# JEE Main (AIEEE) 2012 Question Paper

JEE Main (AIEEE) 2012 Question Paper

Physics

1. A wooden wheel of radius R is made of two semicircular parts (see figure). The two parts are held together by a ring made of a metal strip of cross-sectional area s and length L. L is slightly less than 2πR. To fit the ring on the wheel, it is heated so that its temperature rises by ∆T and it just steps over the wheel. As it cools down to surrounding temperature, it presses the semicircular parts together. If the coefficient of linear expansion of the metal is α and its Young’s modulus is Y, the force that one part to the wheel applies on the other part is:

(1)  2πSYα∆T

(2)  SYα∆T

(3)  πSYα∆T

(4)  2SYα∆T

2. The figure shows an experimental plot discharging of a capacitor in an R-C circuit. The time constant τ of this circuit lies between

(1)  150s and 200s

(2)  0 and 50s

(3)  50s and 100s

(4)  100s and 150s

3. In a uniformly charged sphere of total charge Q and radius R, the electric field E is plotted as function of distance from the centre. The graph which would correspond to the above will be

(1)

(2)

(3)

(4)

4. An electromagnetic wave in vacuum has the electric and magnetic fields E and B, which are always perpendicular to each other. The direction polarization is given by X and that of wave propagation by k. Then,

(1)  X || B and k || B × E

(2)  X || E and k || E × B

(3)  X || B and k || E×B

(4)  X || E and k || B×E

5. If a simple pendulum has significant amplitude (upto a factor of 1/e of original) only in the period between t = 0 s to t = τs, then τ may be called the average life of the pendulum. When the spherical bob of the pendulum suffers a retardation (due to viscous drag) proportional to its velocity with b as the constant of proportionality, the average life time of the pendulum is (assuming damping is small) in seconds.

(1)  0.693/b

(2)  b

(3)  1/b

(4)  2/b

6. Hydrogen atom is excited from ground state to another state with principal quantum number equal to 4. Then, the number of spectral lines in the emission spectra will be

(1)  2

(2)  3

(3)  5

(4)  6

7. A coil is suspended in a uniform magnetic filed with the plane of the coil parallel to the magnetic lines of force. When a current is passed through the coil, it starts oscillating; it is very difficult to stop. But if an aluminium plate is placed near to the coil, it stops. This is due to

(1)  development of air current when the plate is placed

(2)  induction of electrical charge on the plate

(3)  shielding of magnetic lines of force as aluminium is a paramagnetic material

(4)  electromagnetic induction in the aluminium plate giving rise to electromagnetic damping

8. The mass of a spaceship is 1000 kg. It is to be launched from the earth’s surface out into free space. The value of g and R (radius of earth) are 10 m/s2 and 6400 km respectively. The required energy for this work will be

(1)  6.4 × 1011 J

(2)  6.4 × 108 J

(3)  6.4 × 109 J

(4)  6.4 × 1010 J

9. Helium gas goes through a cycle ABCDA (consisting of two isochoric and isobaric lines) as shown in figure. Efficiency of this cycle is nearly (Assume the gas to be close to ideal gas)

(1)  15.4%

(2)  9.1%

(3)  10.5%

(4)  12.5%

10. In Young’s double slit experiment, one of the slit is wider than other, so that amplitude of the light from one slit is double of that form other slit. If Im be the maximum intensity, the resultant intensity I when they interfere at phase difference ϕ, is given by

(1)

(2)

(3)

(4)

11. A liquid in a beaker has temperature θ(t) at time t and θ0 is temperature of surroundings, then according to Newton’s law of cooling, the correct graph between loge(θ – θ0) and t is

(1)

(2)

(3)

(4)

12. A particle of mass m is at rest at the origin at time t = 0. It is subjected to a force F(t) = F0e−bt in the x direction. Its speed v(t) is depicted by which of the following curves?

(1)

(2)

(3)

(4)

13. Two electric bulbs marked 25 W-220 V and 100 W-220 V are connected in series to a 440 V supply. Which of the bulbs will fuse?

(1)  Both

(2)  100 W

(3)  25 W

(4)  Neither

14. Resistance of a given wire is obtained by measuring the current flowing in it and the voltage difference applied across it. If the percentage errors in the measurement of the current and the voltage difference are 3% each, then error in the value of resistance of the wire is

(1)  6%

(2)  zero

(3)  1%

(4)  3%

15. A boy can throw a stone up to a maximum height of 10 m. The maximum horizontal distance that the boy can throw the same stone up to will be

(1)  20√2 m

(2)  10 m

(3)  10√2 m

(4)  20 m

16. This question has statement 1 and statement 2. Of the four choices given the statements m choose the one that describes the two statements.

Statement 1: Davisson-Germer experiment established the wave nature of electrons.

Statement 2: If electrons have wave nature, they can interfere and show diffraction.

