full test

CHEMISTRY1. Which one of the following is an example of co-ordination

isomerism ?(a) [Co (NH3)5 Br] SO4 and [Co (NH3)5 SO4] Br(b) [Co (NH3)5 NO2] Cl2 and [Co (NH3)5 ONO]Cl2(c) [Cr (H2O)6] Cl3 and [Cr (H2O)5 Cl] Cl2 H2O(d) [Co (NH3)6] [Cr (CN)6] and (Cr (NH3)6] [Co (CN)6]

2. The z - component of angular momentum of an electron inan atomic orbital is governed by the(a) azimuthal quantum number(b) principal quantum number(c) magnetic quantum number(d) spin quantum number

3. Two containers P and Q of equal volume (1 litre each) contain6 g of O2 and SO2 respectively at 300 K and 1 atmosphere.then(a) Number of molecules in P is less than that

in Q(b) Number of molecules in P and Q is same(c) Number of molecules in Q is less than that in P(d) Either (a) or (b)

4. A wedding ring presented to a bride contains 788 mg ofgold and rest diamond. If the ring weighs 1 g the bridereceives (At wt : Au = 197, C = 12)(a) more number of gold atoms

(b) more number of carbon atoms(c) equal number of gold and carbon atoms(d) gold and carbon atoms in ratio of 4:1 approximately

5. The solution containing 4.0 % PVC in one litre of dioxanewas found to have osmotic pressure of 6.0 104 atm at 300K. The molecular mass of polymer is

(a) 6.4 102 (b) 3.0 102

(c) 1.6 105 (d) 5.6104

6. Urea and hydrazine react to form ammonia gas along withcompound X which reacts with aldehydes and ketones toform specific crystalline derivatives. X is

(a) phenylhydrazine (b) semicarbazide

(c) biuret (d) acetylurea

7. The I.U.P.A.C. name of 2||

2|

H

CN

CH

CN

CH

CN

C -- is

(a) 1, 2, 3-propanetrinitrile

(b) 1, 2, 3 tricyanopropane

(c) 3-cyano-1, 5-dinitrilepentane

(d) 3-cyano 1, 5-pentanedinitrile

INSTRUCTIONS

Chemistry (120 marks) : Question No. 1 to 30 are of 4 marks each.

Physics (120 marks) : Question No. 31 to 60 are of 4 marks each.

Mathematics (120 marks) : Question No. 61 to 90 are of 4 marks each.

Negative Marking : One fourth () marks will be deducted for indicating incorrect response of each question.. . . Best of Luck

Time : 3 hrs. Max. Marks : 360

FULL TEST

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28. Diborane has

(a) 2 two - centre - two - electron bonds and 4 three -centre - three - electron bonds.

(b) 4 two - centre - two - electron bonds and 2 three -centre - three - electron bonds

(c) 3 two - centre - two - electron bonds and 3 three -centre - three - electron bonds.

(d) all six identical bonds.

9. The repeating units of acrilan are

(a) 32 COOCH

H|CCH -= (b) 522 HCOOC

H|CCH -=

(c) CN

H|CCH 2 -= (d)

3

2 3

CH|

H C C COOCH= -

10. A metal X on heating in nitrogen gas gives Y.Y ontreatment with H2O gives a colourless gas which whenpassed through CuSO4 solution gives a blue colour. Y is

(a) Mg(NO3)2 (b) Mg3N2(c) NH3 (d) MgO

11. During bromination of benzene in the presence of FeBr3 thespecies that attacks the ring to initiate the process is

(a) FeBr4 (b) Br+

(c) Br (d) Br2 molecule

12. Which of the following has a shape different from others?

(a) [Zn(NH3)4]+2 (b) Ni(CO)4

(c) [Cd(CN)4]-2 (d) [Cu(NH3)4]

+2

13. Di-n-propyl ether and diallyl ether can be distinguished by

(a) Acetic acid

(b) Sodium Metal

(c) Cold dilute KMnO4 solution

(d) PCl514. In which of the following cases, the stability of two oxidation

states is correctly represented

(a) Ti3+ > Ti4+ (b) Mn2+ > Mn3+

(c) Fe2+ > Fe3+ (d) Cu+ > Cu2+

15. A and B in the reaction,

BAClCHCHCHO3H.2

HCHO.1Mgether223 +

(a) CH3CH2CH2MgCl, CH3CH2CH2CH2OH

(b) CH3CHMgCH2Cl, CHOH

CH CH3 2H C3

(c) CH3CH2CH2MgCl,

H C3

H C3H C C OH3

(d) CH3CH2CH2MgCl, CH3CH2OCH2CH316. A gaseous mixture of three gases A, B and C has a pressure

of 10 atm. The total number of moles of all the gases is 10. Ifthe partial pressures of A and B are 3.0 and 1.0 atmrespectively and if C has mol. wt of 2.0 what is the weightof C in g present in the mixture.

