Ai Iitjee Modeltest 06

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Model Test - 06

Take This test comprising 3 sections with a duration of 2 hours eachsection.Each section contains 50 questions. Observe the given weightage toeach question. 1/3 rd negative marking do exist for the wrong answers.Maintaining the time limit of 2 hours for each section is of atmost im-portance, while taking the test.

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Questions 1 - 5 are based on the following passageThe photoconductor is the central component of a photocopy machine . In the first step of the copuing process ,charged gaseous ions are adsorbed onto the surface of the photoconductor . Opposite charges flow from the groundto the ground plane to maintain overall neutrality . An electric field is set up within the photoconductor and apotential known as the “ dark voltage” is estalished on the surface . The photoconductor can be through of as a

parallel- plate capacitor , with a capacitance C given by dAkeC 0= where k is the dielectric constant characterizing

the material from which the photoconductor is made , A is the area of the photoconductor surface ,and d is itsthickness .An image of the photocopied page is projected onto the photoconductor by a series of mirrors and lenses , causingselect areas of the photoconductor to be exposed to light . When light of the right wavelength hits the photoconductor, electrons may be given enough energy to overcome the forces holding them in place and become energizedto movefreely through the bulk of the photoconductor , leaving behind positively charged “holes” where they once were .In the presence of an external electric field , the electrons and the holes move in opposite directions .

The creation of electron - hole pairs occurs only in the areas of the photoconductor exposed to light , in a layerknown as the charge generation layer that typically lies near the bottom . The holes will migrate to the surface underthe attractive force of the adsorbed surface ions , while the electrons flow to the ground plane . The process isschematized in figure 1 .

Absorbed ions

Photon

Groundplane

Charge generationlayer Ground

m15µ

)m1.0( µ

The migration of the holes and their interaction with the surface ions lead to a decrease in the magnitude of thepotential at the surface . The new value is known as the “ light voltage,” since only those areas exposed to lightexperience this change in potential . The surface potential ( relative to the potential of the ground plane ) is plottedFigure 2.

Light of

Time

Light ofLight of

Light ofVoltage

Dark Voltage

Figure - 2

0

Surfa

c e p

ote n

tial

Toner particles ( ink particles are released from an ink gunand electrodes are used to direct and accelerate the tonerparticles onto the page . An electrical charge is imparted to the particles as they leave the gun . These charged tonerparticles adhere to the photoconductor surface in a manner that reflects the contrast between the light and darkvoltages , and the toner is then transferred onto paper. Finally , the paper is heated to bake the toner onto the pageand the copy is complete.

Paper - I : Physics - 1 to 50




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1. As can be seen from figure2 , thetransition from dark to light voltage is not abrupt , but exhibits a finite duration thatcorresponds roughly to the time the hole takes to that move to the surface . This is characterized by the transit time

which is approximated by the relation |V|mdT

0

2

L = where d is the thickness of the photoconductor, µ is thedrift

mobility of the charge and 0V is thedark voltage . Which of the following will decrease the transit time ?

a) increasing the thickness of the photocoductorb) increasing the intensity of the light projected onto the photocoductorc) increasing the wavelength of the light projected onto the photocoductord) increasing the number of gaseous ions adsorbed per unit area

2. Find the units of 0ε from Coulomb’s law .

= 2

21

0 rqq

4ππ1F Where F is the force of electrostatic attraction

1q and 2q are charges ,and r is the distance between the charged particles.

a) C / ( Nm) b) 22/CNm c) )Nm/(C 22 d) C / J

3. If the dark voltage is - 80V ,and if the photoconductor can be thought of as a m15 µ− thick dielectric with dielectricconstant 3 ,what is the magnitude of the surface charge density due to the adsorbed ions ?

a) 20 cm/C500ε b) 2

0 cm/C1600ε c) 20 cm/C5000ε d) 2

0 cm/C16000ε

4. What would happen if light of the correct wavelength is direct onto the photoconductor but no ions are adsorbedonto its surface?a) The electron-hole pairs created would recombine among themselves.b) No electron-hole pairs would be created.c) The difference between the dark and light voltages would increase.d) Positive holes would not be annihilated when they reach the surface, thus reversing the polarity of the

photoconductor.5. Electron-hole pairs are generated only when there sufficient energy to promote an electron to a higher energy state.

If light of 600 nm hits the photoconductor and produces no electron-hole pairs, which of the following actions maylead to their generation?a) increasing the intensity of the light.b) increasing the number of gaseous ions adsorbed per unit area.c) decreasing the frequency of the light.d) decrerasing the wavelength of the light.

Questions 6 to 45 are independent Question

6. The diagram shows a cell E, connected to a pure resistor R and pure self induction L of negligible resistance. If attime, t = 0, the switch S is closed, the self induced emf ‘e’ across the self inductor varies with time as shown by thegraph :

LR

E S

a)

e

to

b)

e

t

c)

e

t

d)

e

t

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7. Two masses m and M are connected by a massless inextensible string which is passing over a fixed pulley. At someinstant V1 and V2 are their velocities, shown in the figure. The ratio V1/V2 is ——

V2

V1

m

M

a) 1 b) cosθ c) cosec θ d) sin θ

8. n-identical spheres are placed on a smooth surface touching each other in a straight line. An another identical sphere

moving with velocity ‘u’ along the same straight line hits the sphere. If 12

e = , is the coefficient of restitution

between any 2 spheres, the velocity of nth sphere is ——

a)13

4

n

u− ⋅

b)34

n

u ⋅ c)

112

n

u− ⋅

d) zero

9. The figure shows the net power (P) versus current intensity (I) in a circuit. The emf of the cell φ when the powerconsumed is maximum.

