mains practice paper

23
PRACTICE TEST – 2 (MAIN) CENTERS: MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW / NASHIK / GOA / PUNE # 1 PHYSICS (PART – A) SECTION I: (SINGLE CHOICE QUESTIONS) This section contains 30 Multiple Choice Questions. Each question has four choices (1), (2), (3) and (4) out of which ONLY ONE is correct. 1. A person sitting in the rear end of the compartment throws a ball towards the front end. The ball follows a parabolic path. The train is moving with velocity of 1 20 ms . A person standing outside on the ground also observes the ball. How will the maximum heights m y attained and the ranges R seen by the thrower and the outside observer compare with each other? (1) same m y different R (2) Same m y and R (3) Different m y same R (4) Different m y and R 2. Two trains are moving with equal speed in opposite directions along two parallel railway tracks, if the wind is blowing with speed u along the track so that the relative velocities of the trains with respect to the wind are in the ratio 1:2, then the speed of each train must be. (1) 3 u (2 ) 2 u (3) 5 u (4) 4 u 3. A person moves along OABC as shown in the figure. What is the magnitude of displacement from the initial position? OA 30 m, AB 20 m and BC 30 2m x' B' x B A C Y' Y O (1) 10 m (2) 20 m (3) 30 m (4) 30 2m SPACE FOR ROUGH WORK

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Page 1: Mains Practice Paper

PRACTICE TEST – 2 (MAIN)

CENTERS: MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW / NASHIK / GOA / PUNE # 1

PHYSICS (PART – A)

SECTION I: (SINGLE CHOICE QUESTIONS)

This section contains 30 Multiple Choice Questions. Each question has four choices (1), (2), (3)

and (4) out of which ONLY ONE is correct.

1. A person sitting in the rear end of the compartment throws a ball towards the front end. The ball

follows a parabolic path. The train is moving with velocity of 120ms . A person standing outside on

the ground also observes the ball. How will the maximum heights my attained and the ranges R

seen by the thrower and the outside observer compare with each other?

(1) same my different R (2) Same my and R

(3) Different my same R (4) Different my and R

2. Two trains are moving with equal speed in opposite directions along two parallel railway tracks, if

the wind is blowing with speed u along the track so that the relative velocities of the trains with

respect to the wind are in the ratio 1:2, then the speed of each train must be.

(1) 3 u (2 ) 2 u (3) 5 u (4) 4 u

3. A person moves along OABC as shown in the figure. What is the magnitude of displacement from

the initial position? OA 30m,AB 20m and BC 30 2 m

x'B' x

BA

C

Y'

Y

O

(1) 10 m (2) 20 m (3) 30 m (4) 30 2 m

SPACE FOR ROUGH WORK

Page 2: Mains Practice Paper

PRACTICE TEST – 2 (MAIN)

CENTERS: MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW / NASHIK / GOA / PUNE # 2

4. Graph below shows the force F in N versus time t in s curve. The impulse of the force in the

interval 0 to 1.2 is nearest to

(1) 16 Ns (2) 8 Ns (3) 7 Ns (4) 5 Ns

5. A ladder is leaned against a smooth wall and it is allowed to slip on a frictionless floor. Which figure

represents path of its centre of mass?

(1) (2)

(3) (4)

6. Figure shows the force versus displacement for a particle in motion. The increase of kinetic energy in

travelling over s 0 to 20 m will be

F in N

15

10

5

0 10 20s in m

(1) 250 J (2) 200 J (3) 150 J (4) 10 J

SPACE FOR ROUGH WORK

Page 3: Mains Practice Paper

PRACTICE TEST – 2 (MAIN)

CENTERS: MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW / NASHIK / GOA / PUNE # 3

7. Two balls of same mass each „m‟ are moving with same velocities v on a smooth surface as shown in

figure. If all colissions between the masses and with the wall are perfectly elastic the possible

number of collisions between the bodies and wall together is

(1) 1 (2) 2 (3) 3 (4) inifnity

8. A thin disc of mass 9M and radius R from which a disc of radius R/3 is cut. Moment of inertia of the

disc about an axis passing through O and perpendicular to the plane of disc is

2R

3

OR

(A) 240MR

9 (2)