(a) 8 (b) 12

(c) 3 (d) 6

17. Alcohols can be obtained from all methods except

(a) hydroboration-oxidation

(b) oxymercuration-demercuration

(c) reduction of aldehyde/ketones with ZnHg/HCl

(d) fermentation of starch

18. Which of the following expressions is not true ?

(a) For a netural solution [H+] = [OH] = wK

(b) For an acidic solution

[H+] > ww K]OH[andK -

(d) For a neutral solution [H+] = [OH] = 107 M at alltemperatures

19. For a chemical reaction X Y, the rate of reaction increasesby a factor of 1.837 when the concentration of X is increasedby 1.5 times. The order of the reaction with respect to X is

(a) 1 (b) 1.5

(c) 2 (d) 2.5

20. The final product of the following sequence of reactions

H D OH,NaOH 2 B

OH

)CH(LiCu

2

23 C

OH

LiAlH

2

4 D

(a) 3-ethyl-2-methyl-1-hexanol

3(b) 2, 3-dimethyl-1-pentanol

(c) 2-ethyl-3-methyl-1-hexanol

(d) 3, 3-dimethyl-1-pentanol

21. A solution containing As3+, Cd2+, Ni2+ and Zn2+ is madealkaline with dilute NH4OH and treated with H2S. Theprecipitate obtained will consist of

(a) As2S3 and CdS (b) CdS, NiS and ZnS(c) NiS and ZnS (d) Sulphide of all ions

22. M2SO4 (M+ is a monovalent metal ion) has a Ksp of 3.2 10

5

at 298 K. The maximum concentration of SO42 ion that

could be attained in a saturated solution of this solid at 298K is

(a) 3 103 M (b) 7 102 M(c) 2.89 104 M (d) 2102M

23. The intermediate formad during the addition of HCl topropene in the presence of peroxide is

(a) ClCHCHCH 23

(b)33CHCHCH

(c) 223 CHCHCH

(d) 3 2 2CH CH CH

24. Which of the following is not an actinide?

(a) Curium (b) Californium

(c) Uranium (d) Terbium

25. In the following compounds,

OH OH OHOH

CH3

NO2NO2

(I) (II) (III) (IV)

the order of acidity is(a) III > IV > I > II (b) I > IV > III > II

(c) II > I > III > IV (d) IV > III > I < II26. The electrolyte most effective for bringing about coagulation

of Fe (OH)3 sol is(a) K4[Fe(CN)6] (b) BaCl2(c) NaCl (d) None

27. The reaction of CH3 CH = CH OH with HBr

gives

(a) CH3CH = CH Br..

(b) CH3CH2CHBr OH

(c) CH3CHBrCH2 Br

(d) CH3CH2CHBr Br

28. The standard reduction potential for Cu2+/Cu is + 0.34.Calculate the reduction potential at pH = 14 for the abovecouple. (Ksp Cu(OH)2 = 1 10

19)(a) 0.22 V (b) + 0.22 V(c) 0.44 V (d) + 0.44 V

29. Final product (D) formed in the following reaction sequence is

CH2=CHCH=CH2+

BA NBS2 D

DC )2H(,PdKOH.alc -

(a) Naphthalene (b) Tetralin(c) Benzene (d) Anthracene

30. A current of 2.0A when passed for 5 hours through a moltenmetal salt deposits 22.2g of metal of atomic weight 177.Theoxidation state of the metal in the metal salt is(a) +1 (b) +2(c) +3 (d) +4

PHYSICS31. Drops of water fall from the roof of a building 9m high at

regular intervals of time. The first drop reaching the groundat the same instant fourth drop starts its fall. What are thedistances of second and third drops from roof ?