O

25–

50–

75–

100 W

5A

10 AmpI

a) 40 V b) 30 V c) 20 V d) 10 V

10. When a particle is allowed to Slide down from the top (A) of a hill. It stops at X due to friction. The coefficient offriction is independent of velocity and direction of motion. The work performed by a tangential force to take the sameobject back to the top of the hill is ——

h

x

A

X

a) mgh b) mgh + mgx

c) 2mgh d) depends on the shape of the hill

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11. The graph shows the motion of a ball rebounding from a horizontal surface being released from a point above thesurface. The quantity represented on the y-axis is the ball’s

y

xtime

a) velocity b) acceleration c) kinetic energy d) displacement

12. Suppose the potential energy between electron and proton at a distance ‘r’ is given by 2

33Ker

−. Application of

Bohr’s theory to hydrogen atom in this case shows that :i) energy in nth orbit is proportional to n6

ii) energy is proportional to m–3 (m: mass of electron) Then :a) both (i) and (ii) are correct b) only (i) is correctc) only (ii) is correct d) none are correct

13. If the ground state of hydrogen atom is chosed as zero potential energy level, the value of the total energy in the firstexcited state (in eV) is —

a) 10.2 b) 13.6 c) 3.4 d) 23.8

14. A thermodynamic process is such that VT = constant, the molar heat capacity of such a process is —

a) 2VC C R= + b) 2VC C R= − c) VC C R= + d) zero

15. The given diagram indicates the energy levels of a certain atom. When the system moves from 2E level to E level, a

photon of wavelength λ is emitted. The wavelength of the photon emitted during its transition from 43E

level to E

level is —

2E

4E/3

E

a) 3λ b) 3

4λ c) 4

3λ d) 3λ

16. A tank is filled with water to a height 0.5m. It is placed on a stand of height 1.5m above the ground. At what height‘h’ above the ground a small hole be made in the tank so that the water falls on the ground at the farthest distance‘R’?

h1.5m

0.5m

K R–

a) 0.5 m b) 1.0 m c) 1.5 m d) any value of h

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17. The V – T diagram of an ideal gas for the process A B C→ → as shown. Select the correct alternative

V

T

A

B

C

o

a) pressure of the gas first increases then remains constantb) pressure of the gas first decreases then remains constantc) pressure of the gas remains constant through outd) nothing can be said about the pressure of the gas from this graph

18. Two moles of an ideal gas undergo a cyclic process A B C D→ → → as shown in a U –V diagram. U represents

internal energy. V represents volume of , 500A D BT T O k T k= = = and 300CT k= , the heat absorbed in one cycleis —

U

V

A

D

B

C

V0 2V0o

a) 0 b) 400 R ln 2 c) –600 ln 2 d) 1000 R ln 2

19. One mole of an ideal gas whose pressure changes with volume on P V= α , where α is a constant, is expanded sothat its volume increases twice. The heat capacity of the gas is —

a)( 2)

2( 1)R r

r++ b)

22 2R r

r+

− c)

22 1R r

r+

− d)

12 1R r

r+

−

20. The thermal power of density U (energy/volume) is generated at a constant rate inside a uniform sphere of radius Rand thermal conductivity K. The distribution of temperature inside the sphere when the steady state temperature. T0at the surface is —

a)2 2

0( )

6U R rT

K−+ b)

2 2

0( )

3U R rT

K−+ c)

2 2

0( )

2U R rT

K−+ d)

2 2

0( )U R rT

K−+

21. A beam of parallel rays falling on a converging lens converges at point F of the lens is split into 2 parts along thedotted line and shifted perpendicular to the optical axis in the opposite directions slightly, choose the correctalternative :

F

a) there will be two images above and below Fb) there will be three images on same vertical time passing through F and towards nightc) there will be three images one at F and other two above and below the point F towards rightd) there will be three images one at F and other two above and below the point F towards left

22. In a double slit experiment, the distance between the slits is ‘d’. The screen is at a distance ‘D’ from slits. If abright fringe is formed opposite to a slit on the screen, the order of the fringe is : ( λ –wavelength of light)

a)2

2d

Dλ b) 22 Ddλ

c)dDλ d)

2 dDλ

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23. The figure shows a thin glass converging lens of focal length, 403

f = cm. To the left of the lens is a square of area

100cm2. The base of the square is on the principal axis, and the right side of the square is 20cm from the lens. The areaof the image is —

F

20cm

a) 252 cm2 b) 224 cm2 c) 234 cm2 d) 244 cm2

24. In the stationary wave pattern that forms as a result of the reflection of waves from an obstacle the ratio of theamplitude at an antinode to the amplitude at a node is 1.5. The percentage of energy passes across the obstacle is——

a) 12% b) 24% c) 48% d) 96%

25. A source emitting a sound of frequency f is placed at a large distance from an observer. The source starts movingtowards the observer with a uniform acceleration ‘a’. If the speed of sound in the medium is V. Then the frequencyheard by the observer corresponding to the wave emitted just after the source starts is —

a)2Vf

Vf a− b)2Vf

Vf a+ c)22

2Vf

Vf a− d)22

2Vf

Vf a+

26. A positively charged sphere of mass 5kg is attached by a spring of spring constant 410 Nm . The sphere is tied with

a thread so that spring is in its natural length. Another idential, negatively charged sphere is fixed with floor,

vertically below the positively charged sphere as shown. If initial seperation between the spheres is 012

r m= , and

magnitude of charge on each sphere is q = µθ 100 , the maximum elongation of the spring when the thread is burntis —

r0

a) 15 cm b) 10 cm c) 25 cm d) 22 cm

27. A solid non conducting hemisphere of radius R has a uniformly distributed positive charge of density ρ per unitvolume. A negatively charged particle having ‘q’ is transferred from centre of its base to infinity. Work performed inthe process is —

a)2

0

ρ4

q Rε b)