237MR

9 (3)

29MR (4) 24MR

9. A roller coaster is designed such that riders experience “ weightlessness” as they go round the top of

a hill whose radius of curvature is 20m. The speed of the car at the top of the hill is between

(1) 15m / s and 16m / s (2) 16m / s and 17m / s

(3) 13m / s and 14 m/s (4) 14 m/s and 15 m/s

10. Given that for a gas iB isothermal bulk modulus, aB adiabatic bulk modulus , p pressure and

ratio of the specific heat at constant pressure to that at constant volume. Which of the following

relation is correct?

(1) aB p (2) iB p (3) i aB B (4) None of these

SPACE FOR ROUGH WORK

Page 4: Mains Practice Paper

PRACTICE TEST – 2 (MAIN)

CENTERS: MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW / NASHIK / GOA / PUNE # 4

11. Two rods of different materials having coefficients of linear expansion 1 and 2 . Young‟s

modulus 1Y and 2Y respectively are fixed between two rigid massive walls. The rods are heated

such that they undergo same increase in temperature. There is no bending of the rods. If 1

2

2

3

, the

thermal stresses developed in the two rods are equal provided 1 2Y / Y

(1) 2

3 (2) 1 (3)

3

2 (4)

4

9

12. Which of the following is true of the following P V diagram?

(1) Temperature is constant 1 2T T

(2) Temperature increases 2 1T T

(3) Temperature decreases 2 1T T

(4) Temperature first increases and then decreases

SPACE FOR ROUGH WORK

Page 5: Mains Practice Paper

PRACTICE TEST – 2 (MAIN)

CENTERS: MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW / NASHIK / GOA / PUNE # 5

13. An ideal gas is expanding such that 2PT constant. The co-efficient of volume expansion of the gas

is

(1) 1

T (2)

2

T (3)

3

T (4)

4

T

14. A wooden block performs SHM on a frictionless surface with frequency 0v . The block carries a

charge Q on its surface. If now a uniform electric field E is switched on as shown, then the SHM

of the block will be

(1) of the same frequency and with shifted mean position

(2) of the same frequency and with same mean position

(3) of changed frequency and with shifted mean position

(4) of changed frequency and with same mean position

15. Two gases of different densities are mixed in proportions 1V and 2V by volume. If 1v and 2v be the

velocities of sound in pure gases respectively then, the velocity of sound in mixture is(Assume

same )

(1) 1 21 2 2 2

1 2

V Vv v v

v v

(2) 1 2

1 2

v vv

V V

(3) 1 21 2 2 2

1 2

v vv v v

V V

(4) 1 2

1 2 2 2

1 1 2 1

V Vv v v

V v V v

SPACE FOR ROUGH WORK

Page 6: Mains Practice Paper

PRACTICE TEST – 2 (MAIN)

CENTERS: MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW / NASHIK / GOA / PUNE # 6

16. The figure below shows spherical distribution of the charge density varies as C

rr

for a r b

where C is a constant. A point charge q lies at the centre of the sphere at r 0 . Which of the

following is the condition for the electric field to have a constant magnitude in the region a r b ?

(1) C 0 (2) C q (3) 2

qC

4 a

(4)

2

qC

2 a

17. Two equal positive point charges q are held at a distance a-apart. A point test charge is located in a

plane that is normal to the line joining these charges and midway between them. What is the radius r

of the circle in this plane for which the force on the test particle has maximum value?

(1) a / 2 (2) a / 2 2 (3) a / 2 (4) 2a

SPACE FOR ROUGH WORK

Page 7: Mains Practice Paper

PRACTICE TEST – 2 (MAIN)

CENTERS: MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW / NASHIK / GOA / PUNE # 7

18. A battery of e.m.f V , resistors 1R and 2R , a capacitor C and switches 1S and 2S are connected in

the circuit as shown in figure below. The capacitor will get fully, charged to V volts when