(a) 6m & 3m (b) 4m & 2m

(c) 6m & 2 m (d) 4m & 1m

32. A 55 gm tennis ball strikes the ground and rebounds back. Ifits velocities are 5 m/s and 4 m/s before and after the collisionrespectively then the impulse imparted will be

(a) 0.495 Ns (b) 0.495 Ns

(c) 495 Ns (d) 495 Ns

33. The time taken by a body in sliding down a rough inclinedplane of angle of inclination 45 is n times the time taken bythe same body in slipping down a similar frictionless plane.The coefficient of dynamic friction between the body and

4the plane will be

(a) 1/(1n2) (b) 1 (1/n2)

(c) {1 (1/n2)} (d) {1/(1n2)}34. A small ball of mass m is allowed to slip down from the top of

a hemispherical dome of radius R. At what height h from thelower end the ball will leave contact with the dome ?

R

m

(a) 2R / 3 (b) R / 2

(c) R / 4 (d) R / 6

35. Three particles of equal mass m are situated at the verticesof an equilateral triangle of side L. What should be thevelocity of each particle so that they move on a circular pathwithout changing L?

(a) (GM/2L) (b) (GM/L)(c) (2GM/L) (d) (GM/3L)

36. Two masses M & m situated at infinite distance apart are atrest. Due to mutual interaction they start approaching eachother, the relative velocity of the system when the distancebetween them is r, will be

(a)mM

Gr2+ (b) mM

Gr2+

(c)r

)mM(G2 +(d)

r)mM(G2 +

37. A wheel is rolling straight on ground without slipping. If theaxis of the wheel has speed v, the instantenous velocity of apoint P on the rim, defined by angle q, relative to the groundwill be

qP

(a)

q

21cosv (b)

q

21cosv2

(c) )sin1(v q+ (d) )cos1(v q+38. Statement 1 : The tyres of aircrafts are slightly conducting.

Statement 2 : If a conductor is connected to ground, theextra charge induced on conductor will flow to ground.(a) Statement-1 is true, Statement-2 is true; Statement-2 is

a correct explanation for Statement-1.(b) Statement-1 is true, Statement-2 is true; Statement-2 is

not a correct explanation for Statement-1.(c) Statement-1 is true, Statement-2 is false.(d) Statement-1 is false, Statement-2 is true.

39. Suppose the gravitational force varies inversely as the nthpower of the distance. Then the time period of a planet incircular orbit of radius R around the sun will be proportionalto(a) Rn (b) R-n(c) R (n+1)/2 (d) R (n-1)/2

40. At a specific temperature, the energy densities for a blackbody for three different wavelengths are 10, 19 and 7 units.For these wavelengths if the absorption coefficient of a bodyis respectively 0.8, 0.3 and 0.9 then the emissive powers ofthis body for these wavelengths are in the ratio(a) 10:19:7 (b) 8:3:9(c) 8:5.7:6.3 (d) 8:9.5:4.3

41. A mixture of two gases X and Y is enclosed, at constanttemperature. The relative molecular mass of X, which isdiatomic, is 8 times that of Y which is monoatomic. What isthe ratio of the r.m.s speed of molecules of Y to that ofmolecules of X?(a) 2 (b) 22(c) 4 (d) 8

42. A Carnots engine works as a refrigerator between 250 Kand 300 K. If it receives 750 calories of heat from the reservoirat the lower temperature, the amount of heat rejected at thehigher temperature is(a) 900 cal. (b) 625 cal.(c) 750 cal. (d) 1000 cal.

43. The figure shows a process on a gas in which pressure andvolume both change. The molar heat capacity for this processis C. Then

V

P

(a) C = 0 (b) C = Cv(c) C > Cv (d) C < Cv

44. A charge Q is distributed over two concentric hollow spheresof radii r and R (>r) such that the surface densities are equaland placed on the same axial points. Then the potential atthe common centre is

(a) )rR(4)rR(Q

0

22

+pe+

(b)rR

Q+

(c) Zero (d))rR(4

)rR(Q22

0 +pe+

545. Statement-1 : A paramagnetic sample display greatermagnetisation (for the same magnetic field) when cooled.Statement-2 : The magnetisation does not depend ontemperature.(a) Statement-1 is true, Statement-2 is true; Statement-2 is



46. If a battery is connected across series combination of acapacitor and a resistor, at t = 0. If at an instant t potentialdifference across the capacitor be V and energy stored in itbe U. Then which of the following graph is correct.