2

02q R− ρ

ε c)2

04q Rρ

πε d)2

02q R− ρπ ε

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28. The potential energy U of a particle of mass 1 kg, moving in xy plane is given by U = 2x – y. If the particle is at restat (2, 1), its acceleration at this position is given by —

a) 10 b) 2 10 c) 1; 3x ya a= = d) 3; 1x ya a=− =−

29. A less is coated with a thin film of µ = 1.2 inorder reduce the reflection from its surface at 05000Aλ = . Minimumthickness of the film which will minimize the intensity of the reflected light is —

a) 10–6 m b) 10–1 cm c) 10–5 cm d) 10–6 cm

30. Two thin symmetrical lenses of different nature have equal radii of curvature 20cm. The lenses are put close togetherand immersed in water. The focal length of the system is 24 cm. Then the difference between refractive indices of thetwo lenses will be —

a)29 b)

13 c)

49 d)

59

31. At a certain point on a screen the path difference for the two interfering rays of same amplitude is 18

th of a

wavelength. The raio of the, intensity at this point to that at the centre of a bright fringe is —

a) 0.853 : 1 b) 0.693 : 1 c) 0.492 : 1 d) 0.341 : 1

32. Two sources A and B are sounding notes of frequency 680 Hz. A listener moves from A to B with a constant velocity

u. If the speed of the sound is 340 ms . What must be the value of u so that he hears 10 beats per second?

a) 2.0ms b) 2.5

ms c) 3.0

ms d) 3.5

ms

33. In the circuit shown the cells have no internal resistances the potential difference VA–VB is —

A

+ ++

–– –

B

a) 1 V b) 15 V c) 5 V d)1017 V

34. A charge particle q pases through a region having perpendicular uniform electric field E and magnietic field Bundeviated. Choose the current alternative —a) the charged particle must be at restb) the component of velocity vector along E must be zero

c) the velocity vector V must be perpendicular to E and B togetherd) none of these is correct

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35. Charges Q1, Q2, Q3 and Q4 are given to the plates 1, 2, 3 and 4 respectively. The charge appear at the outer surface ofthe first plate to —

1 2 3 4

a) 1 2 3 4Q + Q + Q +Q2 b) 1 2 3 4Q +Q + Q +Q

4 c) 1 2 3 4Q – Q – Q – Q2 d) 1 2 3 4Q – Q – Q – Q

4

36. A parallel plate capacitor is connected to a battery. The plates are pulled apart is in a uniform speed. If x is theseperation between the plates, then the φ rate of change of the electrostatic energy of the capacitor is proportionalto :

a) x2 b) x c) 2

1x d)

1x

37. For a cell, graph between potential difference (V) across the terminals of the cell and current (I) drawn from cell isshown. The emf and internal resistance of cell—

V (volt)

o

2

S i(amp)

a) 2V,0.4Ω b) 2V,0.5Ω c) 2V,0.4> Ω d) 2V,0.4< Ω

38. The gap between the plates of a parallel plate capacitor is filled with glass of resistivity ρ and dielectric constant K.The capacitance of the capacitor equals C. If a voltage V is applied to it, then the leakage current of the capacitor atthat instant is —

a)02 ρ

CVk ε b)

0

2ρCV

k ε c)0

2ρCV

k ε d)0ρ

CVk ε

39. The annular disc of copper with inner and outer radii as a and b is rotating with a uniform angular speed ω is a regionwher a uniform magnetic field B along the axis of notation exists. Then the emf induced between the inner sides andouter rim of the disc is —

a

b

a) zero b)21 ω

2B a c)

2 21 ω ( )4

B b a− d)2 21 ω ( )

2B b a−

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40. A spring of mass ‘m’ lies on a smooth table. One end of it is clamped to the nearest wall and the other end is pulledwith velocity ‘V’ the kinetic energy of the spring is —

V

a)21

2mv b)

213

mv c)21

6mv d)

219

mv

41. A small sphere of superdense material has a mass half of the mass of the earth. It is situated at a height ‘h’ (muchsmaller than the earth’s radius) from where it falls on the earth’s surface. Then its speed when its height from the

earth’s surface is reduced to 2h

is —

a) 2gh b) gh c)2

3gh

d)3

2gh

42. Two particles of masses M1 and M2 are connected by a light spring of spring constant K as shown M2 is placed onrough table where as M1 is hanging vertically. M1 performs vertical oscillations with amplitude A. If M2 on tablealways remains at rest then minimum value of coefficient of friction between M2 and table is —

M1

M2

a)2

1 2

M KAM M g

+ b)1

2 2

M KAM M g

+ c)2

1 2

M KAM M g

− d)1

2 2

M KAM M g

−

43. A particle moves with an initial velocity V0 and retardation αV , where V is its velocity at any time ‘t’. The particlewill cover a total distance of —

a) 0V4α b) 0V

2α c) 0Vα d) 02V

α

44. A block of mass ‘m’ is allowed to slide down a smooth inclined plane of angle θ as shown. The power developedby the force of gravity at the moment when the block reaches the bottom is —

m u=0

l

a) 2 31 lg sin3

m⋅ θ b) 2 32 lg cos3

m⋅ θ c) 2 32 lg sin3

m⋅ θ d) 2 3 32 lg sinm⋅ θ

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45. Three points are located at the vertices of an equilateral triangle whose sides equals ‘a’. They all start movingsimultaneously at time, t = 0 with velocity V = kt, where k is a positive constant. All the three points move such thatfirst always point heading continually for the second, the second for the third, and the third for the first, how soonwill the points converge?

a)aK b)

2aK c)

2aK d)

4aK

Questions 46 - 50 are based on the following passage

Any object, regardless of its temperature, emits thermal radiation. For most objects, we can’t see this radiation at lowtemperatures, because the emissions are in the infraed part of the psectrum. However, as the temperature increases,the object will eventually glow a dull red. Objects like the filament of a light bulb have high enough temperatures togive off white light. However, even though we see or detect light of one color, the object is, in fact, emitting lightacross a spectrum of frequencies.