(1) 1S and 2S are both open (2) 1S and 2S are both closed

(3) 1S is closed and 2S is open (4) 1S is open and 2S is closed

19. In the circuit given here, the points A,B and C are at 70 V, zero and 10 V respectively. Then

(1) currents in the paths AD, DB and DC are in the ratio of 3:2:1

(2) currents in the paths AD,DB and DC are in the ratio of 1:2:3

(3) the point D will be at a potential of 20 V

(4) the point D will be at potential of 60 V

20. In the circuit the battery is assumed to be ideal (no internal resistance) and has emf of 3.0 V. The

resistor that dissipated most power is

(1) 1R (2) 2R (3) 3R (4) 4R

SPACE FOR ROUGH WORK

Page 8: Mains Practice Paper

PRACTICE TEST – 2 (MAIN)

CENTERS: MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW / NASHIK / GOA / PUNE # 8

21. Circular loop of a wire and a long straight wire carry currents cI and eI respectively as shown in

figure. Assuming that these are placed in the same plane, the magnetic field will be zero at the centre

of the loop when separation H is

(1) e

c

I R

I (2) e

c

I R

I (3) c

e

I

I R

(4) e

c

I

I R

22. In the inductive circuit given in the figure, the currents rises after the switch is closed. At instant

when the currents is 15 mA, then potential difference across the inductor will be

(1) zero (2) 240 V (3) 180 V (4) 60 V

23. A luminous object is lying in front of a screen at a distance D from it. What is the maximum focal

length of a converging lens, which can be mounted between the object and the screen to cast the real

image of the object on the screen?

(1) 2D (2) D (3) D/2 (4) D/4

24. A parallel beam of white light falls on a convex lens. Images of blue, yellow and red light are formed

on other side of the lens at a distance 20 cm, 20.5 cm and 21.4 cm respectively. The dispersive power

of the material of the lens will be

(1) 619

1000 (2)

9

200 (3)

14

205 (4)

5

214

SPACE FOR ROUGH WORK

Page 9: Mains Practice Paper

PRACTICE TEST – 2 (MAIN)

CENTERS: MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW / NASHIK / GOA / PUNE # 9

25. A point object is placed at a distance of 20 cm from a thin plano-convex lens of focal length 15 cm.

If the plane surface is silvered. The system will act as

(1) concave mirror (2) complex mirror (3) convex mirror (4) concave lens

26. The fusion of hydrogen into helium is more likely to take place

(1) at high temperature and high pressure (2) at high temperature and low pressure

(3) at low temperature and low pressure (4) at low temperature and high pressure

27. When 228

90 Th transforms to 212

83 Bi , then the number of the emitted and particles is

respectively

(1) 8 ,7 (2) 4 ,7 (3) 4 ,4 (4) 4 ,1

28. In photoelectric emission process from a metal of work function 1.8eV , the kinetic energy of most

energetic electrons is 0.5eV . The corresponding stopping potential is

(1) 1.8eV (2) 1.3V (3) 0.5V (4) 2.3V

SPACE FOR ROUGH WORK

Page 10: Mains Practice Paper

PRACTICE TEST – 2 (MAIN)

CENTERS: MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW / NASHIK / GOA / PUNE # 10

29. The given plot of binding energy per nucleus bE , against the nuclear mass M; A,B,C,D,E,F

correspond to different nuclei. Consider four reactions

(i) A B C (ii) C A B (iii) D E F (iv) F D E

Where is the energy released in the which reaction is positive

(1)(i) and (iv) (2) (i) and (iii) (3) (iii) and (iv) (4) (ii) and (iii)

30. Which of the following is the graph between the frequency (v) of the incident radiations and the

stopping potential?

(1) (2)

(3) (4)

SPACE FOR ROUGH WORK

Page 11: Mains Practice Paper

PRACTICE TEST – 2 (MAIN)

CENTERS: MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW / NASHIK / GOA / PUNE # 11

CHEMISTRY (PART – B)

SECTION I: (SINGLE CHOICE QUESTIONS)

This section contains 30 Multiple Choice Questions. Each question has four choices (1), (2), (3)

and (4) out of which ONLY ONE is correct.

