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VMC | JEE Mains-2019 1 12th JAN EVENING JEE Entrance Examination

Joint Entrance Exam | Mains-2019

Paper Code -

12th January 2019 | 3.00 PM – 6.00 PM

PHYSICS , CHMISTRY & MATHEMATICS

Important Instructions: 1. Immediately fill in the particulars on this page of the Test Booklet with only Black Ball Point Pen

provided in the examination hall.

2. The Answer Sheet is kept inside this Test Booklet. When you are directed to open the Test Booklet,

take out the Answer Sheet and fill in the particulars carefully.

3. The test is of 3 hours duration.

4. The Test Booklet consists of 90 questions. The maximum marks are 360.

5. There are three parts in the question paper A, B, C consisting of Physics, Mathematics and Chemistry

having 30 questions in each part of equal weightage. Each question is allotted 4 (four) marks for

correct response.

6. Candidate will be awarded marks as stated above in instruction No. 5 for correct response of each

question. 14

(one-fourth) marks of the total marks allotted to the questions (i.e. 1 mark) will be

deducted for indicating incorrect response of each question. No deduction from the total score will be

made if no response is indicated for an item in the answer sheet.

7. There is only one correct response for each question. Filling up more than one response in any

question will be treated as wrong response and marks for wrong response will be deducted

accordingly as per instruction 6 above.

8. For writing particulars/marking responses on Side-1 and Side-2 of the Answer Sheet use only Black

Ball Point Pen provided in the examination hall.

9. No candidate is allowed to carry any textual material, printed or written, bits of papers, pager, mobile

phone, any electronic device, etc. except the Admit Card inside the examination room/hall.

10. Rough work is to be done on the space provided for this purpose in the Test Booklet only. This space is

given at the bottom of each page and in four pages (Page 20-23) at the end of the booklet.

11. On completion of the test, the candidate must hand over the Answer Sheet to the Invigilator on duty

in the Room/Hall. However, the candidates are allowed to take away this Test Booklet with them.

12. The CODE for this Booklet is B. Make sure that the CODE printed on Side-2 of the Answer Sheet is

same as that on this Booklet. Also tally the serial number of the Test Booklet and Answer Sheet are

the same as that on this booklet. In case of discrepancy, the candidate should immediately report the

matter to the Invigilator for replacement of both the Test Booklet and the Answer Sheet.

13. Do not fold or make any stray mark on the Answer Sheet.

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Joint Entrance Exam/IITJEE-2019

PART-A PHYSICS

1. A resonance tube is old and has jagged end. It is still used in the laboratory to determine velocity of

sound in air. A tuning fork of frequency 512 Hz produces first resonance when the tube is filled with

water to a mark 11 cm below a reference mark, near the open end of the tube. The experiment is

repeated with another form of frequency 256 Hz which produced first resonance when water reaches a

mark 27 cm below the reference mark. The velocity of sound in air, obtained in the experiment, is close

to:

(1) 1335 −ms (2) 1341 −ms (3) 1328 −ms (4) 1322 −ms

2. A parallel plate capacitor with plates of area 21m each, are at a separation of 0.1 m. If the electric field

between the plates is 100 N/C, the magnitude of charge on each plate is:

(2

120 2

Take 8.85 10− = −

C

N m)

(1) 108.85 10− C (2)

109.85 10− C (3) 107.85 10− C (4)

106.85 10− C

3. A 10 m long horizontal wire extends from North East to South West. It is falling with a speed of 5.0

1−ms , at right angles to the horizontal component of the earth’s magnetic field, of 4 20.3 10 /− Wb m .

The value of the induced emf in wire is:

(1) 31.5 10− V (2)

31.1 10− V (3) 30.3 10− V (4)

32.5 10− V

4. To double the covering range of a TV transmittion tower, its height should be multiplied by:

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

5. The moment of inertia of a solid sphere, about an axis parallel to its diameter and at a distance of x

from it, is ' ( ) 'I x . Which one of the graphs represents the variation of ( )I x with x correctly?

(1) (2)

(3) (4)

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6. A soap bubble, blown by a mechanical pump at the mouth of a tube, increases in volume, with time, at

a constant rate. The graph that correctly depicts the time dependence of pressure inside the bubble is

given by

(1) (2)

(3) (4)

7. A galvanometer, whose resistance is 50 ohm, has 25 divisions in it. When a current of 44 10− A A

passes through it, its needle (pointer) deflects by one division. To use this galvanometer as a voltmeter

of range 2.5 V, it should be connected to a resistance of:

(1) 6200 ohm (2) 200 ohm (3) 250 ohm (4) 6250 ohm

8. In the circuit shown, find C if the effective capacitance of the whole circuit is to be 0.5F . All values

in the circuit are in F .