Classical theories aren’t enought to explain why this distribution of thermal wavelengths should be Scientists turnedtheir attention to the blackbody, an ideal system whose surface is a perfect absorber and emitter of radiation. Thethermal emission spectra from stars can be approximated closely by the blackbody distribution.

Wien’s displacement law arose from experimental results : as the temperature of a blackbody increases, the wavelengthat which the maximum intensity of thermal radiation occurs decreases.

m.k102.9Tλ 3max

−×=

Max Planck was able to determine the correct formula for the distribution of blackbody emission wavelengths bumaking two assumptions. First, the oscillating molecules of the blackbody that give off the thermal radiation can onlyhave particular, discrete frequencies. The fact that the allowed frequencies of the molecules were quantized meansthat their energy has to satisfy E = nhf, where h is Planck’s constant, and n is a positive interger. Second, the emittedquanta, called photons, have discrete energies that are proportional to their frequency. (Note : h = 4.15 x 10-15 eV-s).

46. As a result of the Big Bang, a cosmic background radiation exists in all of space, at a temperature of 3 K. What is thewavelength of this radiation?

a) 1.1 x 10-15 m b) 8.7 x 10-4 m c) 9.7 x 10-4m d) 1.0 x 103 m

47. Which of the following could be the termperature of a blackbody whose maximum emitted intensity is in the visiblelight spectrum?

a) 500 K b) 2000 K c) 5800 K d) 29000 K

48. How does the frequency of blackbody radiation at maximum intensity vary with the temperature of the emittingbody?a) As the temperature of the object increases, the emitted frequency at maximum intensity increasesb) As the temperature of the object decreases, the emitted frequency at maximum intensity increasesc) As the temperature of the object increases, the emitted frequency at maximum intensity is unchangedd) As the temperature of the object increases, the emitted frequency at maximum intensity decreases.

49. A mass m, connected to a spring with force constank k, undergoes oscillation. If the energy of oscillation isquantized, how big is the energy gap between consecutive energy states?

a)mk

21π

b)mkh c)

km

2hπ

d)mk

2hπ

50. A 3-kg block of lead is heated from a temperature of 400 K to 410 K by adding 3840 J of heat energy. What is thespecific heat of lead?

a) 3.12 J/kg.K b) 128 J/kg. K c) 312 J/kg.K d) 384 J/kg.K

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The locun of the middle points of a system of parallel chords of an ellipse is called a diameter and chords of are called

its double ordinates, the equation is 2

2by x

a m

−= , (where m is slope of chord of ellipse

2 2

2 2 1x ya b

+ = ) which is a st.

line thro’ (0, 0) having slope m1 such that 2

12

bmma

−= . Also if y mx= bisect all chords 11to 1y m x= , where

21

2bmm

a−= . Such a pair of diameters which bisects all chords 11 to the other are called conjugate diameters.

51. If 1 2&θ θ be eccentric angles of the ends A & C of a pair of conjugate diameters, then 1 2,θ θ differ by

(a) 3π

(b) π (c)32π

(d)2π

52. In the above question (1) 2 2AO CO+ = Where O is centre of ellipse 2 2

2 2 1x ya b

+ =

(a) 2 2a b+ (b) ( )22 2a b+ (c) 2 2a b− (d) 1

53. In Question (1) IF AO CO= then θ =

(a)2π

(b) 3π

(c)3or

4 4π π

(d)23π

Questions 54 - 95 are independent questions

54. The equation 06)(4)( 222244 =+−−+ ycxyxbxyyxa represents two pairs of lines at right angles and the twopairs will coincide if

a) acab 32 22 == b) bcba 32 22 == c) acba =+ 22 d) 22 23 baca =−

55. No. of line that can be drawn through the point (4, -5) so that its distance from (-2, 3) will be equal to 12.

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

56. Radius of the circle 08)ˆ2ˆ2ˆ.( ,019)ˆ4ˆ2ˆ2.(2 =++−=−−−+ kjirkjirr

a) 5 b) 4 c) 3 d) 2

57. The equation ax + by + c = 0 reprsents a plane perpendicular to the

a) xy - plane b) yz-plane c) zx - plane d) none of these

58. The period of the function xxxx

xfcossincossin

)(+−

= is

a) 2π

b) π2 c) π d) None of these

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Paper - I I : Mathematics - 51 to 100




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59. The function ],0(]1,(: 5ef →−∞ defined by 233)( +−= xxexf is

a) Many one and onto b) Many one and intoc) One-one and onto d) One-one and into

60. If the normal of f(x) = 0 at x = 0 is given by 3x - y + 3 = 0 then )7(4)4(5)(lim 222

2

0 xfxfxfx

x +−→ is equal to

a)31− b)

