V;thereactionwillnotoccur(d) 0 33. V;thereactionwilloccur

49. Which series of reactions correctlyrepresents chemical relations related toironanditscompound?

(a) Fe FeSO4

Dil. H SO H SO ,O2 4 2 4 2 Fe (SO )

2 4 3

Heat Fe

(b) Fe Fe2 2 4, Heat Dil. H SO

O

O FeSO4

Heat Fe

(c) Fe FeClCl , Heat Heat, air2

3 FeCl2

Zn Fe

(d) Fe Fe OO , Heat CO, 600 C2

3 4 FeO

CO, 700 C Fe

50. The equation which is balanced andrepresentsthecorrectproduct(s)is

(a) Li O + 2KCl 2LiCl + K O2 2(b) [CoCl(NH ) ] + 5H Co3 5

+ 2++

+5NH + Cl4+

(c) [Mg(H O) + (EDTA)2 62+ 4 Excess NaOH]

[Mg(EDTA)] + 6H O2+ 2

(d) CuSO + 4KCN K [Cu(CN) ]4 2 4+ K SO2 4

51. In S 2N reactions, the correct order of

reactivity for the following compoundsCH Cl,CH CH Cl,3 3 2 (CH ) CHCl3 2 and(CH ) CCl3 3 is

(a) CH CI > (CH ) CHCI > CH CH CI3 3 2 3 2> (CH ) CCI3 3

(b) CH CI> CH CH CI> (CH ) CHCI3 3 2 3 2> (CH ) CCI3 3

(c) CH CH Cl > CH Cl > (CH ) CHCl3 2 3 3 2> (CH ) CCI3 3

(d) (CH ) CHCI > CH CH CI > CH CI3 2 3 2 3> (CH )3 3CCI

52. On heating an aliphatic primary aminewith chloroform and ethanolic potassiumhydroxide, the organic compound formedis

(a) an alkanol

(b) an alkanediol

(c) analkylcyanide

(d) analkyl isocyanide

53. The most suitable reagent for the

conversionof

R R CH2 OH CHO is(a) KMnO4(b) K Cr O2 2 7(c) CrO3(d) PCC (pyridinium chlorochromate)

54. The major organic compound formed bythe reaction of 1,1,1-trichloroethane withsilverpowderis

(a) acetylene (b) ethene

(c) 2-butyne (d) 2-butene


55. Sodium phenoxide when heated withCO2under pressure at 125C yields a productwhichon acetylation produces C.

+ +

CO2125

5 atm

H

Ac O2

B C

Themajorproduct C wouldbe

56. Considering the basic strength of aminesin aqueous solution, which one has thesmallest pKb value?

(a) (CH ) NH3 2 (b) CH NH3 2(c) (CH ) N3 3 (d) C H NH6 5 2

57. For which of the following moleculesignificant 0 ?

(a) OnlyI (b) IandII

(c) OnlyIII (d) IIIandIV

58. Which one is classified as a condensationpolymer?

(a) Dacron (b) Neoprene(c) Teflon (d) Acrylonitrile

59. Which one of the following bases is notpresentinDNA?

(a) Quinoline (b) Adenine(c) Cytosine (d) Thymine

60. Inthereaction,

CH COOH3 LiAlH PCl Alc. KOH4 5

A B C

Theproduct C is

(a) acetaldehyde (b) acetylene(c) ethylene (d) acetylchloride


Cl

ClI

CN

CNII

OH

OHIII

SH

SHIV

COOH

OCOCH3

COCH3

OH

COCH3

COOCH3

OH

COOH

OCOCH3

(a) (b)

(d)(c)

Mathematics61. If X n n Nn= { : }4 3 1 and

Y n n N= { ( ) : }9 1 ; where N is the set ofnaturalnumbers,then X Y isequalto(a) N (b) Y X

(c) X (d) Y

62. If z is a complex number such that| |z 2,

thentheminimumvalueof z +12

(a)isequalto52

(b)liesintheinterval(1,2)

(c)isstrictlygreaterthan52

(d)isstrictlygreaterthan32

butlessthan52

63. If a R and the equation

+ 3 22( [ ]) ( [ ])x x x x + =a2 0 (where,[ ]x

denotes the greatest integer x)has nointegral solution, then all possible valuesof a lieintheinterval

(a) ( , ) ( , ) 1 0 0 1 (b) ( , )1 2

(c) ( , ) 2 1 (d) ( , ) ( , ) 2 2

64. Let and be the roots of equationpx qx r2 0+ + = , p 0. If p q, and r are in

AP and1 1

4 + = , then the value of| |

is

(a)619

(b)2 17

9(c)

349

(d)2 13

9

ONa

65. If , 0 and f n n n( ) = + and3 1 1 1 2

1 1 1 2 1 31 2 1 3 1 4

+ ++ + ++ + +

f f

f f f

f f f

( ) ( )( ) ( ) ( )( ) ( ) ( )