31. The amount of energy required to remove an electron from a 2Li ion from its ground state is how

many times the amount of energy required to remove the electron from a hydrogen atom fro

m its ground state

(1) 9 (2) 2 (3) 3 (4) 5

32. In an 2H ion

(1) One electron is bound to two protons

(2) Two electrons are bound to two protons

(3) Three electrons are bound to two protons

(4) Bond does not exist

33. The shape of 4 4PCl ,PCl and 5AsCl are respectively

(1) square planar, tetrahedral and see-saw

(2) tetrahedral, see-saw and trigonal bipyramidal

(3) tetrahedral, square planar and pentagonal bipyramidal

(4) trigonal bipyramidal , tetrahedral and square pyramidal

34. At constant pressure what would be the percentage decrease in the density of an ideal gas for a 10%

increase in the temperature.

(1) 10% (2) 9.1% (3) 11% (4) 12.09%

35. A cylinder with a movable piston is filled at 25 C with a gas that occupies a volume of 330.5cm . If

the maximum capacity of the cylinder is 345.8cm , what is the highest temperature to which the

cylinder can be heated at constant pressure without having the piston come out?

(1) 50 C (2) 147.5 C (3) 174.5 C (4) 120 C

SPACE FOR ROUGH WORK

Page 12: Mains Practice Paper

PRACTICE TEST – 2 (MAIN)

CENTERS: MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW / NASHIK / GOA / PUNE # 12

36. For the following gaseous equilibria X,Y and Z at 300 K

2 2 3X : 2SO O 2SO

5 3 2Y : PCl PCl Cl

2 2Z: 2HI H I

ratio of pK and cK in the increasing order is

(1) X Y Z (2) X Y Z (3) X Z Y (4) Z Y X

37. A weak acid HX has 5

aK 1.0 10 and its forms a salt NaX when treated with caustic soda. The

percentage hydrolysis of a 0.1M NaX solution is

(1) 1% (2) 0.01% (3) 0.001 % (4) 1%

38. 0.1 millimole of 4CdSO is present in 10 mL acid solution of 0.08NHCl . Now 2H S is passed to

precipitate all the 2Cd

ions. The pH of the solution after filtering off precipitate, boiling off 2H S

and making the solution 100 mL by adding 2H O is approximately

(1) 2 (2) 4 (3) 6 (4) 8

39. When poly atomic gas undergoes an adiabatic process, its temperature and volume are related by the

equation nTV constant, the value of n will be

(1) 1.33 (2) 0.33 (3) 2.33 (4) 1

40. The standard heat of formation values of 6SF g ,S g and F g are : 1100,275 and 80kJ 1mol

respectively. Then the average S F bond energy in 6SF

(1) 1301kJ mol (2)

1320kJ mol (3) 1309kJ mol (4)

1280kJ mol

41. A mixture contains 1 mole of volatile liquid AA P 100mmHg and 3 moles of volatile liquid

BB P 80mmHg . If solution behaves ideally, the total vapour pressure of the distillate is

(1) 85 mm Hg (2) 85.88 mm Hg (3) 90 mm Hg (4) 92 mm Hg

SPACE FOR ROUGH WORK

Page 13: Mains Practice Paper

PRACTICE TEST – 2 (MAIN)

CENTERS: MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW / NASHIK / GOA / PUNE # 13

42. In a galvanic cell, if salt bridge is replaced by inert platinum sheet, the cell will

(1) not work further; as positive and negative charges will accumulate on respective sides

(2) not work further; as platinum performs poorly as salt-bridge

(3) work as previous; as platinum performs will as salt-bridge

(4) produce less potential; as platinum performs poorly as salt-bridge

43. The solubility of hydroxide of alkaline earth metal increases down the group because

(1) the decrease in lattice energy is more predominant than the decrease in hydration energy

(2) the decrease in lattice energy is less predominant than the decrease in hydration energy

(3) the salts become less ionic.

(4) lattice energy increases and hydration energy decreases as we move from 2Be

to 2Ba

44. Which one of the following oxides is neutral?

(1) CO (2) 2SnO (3) ZnO (4) 2SiO

45. The true statement for the acids of phosphorus,

3 2 3 3H PO ,H PO and 3 4H PO is

(1) The order of their acidity is 3 4 3 3 3 2H PO H PO H PO

(2) All of them are reducing in nature

(3) All of them are tribasic acids

(4) The geometry of phosphorus is tetrahedral in all the three.