31

c)61

d) 0

61. The following functions are differentiable on (-1, 2)

a) dttx

x22

)(log∫ b) dtt

tx sin2

0∫ c) dtttttx

2

2

0 1 -1

+++∫ d) None of these

62. Between any two real roots of the equation 1sin =xex the equation 1cos −=xex has

a) atleast one root b) exactly one root c) atmost one root d) no root

63. The normal to the curve (cos sin ), (sin cosθ)x yα θ θ θ α θ θ= + = − at any θ is such that :

a) it makes a constant angle with x-axis b) it passes through the originc) it is at a constant distance from the origin d) None of these

64. Hyperbolas are drawn with the same two perpendicular lines as the transverse and conjugate axes of every member,with the property that the reciprocal of the square of the length of the semi-conjugate axios of a hyperbola is lessthan the reciprocal of the square of the length of the semi-transeverse axis by a constant. The hyperbolas have

a) no point in common b) one point in commonc) two points in common d) four points in common

65. If 1e and 2e are the eccentricities of the conic sections 144916 22 =+ yx and 144169 22 =− yx , then

a) 322

21 =+ ee b) 32

221 >+ ee c) 32

221 <+ ee d) 2 2

1 2 1e e+ =

66. A line meets the co-ordinate axes in A and B. A circle is circumscribed about the triangle OAB. If the distances fromA and B of the tangent to the circle at the origin be m and n. Then the diameter of the circle is

a) m (m + n) b) m + n c) n (m + n) d) m2 - n2

67. In the figure OABC is a square of side 8 cm, then the equation of the smallest circle is

A

O

B

A X

Y

a) 4)4()4( 22 =−+− yx b) 8)4()4( 22 =−+− yx

c) 12)4()4( 22 =−+− yx d) None of these

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68 ∆then ,D)(C cisCsin C cosD) (B cosBsin B cosD) (A cosAsin A cos

∆Let ABCD. ralquadrilate aIn +++

=

a) independent of A and B only b) independent of B and C onlyc) independent of A, B and C only d) independent of A, B, C and D all

69. 2 4 6 2[sin sin sin ...... ]logIf e etoα α α+ + + ∞ is a root of equation ,0892 =+− xx where ,2

0 πα << then the principle

value of

−

32sinsin 1 π

is

a) α b) α2 c) α− d) α2−

70.

2 2

7 6 5 4 3 2 2 2

2 2

x 1 x 2 1 2 then

2 1

x xLet ax bx cx dx ex fx gx h x x x x

x x x x

+ + ++ + + + + + + = + + +

+ + +

a) g = 3 and h = -5 b) g = - 3 and h = -5 c) g = -3 and h = -9 d) None of these

71. If in x, y, z the system of linear equations 0 7z7y2x 0; 3z4y x)2 (cos 0, z y - x)3(sin =++=++=+ θθ hasnon-zero solution, then θ =

a) πn b)6

2 ππ ±n c)3ππm d)

2πn

72. If the integers λ and µ are chosen at random between 1 to 100 then the probability that a number of the form

µλ 77 + is divisible by 5 is

a)41

b)71

c)81

d)491

73. Amar, Akbar and Anthony repeatedly take turns tossing a die. Amar begins; Akbar always follows Amar Anthonyalways follows Akbar; and Amar always follows Anthony. Find the probability that Anthony will be the first one totoss a six. (The probability of obtaining a six on any toss is independent of the outcome of any other toss.)

a)31

b)92

c)185

D)9125

74.2

2 2 2 22If sin cos , sin cos tan tan , aa x b x c b y a y d and a x b y then

b+ = + = = is equal to

a) a)-(c d) -(a)( )( bdcb −−

b) b)-(d c) -(b)( )( acda −−

c) b)-(d c) -(b)( )( acad −−

d) d)-(a c) -(a)( )( dbcb −−

75. wvuxzandzy ,, ;cos3sin2 tansin 6 y, cos sin x If === denote respectively zyx 222 sin ,sin ,sin

then the value of the triplet wvu is

a) (1, 0, 0) b) (0, 1, 0) c)

43;

21;

21

d)

21;

43;

21

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76. The greatest and least values of ( ) ( )3131 cossin xx −− + are

a) 2,

2ππ − b)

8,

8

33 ππ − c)32

,8

7 33 ππd) None of these

77. If in triangle ABC, oC 45=∠ then exhaustive range of BA 22 sinsin + is

a) [ 0, 1 ] b) ( 0, 1 ) c)

+2

21,21

d) None of these

78. The line x + 2y = 4 is translated parallel to itself by 3 unit in the sense of increasing x and then rotated by 300 in theanticlockwise direction about the point where the shifted line cuts the x-axis. The equation of the line in the newposition is

a) )53-4-(x )30( tan oy −= θ b) )53-4-(x )30( tan θ−= oy

c) )534(x )30 ( tan +++= oy θ d) )534(x )30 ( tan ++−= oy θ

79. A point moves such that the square of its distance from a straight line is equal to the difference between the squareof its distance from the centre of a circle and the square of the radius of the circle. The locus of the point is

a) a straight line at right angle to the given lineb) a circle concentric with the given circlec) a parabola with its axis parallel to the given lined) a parabola with its axis perpendicular to the given line