= K( ) ( )1 12 2 ( ) 2 ,then K isequalto

(a) (b) 1 (c) 1 (d) 1

66. If A is a 3 3 non-singular matrix such

that AA A AT T= and B A AT= 1 , then BBT

isequalto

(a) I B+ (b) I (c) B1 (d) ( )B T1

67. If the coefficients of x3 and x4 in theexpansion of ( )( )1 1 22 18+ + ax bx x in

powers of x are both zero, then ( , )a b isequalto

(a) 162513

, (b) 14 2513, (c) 14

2723

, (d) 16 2723, 68. The angle between the lines whose

direction cosines satisfy the equationsl m n+ + = 0 and l m n2 2 2= + is

(a)pi3

(b)pi4

(c)pi6

(d)pi2

69. If( ) ( ) ( ) ( ) ( ) ... ( )10 2 11 10 3 11 10 10 119 1 8 2 7 9+ + + +

= k( )10 9 , then k isequalto

(a)12110

(b)441100

(c)100 (d)110

70. Three positive numbers form anincreasing GP. If the middle term in thisGP is doubled, then new numbers are inAP.Then,thecommonratiooftheGPis

(a) 2 3+ (b) 3 2+(c) 2 3 (d) 2 3+

71. limsin( cos )

x

x

x 0

2

2

piisequalto

(a)pi2

(b)1

(c) pi (d) pi

72. If g is the inverse of a function f and

=

+f x

x( )

1

1 5,then g x( ) isequalto

(a) 1 5+ x (b) 5 4x

(c)1

1 5+ { ( )}g x(d) 1 5+ { ( )}g x

73. If f and g are differentiable functions in

(0,1) satisfying f g( ) ( )0 2 1= = , g( )0 0= andf( )1 6= ,thenforsome c ] , [0 1

(a) 2/ = f c g c( ) ( ) (b) 2 3 = f c g c( ) ( )(c) = f c g c( ) ( ) (d) = f c g c( ) ( )2

74. If x = 1 and x = 2 are extreme points of

f x x x x( ) log| |= + +2 ,then(a) = 6, = 1

2(b) = 6, = 1

2

(c) = 2 , = 12

(d) = 2 , = 12

75. The integral 11

1

+ + x x e dxx x is equal

to

(a) ( )x e Cx

x +

+1

1

(b) xe Cx

x+

+1

(c) ( )x e Cx

x+ ++

11

(d) ++

xe Cx

x

1

76. The integral 1 42

42

2

0

+ sin sinx x dxpi isequalto

(a) pi 4 (b)23

4 4 3pi

(c) 4 3 4 (d) 4 3 43

pi

77. The area of the region described byA x y x y= + {( , ) : 2 2 1and y x2 1 } is

(a)pi2

43

+ (b) pi2

43

(c)pi2

23

(d)pi2

23

+

78. Let the population of rabbits surviving at

atime t be governed by the differential

equationdp t

dtp t

( )( )=

12

200. If p( )0 100= ,

then p t( ) isequalto

(a) 400 300 2 et

(b) 300 200 2

et

(c) 600 500 2 et

(d) 400 300 2

et


79. If PS is the median of the triangle withvertices P ( , )2 2 , Q ( , )6 1 and R( , )7 3 , thenequation of the line passing through( , )1 1andparallelto PS is

(a) 4 7 11 0x y = (b) 2 9 7 0x y+ + =(c) 4 7 3 0x y+ + = (d) 2 9 11 0x y =

80. Let a b c, , and d be non-zero numbers. If

the point of intersection of the lines4 2 0ax ay c+ + = and 5 2 0bx by d+ + =lies in the fourth quadrant and isequidistantfromthetwoaxes,then

(a) 2 3 0bc ad = (b) 2 3 0bc ad+ =(c) 2 3 0ad bc = (d) 3 2 0bc ad+ =

81. The locus of the foot of perpendiculardrawn from the centre of the ellipsex y2 23 6+ = onanytangenttoitis

(a) ( )x y x y2 2 2 2 26 2 = +

(b) ( )x y x y2 2 2 2 26 2 =

(c) ( )x y x y2 2 2 2 26 2+ = +

(d) ( )x y x y2 2 2 2 26 2+ =

82. Let C be the circle with centre at (1, 1) andradius 1. If T is the circle centred at ( , )0 ypassing through origin and touching thecircle C externally, then the radius of T isequalto

(a)3

2(b)

