CHEMISTRY1. At certain temperature (T) if conductivity of pure water is

5.5 107 S cm1, then calculate pOH of pure water attemperature T.

Given 2 1 2 1H OH350 S cm eq , 200 S cm eq - -

+ -l = l =(a) 5 (b) 6(c) 7 (d) 8

2. What volume of 0.2 M RNH3Cl solution should be added to100 ml. of 0.1 M RNH2 solution to produce a buffer solutionof pH = 8.7 ? Given : pKb of RNH2 = 5, log 2 = 0.3(a) 50 ml. (b) 100 ml.(c) 200 ml. (d) None of these

3. Natural rubber is a polymer of (a) butadiene (b) ethylene(c) isoprene (d) styrene

4. When N2O5 is heated at temp. T, it dissociates as N2O5(g) N2O3(g) + O2(g), Kc =2.5. At the same time N2O3 alsodecomposes as N2O3(g) N2O(g) + O2(g). If initially 4.0moles of N2O5 are taken in 2.0 litre flask and allowed toattain equilibrium, concentration of O2 was formed to be2.5M. Equilibrium concentration of N2O is (a) 2.0 (b) 1.0(c) 0.334 (d) None of these

5. What mass of H2(g) is needed to reduce 192gm. of MoO3 tometal ? [At. wt. of Mo = 96](a) 8 gm. (b) 16 gm.

(c) 32 gm. (d) None of these6. How many times solubility of CaF2 is decreased in 410

3

M KF (aq). solution as compare to pure water at 25C. GivenKsp (CaF2) = 3.2 10

11.(a) 50 (b) 100(c) 500 (d) 1000

7. Choose the correct options (a) Tranquillizers are called psychotherapeutic drugs.(b) Vitamin C, Vitamin E and b-carotene are antioxidants.(c) Sodium or potassium salt of a long chain fatty acid is

called soap.(d) All of these

8. A sample of U238 (t1/2 = 4.5 109 yrs) is found to contain

11.9gm U238 and 20.6gm Pb206. The age of sample is(Given ln 2 = 0.693, ln 3 = 1.1](a) 1.64 1010 yrs (b) 7.14 109 yrs(c) 2.25 109 yrs (d) None of these

9. 96.5gm. of oxidized as A A3+ + 3e, when 2F charge ispassed through solution with current efficiency of 90%.What is electrochemical equivalent of A ?(a) 53.61 (b) 48.25(c) 5.55 104 (d) 5.55 103

10. pH of the solution :

2

3

3

10 litre of 0.03 N X(OH) (strong diacidic base)+

5 litre of 0.08 M HNO

485 litre of 0.01 M NaNO

+

is

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

MOCK TEST - 2

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ABOUT DISHA PUBLICATION

2(a) 3.7 (b) 11.0(c) 7.0 (d) 10.0

11. What is the name of the following reaction :

COOEt

COOEt

EtONa

(a) Knovengel reaction(b) Perkin reaction(c) Reformatsky reaction(d) Dieckmanns condensation.

12. Choose the correct options (a) The change of atmospheric temperature with altitude

is called the lapse rate.(b) The gases responsible for greenhouse effects are CO2,

Water vapour, CH4 and ozone.(c) The order of contribution to the acid rain is

H2SO4 < HNO3 < HCl(d) Both (a) and (b)

13. What is the EMF of represented cell at 298Ag | Ag+ (aq 0.1M) | | H+ (aq, 0.1 M), H2 (g, 0.1 bar) | Pt

Given : 0Ag/Ag

E 0.8 V+ = -

(a) 0.77 V (b) 0.829 V(c) 0.77 V (d) None of these

14. Which statement is/are correct about ICl3 molecule (a) All I Cl bonds are equivalent(b) Molecule is polar and non-planar(c) All adjacent bond angles are equal(d) All hybrid orbitals of central atom having equal

s-character

15.

OH

CH3 OH

PCC (X) Major

(a) OH

O

(b) O

O

(c)

OH

O (d) OH

16. To obtain silver from silver amalgam it is heated in vesselwhich is made of (a) Cu (b) Fe(c) Ni (d) Zn

17. Which of the following compounds does not give nitrogengas on heating (a) NH4NO2 (b) (NH4)2SO4(c) NH4ClO4 (d) (NH4)2Cr2O7

18. Choose the correct options (a) The size of the colloidal particles is in between the size

of the molecule and the size of the particle of a coarsesuspension.

(b) White portion of the egg is condensed colloidal solutionwhereas the sol prepared from this substance is a dilutecolloidal solution.

(c) Dialysis is a slow process.(d) All of these

19. Choose the correct option for

Cl

I

II

HBr

HBrPerdoxide

(a) I and II are identical(b) I and II are different(c) Mechanism of formation of I and II are same(d) None of these

20. HC CH HgSO dil. OH4H SO 5C2 4

(A) (B)Q

Give the IUPAC name of B is (a) 2-butanal (b) 3-hydroxy butanal(c) 3-formyl-2-propanol (d) 4-oxo-2-propanol

21. Br Alcoholic2 / h

KOH /Major (X)n

D

H Br

PeroxideMajor (Y) Major (Z)-

Major final product (Z) is

(a)

Br

(b)Br

3(c)Br

(d)

Br

22. Choose the incorrect option (a) When a dilute solution of an acid is added to a dilute

solution of a base, neutralization reaction takes place.(b) In acid-base titrations, at the end point, the amount of

acid becomes chemically equivalent to the amount ofbase present.

(c) In the case of a strong acid and a strong base titration,at the end point the solution becomes acidic.

(d) In acid-base titrations the end point is determined bythe hydrogen ion concentration of the solution.

23. Light of wavelength l strikes a metal surface with intensityx and the metal emits y electrons per second of maximumenergy z . What will happen to y and z if x is halved?(a) y will remain same (b) y will be doubled(c) z will remain the same (d) z will halved

24.