80. If CP and CD are semi-conjugate diameters of the ellipse =+=+ 222

2

2

2 1 CDCPthen

by

ax

a) a + b b) 22 ba + c) 22 ba − d) 22 ba +

81. If the line y = mx + 2 cuts the parabola 2y = x2 at points ) ,( 11 yx and ) ,( 22 yx ; where, 21 xx < , then value of m

for which dxxmxx

x

−+∫ 2

222

1 is minimum is

a) 2 b) 8/3 c) 3/1 d) None of these

82.13

2 13 6

1

x

x x xI dx

x

+

+ += ∫ is equal to

a) cxx

++

− 61

132

tan623 b) cx

x+−

− 61

132

tan623

c) cxx

++

− 51

132

tan623 d) None of these

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83. ,......! 7

6! 5

4! 3

21 ∞+++= eeex

then find 1 , log)(0

>∫ yxdyyf y

x

a)2

)]([ 2efb)

21

21

ef c)

2)]([ 22ef

d) None of these

84. )( , and x cos (x) g ,)( 2 βαβα <== xxfLet the roots of the equation 0918 22 =+− ππxx . The area

bounded by the curve y = fog (x), the ordinates βα == xx , and the x-axis is

a) )3(61 −π b) )3(

121 −π c) 12

πd) 6

π

85.1 ( ) tan ,cot , 0, .3 2

Let f x MIN x x x π = ∀ ∈ The area bounded by y = f(x) and the x -axis is

a)363

4 π+

In b) 3123

2 π+

In c)3123

4 π+

In d) 363

2 π+

In

86. The tangent at a point P (x,. y) on a curve meets the axes at P1 and P2 such that P divides P1, P2 internally in the ratio2 :1. The equation of the curve is

a) xy = c b) x2y = c c) xy2 = c d) None of these

87. Unit vectors c and , ba are coplanar. A unit vector d is perpendicular to them. If ( ) ( ) ,ˆ31ˆ

31ˆ

61dc kjiba +−=×××

and the angle between a and b is 300, then c is

1) )ˆˆ2ˆ(31 kji +− 2) )ˆˆˆ2(

31 kji +− 3) )ˆˆ2ˆ2(

31 kji +−− 4) )ˆ2ˆ2ˆ(

31 kji −+−

Which of the given statement(s) match with the main statement.a) 1, 4 b) 1, 2 c) 1, 3 d) None of these

88. Consider a tetrahedron with faces . , , , 4321 FFFF Let 4321 , , , VVVV be the vectors whose magnitudes are

respectively equal to areas of 4,32 1, ,F FFF and whose directions are perpendicular to their faces in outward

direction. Then 4321 VVVV +++ equals

a) 1 b) 4 c) 0 d) None of these

89. OA, OB, OC are the sides of a rectangular parallelopiped whose diagonal are OO’, AA’, BB’ and CC’. D is the centreof the rectangel AC’O’B’ and D’ is the centre of the rectangle O’A’CB’. If the sides OA, OB, OC are in the ratio1 : 2 : 3, the angle DOD’ is equal to

a) 69724cos 1−

b) 61922cos 1−

c) 69724sin 1−

d) 69122sin 1−

90. The least value of 100 a2 log a-log 0.0001, 1a >

a) 2 b) 3 c) 4 d) None of these

91. If α is an imaginary fifth root of unity, then =−+++α

ααα 11log 322

a) 1 b) 0 c) 2 d) -1

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92. Which of the following statements is ture?

i) The amplitude of the product of the two complex numbers is equal to product of their amplitudes

ii) For any polynomial f(x) with real coefficients imaginary rosts always occur in conjugate pairs

iii) Order relation exists in complex numbers whereas it does not exist in real numbers.

iv) The values of ω used as a cube root of unity and as a fourth root of unity are different

Which of the given statement(s) match with the main statement.

a) i and ii only b) ii and iv only c) iii and iv only d) i, ii, and iv only

93. In the expansion of (1 + x)n ( 1+y)n (1+ z)n, the sum of the coefficients of the terms of degree r is

a) ( )3rn C b) 3.n

rC c) rn C3 d) r

n C3

94. If yxxx =∞+++ ......logloglog 41

1021

1010 and 'log720

)13....(1074)12(....531

10 xyy =

++++−++++

then x =

a) 510 b) 410 c) 310 d) 210

95. The solution set of the inequality 21

||1||1 ≥

+−

xx

is

a) ]1,1[−∈x b) )1,1(−∈x c) )1,0(∈x d) none of these


Suppose that OX be a fixed straight line drawn through a fixed point O. The position of any point P can be expresseduniquely by its distance OP from O and the angle that the radius vector. OP forms with line OX. If OP=r, XOP∠ = θthen we say (r, θ ) are polar coordinates of the point P. O is called pole & OX is the initial line. If cartesian coordinates

of P be (x, y) as shown in figure then we have cos , sinx r y r= θ = θ

y

y

xr

P(r, )θ

θ0

We know that cartesian equation line is

0 or ( cos ) ( sin ) 0ax by c a r b r c+ + = θ + θ + =

cos sin or cos sinc Ka b a br r

⇒ − = θ + θ = θ + θ

which is equation of line in polar form where a, K, b are constants.