(a)

U

VO

(b)

U

VO

(c)

U

VO

(d)

U

VO

47. A cell of e.m.f. E is connected across a reistance R. Thepotential difference between the terminals of the cell is foundto be V. The internal resistance of the cell must be(a) [2(EV)V]/R (b) [2(EV)/V]R(c) [(EV)/V]R (d) (EV)R

48. A charge +q is taken around a long charged cylinder carryinga charge of l per unit length. The total work done is(a) zero (b) 4pl/e0 (c) l/e0 (d) l/4pe0

49. Two straight long conductors AOB and COD areperpendicular to each other and carry currents I1 and I2respectively. The magnitude of the magnetic induction at apoint P at a distance a from the point O in a directionperpendicular to the plane ABCD is

(a) )II(a2 210 +pm

(b) )II(a2 210pm

(c)1/ 22 20

1 2I I2 am + p (d)

+pm

21

210II

IIa2

50. The mutual inductance of a pair of coils, each of N turns, isM henry. If a current of I ampere in one of the coils isbrought to zero in t second, the emf induced per turn in theother coil, in volt, will be

(a)t

MI (b) t

NMI (c) It

MN (d)

NtMI

51. In a series resonant circuit, having L, C and R as its elements,the resonant current is i. The power dissipated in the circuitat resonance is

(a)

w-w C1L

R2i(b) zero

(c) i2wL (d) i2R52. Two different coils have self inductances L1 = 8 mH and

L2 = 2 mH. The current in one coil is increased at a constantrate. The current in the second coils is also increased at thesame constant rate. At a certain instant, the power given tothe two coil is the same. At that instant of time, if W1 and W2are the energies stored in the first and the second coil,respectively, the W1 : W2 is(a) 1 : 2 (b) 2 : 1 (c) 4 : 1 (d) 1 : 4

53. 56 tuning forks are so arranged in series that each fork gives4 beats per sec with the previous one. The frequency of thelast fork is 3 times that of the first. The frequency of the firstfork is(a) 110 (b) 56(c) 60 (d) 52

54. Two slits separated by a distance of 1 mm are illuminatedwith red light of wavelength 6.5 107 m. The interferencefringes are observed on a screen placed 1 m from the slits.The distance between third dark fringe & the fifth brightfringe is equal to(a) 0.65 mm (b) 1.63 mm(c) 3.25 mm (d) 4.88 mm

55. Statement 1 : A person is standing near a railway track. Atrain is moving on the track. As train is approaching theperson, apparent frequency keeps on increasing and whentrain has passed the person then apparent frequency keepson decreasing.Statement 2 : When train is approaching the person then

0c

f fc u = - and when train is moving away from the

person 0c

f fc u = +

, c is velocity of sound, u is velocity

of train and f0 is original frequency of whistle.(a) Statement-1 is true, Statement-2 is true; Statement-2 is



656. A lens is placed between a source of light and a wall. Itforms images of area A1 and A2 on the wall, for its twodifferent positions. The area of the source of light is

(a) ( )2A1A (b) (A1+A2)/2

(c) ( )2

AA2

21 + (d)1

21 A1

A1

+

57. For sky wave propagation of a 10 MHz signal, what shouldbe the minimum electron density in ionosphere

(a) 12 3~ 1.2 10 m- (b) 6 3~ 10 m-

(c) 14 3~ 10 m- (d) 22 3~ 10 m-

58. The counting rate observed from a radioactive source att = 0 second was 1600 counts per second and at t = 8 secondsit was 100 counts per second. The counting rate observed,as counts per second at t = 6 seconds will be(a) 400 (b) 300(c) 200 (d) 150

59. In a npn transistor 1010 electrons enter the emitter in 106 s.4% of the electrons are lost in the base. The current transferratio will be(a) 0.98 (b) 0.97(c) 0.96 (d) 0.94

60. The vernier constant of two vernier callipers A and B are0.01 cm and 0.01 mm respectively. Which one can measurethe length of an object more accurately?(a) Vernier A(b) Vernier B(c) Accuracy in measurement does not depend on vernier

constant(d) Both A and B are equally accurate.

MATHEMATICS61. n+4Cr

nCr 3.nCr1 3.

nCr2 nCr3 is equal to

(a) n+1Cr1 (b)n+2 Cr1

(c) n+3Cr1 (d)n+4Cr1

62. The sum of the first three consecutive terms of an A.P. is 9 and the sum of their squares is 35 . The sum to n terms of theseries is(a) 3n2 (b) 2n2(c) 6n 2n2 (d) 6n n2

63. Let a/(a1) and b/(b1) be the roots of x2 + ax + b = 0. Then1/a and 1/b are the roots of(a) bx2 + ax + 1 = 0(b) bx2 ax + 1 = 0(c) bx2 + (a + 2b)x + a + b + 1 = 0(d) bx2 (a + 2b) x + a + b + 1 = 0