(1) 7

11F (2)

6

5F (3) 4F (4)

7

10F

9. Formation of real image using a biconvex lens is shown below:

If the whole set up is immersed in water without disturbing the object and the screen positions, what

will one observe on the screen?

(1) No change (2) Image disappears

(3) Magnified image (4) Erect real image

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10. In the above circuit, 2

3, 20 ,

2= = C F R

3

10=L H and

1 10= R . Current in L- 1R path is 1I and in C- 2R path it is 2I .

The voltage of A.C. source is given by, ( )200 2 sin 100=V t volts.

The phase difference between 1I and 2I is:

(1) 90° (2) 60°

(3) 30° (4) 0°

11. A particle of mass 20 g is released with an initial

velocity 5 m/s along the curve from the point A, as

shown in the figure. The point A is at height h from

point B. The particle slide along the frictionless surface.

When the particle reaches point B, its angular

momentum about O will be: (Take 210 /=g m s )

(1) 26 /−kg m s (2) 23 /−kg m s

(3) 28 /−kg m s (4) 22 /−kg m s

12. The charge on a capacitor plate in a circuit, as a function of time, is shown in the figure:

What is the value of current at 4=t s ?

(1) 1.5A (2) zero (3) 3A (4) 2A

13. A vertical closed cylinder is separated into two parts by a frictionless piston of mass m and of

negligible thickness. The piston is free to move along the length of the cylinder. The length of the

cylinder above the piston is 1l , and that below the piston is 2l , such that 1 2l l . Each part of the

cylinder contains n moles of an ideal gas at equal temperature T. If the piston is stationary, its mass, m,

will be given by: (R is universal gas constant and g is the acceleration due to gravity)

(1) 1 2

1 2

2 +

RT l l

g l l (2)

2 1

1 1 +

nRT

g l l (3) 1 2

2 1

3 −

RT l l

ng l l(4) 1 2

2 1

−

nRT l l

g l l

14. Two satellites, A and B, have masses m and 2m respectively. A is in a circular orbit of radius R, and B is

in a circular orbit of radius 2R around the earth. The ratio of their kinetic energies, /A BT T , is:

(1) 1

2 (2) 1 (3)

1

2 (4) 2

15. An ideal gas is enclosed in a cylinder at pressure of 2 atm and temperature, 300 K. The mean time

between two successive collisions is 86 10− s . If the pressure is doubled and temperature is increased

to 500 K, the mean time between two successive collisions will be close to:

(1) 84 10− s (2)

63 10− s (3) 72 10− s (4)

80.5 10− s

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16. Two particles A, B and moving on two concentric circles of radii

1 2andR R with equal angular speed . At 0=t , their positions

and direction of motion are shown in the figure. The relative

velocity −A Bv v at 2

=

t is given by:

(1) ( )1 2 −R R i (2) ( )1 2 +R R i

(3) ( )1 2− +R R i (4) ( )2 1 −R R i

17. A load of mass M kg is suspended from a steel wire of length 2 m and radius 1.0 mm in Searle’s

apparatus experiment. The increase in length produced in the wire is 4.0 mm. Now the load is fully

immersed in a liquid of relative density 2. The relative density of the material of load is 8. The new

value of increase in length of the steel wire is :

(1) zero (2) 4.0 mm (3) 3.0 mm (4) 5.0 mm

18. An alpha-particle of mass m suffers 1-dimensional elastic collision with a nucleus at rest of unknown

mass. It is scattered directly backwards losing, 64% of its initial kinetic energy. The mass of the

nucleus is:

(1) 3.5 m (2) 2 m (3) 4 m (4) 1.5 m

19. Let , ,l r c and v represent inductance, resistance, capacitance and voltage, respectively. The dimension

of l

rcv in SI units will be:

(1) 2 LT (2) 2−

LA (3) 1−

A (4) LTA

20. A plano-convex lens (focal length 2 ,f refractive index 2, radius of curvature R) fits exactly into a

plano-concave lens (focal length 1f , refractive index 1 , radius of curvature R). Their plane surfaces

are parallel to each other. Then, the focal length of the combination will be:

(1) 1 2+f f (2) 2 1 −

R (3) 1 2−f f (4) 1 2

1 2

2

+

f f

f f

21. A block kept on a rough inclined plane, as shown in the figure,

remains at rest upto a maximum force 2N down the inclined plane.