46. The number of S-S bonds in cyclic trimer of sulphur trioxide 3 9S O is

(1) three (2) two (3) one (4) zero

SPACE FOR ROUGH WORK

Page 14: Mains Practice Paper

PRACTICE TEST – 2 (MAIN)

CENTERS: MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW / NASHIK / GOA / PUNE # 14

47. Which of the given statements in not true for the following reaction?

2 2

2 3 3 24 4Cu H O 4NH Cu NH 4H O

(1) It is a ligand- substitution reaction.

(2) 3NH is a relatively strong field ligand while 2H O is a weak field ligand

(3) During the reaction, there is a change in colour from light blue to dark blue.

(4) 2

3 4Cu NH

has a tetrahedral structure and is paramagnetic.

48. The IUPAC name of the complex 4 7 2 2 2Ni C H O N formed from the reaction of

2Ni with

dimethyl glyoxime is

(1) Bis methylgloxal nickel(II) (2) Bis dimethyloxime nickelate(IV)

(3) Bis 2,3- butanedioldioximato nickel(II) (4) Bis bu tan e 2,3-dionedioximato nickel(II)

49. The actual reducing agent of haematite in blast furnace is

(1) C (2) CO (3) Al (4) 2CO

50. Forth floatation process for the concentration of ores is a practical application of

(1) Adsorption (2) Absorption (3) Coagulation (4) Sedimentation

51.

The value of „N‟ and „n‟ are

(1) 6, 6 (2) 6,4 (3) 4,4 (4) 3,3

SPACE FOR ROUGH WORK

Page 15: Mains Practice Paper

PRACTICE TEST – 2 (MAIN)

CENTERS: MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW / NASHIK / GOA / PUNE # 15

52. During debromination of meso-dibromobutane, the major compound formed is

(1) n- butane (2) 1 butene (3) cis 2 butene (4) trans 2 butene

53. on hydrolysis in presence of aqueous

acetone gives

(1) (K) and (L) (2) (K) only (3) (K) and (M) (4) (M) only

54. Heating a particular ether with HBr yielded a single organic product. Which of the following

conclusions may be reached?

(1) The reactant was a methyl ether (2) The reactant was a symmetrical ether

(3) The reactant was a cyclic ether (4) Both (2) and (3) may be correct.

SPACE FOR ROUGH WORK

Page 16: Mains Practice Paper

PRACTICE TEST – 2 (MAIN)

CENTERS: MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW / NASHIK / GOA / PUNE # 16

55. In the reaction, dil.NaOH

2HO CH CHO how many distinct products(saturated) are possible?

(1) 1 (2) 2 (3) 3 (4) 4

56.

(A) (2)

(3) (4)

SPACE FOR ROUGH WORK

Page 17: Mains Practice Paper

PRACTICE TEST – 2 (MAIN)

CENTERS: MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW / NASHIK / GOA / PUNE # 17

57. Consider the following reaction,

2H /H O

6 5 2 3C H NH CHCl KOH A B C

The compounds (B) and (C) are

(A) 6 5C H COOH and 3NH respectively (2) 6 5 2C H NH and HCOOH respectively

(3) 6 5 2C H NH and 2H O respectively (4) none of these

58. When aniline is treated with benzene diazonium chloride at low temperature in weakly acidic

medium, the final product obtained is

(1)

N=N

2NH

(2)

N=N 2NH

(3)

N=N NH

(4)

N=N

2NH

59. Consider following reagents

(I) 2Br water (II) Tollen‟s reagent and (III) Fehling‟s solution.

Which reagent (s) can be used to make distinction between an aldose and a ketose?

