Mock Test-2
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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
One of the leading publishers in India, Disha Publication provides books and study materials for
schools and various competitive exams being continuously held across the country. Disha's sole
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online bookstore to help students buy exam books online with ease. We, at Disha provide a wide
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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]- + + = -