The maximum external force up the inclined plane that does not

move the block is 10 N. The coefficient of static friction between

the block and the plane is:

[Take 210 /=g m s ]

(1) 1

2 (2)

3

4 (3)

2

3 (4)

3

2

22. When a certain photosensistive surface is illuminated with monochromatic light of frequency v, the

stopping potential for the photo current is 0

2−

V. When the surface is illuminated by monochromatic

light of frequency 2

v, the stopping potential is 0−V . The threshold frequency for photoelectric emission

is:

(1) 3

2

v (2) 2v (3)

5

3

v (4)

4

3

v

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23. In a radioactive decay chain, the initial nucleus is 23290 Th . At the end there are 6− particles and

4−particles which are emitted. If the end nucleus is AZ X , A and Z are given by:

(1) 202; 80= =A Z (2) 200; 81= =A Z

(3) 208; 82= =A Z (4) 208; 80= =A Z

24. A long cylindrical vessel is half filled with a liquid. When the vessel is rotated about its own vertical

axis, the liquid rises up near the wall. If the radius of vessel is 5 cm and its rotational speed is 2

rotations per second, then the difference in the heights between the centre and the sides, in cm, will be:

(1) 2.0 (2) 0.4 (3) 1.2 (4) 0.1

25. A paramagnetic material has 28 310 atoms/m . Its magnetic susceptibility at temperature 350 K is

42.8 10− . Its susceptibility at 300 K is :

(1) 43.726 10− (2)

42.672 10− (3) 43.672 10− (4)

43.267 10−

26. The mean intensity of radiation on the surface of the Sun is about 8 210 /W m . The rms value of the

corresponding magnetic field is closest to:

(1) 410− T (2)

210− T (3) 210 T (4) 1 T

27. In a Frank-Hertz experiment, an electron of energy 5.6 eV passes through mercury vapour and emerges

with an energy 0.7 eV. The minimum wavelength of photons emitted by mercury atoms is close to:

(1) 2020 nm (2) 1700 nm (3) 220 nm (4) 250 nm

28. A simple harmonic motion is represented by: ( )5 sin3 3 cos3 cm= + y t t

The amplitude and time period of the motion are:

(1) 3

10 ,2

cm s (2) 3

5 ,2

cm s (3) 2

10 ,3

cm s (4) 2

5 ,3

cm s

29. In the figure, given that BBV supply can vary

from 0 to 5.0 V, 5 , 200,= =CC dcV V

100 ,= BR k 1= CR k and 1.0 ,=BEV V The

minimum base current and the input voltage

at which the transistor will go to saturation,

will be, respectively:

(1) 25 and 2.8A V (2) 20 and2.8A V

(3) 25 and3.5A V (4) 20 and3.5A V

30. In the given circuit diagram, the currents, 1 0.3 ,= −I A

4 50.8 and 0.4 ,= =I A I A are flowing as shown. The

currents 2 3 6, andI I I , respectively, are :

(1) 0.4 ,0.4 ,1.1− A A A

(2) 0.4 ,1.1 ,0.4A A A

(3) 1.1 , 0.4 ,0.4−A A A

(4) 1.1 ,0.4 ,0.4A A A

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PART-B CHEMISTRY

31. The major product of the following reaction is:

(1) 3 2CH CH C CH (2) 3 2CH CH C CH= =

(3) (4) 3 2 2CH CH CHCH NH=

32. Molecules of benzoic acid 6 5(C H COOH) dimerise in benzene. ‘w’ g of the acid dissolved in 30 g of

benzene shows a depression in freezing point equal to 2K. If the percentage association of the acid to

form dimer in the solution is 80, then w is :

(Given that 1fK 5Kkg mol−= , Molar mass of benzoic acid = 122 g 1mol− )

(1) 2.4g (2) 1.5g (3) 1.0g (4) 1.8g

33. The combination of plots which does not represent isothermal expansion of an ideal gas is:

(1) (2)

(3) (4)

34. An open vessel at 27°C is heated until two fifth of the air(assumed as an ideal gas) in it has escaped

from the vessel. Assuming that the volume of the vessel remains constant, the temperature at which the

vessel has been heated is:

