vmc aits

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VMC/2013 1 Mock IIT Advanced Test-3/Paper-1

Mock IIT Advanced Test - 3/2013/Paper-1

12/05/2013

TEST CODE : ACEG

M.M. : 240

10:00 AM - 01:00 PM TIME : 3.00 Hrs

Read the following Instructions very carefully before you proceed.

1. Blank spaces and blank pages are provided in this booklet for your rough work.

2. Using a black ball point pen, darken the bubbles on the upper original sheet. Apply sufficient pressure so that the impression is created on the bottom sheet.

3. Write your Name, Registration Number and the name of examination centre and sign with pen in the boxes provided on the right part of the ORS. Do not write any of this information anywhere else. Darken the appropriate bubble UNDER each digit of your Registration Number.

4. The question paper consists of 3 parts (Chemistry, Physics and Mathematics). Each part consists of three sections.

5. Section I contains 12 Straight objective type questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

6. Section II contains 4 Multiple type questions. Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE are correct.

7. Section III contains 7 Integer (Subjective) type questions. Each question has an integer answer lying between 0 and 9.

8. For each question in Section I, you will be given 3 Marks if you have darkened only the bubble

corresponding to the correct answer and zero mark if no bubble is darkened. In all other cases,

minus ONE (–1) marks (NEGATIVE MARKING) will be given.

9. For each question in Section II, you will be given 4 Marks if you have darkened only the bubbles corresponding to the correct answers and zero mark if no bubble is darkened. There is NO Negative

Marking.

10. For each question in Section III, you will be given 4 Marks if you have darkened only the bubble corresponding to the correct answer and zero mark if no bubble is darkened. In all other cases,

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PART - I (CHEMISTRY)

80 MARKS

SECTION - I

STRAIGHT OBJECTIVE TYPE

This Section contains 12 Multiple Choice Questions. Each Question has 4 choices A, B, C & D, out of which Only One is Correct :

1. Acetic acid (CH3COOH) can form a dimer (CH3COOH)2 in the gas phase.

The dimer is held together by two H-bonds with a total strength of 66.6 kJ per mol of dimer (under standard conditions). If at 27°C, the equilibrium constant for dimerisation is 103, calculate ∆S° (approximately) for the reaction given below:

3 3 2

2CH COOH(g) _↽⇀(CH COOH) (g)

(A) − 165 J/K (B) +279 J/K (C) −387 J/K (D) 501 J/K

2. A plot of PVm vs P is drawn for three gases as shown below:

Which of the following statement is correct?

(A) Gas 1 has minimum critical temperature and is the easiest to liquify

(B) Gas 1 has maximum critical temperature and is the easiest to liquify

(C) Gas 3 has maximum critical temperature and is the easiest to liquify

(D) Gas 3 has minimum critical temperature and is the easiest to liquify

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3. Three different solutions of oxidising agents K2Cr2O7 (M0 = 294), I2 (M0 = 254) and KMnO4 (M0 = 158)

is titrated separately with 0.19 g of K2S2O3 (M0 = 190). The molarity of each oxidising agent is 0.1 M and

the reactions are:

Cr O2 72− +S O2 23− → Cr3++ SO24− I + S O2 2 23 I S O4 26 −___→ −₊ − MnO + S O4 2 32 MnO + SO2 24 − −_→ − Which of the following statements is(are) correct?

I. All three oxidising agents can act as self-indicators.

II. Volume of I2 used is minimum.

III. Volume of K2Cr2O7 used is maximum.

IV. Weight of KMnO4 used in the titration is maximum.

(A) I, IV (B) I, II, IV (C) I, III, IV (D) I, III

4. Which of the following is true about Castner Kellner Cell ?

I. Na+ ions are discharged instead of H+ ions due to high over voltage

II. Sodium metal discharged at the anode combines with mercury to form sodium amalgam

III. The sodium-mercury amalgam is treated with water to give sodium hydroxide and hydrogen gas

IV. The concentration of the brine solution used remains constant during the whole process The correct choice is :

(A) I, II, III, IV (B) I, II, III (C) I, III (D) II, III

5. 4 HNO + P O₃ ₄ ₁₀ →4 HPO + Product(s), the product(s) is(are) : ₃

(A) NO2, N2O3 and O2 (B) N2O3 and O2

(C) N2O3, O2 and N2O5 (D) Only N2O5

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For Question 6 - 7: 2 4 MeOH MeOH dry HCl H SO (140 C) D -Glucopyranose→ (A) _° → (B) 6. Compound (A) is : (A) (B)

(C) (D) Both (A) and (B)

7. Compound (B) is :

(A) (B)

(C) (D) None of these

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8. Which of the following compounds show optical isomerism?