64. For any real q, the maximum value ofcos2(cos q) + sin2(sin q) is(a) 1 (b) 1+sin21

(c) 1+cos21 (d) 1/265. If tan(A/2), tan(B/2), tan(C/2) are in A.P. then sec A, sec B, sec

C are in(a) A.P. (b) G..P.(c) H.P. (d) None

66. The straight line y = mx + c is a tangent to the parabolax2 = 4by if(a) c = b/m (b) c = bm

(c) c = bm2 (d) c = 2mb

67. If tangents be drawn to the cricle x2 + y2 = 12 at its points ofintersection with the circle x2 + y2 5x + 3y 2 = 0, then thetangents intersect at the point

(a)

518,6 (b)

518,6

(c)

518,6 (d)

518,6

68. The locus of the mid points of the chords of the ellipsex2/a2 + y2/b2 = k, k > 0, making equal intercepts on thecoordinate axes, is

(a) x = y (b) x + y = 0(c) x/a2 = y /b2 (d) x/a2 + y/b2 = 0

69. I= n/m means that n is a factor of m, then the relation I, is

(a) Reflexive and symmetric(b) Transitive and symmetric(c) Reflexive, transitive and symmetric

(d) Reflexive, transitive and not symmetric

70. Matrix rM is defined as rM =

r1r1rr

, r N value

of det( 1M ) + det( 2M ) + det( 3M ) + .... + det(M2007) is

(a) 2007 (b) 2008

(c) 22008 (d) 20072

71. Let .111312

111A

-

-= and .

32105

224B)10(

-a-=

If B is the inverse of matrix A, then a is(a) 5 (b) 1 (c) 2 (d) 2

72. The value of 1xsin21

26/x

)xcos2xsin34(lim -p

-- is

(a) 1 (b) e (c) e (d) e1/2

773. Let f(x) = x32 e)2x()1x( -- . Then

(a) f(x) has local maximum at x = 1

(b) f(x) has point of inflexion at x = 1

(c) f(x) has local minimum at x = 2

(d) all of these

74. The derivative of tan1[(3x2 1)/(3x x3)] with respect tosin1[(x2 1)/( x2 + 1)] is

(a) 2/3 (b) 2/3 (c) 3/2 (d) 3/23

75. If 0c6b3a2 =++ then at least one root of the equation

0cbxax2 =++ lies in the interval

(a) (0, 1) (b) (1, 3) (c) (2, 3) (d) (1, 2)

76. A curve y = f(x) passes through the point P(1, 1). The normalto the curve at P is a (y 1) + (x 1) = 0. If the slope of thetangent at any point on the curve is proportional to theordinate of the point, then the equation of the curve is

(a) y = ea (x 1) (b) y = ea(1 x)

(c) y = ea/2 (x 1) (d) ea/2 (x + 1)

77. Value of n 2m 10

sin x cos x dxp +

( where m, n N) is

(a) !n)!1m2( +

(b) 2n)!1m2( +

(c) 2m 10 cos x dxp + (d) none of these

78. If )x(f)x(f = and ,2)0(f = then =+ dx)x(f43)x(f

(a) C)e83log( x ++ (b) C)e83log(41 x ++

(c) C)e83log(21 x ++ (d) none of these

79. The primitive of x4sinxsin

is

(a) C1xsin21xsin2log

21

1xsin1xsinlog

41 +

+--+

-

(b) C1xcos21xcos2log

221

1xcos1xcoslog

81 +

+--+

-

(c) C1xsin21xsin2log

241

1xsin1xsinlog

81 +

+--+

-

(d) none of these80. Let coordinates of a point p with respect to the system of

non-coplanar vectors ar , br

and cr is (3, 2, 1). Thencoordinates of p with respect to the system of vectors

ar + br

+ cr , ar br

+ cr and ar + br

cr is

(a)

1,

21,

23

(b)

21,1,

23

(c)

1,

23,

21

(d) None

81. The three lines through the origin with direction cosinesl1, m1, n1; l2, m2, n2; and l3, m3, n3 are coplanar, if(a) l1 (l2 l3) + m1 (m2 m3) + n1 (n2 n3) = 0(b) l1(m2n3 m3n2) + m1(n2l3 n3l2) + n1 (l2m3 l3m2) = 0