(1) 500°C (2) 750°C (3) 750 K (4) 500 K

35. The correct structure of histidine in a strongly acidic solution ( )pH 2= is:

(1) (2)

(3) (4)

36. m for NaCl, HCl and NaA are 126.4, 425.9 and 100.5 S

2 1cm mol− , respectively. If the conductivity

of 0.001 M HA is 5 15 10 Scm− − , degree of dissociation of HA is:

(1) 0.25 (2) 0.125 (3) 0.50 (4) 0.75

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( )

( )( ) ( )

2

3

2 4

i NaNO /H

ii CrO /H

iii H SO conc. ,

+

+

⎯⎯⎯⎯⎯⎯⎯→

(1) (2)

(3) (4)

38. The magnetic moment of an octahedral homoleptic Mn(II) complex is 5.9 BM. The suitable ligand for

this complex is:

(1) NCS− (2) ethylenediamine

(3) CN− (4) CO

39. The pair that does NOT required calcination is:

(1) 2 3Fe O and 2 3 3Ca CO .MgCO (2) 3ZnCO and CaO

(3) ZnO and 2 3 2Fe O .xH O (4) ZnO and MgO

40. The correct order of atomic radii is:

(1) Eu Ce Ho N (2) Ho N Eu Ce

(3) N Ce Eu Ho (4) Ce Eu Ho N

41. The aldehydes which will not form Grignard product with one equivalent Grignard reagents are:

(A) (B)

(C) (D)

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

42. If the de Broglie wavelength of the electron in nth Bohr orbit in a hydrogenic atom is equal to

1.5 0a ( 0a is Bohr radius), then the value of n / z is:

(1) 1.50 (2) 0.75 (3) 1.0 (4) 0.40

43. For a reaction, consider the plot of lnk versus 1/T given

in the figure. If the rate constant of this reaction at 400

K is 5 110 s ,− −

then the rate constant at 500 K is:

(1) 4 110 s− −

(2) 6 110 s− −

(3) 4 14 10 s− − (4)

4 12 10 s− −

44. The two monomers for the synthesis of Nylon 6, 6 are:

(1) 2 4 2 2 4 2HOOC(CH ) COOH,H N(CH ) NH



(4) ( ) ( )2 2 2 26 6HOOC CH COOH,H N CH NH

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

46. The correct statement(s) among I to III with respect to potassium ions that are abundant within the cell

fluids is/are:

I. They activate many enzymes

II. They participate in the oxidation of glucose to produce ATP

III. Along with sodium ions, they are responsible for the transmission of nerve signals

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

(3) III only (4) I and II only


(A) (B)

(C) (D)

48. Among the following, the false statement is:

(1) It is possible to cause artificial rain by throwing electrified sand carrying charge opposite to the

one on clouds from an aeroplane.

(2) Lyophilic sol can be coagulated by adding an electrolyte

(3) Latex is a colloidal solution of rubber particles which are positively charged

(4) Tyndall effect can be used to distinguish between a colloidal solution and a true solution

49. The element that does NOT show catenation is:

(1) Si (2) Pb (3) Sn (4) Ge

50. The increasing order of the reactivity of the following with 4LiAlH is:

A. B. C. D.

(1) (A) < (B) < (D) < (C) (2) (B) < (A) < (C) < (D)

(3) (A) < (B) < (C) < (D) (4) (B) < (A) < (D) < (C)

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51. The major product in the following conversion is:

(1) (2)

(3) (4)

52. If spK of 2 3Ag CO is 128 10− , the molar solubility of 2 3Ag CO in 0.1 M 3AgNO is:

(1) 108 10 M− (2) 128 10 M− (3) 138 10 M− (4) 118 10 M−

53. The element that shows greater ability to form p p− multiple bonds, is:

(1) Si (2) C (3) Ge (4) Sn

54. The compound that is NOT a common component of photochemical smog is:

(1) 3O (2)

(3) 2 2CF Cl (4) 2CH CHCHO=

55. The volume strength of 1 M 2 2H O is : (Molar mass of 2 2H O = 134gmol− )

(1) 16.8 (2) 11.35 (3) 22.4 (4) 5.6

56. 8 g of NaOH is dissolved in 18 g of 2H O . Mole fraction of NaOH in solution and molality (in mol

1kg− ) of the solution respectively are:

(1) 0.2,11.11 (2) 0.167,11.11 (3) 0.2, 22.20 (4) 0.167, 22.20

57. Given:

(i) ( ) ( ) ( )2 2C graphite O g CO g ;+ → 1rH xkJmol− =

(ii) ( ) ( ) ( )2 2

1C graphite O g CO g ;

2+ → 1rH ykJmol− =

(iii) ( ) ( ) ( )2 2

1CO g O g CO g ;

2+ → 1rH zkJmol− =

Based on the above thermochemical equations, find out which one of the following algebraic

relationships is correct?