I. cis-[Co(NH3)4Cl2]+ II. trans-[Co(en)2Cl2]+

III. cis-[Co(en)2Cl2]+ IV. [Co(en)3]3+

The correct choice is :

(A) I and II (B) II and III (C) III and IV (D) I, III and IV

9. Some reactions are given below:

I. FeO + SiO₂ → FeSiO₃

II. FeS + 3O₂ → 2FeO + 2SO₂

III. 2Cu S + 3O₂ ₂ → 2Cu O + 2SO₂ ₂

IV. 2Cu O + Cu SO₂ ₂ ₂ →6Cu + SO₂

The reactions actually taking place during bessemerisation are:

(A) I, II, III, IV (B) I, III (C) I, III, IV (D) III, IV

10. Which of the following is correct :

I. B is O O || || Ph C C Ph II. D is OH | Ph C COOH | Ph   III. F is ₂ O || PhCCH Ph

(A) I, II, III (B) I, II (C) II, III (D) None of these

SPACE FOR ROUGH WORK

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11. The major product of the reaction :

(A) (B) (C) (D) 12. Relate the following compounds :

(A) Identical (B) Enantiomers (C) Diastereomers (D) Different compounds

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SECTION - II

MULTIPLE CORRECT ANSWERS TYPE

This Section contains 4 Multiple Choice Questions. Each Question has 4 choices A, B, C & D, out of which one or More Choices may be Correct:

13. The conductivity (κ) of 10−3 M Na2SO4 solution is 2.6 × 10 −4

Scm−1 and increases to 8 × 10−4 Scm−1 when the solution is saturated with CaSO4. The molar conductivity/conductance values of Na

+

and Ca2+ are 50 Scm2 mol−1 and 120 Scm2 mol−1 respectively and conductivity of water is

2

H O

(κ

)

is 10−5 Scm−1. Neglecting the distance factor answer the following:

(A) Molar conductance of sulphate ions is 150 Scm2 mol−1

(B) Solubility of CaSO4 in presence of 10 −3

M Na2SO4 is 2 × 10 –3

mol / L

(C) Solubility product of CaSO4 is 6 × 10 −6

(D) If distance factor is neglected then molar conductance of a weak electrolyte is independent of dilution

14. Thallium sulphide (Tl2S) is sparingly soluble in water, however whatever amount of sulphide ions get

dissolved, are almost completely hydrolysed to HS− (assume that the further hydrolysis of HS− to H2S is

neglected). The solubility of thallium sulphite (Tl2S) in water (under given conditions) is 10 −6

M. Which of the following of true? (Given : (K )_{2 H S}₂ = 10−14)

(A) pH of solution is 6

(B) Ksp of Tl2S is 4×10−24

(C) Concentration of sulphide ions ([S2−]) is 10−12 M

(D) Tl exists in +1 oxidation state due to inert pair effect

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15. Which of the following is true about Mn2O7 ?

(A) It is a covalent green oil

(B) Each Mn is tetrahedrally surrounded by O's and it has a Mn−O−Mnbridge

(C) It is iso-structural w.r.t Cl2O7

(D) The ability of oxygen to stabilise high oxidation states exceeds that of fluorine which explains that the highest fluoride which exists is MnF4 whereas the highest oxide which exists is Mn2O7

16. Styrene undergoes following reactions in acidic medium:

The various intermediate formed are:

(A) (B) (C) (D)

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SECTION - III

SUBJECTIVE TYPE

This Section contains 7 Subjective Questions. Each question has an integer answer between 0 and 9. Fill the answer bubbles in OMR Sheet appropriately and CAREFULLY. [Please note that an answer ‘5’ should be filled as ‘5’ in the OMR sheet]

17. A straight line was obtained on plotting n

10 10

log dx vs log (a x)

dt

  ₋

 

  with an intercept on log

dx dt

      axis equal to 0.6021. The rate constant for the reaction is ...litren−1mole1− n t −1.

18. The quantum number n, corresponding to the excited state of He+ ion, if on transition from that state to the ground state it emits two photons in succession with wave lengths 121.5 nm and 30.4 nm, is ...

19. The total number of iron atoms present in the Prussian Blue coloured precipitate is _______

20. 20. 20.