(c) 0nmlnmlnml

333

222

111=

(d) None of these82. A plane makes intercepts OA, OB, OC whose measures are

a, b, c on the axes x, y, z respectively. The area of the triangleABC is

(a) 222 cba21 ++ (b) 222222 baaccb

21 ++

(c) acbcab21 ++ (d) accbba

21 222 ++

83. If x + 2 = 6, then x = 4. So, which statement is its converse?(a) If x 4, then x 2 6 + (b) If x 4, then x 2 6= + (c) If x 4, then x 2 6= + =(d) If, x 4 then x + 2 = 6

84. If y = f(x) be a monotonically increasing or decreasingfunction of x and M is the median of variable x, then themedian of y is(a) f (M) (b) M/2(c) f 1 (M) (d) None

85. A fair die is tossed eight times. The probability that a thirdsix is observed on the eighth throw is

(a) 85

27

65C (b)

8

3

37

65C

(c) 86

67

65C (d) None

886. Statement-1 : If a i j= +r and b j k= -r

then

( a b, a b+ -r r

r r ) = 90

Statement-2 : Projection of a b+r

r on a b-r

r is zero.(a) Statement-1 is true, Statement-2 is true; Statement-2 is



87. Let A and B are two events such that P(A) = 3/5 andP(B)= 2/3, then

Statement1 : 4

15 P ( ) 3A B

5 .

Statement2 : 2 A 9

P5 B 10

(a) Statement-1 is true, Statement-2 is true; Statement-2 is



88. Let 1X and 2X are means of two distributions such that

1 2X X and X< is the mean of the combined distribution.

Statement-1 : 1 2X X X< Cv

44. (d) Let the charges on spheres of radii r and R be q1 and q2respectively. Then, Q = q1+q2. The potential at thecentre will be :

)Rq

rq(

41V 21

0+

pe=

As surface densities are equal, hence

s = 22

21

R4q

r4q

p=

p or q2 = q1

2

2

rR

V = )rRq

rq(

41

211

o+

pe = )Rr(

rq

41

21

o+

pe

Now, q1 + q2 = Q or q1 + q1 22

rR

= Q

2221

rRQ

rq

+= ; V =

2 204

R rQR r

+ pe +

45. (c) A paramagnetic sample display greater magnetisationwhen cooled, this is because at lower temperature, thetendency to disrupt the alignment of dipoles (due tomagnetising field) decreases on account of reducedrandom thermal motion.

46. (a) U = 21

CV2

From this, we get U V2

U

V2

U

V

47. (c) Let r be the internal resistance of the cell.Terminal potential difference (V) < EMF of a cell (E).

V = E Ir or E V E Vr R

I V- - = =

48. (a) The workdone in moving a charge along anequipotential surface is zero.

49. (c) B due to AOB and COD are ^ to each other. Hence,

net 2221

2BB B +=

B = 2/12221

0 )II(a2

+pm

( ^ to plane ABCD)

50. (a) dtdINME)NMI(

dtdE ==

TNMIE =

Emf induced per unit turn = tMI

NE =

51. (d) At resonance Lw = wC1

, w = LC1

Current through circuit i = RE

Power dissipated at Resonance = i2R

13

52. (d)dtdILE -= ; 2LI

21W =

We know that dt

LdIE -= As dtdI is same,

Hence, 4LL

EE

2

1

2

1 ==

Power is constant so 2211 IEIE = ;

4II

EE

1

2

2

1 == We know that 2LI21W = ;

222

211

2

1

IL

ILWW =\

41

1614 ==

53. (a) ( )1= + - last firstn n n d ( )3 56 1 4= + - n nor 2 55 4= n 110Hz=n

54. (b) 3D 1

2d 10-l b = =

76.5 10 = 0.65 103 = 0.65 mm

The distance between fifth bright fringe from third darkfringe = 2.5 b = 2.5 .65 = 1.63 mm.

55. (d) Statement-2 is correct and statement-1 is false.56. (a) By displacement method, the size of object is given by

O = 21 II ;

21 AAA =\57. (a) The critical frequency of a sky wave for reflection from

a layer of atmosphere is given by 1/ 2c maxf 9(N )=6 1/ 2

max10 10 9(N ) =

2612 3

max10 10N 1.2 10 m

6- =

;

58. (c) After 8 sec., the counting rate falls to

Part th21

161

1600100 4

== ;

Time period = 48 sec. = 2 sec.

Therefore, after 6 seconds, the counting rate should be3

21

th part of 1600, i.e. 20016008

1 = .

59. (c) No. of electrons reaching the collector,

1010C 1096.010100

96n ==

Emitter current, t

enEE

=I

Collector current, t

enCC

=I

\ Current transfer ratio,

E

C

E

Cnn==a

II

= 96.010

1096.010

10=

60. (b) Lower the vernier constant, more accurate measurementis possible by it.