NO2

Cl

CH ONaBr 32

Fe, CH OH,3X Y

D D

Product (Y) of this reaction is

(a)

NO2

ClOCH3

(b)

NO2

BrOCH3

(c)

NO2

OCH3

(d)

NO2

OCH3

25. C H NHSOCl(i) NaOI 6 5 22NaOH(ii) H

A B C

Conc. HNO3Conc. H SO2 4(1 equivalent)

D E (Major) E can be

(a)

NO2

NH

O

(b)

NO2

NH

O

(c)NO2

NH

O

(d) O2N NH2

H

Cl26. The DfH(N2O5, g) in kJ/mol on the basis of the following

data is 2NO(g) + O2(g) 2NO2(g) ; DrH = 114 kJ/mol4NO2(g) + O2(g) 2N2O5(g) ; DrH = 102.6 kJ/mol

DfH(NO, g) = 90.2 kJ/mol(a) 15.1 (b) 30.2(c) 36.2 (d) None of these

27. Which graph represents zero order reaction [A(g) B(g)]

(a)

t

[B] (b)

t

d[B] dt

(c)

[A]0

t1/2 (d)

[A]0

t3/4

28. This question contains Statement-1 and Statement-2. Of thefour choices given after the statements, choose the one that

4best describes the two statements.Statement 1 : Anhydrous copper (II) chloride is a covalentwhile anhydrous copper (II) fluoride is ionic in nature.Statement 2 : In halides of transition metals, the ioniccharacter decreases with increase in atomic mass of thehalogen.(a) Statement-1 is false, Statement-2 is true.(b) Statement-1 is true, statement-2 is true and statement-

2 is correct explanation for statement-1(c) Statement-1 is true, statement-2 is true and statement-

2 is NOT correct explanation for statement-1(d) Statement-1 is true, Statement-2 is false.

29. At 300K, the vapour pressure of an ideal solution containing3 mole of A and 2 mole of B is 600 torr. At the sametemperature, if 1.5 mole of A and 0.5 mole of C (non-volatile)are added to this solution the vapour pressure of solutionincreases by 30 torr. What is the value of 0BP ?(a) 940 (b) 405(c) 90 (d) None of these

30. The standard redox potentials E of the following systemsare :

System E (volts)(i) MnO4

+ 8H+ + 5e Mn2+ + 4H2O 1.51(ii) Sn4+ + 2e Sn2+ 0.15(iii) Cr2O7

2 + 14H+ + 6e 2Cr3+ + 7H2O 1.33(iv) Ce4+ + e Ce3+ 1.61The oxidising power of the various species decreases in theorder-(a) Ce4+ > Cr2O7

2 > Sn4+ > MnO4

(b) Ce4+ > MnO4 > Cr2O7

2 > Sn4+(c) Cr2O7

2 > Sn4+ > Ce4+ + MnO4

(d) MnO4 > Ce4+ > Sn4+ > Cr2O7

2

PHYSICS31. The horizontal component of impulse provided by string

tension in moving from extreme point to lower most point

///////////////

h

q

(a) m 2gh

(b) m 2gh (1 cos )- q

(c) m 2gh (cos )q(d) cant be found without knowing tension as a function

of time32. N divisions on the main scale of a vernier callipers coincide

with N + 1 divisions on the vernier scale. If each division onthe main scale is of a units, determine the least count of theinstrument.

(a)a

N 1+ (b)a

N 1-

(c)2a

N 1- (d)2a

N 1+33. A current of 4A produces a deflection of 30 in the

galvanometer. The figure of merit is(a) 6.5 A/rad (b) 7.6 A/rad(c) 7.5 A/rad (d) 8.0 A/rad

34. A uniform disc of radius R and mass m is connected to a wallby string of length 2R. The normal reaction of wall is

(a) mg (b) mg/2

(c) mg / 3 (d) 2mg35. Consider the following statements

S1 : Non zero work has to be done on a moving particle tochange its momentum.

S2 : To change momentum of a particle a non zero net forceshould act on it.

S3 : Two particles undergo rectilinear motion along differentstraight lines. Then the centre of mass of system ofgiven two particles also always moves along a straightline.

S4 : If direction of net momentum of a system of particles(having nonzero net momentum) is fixed, the centre ofmass of given system moves along a straight line.

State, in order, whether S1, S2, S3, S4 are true or false.(a) TTFT (b) FTFT(c) FTTT (d) FFTF

36. A particle moves along x-axis with initial position x = 0. Itsvelocity varies with x-coordinate as shown in graph. Theacceleration a of this particle varies with x as

v

x

(a)

a

x (b)

a

x

5(c)

a

x (d)

a

x

37. Which of these nuclear reactions is possible

(a) 23 1 20 411 1 10 2Na H Ne He+ +

(b) 10 4 13 15 2 7 1B He N H+ +

(c) 10 1 115 0 5B n B-+ + b + n

(d) 14 1 127 1 6N H C-+ + b + n

38. For the determination of the focal length of a convex mirror,a convex lens is required because(a) it is not possible to obtain the image produced by a

convex mirror on the screen(b) a convex lens has high resolving power so it helps to

measure the focal length correctly(c) a convex mirror always forms a real image which is

diminished by the convex lens(d) none of these

39. Choose the wrong statement regarding the experiment toidentify a diode, an LED, a resistor and a capacitor(a) when pointer moves in one way when voltage is applied

and doesn't move when reversed and there is lightemission, the item is LED

(b) when pointer moves in one way when voltage is appliedand doesn't move when reversed and there is noemission of light, the item is a diode

(c) when pointer moves in one way when voltage is appliedand not when reversed, the item is a resistor

(d) none of these40. The activity of a radioactive substance drops to 1/32 of its

initial value in 7.5 h. Find the half life.(a) 2 hours (b) 1.5 hours(c) 1 hours (d) 2.5 hours

41. A ring of mass M and radius R lies in x-y plane with itscentre at origin as shown. The mass distribution of ring isnun-uniform such that at any point P on the ring, the massper unit length is given by l = l0 cos2q (where l0 is apositive constant). Then the moment of inertia of the ringabout z-axis is

x

y

q

MR

P

(a) MR2 (b) 21

MR2

(c)0

1 M R2 l (d) 0

1 M Rp l42. An electron and a proton each travels with equal speed

around circular orbits in the same uniform magnetic field asindicated (not to scale) in the diagram. The field is into thepage on the diagram. The electron travels ............... aroundthe ............ circle and the proton travels ............... around the............... circle.

B

(a) clockwise, smaller, counter-clockwise, larger(b) counter-clockwise, smaller, clockwise, larger(c) clockwise, larger, counter-clockwise, smaller(d) counter-clockwise, larger, clockwise, smaller

43. In the screw gauge, the screw should be moved in the samedirection to avoid which of the following error?(a) Zero error (b) Parallel error(c) Back lash error (d) Index error

44. Consider Gausss law 0

qE.dA =

uuurr

. Which of the following

is true ?