32

(c)12

(d)14

83. The slope of the line touching both theparabolas y x2 4= and x y2 32= is

(a)12

(b)32

(c)18

(d)23

84. The image of the linex y z

=

=

1

3

3

1

4

5

in the plane 2 3 0x y z + + = is the line

(a)x y z+

=

=

33

51

25

(b)x y z+

=

=+3

35

12

5

(c)x y z

=+

=

33

51

25

(d)x y z

=+

=3

35

12

5

85. If [ ]a b b c c a = [ ]a bc 2 , then isequalto

(a)0 (b)1

(c)2 (d)3

86. Let A and B be two events such that

P A B( ) =16

, P A B( ) =14

and P A( ) =14

,

where A stands for the complement of the

event A.Then,theevents A and B are

(a)independentbutnotequallylikely(b)independentandequallylikely

(c)mutuallyexclusiveandindependent(d)equallylikelybutnotindependent

87. The variance of first 50 even naturalnumbersis

(a)8334

(b)833

(c)437 (d)437

4

88. If f x k x xkk k( ) / (sin cos )= +1 , where x R

and k 1,then f x f x4 6( ) ( ) isequalto(a)

16

(b)13

(c)14

(d)1

12

89. A bird is sitting on the top of a verticalpole 20m high and its elevation from apoint O on the ground is 45. It flies offhorizontally straight away from the pointO. After 1s, the elevation of the bird fromOis reduced to 30. Then, the speed (in m/s)ofthebirdis

(a) 40 2 1( )

(b) 40 3 2( )

(c) 20 2

(d) 20 3 1( )

90. Thestatement ~( ~ )p q is

(a)equivalentto p q(b)equivalentto ~ p q(c)atautology

(d)afallacy


Solutions

Physics

1. Given, I e V T= ( )/1000 1 mA,dV = 0 0. 1 VT = 300 K

I = 5 mA

So, I e V T= 1000 1/

I e V T+ =1 1000 /

Takinglogonbothsides, weget

log( )IV

T+ =1

1000

Ondifferentiating,dI

I T+=

11000

dV

dIT

I= +1000 1( )dV

dI = + 1000300

5 1 0 01( ) . = 0.2 mA

So, error in the value of current is 0.2 mA.

2. Time taken to reach the maximum height t ug1

=

If t 2 isthetimetakentohittheground,

i.e., = H ut gt2 221

2

But t nt2 1= [Given]

So, = H unug

gn u

g

12

2 2

2

= Hnug

n ug

2 2 212

2 22gH nu n= ( )

3. For the mass m,mg T ma =

Asweknow, a R=

So, mg T mR = (i)

Torqueaboutcentreofpully

T R mR = 2 ...(ii)

FromEqs.(i)and(ii),weget

ag

=

2

Hence, the acceleration with the mass of a body fall isg2

.

4. A block of mass m is placed on a surface with avertical cross-section, then

tan =

=dydx

dx

dxx

3

26

2

H

t1

t2

u

Answers

1. (a) 2. (c) 3. (b) 4. (a) 5. (c) 6. (c) 7. (d) 8. (a) 9. (c) 10. (*)

11. (c) 12. (d) 13. (a) 14. (d) 15. (c) 16. (c) 17. (a) 18. (c) 19. (b) 20. (c)

21. (c) 22. (c) 23. (b) 24. (d) 25. (d) 26. (b) 27. (c) 28. (a) 29. (b) 30. (d)

31. (a) 32. (b) 33. (c) 34. (b) 35. (a) 36. (a) 37. (c) 38. (a) 39. (b) 40. (d)

41. (c) 42. (b) 43. (b) 44. (a) 45. (d) 46. (b) 47. (b) 48. (a) 49. (d) 50. (b)

51. (b) 52. (d) 53. (d) 54. (c) 55. (a) 56. (a) 57. (d) 58. (a) 59. (a) 60. (c)

61. (d) 62. (b) 63. (a) 64. (d) 65. (c) 66. (b) 67. (d) 68. (a) 69. (c) 70. (d)

71. (d) 72. (d) 73. (d) 74. (c) 75. (b) 76. (d) 77. (a) 78. (a) 79. (b) 80. (c)

81. (c) 82. (d) 83. (a) 84. (a) 85. (b) 86. (a) 87. (b) 88. (d) 89. (d) 90. (a)

NOTE *Means,noneoptionsarenotcorrect.

R

m a

m

mg

T

T

Y

y

X

m

Atlimiting equilibrium, weget

= tan

0 52

2

. =x x2 1= x = 1

Now,puttingthevalueof x in yx

=

3

6,weget

When x = 1

y = =( )16

16

3

When x = 1

y =

=( )1

61

6

3

So, the maximum height above the ground atwhich the block can be placed without slipping is16

m .