(1) (I), (II) and (III) (2) (II) and (III) (3) (I) only (4) (II) only

60. This structure of glycine (amino acid) is 3 2H NCH CO

O

(zwitter ion)

Select the correct statement of the following

(A) Glycine, as well as other amino acids are amphoteric

(B) The acidic functional group in amino acids is 3N H

(C) The basic functional group in amino acids is

(D) All the statements are correct

SPACE FOR ROUGH WORK

2CO

Page 18: Mains Practice Paper

PRACTICE TEST – 2 (MAIN)

CENTERS: MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW / NASHIK / GOA / PUNE # 18

MATHEMATICS (PART – C)

SECTION I: (SINGLE CHOICE QUESTIONS)

This section contains 30 Multiple Choice Questions. Each question has four choices (1), (2), (3)

and (4) out of which ONLY ONE is correct.

61. Total number of positive integral solutions of 1 2 3x x 3x 15 is equal to…..

(1) 63 (2) 64 (3) 65 (4) None of these

62. If A 1,1,1 and C 0,1 1 are given vectors, then a vectors B satisfying the equation A B C

and A.B 3 is

(1) 5 2 2

, ,3 3 3

(2) 5 2 2

, ,3 3 3

(3) 5 2 2

, ,3 3 3

(4) none of these

63. If f x 0 be a quadratic equation such that f f 0 and 23

f2 4

, then

x

f xlim

sin sin x is equal to

(1) 0 (2) (3) 2 (4) none of these

64.

2 2 2 2 2n 2

1 1 1 1lim ....

n n 1 n 2 n n 1

is

(1) 4

(2)

2

(3) does not exist (4) none of these

65. The value of 3

4

4xsin x dx

where . denotes greatest integer function

(1) 0 (2) 5

sq units4

(3)

7sq units

4

(4)

3sq units

4

SPACE FOR ROUGH WORK

Page 19: Mains Practice Paper

PRACTICE TEST – 2 (MAIN)

CENTERS: MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW / NASHIK / GOA / PUNE # 19

66. The system of equations a b x ay bz 0 , b c x by cz 0 ,

a b y b c z 0 has non-trivial solution, if

(1) a,b,c are in A.P. (2) a,b,c are in G.P.

(3) a,b,c are in H.P. (4) None of these

67. If z 1 z 3 8 , then the range of values of z 4 is

(1) 0,8 (2) 0,8 (3) 1,9 (4) 5,9

68. In the binomial 2

2 27 2 14exp 1 x x ......x 1 x x ...... x , the coefficient of 28x

(1) 220 (2) 222 (3) 224 (4) 226

69. How many real solutions does the system 2 2x y 25 and xy 10 have

(1) 3 (2) 1 (3) 2 (4) none of these

70. If the family of lines x a 2b y a 3b a b passes through the point for all values of a and b,

then the coordinates of the point are

(1) 2,1 (2) 2, 1 (3) 2,1 (4) None of these

71. The diagonals of a parallelogram PQRS are along the lines x 3y 4 and 6x 2y 7 . Then PQRS

must be a

(1) rectangle (2) square

(3) cyclic quadrilateral (4) rhombus

72. S and T are the foci of an ellipse and B is an end of the minor axis. If STB is an equilateral triangle,

the eccentricity of the ellipse is

(1) 1

4 (2)

1

3 (3)

1

2 (4)

2

3

SPACE FOR ROUGH WORK

Page 20: Mains Practice Paper

PRACTICE TEST – 2 (MAIN)

CENTERS: MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW / NASHIK / GOA / PUNE # 20

73. If t t

P 1 ,22 2

be any point on a line, then the values of t for which the point P lies between

parallel lines x 2y 1 and 2x 4y 15 may belong to the interval

(1) 4 2 5 2

,3 6

(2) 2 5

0,6

(3) 4 2 5 2

,3 6

(4) None of these

74. If 5c 3b

log , loga 5c

and a

log3b

are in AP, where a,b,c are in GP, then a,b,c, are the lengths of

sides of

(1) an isosceles triangle (2) an equilateral triangle

(3) a scalene triangle (4) None of the above

75. The integral value of k for which the inequality 2 2x 2 4k 1 x 15k 2k 7 0 holds for all

x R is

(1) 0 (2) 1 (3) 2 (4) 3

76. If 1 2 3 na ,a ,a ,....a are in HP and n

r k

r 1

f k a a

, then

1 2 na a a

, ,......,f 1 f 2 f n

are in

(1) AP (2) GP (3) HP (4) None of the above

77. General solution of 2

a Rsin x cos x min 1,a 4a 6

is

(1) nn

12 4

(2)

n2n 1

4

(3) n 1

n 14

(4)

nn 1

4 4

SPACE FOR ROUGH WORK

Page 21: Mains Practice Paper

PRACTICE TEST – 2 (MAIN)