+ +

(a) Er

must be the electric field due to the enclosed charge

(b) If net charge inside the Gaussian surface = 0, then Er

must be zero everywhere over the Gaussian surface.(c) If the only charge inside the Gaussian surface is an

electric dipole, then the integral is zero.(d) E

r

is parallel to dAuuur

everywhere over the Gaussiansurface.

45. Six resistors of 10W each are connected as shown. Theequivalent resistance between points X and Y is

X

Y(a) 20 W (b) 5 W(c) 25/3 W (d) 10 W

46. What happens when we do not stir the mixture continuouslyin the experiment to determine the specific heat capacity ofa given solid ?(a) Temperatue of mixture is increased

6(b) Temperature of mixture is not constant(c) Temperature of mixture is constant(d) None of these

ASSERTION & REASON DIRECTIONS (Qs. 47 - 49) : Each of these questionscontains two statements: Statement-1(Assertion) andStatement-2 (Reason). Choose the correct answer (ONLYONE option is correct) from the following -(a) Statement-1 is false, Statement-2 is true.(b) Statement-1 is true, Statement-2 is true; Statement-2

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

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

47. This question contains Statement-1 and Statement-2. Of thefour choices given after the statements, choose the one thatbest describes the two statements.Statement 1 : When an electrical heater is switched on, thecolour of the filament gradually changes from red to yellowto almost white.Statement 2 : Wiens displacement law states that lmT = b(constant) where symbols have their usual meaning.

48. This question contains Statement-1 and Statement-2. Of thefour choices given after the statements, choose the one thatbest describes the two statements.Gas in cylinder with light piston is slowly heated till all themercury spils over. (Patm = 76cm of Hg column)

Hg

76cm.

76cm.

Statement 1 : The temperature of gas continuouslyincreases.

Statement 2 : According to first law of thermodynamics dQ= dU + dW where symbols have their usual meaning.

49. Statement 1 : In YDSE, as shown in figure, central brightfringe is formed at O. If a liquid is filled between plane ofslits and screen, the central bright fringe is shifted in upwarddirection.

S1

S2

O

Statement 2 : If path difference at O increases, y-coordinateof central bright fringe will change.

50. A dipole consisting of two charges +q and q joined by amassless rod of length l, is seen oscillating with a smallamplitude in a uniform electric field of magnitude E. Theperiod of oscillation

(a) is proportional to E

(b) is proportional to 1/E

(c) is qE3mlp

(d) is proportional toE1

but qE3mlp

51. The electric potential V is given as a function of distance xby V = (5x2 + 10x 9) volt. Value of electric field at x = 1m is

(a) 20 V/m (b) 6 V/m

(c) 11 V/m (d) +20 V/m

DIRECTIONS (Qs. 52 - 54) : Read the following passageand answer the questions that follows :

A sinusoidal wave is propagating in negative x-direction ina string stretched along x-axis. A particle of string atx = 2m is found at its mean position and it is moving inpositive y-direction at t = 1 sec. The amplitude of the wave,the wavelength and the angular frequency of the wave are0.1 meter, p/2 meter and 2p rad/sec respectively.

52. The equation of the wave is(a) y = 0.1 sin (4p (t 1) + 8 (x 2))(b) y = 0.1 sin ((t 1) (x 2))(c) y = 0.1 sin (2p (t 1) + 4 (x 2))(d) None of these

53. The speed of particle at x = 2m and t = 1 sec is(a) 0.2p m/s (b) 0.6p m/s(c) 0.4p m/s (d) 0

54. The instantaneous power transfer through x = 2m andt = 1.25 sec, is

7(a) 10 J/s (b) 4p/3 J/s(c) 2p/3 J/s (d) 0

55. A block of mass m is attached with massless spring of forceconstant k. The block is placed over a fixed rough inclinedsurface for which the coefficient of friction is = 3/4. The block of mass m is initially at rest. The block ofmass M is released from rest with spring in unstretchedstate. The minimum value of M required to move the blockup the plane is (neglect mass of string and pulley and frictionin pulley.)

/////////////////////////////////////////

Mm

37

(a)3

m5

(b)4

m5

(c)6

m5

(d)3

m2

56. A charged particle of mass m and having a charge Q is placedin an electric field E which varies with time as E = E0 sin wt.What is the amplitude of the S.H.M. executed by the particle?

(a) 02QEmw (b)

02

QE12 mw

(c) 022QEmw (d) None of these

57. In a photoelectric experiment, with light of wavelength l,the fastest electron has speed v. If the exciting wavelengthis changed to 3l/4, the speed of the fastest emitted electronwill become

(a)3v4

(b)4v3

(c) less than 4v3

(d) greater than 4v3

58. Three moles of an ideal monoatomic gas perform a cycleshown in figure. The gas temperatures in different states areT1 = 200K, T2 = 400K, T3 = 1600 K, and T4 = 800K. The workdone by the gas during the cycle is(Take R = 25/3 J/mol-K)

2 3

1 4

P

T

(a) 5 kJ (b) 25 kJ(c) 15 kJ (d) 20 kJ

59. A body is executing simple harmonic motion. At adisplacement x from mean position, its potential energy isE1 = 2J and at a displacement y from mean position, itspotential energy is E2 = 8J. The potential energy E at adisplacement (x + y) from mean position is(a) 10J (b) 14J(c) 18J (d) 4J

60. A thin convex lens of focal length 10cm and refractive index1.5 is cut vertically into two equal pieces. They are placed asshown with a liquid of refractive index 3 between them. Whatis the focal length of the combination ?