20. The freezing point of 0.05 molal NaHSO4 is −0.198°C. The percentage of HSO −

4 ions that transfers a

proton to water is 10x %. The value of x is _____. [Assume 100% ionization of NaHSO4

and

2

1 f H O

(K )

= 1.8 K molality

− ]

21. If 1-bromo-2-butene is treated with Br2/CCl4, how many pairs of diastereomers would result?

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22. Number of amphoteric species out of Na2O, MgO, PbO, Al2O3, SnO2, BeO2, SiO2 ZnO, Al(OH)3, CaO,

SO3 and NaHCO3 is _______ .

23. How many moles of Grignard Reagent (CH3MgBr) will be required to react completely with 1 mole of

the following compound?

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PART - II (PHYSICS)

80 MARKS

SECTION - I

STRAIGHT OBJECTIVE TYPE

24. If an object is placed 20 cm in front of a half thin convex lens of focal

length 10 cm, as shown in figure, then co-ordinates of image taking p

as origin are:

(A) [20 cm, 0.2 cm] (B) [40 cm, 0.4 cm] (C) [40 cm, − 0.2 cm] (D) [20 cm, 0.4 cm] 25. The voltage shown in the figure is applied

to a 0.5 H inductor. The graph of current

in inductor (iL)versus time (t) is represented best in:

(A) (B) (C) (D)

26. A rectangular bar of soap having density 800 kg/m3floats in water of density 1000 kg/m3. Oil of density 300 kg/m3 is slowly added, forming a layer that does not mix with water. When top surface of the oil is at the same level as the top surface of the soap, what is the ratio of the oil layer thickness to the soap’s thickness, x/L?

(A) 2 10 (B) 2 7 (C) 3 10 (D) 3 8

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27. A particle performs simple harmonic oscillations along a straight line with a period T and amplitude a.

Magnitude of the mean velocity of the particle averaged over the time interval during which it travels a distance

2

a

starting from an extreme position is :

(A) a T (B) 2a T (C) 3a T (D) 2 a T

28. A wire carrying a current of 3 A is bent in the form of a parabola y2 = −4 x

as shown in figure, where x and y are in meter. The wire is placed in a uniform magnetic field B = 5ˆk tesla. The force acting on the wire is :

(A) 60ˆi N (B) − ˆi N 60

(C) 30ˆi+ N (D) −30ˆi N

29. Figure shows a system of two blocks of masses 10 kg and 5 kg, connected by ideal strings and pulleys. Here ground is smooth and friction coefficient between the two blocks is

µ

=0 5. . A horizontal force F is applied on lower block as shown. The minimum value of F required to start sliding between the blocks is: (Take g = 10 m/s2)

(A) 12.5 N (B) 25 N

(C) 50 N (D) 100 N

30. A screw gauge has 100 parts on circular scale and one rotation of a circular scale the screw advances a distance of 1 mm. The L.C. of screw gauge is:

(A) 0.1 cm (B) 0.01 cm (C) 0.001 cm (D) 0.02 cm

31. A boy of mass 30 kg starts running from rest along a circular path of radius 6 m with constant tangential acceleration of magnitude 2 m/s2. After 2 sec from start he feels that his shoes started slipping on ground. The friction coefficient between his shoes and ground is: (Take g = 10 m/s2)

(A) 1 2 (B) 1 3 (C) 1 4 (D) 1 5

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32. The current in a circuit varies with time as I = 2 t . Then the rms value of the current for the interval

t = 2 to t = 4 sec is:

(A) 3 A (B) 2 3 A (C) 3 2/ A (D) (4−2 2) A

Paragraph Question 33 - 35

A uniform rod is lying at rest on a frictionless horizontal surface. A particle of same mass as of the rod collides with the rod at its end at an angle θ from the normal as shown in the figure. Assume that there is no friction between the rod and particle and the collision is perfectly elastic.

33. The magnitude of the velocity of the centre of mass of the rod after collision is:

(A) v cos θ (B) v sin θ (C) 2

5v cosθ (D) 2

5v sinθ

34. The magnitude of change in angular momentum of the rod about the point O after collision is:

(A) zero (B) 1 5m v cosℓ θ (C) 1 5m v sinℓ θ (D) 6 5m v sinℓ θ

35. The magnitude of the change in velocity of the particle after collision is:

(A) 8 5v cosθ (B) 2 5v cosθ (C) 8 5v sinθ (D) 2 5v sinθ

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SECTION - II

MULTIPLE CORRECT ANSWERS TYPE

36. The container containing some liquid shown in the diagram is given some small acceleration a. PA and PB represent pressures at A and B (both are fixed in the reference from of vessel, inside the vessel)

respectively, and P0 = PA − PB when the vessel is kept at rest in equilibrium:

(A) If a is directed upwards, P_A−P_B>P₀ (B) If a is directed towards right, P_A−P_B <P₀

(C) If a is directed down wards, P_A−P_B = P0

(D) If a is directed towards left, P_A−P_B= P0

37. A planoconvex lens

(

µ=1 5.