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Normal form of a straight line:Let P be length of ⊥ r from pole upon a straight line, α the angle which this ⊥ r makes with initial line & (r, θ ) thecoordinates of any movable point on the line, then cos( )P r= θ − α represents equation of line in normal form.

xP

P(r, )θ

α0

96. The general equation of a line is 3x+4y=5. Then the polar form is

(a)53sin 4cosr

θ + θ = (b) 3cos 4sin 5θ + θ =

(c)53cos 4sinr

θ + θ = (d)53cos( ) 4sin( )r

π − θ + π − θ =

97. The equation of line 11 to 2cos sinr

θ + θ = & passing thro´ (1, / 2)π is

(a)1cos sinr

θ + θ = (b)1cos sinr

θ − θ = (c)2cos sinr

θ − θ = (d)2cos sinr

− θ + θ =

98. The equation of line to cos sin la br

θ + θ = , passing thro´ pole is

(a)1

sin cos la br

θ − θ = (b) 1Tan ba

−θ = (c) cosr a= θ (d) sinr a= θ

99. If length of ⊥ r from origin to a straight line is 3 units & angle made by it ( ⊥ r) with initial line is 4π

then equation

of the straight line is

(a)2 cos sin

r= θ + θ (b) 3cos

4r π = θ − θ

(c)3 2 cos

r= θ (d) 3 2 cos sin

r= θ + θ

100. The coordinates of foot of ⊥ r drawn from origin upon the line cos sinl a br

= θ + θ are

(a)1, Tan bl

a−

(b)1

2 2, Sinl b

aa b−

+ (c)

1, cos bla

− (d)

12 2

, Tanl baa b

− +

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Questions 101 to 105 are based on the following passageThe concentration of metallic ions in solution may be determined by photometric titration, which usesspectrophotometric methods to measure the change in absorbance during the titration. The absorbance (A) isdefined by:

bcA ε= Equation

where ε is the molar absorptivity in liters per mole per centimeter, b is the length of the path through the solute incentimeters, and c is the molar concentration. EDTA (ethylenediaminetetraacetic acid) is a nnabsorbing compoundthat is often used in photometric titrations, since it forms complexes with many metallic ions, and each mole of EDTAalways reacts with one mole of metal, regardless of the charge on the cation. The ion Bi3+ reacts with EDTA to forma nonabsorbing complex. However, the concentration of bismuth can be measured by adding to the solution a smallamount of Cu2+, which does combine with EDTA to form an absorbing complex. the reactions are:

∫−+ + )EDTAY(HBiYYBi 4-43 (Reaction 1)

-242 CuYYCu −+ + (Reacton 2)

The stability of each complex is indicated by a characteristic equilibrium constant K, where

23-43 10 ][Y ]Bi[

]BiY[)BiY(K ≈= +

−−

(Equation 2) and

19-42

210

][Y ]Cu[]CuY[)CuY(K ≈= +

−−

(Equation 3)

Figure 1 shows the results of a titration of a bismuth and copper solution with EDTA. Because the K values of thebismuth and copper complexes are sp ;arge. ot can be assumed that the two reactions, which occir consecutively,bothgo to completion.

1 2 3 4 5 6

0.6

A

addedYmL -4

Figure -1

101. How many mL of EDTA must be added befiore the bismuth endpoint is reached?

a) 2 . 5 mL b) 3 . 0 mL c) 4 . 0 mL d) 5 . 0 mL

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102. Why is Cu2+ a good chice to use to locate the Bi3+ endpoint?a) Because the copper complex is less stable than the bismuth complexb) Because the copper complex is more stable than the bismuth complex.c) Because copper and bismuth produce a positive absorbance slope.d) Because the copper complex produces an especially intense color.

103. If the EDTA solution is 0.1 M, approximately how many grams of Br3+ were present in the slution whose titration isshown in Figure 1?

a) 1.9 x 10-1 g b) 2 . 5 x 10-1 g c) 1.9 x 10-2 g d) 6. 2 x 10-2 g

104. Some metal-EDTA complexes can be titrated directly by themselves. Which of the following photometric titrationcurves will be produced when the substance tirated and the titrant absorb light, but the product does not?

a)

titrant added

A

b)

titrant added

A

c)

titrant added

A

d)

titrant added

A

105. What would be the effect of placing the solution in a container that was twice as long?a) There would be no change in the slope of the absorbance curve.b) The slope of the absorbance curve would doublec) The slope of the absorbance curve would decrease by half.d) The slope of the absorbance curve would change in a manner that cannot be determined

Questions 106 to 146 are independent Question

Refer to the hypothetical element E shown below to answer question 106-107CA

Z

106. If E represents any real element, which of the following statements must be true concerning the element E as shownabove?a) The value of A is larger than Z.b) The value of Z is larger than A.c) If E is an ion, the value of Z is negative.d) The value of Z cannot be larger than the value of A.

107. Which of the following is always true of the relationship between A and Z on any given element E ?a) There is more than one possible value for A.b) There is more than one possible value for C.c) There is more than one possible value for Z.d) There is more than one possible value for A + Z.

108. Aspirin is the common name for acetylsalicylic acid. The structure is shown below, what three functional groups arepresent in this molecule?

O

O O

CH3

OHa) ether, aromatic ring, carboxylic acid b) ester, phenul, carboxylic acidc) alkene, aromatic ring, carboxylic acid d) alcohol, ether, carboxylic acid

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109. A sample of hydrazine sulphate (N2H6SO4) was dissolved in 100 ml. of water, 10 ml. of this solution was reacted withexcess of ferric chloride solution and warmed to complete the reaction. Ferrous ion formed was estimated and itrequired 20 ml. of M/50 potassium permanganate solution. Estimate the amount of hydrazine sulphate in one litre ofthe solution.

Reaction : 3 22 4 24 4 4Fe N H N Fe H+ + −+ → + +

2 2 34 25 8 5 4MnO FE H MN FE H O− + + ++ + → + + .

a) .065 g b) .325 g c) 6.5 g d) 3.25 g

110. Consider the reaction X Y Z+ → .

The data below was obtained for the reaction after a kinetic study.

Run [X] [Y] Rate (mole/liter–min.)