(a) 10 cm. (b) 10/4 cm.(c) 10/3 cm. (d) None

MATHEMATICS61. Equation of straight line ax + by + c = 0 where 3a+4b+ c = 0,

which is at maximum distance from (1, 2), is

(a) 3x + y 17 = 0 (b) 4x + 3y 24 = 0(c) 3x + 4y 25 = 0 (d) x + 3y 15 = 0

62. Given f (x) =

2

x 3

10 x if 3 x 3

2 e if x 3-

- - <

863. The solution of the equation 2z = | z | + 2i, where z is acomplex number, is

(a)3

z i3

= - (b) 3z i3

= +

(c)3

z i3

= (d) None of these

64. If x 2, y 2, z 2 and 2 y zx 2 z 0x y 2

= , then the value of

2 y z2 x 2 y 2 z

+ + =- - -

(a) 1 (b) 0(c) 3 (d) 4

65. Box contains 2 one rupee, 2 five rupee, 2 ten rupee and 2 twentyrupee coin. Two coins are drawn at random simultaneously.The probability that their sum is Rs. 20 or more, is

(a) 1/4 (b) 1/2

(c) 3/4 (d) 1/8

66. The equation (5x 1)2 + (5y 2)2 = (l2 4l + 4) (3x + 4y 1)2represents an ellipse if l (a) (0, 1] (b) (1, 2)(c) (2, 3) (d) (1, 0)

67. The value of the definite integral, 2

1

d 5011 tan K

q

q

q p=+ q

where 210032008

pq = and 1 2008pq = . The value of K equals

(a) 2007 (b) 2006

(c) 2009 (d) 2008

68. The straight line y = m (x a) meets the parabola y2 = 4ax intwo distinct points for

(a) all m R (b) all m [1, 1](c) all m R {0} (d) None of these

69. The expansion of (1 + x)n has 3 consecutive terms withcoefficients in the ratio 1 : 2 : 3 and can be written in the formnCk :

nCk+1 : nCk+2. The sum of all possible values of (n + k)

is

(a) 18 (b) 21(c) 28 (d) 32

70. The mean and standard deviation of 6 observations are 8 and

4 respectively. If each observation is multiplied by 3, find thenew standard deviation of the resulting observations.

(a) 12 (b) 18

(c) 24 (d) 144

71. p (p q) is equivalent to (a) q (b) p

(c) ~p (d) ~q

72. Find sin xe 3

2x cos x sin x

cos x

-

dx

(a) x esin x esin x sec x + C

(b) x ecos x esin x sec x + C

(c) x2 esin x + esin x sec x + C

(d) 2x esin x esin x tan x + C

73. The function f : [2, ) (0, ) defined by

f (x) = x2 4x + a, then the set of values of a for which f(x)becomes onto is

(a) (4,) (b) [4, )(c) {4} (d) f

74. If a and b are the real roots of the equation

x2 (k 2) x + (k2 + 3k + 5) = 0 (k R) .

Find the maximum and minimum values of 2 2( )a + b .

(a) 18, 50/9 (b) 18, 25/9

(c) 27, 50/9 (d) None of these

75. The sum of the coefficient of all the terms in the expansionof (2x y + z)20 in which y do not appear at all while xappears in even powers and z appears in odd powers is

(a) 0 (b)202 1

2-

(c) 219 (d)203 1

2-

76. All the five digit numbers in which each successive digitexceeds is predecessor are arranged in the increasing order.The (105)th number does not contain the digit

(a) 1 (b) 2

(c) 6 (d) All of these

77. 3x 0tan x tan x sin x sin xlim

x . x-

equals

(a) 1/4 (b) 3/4

9(c) 1/2 (d) 1

78. Three people each flip two fair coins. The probability thatexactly two of the people flipped one head and one tail, is

(a) 1/2 (b) 3/8(c) 5/8 (d) 3/4

79. If a, b, cr

r r are non-coplanar unit vector such that1

a (b c) (b c)2

= +r r

r r r then the angle between the vectors

a, br

r is

(a) 3p/4 (b) p/4(c) p/8 (d) p/2

ASSERTION & REASON

DIRECTIONS (Qs. 80 - 84) : Each of these questionscontains two statements: Statement-1(Assertion) andStatement-2 (Reason). Choose the correct answer(ONLY ONE option is correct) from the following -

(a) Statement-1 is false, Statement-2 is true.

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

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

(d) Statement-1 is true, Statement-2 is false.

80. Statement-1 : If | z1 | = 30, | z2 (12 + 5i) | = 6, then maximumvalue of | z1 z2 | is 49.

Statement-2 : If z1, z2 are two complex numbers, then| z1 z2 | | z1 | + | z2 | and equality holds when origin, z1 andz2 are collinear and z1, z2 are on the opposite side of theorigin.

81. Consider the family of straight lines 2x sin2q + y cos2q = 2cos 2q

Statement-1 : All the lines of the given family pass throughthe point (3, 2).

Statement-2 : All the lines of the given family pass througha fixed point.

82. Consider I = /4

/4

dx1 sin x

p

-p -

Statement-1 : I = 0

because

Statement-2 : a

a

f (x) dx 0-

= , wherever f (x) is an odd

function.

83. Statement-1 : Let f : R R be a function such thatf (x) = x3 + x2 + 3x + sin x. Then f is one-one.

Statement-2 : f (x) neither increasing nor decreasing function.

84. Statement-1 : If the lengths of subtangent and subnormal at

point (x, y) on y = f (x) are respectively 9 and 4. Then x = 6

Statement-2 : Product of sub tangent and sub normal issquare of the ordinate of the point.

85. Consider the following statements :

S1 : Number of integrals values of a for which the roots ofthe equation x2 + ax + 7 = 0 are imaginary with positive realparts is 5.

S2 : Let a, b are roots x2 (a + 3) x + 5 = 0 and a, a, b are inA.P. then roots are 2 and 5/2

S3 : Solution set of logx (2 + x) logx (6 x) is (1, 2]

State, in order, whether S1, S2, S3 are true or false.

(a) FFT (b) TFT

(c) TFF (d) TTT

86. If the substitution x = tan1 (t) transforms the differential

equation 2

22

d y dyxy sec x 0

dxdx+ + = into a differential

equation 2

2 12

d y dy(1 t ) (2t y tan (t)) kdtdt

-+ + + = then k is

equal to

(a) 2 (b) 2

(c) 1 (d) 0

87. Let f : R R and fn (x) = f (fn1 (x)) n 2, n N,the roots of equation f3(x) f2(x) f (x) 25f2(x) f (x) + 175 f (x)= 375. Which also satisfy equation f (x) = x will be

(a) 5 (b) 15

(c) 10 (d) Both (a) and (b)

88. A triangle ABC satisfies the relation 2 sec 4C + sin2 2A +

sin B = 0 and a point P is taken on the longest side of the

10

triangle such that it divides the side in the ratio 1 : 3. Let Qand R be the circumcentre and orthocentre of D ABC. If PQ :QR : RP = 1 : a : b, then the value of a2 + b2.(a) 9 (b) 8

(c) 6 (d) 7

89. The value of -- + xcos0 1xsin

01

22dttcosdttsin is

(a) p (b) 2p

(c)4p

(d) 1

90. If a is real and 2ax sin By cos Bz 0+ + = ,

x cos By sin Bz 0+ + = , x + sin By cos Bz = 0, then theset of all values of a for which the system of linear equationshas a non-trivial solution, is

(a) [1, 2] (b) [1, 1](c) [1, ] (d) [21/2, 21/2]

11

1. (b) H O2k 1000

N L = ,

765.5 10 1000N 10

550

-- = =

Kw = N2 or M2, pH = pOH = 6

2. (b) Let V ml of RNH3Cl added into RNH2 solution

[RNH3Cl] in resultant solution = 0.2 V100 V

+

2100 0.1[RNH ]100 V

= + ; 3

b2

[RNH ]pOH pK log

[RNH ]

+= +

0.2 V100 V5.3 5 log 100 0.1100 V

+ = +

+

\ 0.2 V210

= V = 100 ml.