)

of focal length 20 cm has its plane side silvered.

(A) The radius of curvature of its curved surface is half that of a surface of equiconvex lens of focal length 20 cm made of same material

(B) An object placed at 15 cm on the axis on the convex side gives rise to a final image at a distance of 30 cm from it

(C) An object placed at a distance of 20 cm on the axis on the convex side gives rise to an image at 40 cm from it

(D) It acts as a convex mirror

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38. A small block of mass m is fixed at upper end of a massless vertical spring of spring constant K 4mg L

= and natural length 10L. The lower end of spring is free and is at a height L from fixed horizontal floor as shown. The spring is initially unstretched and the spring block system is released from rest in the shown position.

(A) When the speed of block is maximum, the magnitude of force exerted by spring on the block is mg.

(B) The maximum speed attained by the block is 3 2 gL

(C) The maximum compression of the spring is L /8

(D) The time taken by the mass to move from the equilibrium

position to the maximum compression position is 4

L g

π

39. In the figure shown R is a fixed conducting ring of negligible resistance and radius ‘a’. PQ is a uniform rod of resistance r. It is hinged at the centre of the ring and rotated about this point in clockwise direction with a uniform angular velocity ω. There is a uniform magnetic field of strength ‘B’ pointing inwards. ‘r’ is a stationary resistance

(A) Current through ‘r’ is zero.

(B) Current through ‘r’ is 2 2 ω 5 B a r        

(C) Direction of current in external ‘r’ is from centre to circumference

(D) Direction of current in external ‘r’ is from circumference to centre

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SECTION - III

SUBJECTIVE TYPE

40. A thin double convex lens forms a real image of an object on a screen which is fixed. Now the lens is given a constant velocity v₀along its axis and away from the screen. For the purpose of forming the image always on the screen, the object is also required to be given an appropriate velocity. Find the speed of the object (in m/s) at the instant its size is

n

_{times the size of image.} (Taken=1 2 and v₀=4m s)

41. A circuits for the measurement of resistance by potentiometer is shown. The galvanometer is connected at point

A

and zero deflection is observed at

length PJ=30cm. In second case the secondary cell is changed.

Take E_s=10V and r = Ω1 in 1streading and E_s=5Vand r = Ω2 in 2nd

reading. In second case, the zero deflection is observed at length PJ =10cm. What is the resistance R in ohm

(

)

?

42. A square loop of area 2 5 10. × −3m2 and having 100 turns with a total resistance of 10,000 Ω is moved out of a uniform magnetic field of 0.40 T in 1 sec with a constant speed. Then what is the work done, in pulling the loop (in Jµ ).

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43. Figure shows, in cross section, two solid spheres with uniformly distributed charge throughout their volumes. Each has radius R. Point P lies on a line connecting the centres of the spheres, at radial distance R/2 from the center of sphere 1. If the net electric field at point P is zero and Q1 is 8 Cµ , what is Q2 (in Cµ ).

44. Two bodies with a mass of 1kg each is hung on a spring of force constant 100 N/m using an inextensible thread of length 10 cm shown. We burn the thread. If the distance (in cm) between the two bodies when the top body first arrives at its highest position is 10x, then find x? _π2 =10_

45. For hydrogen like atom, determine the number of lines in Paschen series which have a wavelength greater than 1000 nm: [R=1 097. ×107m−1]

46. A coil is connected to an alternating emf of voltage 24 V and of frequency 50 Hz. The reading on the ammeter connected to the coil in series is 10 mA. If a 1 Fµ capacitor is connected to the coil in series the ammeter shows 10 mA again. What would be the approx reading on a dc ammeter (in A) if the coil was connected to a 180 V dc voltage supply? (Take π2 =10)

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PART - III (MATHEMATICS)