5. / We know that change in potential energy of asystem corresponding to a conservativeinternal force as

U U W df i

f

i = = F r.Given, F ax bx= + 2

We know that work done in stretching the rubberband by L is

| | | |dW Fdx=

| | ( )W ax bx dx

O

L

= + 2

=

+

ax bx

O

L

O

L2 3

2 3

=

+

aL a b L b22 3 3

20

2 30

3( ) ( )

| |WaL bL

= +2 3

2 3

6. Angular momentum of the pendulum about thesuspension point O is

L r v= ( )m

Then, v can be resolved into two components,radial component rrad and tangential componentrtan. Due to vrad, L will be tangential and due to v tan,L will be radially outwards as shown.

So, net angular momentum will be as shown infigure whose magnitude will be constant(| | )L mvl= . But its direction will change as shown inthe figure.

L r v= ( )m

where, r = radiusofcircle

7. Net force acting on any one particle M,= + GM

R

GM

R

2

2

2

22 245

( ) ( )cos + GM

R

2

2245

( )cos

= + GMR2

2

14

1

2

This force will equal to centripetal force.

So,Mv

RGM

R

2 2

2

14

1

2= +

vGM

R= +

41 2 2( ) = +1

22 2 1

GMR

( )

Hence, speed of each particle in a circular motionis

12

2 2 1GMR

( )+ .

8. / If the deformation is small, then the stress in abody is directly proportional to the corres-ponding strain.

According to Hooke's law i.e.,

Youngs modulus ( )Tensile stressTensile strain

Y =

So, YF AL L

FLA L

= =//

If the rod is compressed, then compressive stressand strain appear. Their ratio Y is same as that fortensile case.

Given, length of a steel wire (L) = 10cm,

Temperature ( ) = 100 CAslengthisconstant.

Strain = = LL


O

L L

O

R

M

m

v

vv

m

M

RR

45

45

R

v

Now, pressure = stress = Y strain

[Given, Y = 2 1011N / m2 and = 1.1 K. 10 5 1]

= 2 10 11 10 10011 5.

= 2 2 108. Pa

9. Equating pressure at A, we getR d R d R dsin cos ( cos ) 2 2 11+ +

= R d( sin )1 1

(sin cos ) (cos sin ) + = d d2 1

d

d1

2

11

=+

tantan

10. The bubble will detach if,Buoyant force Surface tension force

43

3pi R g T dlw sin

( ) ( ) ( )sin pi pi w R g T r43

23 sin = r

R

Solving, rR g

Tw

=2

3

4= R

g

Tw2 2

3

No option matches with the correct answer.

11. / In thermal conduction, it is found that insteady state the heat current is directlyproportional to the area of cross-section Awhich is proportional to the change intemperature ( )T T1 2 .

Then,Qt

KA T T

x=

( )1 2

According to thermal conductivity, we getdQ

dt

dQ

dt

dQ

dt1 2 3

= +

0 92 10046

026 013

. ( ) . ( )=

T T+

012 012

. ( )T

T = 40 C

dQ

dt1 0 92 4 100 40

404 8=

=

. ( ). cal/s

12. According to first law of thermodynamics, we get(i) Change in internal energy from A to B i.e., UAB

U nC T TAB V B A= ( )= =1

52

800 400 1000R

( ) R

(ii) Change in internal energy from B to C

U nC T TBC V C B= ( )= = 1

52

600 800 500R

( ) R

(iii) Uisothermal = 0(iv) Change in internal energy from C to A i.e., UCA

U nC T TCA V A C= ( )= = 1

52

400 600 500R

( ) R

13. / In this question, the system is acceleratinghorizontally i.e., no component of accelerationin vertical direction. Hence, the pressure in thevertical direction will remain unaffected.i.e., p p gh1 0= +

Again, we have to use the concept that thepressure in the same level will be same.

For air trapped in tube, p V p V1 1 2 2=

p p g1 76= =atm V A1 8= [A = areaofcross-section]

p p g x g x2 54 22= = +atm ( ) ( )V A x2 =

g A g x A x76 8 22 = +( )x x2 22 78 8 0+ =

x = 16 cm


K2

l2

K3

l3

K1

l1

100C

0C0C

dQ3dQ2dtdt

dQ1dt

8 cm 54x

54x

p2

d1

RR R sin

d2

A

90 R

r

= sin (2 ) sinT dl T pr

R

14. In SHM, a particle starts from rest, we havei.e., x A t= cos ,at t = 0, x A=

When t = ,then

x A a= ....(i)

When t = 2,then

x A a= 3 ....(ii)

OncomparingEqs.(i)and(ii),weget

A a A = cos

A a A =3 2cos

As cos cos2 2 12 =

A a

A

A a

A

= 3 2 1

2

A a

A

A a Aa A

A

=+ 3 2 2 42 2 2

2

A aA A a Aa2 2 23 2 4 = +

a aA2 2=

A a= 2

Now, A a A = cos

cos / = 1 22

3pi

piT

=

T = 6 pi

15. For closed organ pipe=

+( )2 14

n v

l[ , , , ...]n = 0 1 2

( )2 14

1250n v

l

+

solved paper 2014

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