3. (c) Natural rubber is a coiled linear polymer of isoprene(2-methyl-1,3-butadiene).

4. (b) N2O5(g) N2O3(g) + O2(g)conn at eqm 2 x x y x + y

N2O3(g) N2O(g) + O2(g)conn at eqm x y y y + x

2.5 (x y)2.5(2 x) -= -

x y = 2 x or 2x y = 2 & as per given[O2(g)] = x + y = 2.5x = 1.5 and [N2O(g)] = y = 1.0

5. (a) Equivalent of MoO3 = M 144 246 6

= =

equivalent of H2 : H2

H2

W192W 8gm.

24 1= =

6. (b) 4S3 = 3.2 1011

S = 2 104

Let solubility of CaF2 is x in KF solution.Ksp = x (2x + 4 10

3)2 ; 3.2 1011 = x 106 16x = 2 106

7. (d) (1) Tranquillizers are the substances used for thetreatment of mental diseases. These act on highercentres of the central nervous system. These are alsocalled psychotherapeutic drugs.(2) Vitamin C, Vitamin E and b-carotene are antioxidants(substances that act against oxidants).(3) Sodium or potassium salt of a long chain fatty acidis called soap.

8. (b)U Pb

U

n n1t lnn+= l ; nU = no. of mole of U

238

1 0.15t ln

0.05 = l ; nPb = no. of mole of Pb

206

91 1.1t ln (3) 4.5 100.693

= = l = 7.14 109 years

1 (b) 16 (b) 31 (a) 46 (b) 61 (d) 76 (a)2 (b) 17 (b) 32 (a) 47 (b) 62 (b) 77 (b)3 (c) 18 (d) 33 (b) 48 (a) 63 (b) 78 (b)4 (b) 19 (a) 34 (c) 49 (a) 64 (b) 79 (a)5 (a) 20 (b) 35 (b) 50 (c) 65 (b) 80 (c)6 (b) 21 (c) 36 (a) 51 (a) 66 (c) 81 (a)7 (d) 22 (c) 37 (a) 52 (c) 67 (d) 82 (a)8 (b) 23 (c) 38 (a) 53 (a) 68 (c) 83 (d)9 (c) 24 (b) 39 (c) 54 (d) 69 (a) 84 (a)

10 (a) 25 (b) 40 (b) 55 (a) 70 (a) 85 (b)11 (d) 26 (a) 41 (a) 56 (a) 71 (b) 86 (c)12 (d) 27 (d) 42 (a) 57 (d) 72 (a) 87 (d)13 (c) 28 (b) 43 (c) 58 (c) 73 (d) 88 (a)14 (c) 29 (c) 44 (c) 59 (c) 74 (a) 89 (c)15 (c) 30 (b) 45 (b) 60 (c) 75 (a) 90 (b)

ANSWER KEY

HINTS AND EXPLANATIONS

12

9. (c) w = Z n q496.5Z 5.55 10

0.9 2 96500-= =

10. (a) 4remaining5 0.08 10 0.03

[H ] 2 1010 5 485

+ - - = = + +pH = 3.7

11. (d) Ester of dicarboxylic acids undergo an intramolecularversion of the Classian condensation when a five or sixmembered ring can be formed. This reaction is anexample of a Dieckmanns condensation

12. (d) (a) The change of atmospheric temperature withaltitude is called the lapse rate.

(b) The gases responsible for greenhouse effects areCO2, Water vapour, CH4 and ozone.

(c) The order of contribution to the acid rain isH2SO4 > HNO3 > HCl

13. (c) Cell reaction is 2Ag (s) + 2H+(aq) 2Ag+(aq) + H2(g)

Ecell = 0.80 2

20.0591 (0.1) 0.1log

2 (0.1)

Ecell = 0.80 + 0.02955 = 0.77 V14. (c)

I

Cl

Cl

Cl1

2

2

3

3

x

x

(D 0) x > yMolecule is polar and planar.Both Cl I Cl are equalEquatorial I Cl bond has more s-character than axialICl bond.

15. (c) Oxidation of secondary alcohol.16. (b) Fe and Pt do not form amalgam

Fe vessel

Silver amalgamAg Hg Ag Hg

D- +

Vessel made of other metal will form amalgam withliberated mercury.

17. (b) (a) NH4NO2 D N2 + 2H2O

(b) (NH4)2SO4D NH3 + H2SO4

(c) 2NH4ClO4D N2 + Cl2 + 2O2 + 4H2O

(d) (NH4)2Cr2O7D N2 + Cr2O3 + 4H2O

18. (d) (a) The size of the colloidal particles is in between thesize of the molecule and the size of the particle ofa coarse suspension.

(b) White portion of the egg is condensed colloidalsolution whereas the sol prepared from thissubstance is a dilute colloidal solution.

(c) Dialysis is a slow process. It may some times takeseveral days for its completion.

19. (a)

Cl

HBr I

Cl Br

HBr

Peroxide

II

ClBr

20. (b)HgSO aldol4

3H SO2 4HC CH CH CHO -

3 2

OH|

CH CH CH CHO (3-hydroxy butanal)3 2 1

- - -

21. (c) Br2

(X)

AlcoholicKOH/

(Y)

Br

BrH BrPeroxide

H

(Z)

HBr

22. (c) (a) When a dilute solution of an acid is added to adilute solution of a base, neutralization reactiontakes place.

(b) In acid-base titrations, at the end point, the amountof acid becomes chemically equilvalent to theamount of base present.