1 2.0M 2.0M 1.5

2 6.0M 2.0M 4.5

3 2.0M 6.0M 1.5

Determine the rate expression for the reaction.

a) rate = k[X] [Y] b) rate = k[X]2 c) rate = k [X] d) rate = k [Y]

111. A researcher wishes to have two samples of oxygen gas with the same pressure, volume, and temperature, but withdifferent densitites. Is this possible?

a) Yes, because density is not given in the ideal gas law.

b) Yes, if different isotopes of oxygen are used.

c) No, because once pressure, volume, and temperature of an ideal gas are known, all other parameters aredetermined.

d) No, because density is inversely proportional to volume.

112. A crystal sublimes according to the reaction

2 2(s) (g)X X

Which of the following expressions represents the temperature at which solid crystal is in equilibrium with gaseouscrystal? ( H∆ =change in enthalpy, G∆ =change in free energy, S∆ =entropy change.)

a) equibHTS

∆=∆ b) equib

G HTS

∆ − ∆=∆ c) equib

G HTS

∆ + ∆=∆ d) equib

STH

∆=∆

113. What is the correct IUPAC name for the alkene shown below?

a) 2-metylbutyl-2-butene b) 2-methylbutyl-3-butenec) 3-ethyl-5-methyl-2-heptene d) 3-ethyl-5-methyl-3-heptene

114. Acording to Bohr’s theory, the electronic energy of hydrogen atom in the nth Bohr’s orbit given by19

2

21.76 10 JnEn

−− ×= . Calculate the longest wavelength of light that will be needed to remove an electron from

the third Bohr orbit of the He+ ion.

a) 2176Å b) 3315Å c) 2055Å d) 3017Å

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115. For solids and liquids :2

p vVTC C α− =β ,

pC and vC representing heat capacities at constant pressure (P) and volume (V), respectively. β is a constantcalled the : (Note : T=temperature)

a) coefficient of thermal expansion b) virial coefficient

c) Benedict-Webb-Rubin constant d) isothermal compressibility factor

116. According to the collision model, what three conditions are required for a reaction to occur?a) 1) a molecular collision; 2) sufficient temperature; 3) proper spatial orientation of the moleculesb) 1) a molecular collision; 2) suffieient temperature; 3) sufficient duration of molecular contactc) 1) a molecular collision; 2) sufficient energy of collision; 3) proper spatial orientation of the moleculesd) 1) a molecular collision; 2) sufficient energy of collision; 3) sufficient duration of molecular contact

117. Consider the plot below, which illustrates how Gº∆ varies with the temperature (T). Which of the followingstatements is/are true?

Gº

T1

T2

T3

Temperature0

–

+

I. Gº∆ at 1T is positive

II. 3Gº 0 at T P

T ∂ ∆ = ∆

III. Hº∆ = positive value, at 2T

Which of the given statement(s) match with the main statement.

a) I only b) I and II only c) III only d) I, II and III

118. Which of the following will convert a primary alcohol to an carboxylic acid?

a) LiAlH4 b) dilute cold KMnO4 c) K2CrO4 d) O3 and (CH3)2S

119. A spherical balloon of 21 cm diameter is to be filled up with hydrogen at N.T.P. from a cylinder containing the gas at20 atmospheres at 27ºC. If the cylinder can hold 2.82 litres of water, calculate the number of balloons that can be filledup.

a) 10 b) 8 c) 5 d) 4

120. Once an object falls into a black hole, scientists consider it to have left our universe. That means our universe is:

a) Open b) Closed c) Isolated d) Entropic

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121. The apparatus below is used to measure the electrode potential of zinc. Indicate which of the statement(s) below is/

are true: (Note the following half reactions : 2 2 ;Zn Zn e+→ + 22 2H e H+ −+ → take place in this system)

Pt

V

– Anode–Io

ns +Ions

Zn(1M)

2+ H(1M)

+

1 atmH o2

H2

Cathode+

Zn

Zinc electrode Hydrogen electrode

e–

e–

Salt bridge,KCl-agar

I. The two halves of the cell are connected by the salt bridge. It permits a current to flow and the two solutions tomix.

II. The hydrogen electrode serves as a reference electrode.

III. The reactions occuring at the zinc electrode as current flows are : 2 2Zn Zn e+→ + and 22 2H e H+ −+ → .Which of the given statement(s) match with the main statement.a) I only b) II only c) I and II only d) II and III only

122. Direction : Refer to the structures below to answer the following question.

O OH

Structure A Structure B

What is the relatioship between structure A and B?

a) geometric isomers b) stereoisomers c) tautomers d) There is no relationship

123. At a certain temperature equilibrium constant (Kc) is 16 for the reaction.

2 2 3( ) ( ) ( ) ( )SO g NO g SO g NO g+ +

If we take one mole each of all the four gases in a one litre container, what would be the equilibrium concentrations

of ( )NO g and 2 ( )NO g ?

a) 0.4 moles, 1.6 moles b) 1.6 moles, 0.4 moles

c) 0.6 moles, 2.4 moles d) 2.4 moles, 0.6 moles

124. A heat engine is designed using an ideal gas as a working fluid. At the completion of each cycle, the engine returnsto its original state. If the net flow of heat into the heat engine is 60,000 kJ/cycle, what can be said about the net workdone each cycle? Neglect losses due to friction.

a) It is less than 60,000kJ

b) It is equal to 60,000kJ

c) It is greater than 60,000kJ

d) It begins as 60,000kJ, and decreases on each subsequent cycle

125. An ionic bond is formed between two ions. Which of the following has no effect on the strength of the bond?I doubling the charge on both ionsII doubling the temperatureIII doubling the radii of both ionsWhich of the given statement(s) match with the main statement.a) I only b) II only c) I and II d) I, II and III

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