61. (c) n 4 n n n nr r r 1 r 2 r 3C ( C 3. C 3. C C )+

- - -- + + +

= n 4 n n n nr r r 1 r 1 r 2C ( C C ) ( C C )+

- - -- + + +n n n n

r 1 r 2 r 2 r 3( C C ) ( C C )- - - -+ + + +

= n 4 n 1 n 1 n 1 n 1r r r 1 r 1 r 2C C C C C+ + + + +

- - -- + + +

= n 4 n 2 n 2r r r 1C C C+ + +

-- +

= n 4 n 3 n 3r r r 1C C C+ + +

-- =

62. (d) a d a a d 9- + + + = 3a =

35)da(a)da( 222 =+++-

359d218 2 =++ 2d4d2 ==

( )nS 2a 2d nd d2

= - + -

Taking d = 2, we get(Q d = 2 will not give any option)

S n [6 2n 6]2

= - + = 6n n2

63. (d) Let a' = a/(a 1), b' = b/(b 1) a = a'/(a' 1), b = b'/(b' 1) 1/a = (a' 1)/a', 1/b = (b' 1)/b'Then equation whose roots are 1/a, 1/b, isx2 (1/a + 1/b)x + 1/ab = 0

x2 [(a' 1)/a' + (b' 1)/b'] x + [(a' 1) (b' 1)] / a'b' = 0 a'b'x2 [2a'b' (a' + b')]x + a'b' (a' + b') + 1 = 0 bx2 (a + 2b)x + a + b + 1 = 0

64. (b) )(sinsin)(coscos)(f 22 q+q=qNow, put q = p /2 & 0

1cos01cos)0(f 22 =+=

1sin1)2/(f 2+=p

Since, 1cos1sin 22 >

]

41&

24cossin[ p>p

14

65. (c) tanA/2, tanB/2, tanC/2 are in A.P. tanA/2 tanB/2 = tanB/2 tanC/2

2/Ccos2/Csin

2/Bcos2/Bsin

2/Bcos2/Bsin

2/Acos2/Asin -=-

cos(C/2)sin(A/2 B/2) = cosA/2 sin (B/2 C/2)cos B cos A cosC cosB - = -2cos B cos A cos C = +

secA, secB, secC are in H.P.66. (c) Solving y = mx + c with x2 = 4by, we get,

( )2 2x 4b mx c x 4bmx 4bc 0= + - - =Hence, discriminant = 0

222 bmc0bc16m16b -==+67. (d) Let the point of contact be (h,k); equation of chord of

contact is T = 0 xh yk 12 = 0 ..(1)Equation of common chord C1 C2 = 0 5x 3y 10 = 0 . (2)(1) and (2) represent the same line h/k = k/(3)= 12/(10) h = 6.k = 18/5

68. (c) The equation of chord having (x1,y1) as its mid-pointis(xx1/a

2)+ (yy1/b2 ) k = (x1

2/a2) + (y12/b2) k

(xx1/a2) + (yy1/b2 ) = (x12/a2) + (y12/b2)...(1)It makes equal intercepts on the axes when(a2/x1) = (b

2/y1) (x1/a2) (y1/b2 ) =0Hence the locus of (x1, y1) is (x/a

2) = (y/b2 )69. (d) I is reflexive since every natural number is a factor of

itself, that is n/n for n N. I is transitive since if n is afactor of m and m is a factor of p, then n is surely afactor of p. thus n/m and m/p n/p. However I isnot symmetric since, for example, 2 is factor of 4 but 4 isnot a factor of 2.

70. (d) det (Mr) = 1r2r1r1rr -=-

-

2007 2007

rr 1 r 1

det (M ) 2 r 2007= =

= - 22007 20082 2007 (2007)

2= - =

71. (a)

-

-=

111312

111A

Cofactors of various entries are3,5,2;2,0,2;1,5,4 --

10115)1(41|A| =+--+=

Cofactor Matrix 4 5 12 0 22 5 3

C-

= -

--==\

321505224

CAAdj T

--==\ -

321505224

101

|A|AdjAA 1

Comparing , we get a = 5

72. (d) ( )1/(2sin x 1)2x / 6lim 4 3sin x 2 cos x - - -p{ }

sin x 1

1/ 2x / 6

1 (2sin x 1)(sin x 1)lim e1(2sin x 1)(sin x 1)

--

+ - - = - - p

73. (a) f(x) = (x 1)2 (x 2)3 ex

f '(x) = (x 1) (x 2)2(x2 + 2x 5)exf(x) has local maximum at x = 1, since f '(x) changesfrom positive to negative.f(x) has a point of inflexion at x = 2.