(c) In the case of a strong acid and a strong basetitration, at the end point the solution becomesneutral (i.e., pH = 7)

(d) In acid-base titrations the end point is determinedby the hydrogen ion concentration of the solution.

23. (c) Maximum kinetic energy of the photoelectron isindependent on intensity of radiation. No. of electronsemitted per second will be halved if the intensity oflight striking on the surface halves.

24. (b) NO2

X =

ClBr

CH ONa3

S ArN

NO2

OCH3Br

25. (b)

3

O||

X C CH- - NaOH

H

O||

X C OH- - SOCl2

O||

X C Cl- - Ph NH2-

O||

X C NH Ph- - -

(HNO H SO )3 2 4+ NO2X C NHO

13

26. (a) 2 21 1N (g) O (g) NO(g)2 2

+ ; DfH = 90.2

2 2N (g) O (g) 2NO(g)+ ; DfH = 90.2 2 ...... (1)

2 22NO(g) O (g) 2NO (g)+ ; DfH = 114....... (2)

2 2 2 51

2NO (g) O (g) N O (g)2

+ ;

DfH = 102.6

51.32

- = - ....... (3)Eq. (1) + (2) + (3)

2 2 2 55

N (g) O (g) N O (g)2

+ DfH (N2O5, g) = 15.1 kJ/mol

27. (d)x

kt

= ; 1t x.k

= ; 3/43

t .a4k

=28. (b) Anhydrous copper (II) chloride is a covalent while

anhydrous copper (II) fluoride is ionic in nature becausein halides of transition metals, the ionic characterdecreases with increase in atomic mass of the halogen.Fluorine being the most electronegative, forms ionicsalt with copper.

29. (c) o oA A B BP P x P x= +

o oA B

3 2600 P P

3 2 2 3 = + + + ; A B3P 2P 3000 + =

o oA B

4.5 2630 P P

4.5 2 0.5 4.5 2 0.5 = + + + + +

o oA B4.5 P 2 P 4410+ =

A A B1.5P 1410 ; P 940 and P 90 = = =30. (b) The redox system having the highest redox potential in

the most powerful oxidising agent followed by the onewith the lower redox potential.

31. (a) p mv m 2ghD = =32. (a) Least count of vernier callipers

= value of one division of main scale value of onedivision of vernier scaleNow, N a = (N + 1) a'(a' = value of one division of vernier scale)

Na a

N 1= +

\ Least count = aa aN 1

- = +33. (b) Here I = 4A

c c3030180 6

p p q= = =

Now, I 4 4 6 7 2 6 7k22 11

6

= = = =pq

84 7.6 A / rad11

= =

34. (c)mg = T cos 30 R

R

30

mg

N

T

N = T sin 30

mgN3

=

35. (b) S1 : No work is done by net force, it only changesdirection of momentum of particle. Hence S1 is false.S2 : True by definition.S3 : Nothing is said about acceleration of both particles.Hence angle between velocity and acceleration ofcentre of mass may not be zero. Consequently centreof mass may not move along a straight line. Hence S3 isfalse.

S4 : net1 1 2 2 n n

cm1 2 n 1 2 n

Fm v m v ..... m vV

m m ..... m (m m ..... m )+ + += =+ + + + + +

rr r r

r

Direction at netPr

is fixed so cmVr

is also constant in thedirection. So path of CM will be straight line.

36. (a) The linear relationship between v and x isv = mx + C where m and C are positive constants.

\ Acceleration, dva v m ( mx C)dx

= = - - +

\ a = m2x mC

Hence the graph relating a to x is

a

x

37. (a) Charge conservation is violated in [2, 3, 4], nucleonconservation is violated in (4), (1) works.

38. (a) A convex mirror always forms a virtual image whichcan't be cast on a screen. Hence, a convex lens is usedto find the focal lenght of a convex mirror.

39. (c) For a resistor the pointer moves in both ways whenvoltage is applied.

40. (b) Using t /T

0

A 1A 2

= or 7.5/T1 1

32 2 =

or 5 7.5/T1 1

2 2 = or

7.55

T= i.e., T = 1.5 hours

41. (a) Divide the ring into infinitely small lengths of massdm1. Even though mass distribution is non-uniform,each mass dm1 is at same distance R from origin.\ MI of ring about z-axis is= dm1R

2 + dm2 R2 + ..............+ dmnR

2 = MR2

42. (a) mv

rqB

= r m (v = same for both charges)

43. (c) Back lash error is occurred when the screw is rotatedboth in clockwise and anticlockwise direction.

44. (c) net0

Qf = For dipole Qnet = 0 f = 0

14

45. (b)

C

D

X Y

Equivalent resistance = 5W46. (b) If temp of mixture (T) is not constant then according to

calculation we find wrong value of specific heat. If westir continuously then temperature of whole mixturebecomes same.

47. (b) r yT ; red yellow, l l > l48. (a) Initial condition

Volume = 76cm APressure = (76 + 76) dg = 152dgFinal conditionVolume = 152 cm APressure = 76dg\ P1V1 = P2V2\ Both points lie on the same isothermal linei.e. both have same T.

49. (a) Filling liquid will increase the optical path length bysame amount.\ CBF will not shift.

50. (c) ;12

mI2

l=

Restoring troque = qElsin q qEl q 22

dtd q=

T2

m12qE

IqE

2p===w\

l

ll

qE3mT lp=\

51. (a) E = dxdV-

V = 5x2 + 10x 9,

E = dxd-

(5x2 + 10x 9) = (10x + 10)

On putting value of x in it we get,E = (10 1 + 10) = 20 V/m

52. (c) The equation of wave moving in negative x-direction,assuming origin of position at x = 2 and origin of time(i.e. initial time) at t = 1 sec.

y = 0.1 sin (2pt + 4x)Shifting the origin of position to left by 2m, that is, to x= 0. Also shifting the origin of time backwards by 1 sec,that is to t = 0 sec.

y = 0.1 sin (2p (t 1) + 4 (x 2))53. (a) As given the particle at x = 2 is at mean position at

t = 1 sec.\ Its velocity v = wA = 2p 0.1 = 0.2p m/s.

54. (d) Time period of oscillation 2 2T 1 sec.

2p p= = =w p

Hence, t = 1.25 sec, that is, at T/4 seconds after t = 1

second, the particle is at rest at extreme position. Henceinstantaneous power at x = 2 at t = 1.25 sec is zero.