CENTERS: MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW / NASHIK / GOA / PUNE # 21

78. 31 2 n

0 1 2 n 1

CC C C1 1 1 ..... 1

C C C C

is equal to

(1) n 1

n!

(2)

nn 1

n 1 !

(3)

n

n 1

n!

(4)

n

n 1

n!

79. If

3

sin x,

x 0x

f x 2, x 0

x 0sin x x

x

( where . denotes the greatest integer function) if f x is

continuous at x 0 , then is equal to

(1) 1 (2) 1 (3) 2 (4) 2

80. The set of all values of a for which the function 5a 4f x 1 x 3x log5

1 a

decreases for all

real x, is

(1) , (2) 3 21 3 21

, ,2 2

(3) 5 27

3, 2,2

(4) 3 21

4, 1,2

SPACE FOR ROUGH WORK

Page 22: Mains Practice Paper

PRACTICE TEST – 2 (MAIN)

CENTERS: MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW / NASHIK / GOA / PUNE # 22

81. x

2

e x 1 x ln xdx

x

is equal to

(1) x x ln xe c

x

(2) x x ln x 1

e cx

(3) x

2

x ln xe c

x

(4) x x ln x 1

e cx

82. The solution of the differential equation dy sin y x

dx sin 2y x cos y

is

(1) 2

2 xsin y x sin y c

2 (2)

22 x

sin y x sin y c2

(3) 2

2 xsin y x sin y c

2 (4)

22 x

sin y x sin y c2

83. Let CA denotes the complement of an event A. Let A,B,C be three pair wise dependent events.

Given P A / C =0.2, P B / C 0.4 and P A B / C 0.1 , then C CP A B / C is

(1) 0.5 (2) 0.6 (3) 0.8 (4) 0.4

84. If 1 2 n 11,z ,z ,....., z are the nth roots of unity, then the value of 1 2 n 1

1 1 1.......

3 z 3 z 3 z

is equal

to

(1) n 1

n

n.3 1

3 1 2

(2) n 1

n

n.31

3 1

(3) n 1

n

n.31

3 1

(4) none of these

SPACE FOR ROUGH WORK

Page 23: Mains Practice Paper

PRACTICE TEST – 2 (MAIN)

CENTERS: MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW / NASHIK / GOA / PUNE # 23

85. If 2 2 2a b c 2 ,and

2 2 2

2 2 2

2 2 2

1 a x 1 b x 1 c x

f x 1 a x 1 b x 1 c x

1 a x 1 b x 1 c x

then f x is a polynomial of degree

(1) 0 (2) 1 (3) 2 (4) 3

86. If f x and g x are continuous functions, then

2

1ln

2ln

xf f x f x

4dx

xg g x g x

4

is 0

(1) depend on (2) a non-zero constant

(3) zero (4) None of these

87.

3x1 1 1 1

x x x x

x

p q r slim

4

where p,q, r,s 0 is equal to

(1) pqrs (2) 3

pqrs (3) 3

2pqrs (4) 3

4pqrs

88. If

1000

m

k 1

m 1000x

x k

k limx 10

, then k is m 101

(1) 10 (2) 210 (3)

310 (4) 410

89. The line x y z

k 2 12

makes an isosceles triangle with the planes 2x y 3z 1 0 and

x 2y 3z 1 0 , then value of k is

(1) 3 (2) 2 (3) 5 (4) 0

90. If A and B are two events such that 1

P A2

and 2

P B3

, then which of the following is correct?

(1) 2

P A B3

(2) 1

P A B3

(3) Both (1) & (2) (4) Only (1)

SPACE FOR ROUGH WORK