80 MARKS

SECTION - I

STRAIGHT OBJECTIVE TYPE

47. O is the vertex, P and Q are end points of focal chord of parabola y2=4x. Circles with OP and OQ as diameter intersect at R. If equation of line OR is 2y = x then the point of intersection of tangents at P and

Q is :

(A)

(

−1 0,

)

(B)

(

− − 1, 1

)

(C) (1, 0) (D)

(

−1 1,

)

48. If A is a square matrix such that :

(

)

4 0 0 0 4 0 0 0 4 A adj A     =      

, then Adj Adj A

(

)

Adj A is equal to : (A) 256 (B) 64 (C) 32 (D) 16 49. 2 0 1 1 2 3 x lim x . . . x →    + + +     __ __  

is equal to, where [.] is greatest integer function :

(A) 1 (B) 3

2 (C)

1

2 (D) 2

50. a, b, c are all different and non-zero real numbers which are in arithmetic progression. If the roots of the quadratic equation ax2+bx+ = are c 0 α and β such that 1 1,α β

α + β + and

2 2

α +β are in geometric progression, then the value of a

c is :

(A) 1 (B) 2 (C) 3 (D) 4

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VMC/2013 19 Mock IIT Advanced Test-3/Paper-1 51. 2 5 2 3 2 4 3 2    ₊     

∫

x x a b x x x x ln a ln b x dx a b a b is equal to : (where a, b R + ∈ ) (A) 2 3 2 3 2 3 1 6 x x x x a b a b ln k e ln a b + (B) 2 3 2 3 2 3 1 1 1 6ln a b a xb x ln e a bx x + k (C) 1_{2 3} ₂1 ₃

(

2 3

)

6 x x x x ln a b k ln a b a b + (D)

(

)

2 3 2 3 2 3 1 1 6 x x x x ln a b k ln a b a b − +

52. A coin is tossed 2n times. The chance that the number of times one gets head is not equal to the number of times one gets tail is :

(A)

( )

2 2 2 ! 1 2 ! n n . n       (B)

( )

2 2 ! 1 ! n n − (C)

( )

2 2 ! 1 1 4 ! n n . n − (D) None of these 53.

( )

(

)

6 5 25 25 3 1 3 2 9 6 7 81 3 7 125 409 log log log log N = +  −     

, then log N has the value : ₂

(A) 0 (B) 1 (C) − 1 (D) None of these

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Paragraph for Q.54 - 55

The general solution of a differential equation of the form :

1 2 1 ₁ 2 ₂ 0 n n n n n n n d y d y d y a a . . . a y dx dx dx − − − −

+ + + + = (ais being constants) is given by the following rules :

(i) If the roots of the corresponding auxiliary equation

1 2

1 2 0

n n n

n D +a D − +a D − +. . .+a =

in D has unequal real roots α α α₁, ₂, ₃, . . .,α_nthen the general solution will be :

1 2

1 2 n

x

x x

n

y=c eα +c eα +. . . c e+ α , where ci s are arbitrary constants.

(ii) If the corresponding auxiliary equation has two equal roots (say, α₁=α₂) and the rest are unequal real roots then the general solution will be :

(

)

1 3

1 2 3

= + x + x+ + nx

n y c x c eα c eα . . . c eα .

(iii) If two roots of the corresponding auxiliary equation are α₁= +α i ,β α₂ = −α iβ and the rest are real and unequal then the general solution will be :

(

)

3

3 n

x x

x

n y= A cosβx+B sin x eβ α +c eα +. . . c e+ α .

54. The general solution of the equation

3 2 3 7 2 16 12 0 d y d y dy y dx dx − dx + − = is : (A) 1 2 2 2 3 3 x x x c e +c e− +c e− (B)

(

1 2

)

2 3 3 x x c x+c e +c e (C)

(

c x₁ +c₂

)

e−3x+c e₃ 2x (D)

(

A cos x+B sin x e

)

2x+c e₃ 3x 55. y=

(

c cos x₁ +c sin x e₂

)

−x+c e₃ x is the general solution of the equation :

(A) 3 2 3 2 2 0 d y d y y dx +dx + = (B) 3 3 0 d y y dx − = (C) 3 2 3 2 2 0 d y d y y dx +dx − = (D) 2 2 0 d y y dx − = SPACE FOR ROUGH WORK

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56. Statement 1 : A triangle ABC right angled at A moves so that its perpendicular sides touch the curve

2 2

2 2 1

x y

a +b = all the time. Then loci of the points A, B and C are circles.

Statement 2 : Locus of point of intersection of two perpendicular tangents to the conic is director circle.