55. (a) As long as the block of mass m remains stationary, theblock of mass M released from rest comes down by2Mg

K (before coming it rest momentarily again). Thus

the maximum extension in the spring is2Mgx

k= .......... (1)

For block of mass m to just move up the inclinekx = mg sin q + mg cos q .......... (2)

3 3 42Mg mg mg

5 4 5= + or 3M m

5=

56. (a) For a particle undergoing SHM with an amplitude A andangular frequency w, the maximum acceleration = w2AHere the maximum force on the particle = QE0

\ maximum acceleration = 20QE Am

= w

\ 02QE

Am

= w

57. (d) 21 hc

mv2

= - fl21 hc 4hcm '

2 (3 / 4) 3n = - f = - f

l l . Clearly, 4v v3

>

58. (c) Process12 and Process34 are isochoric processes.W12 = 0, W34 = 0, W23 = nR (T3 T2) = 3R (1600 400)

= 3600 RW41 = nR (T1 T4) = 3R (200 800) = 1800 RW = (3600 1800) R = 1800R = 15 kJ

59. (c) 2 21 21 1E kx , E ky ,2 2

= =

21 2 1 2

1E k(x y) E E 2 E E

2= + = + +

2 8 2 16 18J= + + =

60. (c)1 2

1 1 1( 1)f R R

= m - - R = 10 cm.

1 1 10f (3 1)

10 10 4 = - - = - -

f=20 f=20f '

eq

1 1 4 1 2 4f 20 10 20 20 10

= - + = -

eq10

f3

= - cm61. (d) It passes through a fixed point (3, 4)

Slope of line joining (3, 4) and (1, 2) is 6/2 = 3\ Slope of required line = 1/3

Equation is y 4 = 1 (x 3)3

- -x + 3y 15 = 0

62. (b) h 0

f (3 h) f (3)f (3 ) limh

+

+ -=

15

h h

h 0 h 0

(2 e ) 1 e 1lim lim 1h h

- - -= = - = -

h 0

f (3 h) f (3)f (3 ) limh

-

- -= -2 2

h 0 h 0

10 (3 h) 1 1 (6h h ) 1lim lim

h h - - - + - -= = -- -

= 2

2h 0

6h hlim

h( 1 6h h 1)-

- + - +

2h 0

h (h 6) 6lim 32h( 1 6h h 1)

- -= = = -+ - +

Hence, f ' (3+) f (3)63. (b) 2 22(x iy) x y 2i+ = + +

2 22x x y= + and 2y = 2 i.e., y = 1

4x2 = x2 + 1 i.e., 3x2 = 1 i.e., 1x3

=

1x ( x 0)

3= Q \ 1 3z i i33

= + = +

64. (b) 2 y z 2 y z

0 x 2 z x 2 2 y 0x y 2 x 2 0 2 z

= = - -- -

2 y zx 2 2 y 2 z

(x 2) (2 y) (2 z) 1 1 01 0 1

- - -= - - -

2 y z0x 2 2 y 2 z

= - -- - - 2 y z 0

2 x 2 y 2 z+ + =

- - -65. (b) Let A be the event such that sum is Rs. 20 or more

\ P (1) = 1 P (Total value is < 20)

= 6 2

2 28

2

C C 14 1 11 1 128 2 2C

-- = - = - =

66. (c) 2 2 2

21 2 3x 4y 1x y ( 4 4)5 5 5

+ - - + - = l - l +

i.e., 2 21 2 3x 4y 1x y | 2 |

5 5 5+ - - + - = l -

is an ellipse.If 0 < | l 2 | < 1 i.e., l (1, 2) (2, 3)

67. (d) 1 2 2pq + q =

\ 2 2

1 1

d tan dI

1 tan1 tan2

q q

q q

q q q= =p + q + - q

and also 21

dI1 tan

qq

q=+ q

\ 2

2 11

1002 5012I d I2008 2008

q

q

p p= q = q - q = =Hence, K = 2008.

68. (c)2 y amy 4a

m+ = i.e., my2 4ay 4a2m = 0

m 0; 16a2 + 16a2m2 > 0 which is true m.\ m R {0}

69. (a)n

kn

k 1

C 12C +

= n! (k 1)!(n k 1)! 1k! (n k)! n! 2

+ - - =-

or k 1 1n k 2

+ =-2k + 2 = n kn 3k = 2 .......... (1)

Similarly, n

k 1n

k 2

C 23C

+

+=

n! (k 2)!(n k 2)! 2.(k 1)! (n k 1)! n! 3

+ - - =+ - -k 2 2

n k 1 3+ =- -

3k + 6 = 2n 2k 22n 5k = 8 ............. (2)From (1) and (2)n = 14 and k = 4\ n + k = 18

70. (a) Let the observations be x1, x2, x3, x4, x5 and x6, so

their mean

6

ii 1

xx 8

6== =

6 6

i ii 1 i 1

x 8 6 x 48= =

= = On multiplying each observation by 3, we get the newobservations as 3x1, 3x2, 3x3, 3x4, 3x5 and 3x6.

Now, their mean =

6 6

i ii 1 i 1

3x 3 xx

6 6= == =

3 48x 246

= =Variance of new observations

=

6 62 2 2

i ii 1 i 1

(3x 24) 3 (x 8)

6 6= =

- -=

= 91

Variance of old observations = 9 42 = 144Thus, standard deviation of new observations= Variance 144 12= =

16

71. (b) p (p q) is equivalent to p.

72. (a)3

sin x2

x cos x sin xecos x

- dx= sin xe x cos x dx sin xe tan x sec x dx= ( )sin xx d e sin xe d (sec x)= { }sin x sin xx e e dx-

{ }sin x sin xe secx e sec x cos x dx- = x esin x esin x sec x + C

73. (d) f (x) = x2 4x + a always attains its minimum value.So its range must be closed.So, a = {f}

74. (a) For real roots D 0(k 2)2 4 (k2+ 3k + 5) 0(k2 + 4 4k) 4k2 12k 20 0 3k2 16k 16 0 ; 3k2 + 16k + 16 0

4k (k 4) 0

3 + +

Now 2 2E = a + b ; 2E ( ) 2= a + b - abE = (k 2)2 2 (k2 + 3k + 5) = k2 10k 6E = (k2 + 10k + 6) = [(k + 5)2 19] = 19 (k + 5)2\ Emin occurs when k = 4/3

\ Emin = 121 171 121 5019

9 9 9-- = =

Emax occurs when k = 4Emax = 19 1 = 18

75. (a)p q r p q p q r20! 20!(2x) ( y) (z) 2 ( 1) x y z

p!q!r! p!q!r!- = -

p + q + r = 20, q = 0p + r = 20 (p is even and r is odd).even + odd = even (never possible)Coefficient of such power never occur\ coefficient is zero

76. (a)

Starting with 1 2 3 4 5 6 7 8 9= 8C4 = 70

Starting with 2 3 4 5 6 7 8 9= 7C4 = 35Total = 105

(105)th number 26789

77. (b) 3/2 3/2

3/2 2x 0

(tan x) [1 (cos x) ]limx .x

-.