74. (c)

-

-= -3

21

xx31x3tany

xtan321x3

xx3tan2

12

31 -- +p=

-

--p=

2dy 3dx 1 x

=+

+--p=

+-= --

1x

1xcos21x

1xsinu2

21

2

21

xtan22

1-+p-=

2du 2dx 1 x

=+

dy dy du 3dx du dx 2

= =

75. (a) Let f(x) cbxax 2 ++=3 2

( ) ( )3 2

ax bxg x f x cx= = + +a bg(1) c 2a 3b c 03 2

\ = + + = + + =

)1(g0)0(g ==

15

\ g(x) is continuous apply Rolles theorem0)x(g = for some value

)1,0(x 0cbxax)x(f)x('g 2 =++==has a root in (0, 1)

76. (a) Slope of the normal at (1, 1) is 1/a Slope of the tangent at (1, 1) is a

i.e., (1, 1)

dydx

= a ...... (1)

We are given that dydx y;

dydx = ky,,

where k is some constant dyy = k dx

log | y | = kx + c, where c is a constant| y | = ekx + cy = ec ekx = Aekx, where A is a constant.Since the curve passes through (1, 1),therefore, 1 = Aek A = ekTherefore, y = ek . ekx = ek (x 1)

dydx = kek (x 1)

(1, 1)

dydx

= k a = k [Using (1)]

Thus, the required curve is y = ea (x 1).

77. (d) n 2m 10

I sin x cos x dxp +=

n 2m 10

sin ( x).cos ( x)dxp += p - p -

n 2m 10

sin x cos x dxp += - I I 0= - =

Similarly, one can show that2m 1

0cos x dx 0

p + =78. (b) Since, f ' (x) = f (x), therefore, f (x) = aex.

Since, f (0) = 2, therefore, f (x) = 2ex.

dxe83

e2Ix

x

+=\Put ex = t, \ dt = ex dx

\ += t83dt2I C)e83log(

41I x ++=

79. (c) = dxx4sinxsinI

== x2cosxcos4dxdx

x2cosx2sin2xsinI

--= dx)xsin21)(xsin1(4xcos

22

Put dtdxxcostxsin ==

--=\ )t21)(t1(4dtI 22 2 2

1 1 1dt14 t 1 t

2

= - - -

1t1 1 t 1 1 2log log C14 2 t 1 2 t

2

- - = - ++ + 80. (a) Any vector (x, y, z) can be represented as a linear

combination of three non coplanar vectors.

czbyaxOPr

rr ++=

But P is (3,2,1)r

cb2a3OP rr

r ++=Now if coordinate axis changes then the vector isrepresented by

( ) ( ) 2( )x a b c y a b c a b c+ + + - + + + -r r r

r r r r r r

3a 2b c= + +r

r r

Comparing coeff of c&b,ar

rr , we get

3, 2, 1x y z x y z x y z+ + = - + = + - =On solving, we get

3 1, , 12 2

x y z= = =

Hence the new co-ordinates of point P are

1,

21,

23

81. (c) If the given lines are coplanar, then a line which is normalto the plane in which they lie, will be perpendicular toall the three given lines. Let the direction cosines of thenormal be l, m, u.\ ll1 + mm1 + un1 = 0, ll2 + mm2 + un2 = 0,ll3 + mm3 + un3 = 0. Eliminating l, m, u from theseequations,

we get, 1 1 1

2 2 2

3 3 3

l m nl m n 0l m n

= .

82. (b) Let OP be the normal from O to the plane of DABC,let l, m, n be the directioncosines of OPLet s be the area of DABC.Therefore,OAC = Projection of DABC on zx plane = s.m.But OAC is a right angled triangle, so that area of

DOAC = 21

a.c.

\21 ac = s.m. s2

acm = . Similarly, in DOBC,

16

l = s2bc

and in D OAB, n = s2ab

Since, l2 + m2 + n2 = 1

\ 1s2ab

s2bc

s2ac 222 =

+

+

222222 baaccb21s ++=

83. (c) x + 2 = 6 x = 4, Its converse is x = 4 x + 2 = 6 i.e. ifx = 4 then 4 + 2 = 6

84. (a) y

x

Since, )x(f)x(fxx 2121

full test

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