(1) z x y= + (2) y 2z x= − (3) x y z= − (4) x y z= +

58. Chlorine on reaction with hot and concentrated sodium hydroxide gives:

(1) Cl− and 3ClO− (2) 3 2ClO and ClO− −

(3) Cl and ClO− − (4) 2Cl and ClO− −

59. The upper stratosphere consisting of the ozone layer protects us from the sun’s radiation that falls in the

wavelength region of:

(1) 200-315 nm (2) 0.8-1.5nm (3) 400-550 nm (4) 600-750 nm

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(1) (2)

(3) (4)

PART-C MATHEMATICS

61. In a class of 60 students, 40 opted for NCC, 30 opted for NSS and 20 opted for both NCC and NSS. If

one of these students is selected at random, then the probability that the student selected has opted

neither for NCC nor for NSS is:

(1) 5

6 (2)

1

3 (3)

1

6 (4)

2

3

62. Let S and S be the foci of an ellipse and B be any one of the extremities of its minor axis. If S BS is

a right angled triangle with right angle at B and area ( ) 8S BS sq.units, = then the length of a latus

rectum of the ellipse is:

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

63. If 4 44 2 4 2 0sin cos sin cos ; , , + + = , then ( ) ( )cos cos + − − is equal to:

(1) –1 (2) 2 (3) 2− (4) 0

64. The integral

( )

13 11

44 2

3 2

2 3 1

x xdx

x x

+

+ + is equal to: (where C is a constant of integration)

(1)

( )

12

34 26 2 3 1

+

+ +

xC

x x

(2)

( )

4

34 22 3 1

xC

x x

+

+ +

(3)

( )

4

34 26 2 3 1

xC

x x

+

+ +

(4)

( )

12

34 22 3 1

xC

x x

+

+ +

65. If the function f given by ( ) ( )3 23 2 3 7f x x a x ax= − − + + , for some a R is increasing in ( 0 1, and

decreasing in )1 5, , then a root of the equation, ( )

( )( )

2

140 1

1

f xx

x

−=

− is:

(1) 5 (2) 6 (3) 7 (4) –7

66. If an angle between the line, 1 2 3

2 1 2

x y z+ − −= =

− and the plane, 2 3x y kz− − = is 1 2 2

3cos−

, then

a value of k is

(1) 5

3 (2)

5

3− (3)

3

5 (4)

3

5−

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67. Let f be differentiable function such that ( )1 2f = and ( ) ( )f x f x = for all x R . If

( ) ( )( )h x f f x ,= then ( )1h is equal to:

(1) 22e (2) 2e (3) 4e (4)

24e

68. The set of all values of for which the system of linear equations

2 2x y z x− − =

2x y z y+ + =

x y z− − =

has a non-trivial solution:

(1) is an empty set (2) contains exactly two elements

(3) is a singleton (4) contains more than two elements

69. The tangent to the curve 2 5 5y x x= − + , parallel to the line 2 4 1y x= + , also passes through the point:

(1) 7 1

2 4,

(2) 1

78

,

−

(3) 1 7

4 2,

(4) 1

78

, −

70. If a circle of radius R passes through the origin O and intersects the coordinates axes at A and B, then

the locus of the foot of perpendicular from O on AB is:

(1) ( )( )2 2 2x y x y R xy+ + = (2) ( )3

2 2 2 2 24x y R x y+ =

(3) ( )2

2 2 2 24x y Rx y+ = (4) ( )2

2 2 2 2 24x y R x y+ =

71. The integral

2

1

e x x

e

x elog xdx

e x

−

is equal to:

(1) 2

3 1 1

2 2e e− − (2)

2

3 1

2 2e

e− −

(3) 2

1 1 1

2 2e e− + − (4)