= 1 3

2 3/2x 0

1 cos x 1lim .x 1 (cos x)

-+

= 21 1 3. (1 cos x cos x)2 2 4

+ + =

78. (b) n = 3, P (success) = P (HT or TH) = 1/2

p = q = 12

and r = 2

P (r = 2) = 3C2 21

2 .

1 32 8

=

79. (a) 1 1 (a.c)b (a.b)c b c2 2

- = +

\ 1 a.c2

= and 1a.b2

= -

angle between a and c4p= and angle between

3a and b4p=

80. (c) C1C2 = 13r1 = 30, r2 = 6C1C2 < r1 r2\ The circle | z2 (12 + 5i) | = 6lies with in the circle | z1 | = 30\ max | z1 z2 | = 30 + 13 + 6 = 49\ Statement-1 is true.Statement-2 | z1 z2 | | z1 | + | z2 | is always true.Equality sign holds if z1, z2 origin are collinear and z1and z2 lies on opposite sides of the origin.\ Statement-2 is true.

81. (a) 2 sin2q x + cos2q y = 2 cos 2qStatement-1: The line passes through the point (3, 2)If 6 sin2q 2cos2q = 2 cos 2qi.e. 6 (1 cos2q) 2cos2q = 4cos2q 2i.e. 12cos2q = 8\ Statement-1 is false.Statement : 2 (1 cos2q) x + cos2q y = 4cos2q 2\ cos2q (2x + y 4) + 2x + 2 = 0Family of lines passes through the point of intersectionof line 2x y + 4 = 0 and x = 1\ The point is (1, 2)\ Statement-2 is true.

82. (a)1f (x)

1 sin x=

- and 1f ( x)

1 sin x- =

+

\ /4

/4

dxI1 sin x

p

-p= +

Now, f (x) + f ( x) = /4

2/4

2 dx2I1 sin x

p

-p=

-

/4

2/4

dxIcos x

p

-p= . This is an even function

\ /4

2

0

I 2 sec x dx 0p

= Statement-1 is false.

17

83. (d) Every increasing or decreasing function is one-one2

2 1 8( ) 3 2 3 cos 3 cos 03 3

f x x x x x x = + + + = + + + > 21 8 8[ | cos | 1 and 3 ]

3 3 3x x < + +

Q

\ f (x) is strictly increasing

84. (a)my1 = 9 and my1 = 4

21y = 36 6y1 =Product of subtangent and sub normal is 21y .Statement 1 is false. Statement 2 is true.

85. (b) S1 : if x2 + ax + 7 = 0 has imaginary roots with positive

real parts then D < 0 and sum of roots > 0 a2 28 < 0 and a > 0 28 a 28- < < and a < 0 a = 1, 2, 3, 4, 5S2 : x

2 (a + 3) x + 5 = 0 has roots a, a, bIf a, a, b are in AP. then2a = a + b 2a = a + 3 a = 3The equation becomes x2 6x + 5 = 0 which has roots1 and 5.S3 : Case- I :If 0 < x < 1, then 2 + x 6 x > 0 2x 4 and x < 6 x 2 and x < 6 x [2, 6)\ x (0, 1) [2, 6) = f \ x fCase II : If x > 1, then 0 < 2 + x 6 x x > 2 and x 2 \ x (1, 2]

86. (c) x = tan1 t 2dx 1dt 1 t

=+

2dy dy dt dy. (1 t )dx dt dx dt

= = + ........... (1)2

22

d y d dy dt(1 t ) .dt dt dxdx

= +

2

2 22

dy d y2t (1 t ) (1 t )dt dt

= + + +

........... (2)

Hence the given differential equation2

22

d y dyxy sec x 0dxdx

+ + > , becomes

22 2

2dy d y(1 t ) 2t (1 t )dt dt

+ + +

1 2 2dyy tan t (1 t ) (1 t ) 0

dt- + + + + =

Cancelling (1 + t2) throughout we get2

2 12

d y dy(1 t ) (2t y tan t) 1dtdt

-+ + + = - k = 187. (d) f2 (x) = f (f (x)) = f (x) = x

f3 (x) = f (f2 (x)) = f (x) = x x3 25x2 + 175x 375 = 0(x 5) (x2 20x + 75) = 0(x 5)2 (x 15 ) = 0 x = 5, 15

88. (a) 2 sec 4C + sin22A + sin B = 0A = 45, B = 90 and C = 45

Let AQ = a, then BP = a2

,

PQ = a2

and QR = a

C

P

Q

AR,B

\ 2

2 a 5aPR a4 2

= + =

\ a 5a1: : : a : 1: 2 : 52 2

a b = =

\ a = 2 and 5b = \ 2 2 9a + b =

89. (c) Let -=I xsin0 112

dttsin

Put t = uducosusin2dtusin 2 =udu2sindt =

=I\ x01 duu2sinu

Let I2 = dttcosxcos

01

2

-Put t = vdvsinvcos2dtvcos2 -=

dvv2sindt -=

p p-=-=I\ x2

x

22 vdv2sinvdv)v2sin(v

p-= x2

udu2sinu [change of variable]

\ I = I1 + I2 = p-x0x

2

udu2sinuudu2sinu

x x2

02 2

u sin 2udu u sin 2udu u sin 2udu

p

p p= + -

2

0

u sin 2udu4

p

p= = [Integrate by parts]

90. (b)

For non-trivial solution, 2a sin B cos B1 cos B sin B 01 sin B cos B

D = =- -

2 2a 2 [ cos B sin B] sin B [ cos B sin B]- - - - +cos B [sin B cos B] 0+ + =

a 2 sin 2B cos 2B 0 a [ 1,1]- + + = -