(A) Statement-1 is True, Statement-2 is True and Statement-2 is a correct explanation for Statement-1

(B) Statement-1 is True, Statement-2 is True and Statement-2 is NOT a correct explanation for Statement-1

(C) Statement-1 is True, Statement-2 is False

(D) Statement-1 is False, Statement-2 is True

57. Let f x

( )

=max a, b, c

(

)

, where

(

1

)

2 1 4 2 2 2 n n

c lim cos cos . . . . . cos

n n n n π π π π →∞  −  = _ + + + + _  . Then : Range of f (x) is : (A) [0, 1] (B) 1 1 2,       (C) 1 1 2,       (D) 1 2 2,       Paragraph for Q.58

In a ABC∆ , radii of escribed circles are r1, r2, r3 and inradius of the circle is r.

58. A(z1) ; B(z2) ; C (z3) are vertices of a ΔABC in Argand plane and | z1−α|=|z2 −α| |z= 3−α|= then 8

1 2 3

r +r +r − is : r

(A) 8 (B) 16 (C) 24 (D) 32

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SECTION - II

MULTIPLE CORRECT ANSWERS TYPE

59. A forecast is to be made of the results of five cricket matches, each of which can be a win or a draw or a loss for Indian team. Let,

p = number of forecasts with exactly one error q = number of forecasts with exactly three errors and r = number of forecasts with all five errors

Then the correct statement(s) is(are) :

(A) 2q=5r (B) 8p = q (C) 8p = 5r (D) 2 p

(

+r

)

> q

60. Let a function f satisfy f x f

( ) ( )

′ −x = f

( ) ( )

−x f′ x ∀ ∈x Rand f (0) = 3, which of the following statements is correct ?

(A) The value of f x f

( ) ( )

− for all real x is 9 x (B) The value of f x f

( ) ( )

− for all real is 9x x − (C)

( )

51 51 1 equals 17 3 −

∫

+ dx f x (D)

( )

51 51 1 3 −

∫

+ dx f x equals 34

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61. Which of the following statements is(are) true ?

(A) The number of non zero complex roots of equation z =iz2 is 3

(B) For k = 0, 1, 2, 3…..6 ; zk be the root of (z + 1)7 + z7 = 0 then

6 0 =

∑

k Re(zk) is 7

(C) Ifz1 and z represents adjacent vertices of a polygon of n sides and 1

( )

11 2 1 Im z Re z = − then n is 8 (D) For real

(

,

)

if 3 2 1 2 + ∈ − − i sin i sin θ θ π π

θ is purely real then number of values of θis only one

62. Let P, A, B, C be four collinear points in order, the distances of A, B, C from P being a, b, c respectively.

If the equation :

(

)

2

(

)

0

b−a x + a−c x+ − = has one root double the other then c b (A) B divides AC in the ratio 2 : 1 internally

(B) C divides AB in the ratio 2 : 1 externally

(C) B divides AC in the ratio 1 : 2 internally

(D) None of these

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SECTION - III

SUBJECTIVE TYPE

63. If z ≠0, then

[

]

100 0 x arg | z | dx =

∫

is where [ . ] denotes the greatest integer function :

64. If a+ 2b + 3c = 0 and

(

a × b

) (

+ b × c

) (

+ c×a

)

is equal to λ

(

b×c

)

thenλ is equal to _____. 65. 1 1 1 1 1 − − + →∞ ₌    ₊       

∑

a a a _a _a n a n k k . n k lim n / is equal to _____.

66. If f (x) be a twice differentiable function from R→R such that t f x2

( )

−2tf′

( )

x + f′′

( )

x =0 has two equal values of t for all x and f (0) = 1, f′

( )

0 =2. Find

( )

0 1 2 →  −  −   x  f x t lim x .

67. Let f (x) be a polynomial of degree six divisible by x3, and having a point of extremum at x = 2. If f′

( )

x

is divisible by 1 + x2, find the value of

( )

3 2 4 1

f

f .

68. If r ∈N and z = x + iy, then find number of positive integral solutions of z (i.e. both x and y are positive integers) satisfying the equations | z+8|= 2r and

(

8

)

4 arg z− =π is _______. 69. If 1 2 3 4 5 1 2 3 4 5 10 10 5 4 6 4 n n n n n n r r r r r r n n n n n r r r r r C C C C C C n k C C C C C r k + + + + + + + + + + + + + + + =

+ + + + + . Find the value of k.

(25)