2

1 1

2e

e− −

72. There are m men and two women participating in a chess tournament. Each participant plays two games

with every other participant. If the number of games played by the men between themselves exceeds the

number of games played between the men and the women by 84, then the value of m is:

(1) 12 (2) 9 (3) 7 (4) 11

73. In a game, a man wins Rs. 100 if he gets 5 or 6 on a throw of a fair die and loses Rs. 50 for getting any

other number on the die. If he decides to throw the die either till he gets a five or a six or to a maximum

of three throws, then his expected gain/loss (in rupees) is:

(1) 400

3gain (2)

400

3loss (3)

400

9loss (4) 0

74. 1

1

2

1x

sin xlim

x

−

−

→

−

− is equal to

(1) (2) 2

(3)

1

2 (4)

2

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75. 2 2 2 2 2 2

1

51 2 3n

n n nlim ......

nn n n→

+ + + +

+ + + is equal to:

(1) 2

(2) ( )1 3tan− (3) ( )1 2tan− (4)

4

76. Let S be the set of all real values of such that a plane passing through the points ( ) ( )2 211 1 1− −, , , , ,

and ( )211 −, , also passes through the point ( )1 11, ,− − . Then S is equal to:

(1) 3 3,− (2) 3 3,− (3) 1 1,− (4) 3

77. If a straight line passing through the point P(–3, 4) is such that its intercepted portion between the

coordinate axes is bisected at P, then its equation is:

(1) 4 3 0x y+ = (2) 4 3 24 0x y− + = (3) 3 4 25 0x y− + = (4) 7 0x y− + =

78. The expression ( )~ ~ p q→ is logically equivalent to:

(1) p ^ q (2) ~ p ^ q (3) p^ ~ q (4) ~ p^ ~ q

79. If 4 5 6n n nC , C and C are in A.P., then n can be:

(1) 14 (2) 11 (3) 9 (4) 12

80. The total number of irrational terms in the binomial expansion of

601 1

5 107 3 −

is:

(1) 54 (2) 48 (3) 55 (4) 49

81. The equation of a tangent to the parabola, 2 8x y= , which makes an angle with the positive direction

of x-axis, is:

(1) 2y xtan cot = + (2) 2x ycot tan = −

(3) 2x ycot tan = + (4) 2y xtan cot = −

82. If

1 1

1

1 1

sin

A sin sin

– sin

= − −

; then for all 3 5

4 4,

, det (A) lies in the interval:

(1) 5

42

,

(2) 3

32

,

(3) 3

02

,

(4) 5

12

,

83. Let 1z and 2z be two complex numbers satisfying 1 9z = and 2 3 4 4z i− − = . Then the minimum

value of 1 2z z− is:

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

84. If a curve passes through the point (1,–2) and has slope of the tangent at any point (x, y) on it as

2 2x y

x

−then the curve also passes through the point:

(1) ( )1 2,− (2) ( )3 0, (3) ( )3 0, (4) ( )2 1,−

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85. The number of integral values of m for which the quadratic expression,

( ) ( ) ( )21 2 2 1 3 4 1m x m x m ,x R+ − + + + , is always positive, is:

(1) 8 (2) 3 (3) 7 (4) 6

86. If the angle of elevation of a cloud from a point P which is 25 m above a lake be 30° and the angle of

depression of reflection of the cloud in the lake from P be 60°, then the height of the cloud (in meters)

from the surface of the lake is:

(1) 45 (2) 42 (3) 50 (4) 60

87. If the sum of the first 15 terms of the series

3 3 3 333 1 1 3

1 2 3 34 2 4 4

.......

+ + + + +

is equal to 225k, then k is equal to:

(1) 9 (2) 27 (3) 54 (4) 108

88. The mean and the variance of five observations are 4 and 5.20, respectively. If three of the observations

are 3, 4 and 4; then the absolute value of the difference of the other two observations, is:

(1) 7 (2) 3 (3) 5 (4) 1

89. Let a,b and c be three unit vectors, out of which vectors b and c are non-parallel. If and are

the angles which vector a -makes with vectors b and c -respectively and ( ) 1

2a b c b = , then −

is equal to:

(1) 90° (2) 30° (3) 45° (4) 60°

90. Let Z be the set of integers. If ( )( )22 5 6

2 1x x x

A x Z :+ − +

= =

and 3 2 1 9B x Z : x= − − , then

the number of subsets of the set A B, is:

(1) 182 (2) 152 (3) 122 (4) 102

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