(1)  Statement 1 is false, Statement 2 is true

(2)  Statement 1 is true, Statement 2 is false

(3)  Statement 1 is true, Statement 2 is true. Statement 2 is the correct explanation for Statement 1

(4)  Statement is true, Statement 2 is true, Statement 2 is not the correct explanation of Statement 1

17. A thin liquid film formed between a U-shaped wire and a light slider supports a weight of 1.5 × 10−2 N (see figure). The length of the slider is 30 cm and its weight negligible. The surface tension of the liquid film is

(1)  0.125 Nm1

(2)  0.1 Nm1

(3)  0.05 Nm1

(4)  0.025 Nm1

18. A charge Q is uniformly distributed over the surface of non-conducting disc of radius R. The disc rotates about an axis perpendicular to its plane and passing through its centre with an angular velocity ω. A a result of this rotation, a magnetic field of induction B is obtained at the centre of the disc. If we keep both the amount of charge placed on the disc and its angular velocity to be constant and vary the radius of the disc then the variation of the magnetic induction at the centre of the disc will be represented by the figure

(1)

(2)

(3)

(4)

19. Truth table for system of four NAND gates as shown in figure is

(1)

(2)

(3)

(4)

20. A radar has a power of 1 kw and is operating at a frequency of 10 GHz. It is located on a mountain top of height 500 m. The maximum distance upto which it can detect object located on the surface of the earth (Radius of earth = 6.4 × 106 m) is

(1)  80 km

(2)  16 km

(3)  40 km

(4)  64 km

21. Assume that a neutron breaks into a proton and an electron. The energy released during this process is (mass of neutron = 1.6725 × 10−27 kg, mass of proton = 1.6725 × 10−27 kg, mass of electron = 9 × 10−31 kg)

(1)  0.73 MeV

(2)  7.10 MeV

(3)  6.30 MeV

(4)  5.4 MeV

22. A Carnot engine, whose efficiency is 40%, take in heat from a source maintained at a temperature of 500 K. It is desired to have an engine of efficiency 60%. Then, the intake temperature for the same exhaust (sink) temperature must be

(1)  Efficiency of Carnot engine cannot be made larger than 50%

(2)  1200 K

(3)  750 K

(4)  600 K

23. This question has statement 1 and statement 2. Of the four choices given after the statements, choose the one that best describes the two statements.

If two springs S1 and S2 of force constants k1 and k2, respectively are stretched by the same force, it is found that more work is done on spring S1 than on spring S2.

Statement 1: If stretched by the same amount, work done on S1, will be more than that on S2.

Statement 2:  k1 < k2

(1)  Statement 1 is false, Statement 2 is true

(2)  Statement 1 is true, Statement 2 is false

(3)  Statement is true, Statement 2 is true, Statement 2 is the correct explanation for Statement 1

(4)  Statement is true, Statement 2 is true, Statement 2 is not the correct explanation of Statement 1

24. Two cars of masses m1 and m2 are moving in circles of radii r1 and r2, respectively. Their speeds are such that they make complete circles in the same time t. the ratio of their centripetal acceleration is

(1)  m1r1 : m2r2

(2)  m1 : m2

(3)  r1 : r2

(4)  1 : 1

25. A cylindrical tube, open at both ends, has a fundamental frequency, f, in air. The tube is dipped vertically in water so that half of it is in water. The fundamental frequency of the air-column is now

(1)  f

(2)  f/2

(3)  3f/4

(4)  2f

26. An object 2.4 m in front of a lens forms a sharp image on a film 12 cm behind the lens. A glass plate 1 cm thick, of refractive index 1.50 is interposed between lens and film with its plane faces parallel to film. At what distance (from lens) should object shifted to be in sharp focus on film?

(1)  7.2 m

(2)  2.4 m

(3)  3.2 m

(4)  5.6 m

27. A diatomic molecule is made of two masses m1 and m2 which are separated by a distance r. If we calculate its rotational energy by applying Bohr’s rule of angular momentum quantization, its energy will be given by ( n is integer)

(1)

(2)

(3)

(4)

28. A spectrometer gives the following reading when used to measure the angle of a prism.

Main scale reading : 58.5 degree

Vernier scale reading : 09 divisions

Given that 1 division on main scale corresponds to 0.5 degree. Total divisions on the vernier scale is 30 and match with 29 divisions of the main scale. The angle of the prism from the above data is

(1)  58.59 degree

(2)  58.77 degree

(3)  58.65 degree

(4)  59 degree

29. This question has statement 1 and statement 2. Of the four choices given after the statements, choose the one that best describes the two statement.

An insulating solid sphere of radius R has a uniform positive charge density ρ. As a result of this uniform charge distribution, there is a finite value of electric potential at the centre of the sphere, at the surface of the sphere and also at point outside the sphere. The electric potential at infinite is zero.

Statement 1: When a charge q is taken from the centre of the surface of the sphere its potential energy changes by qρ/3ε0.

Statement 2: The electric field at a distance r(r < R) from the centre of the sphere is ρr/3ε0.

(1)  Statement 1 is false, Statement 2 is true

(2)  Statement 1 is true, Statement 2 is false

(3)  Statement is true, Statement 2 is true, Statement 2 is the correct explanation for Statement 1

(4)  Statement is true, Statement 2 is true, Statement 2 is not the correct explanation of Statement 1

30. Proton, deuteron and alpha particles of same kinetic energy are moving in circular trajectories in a constant magnetic field. The radii of proton, deuteron and alpha particle are respectively rp, rd and rα. Which one of the following relation is correct?

(1)  rα = rp = rd

(2)  rα = rp < rd

(3)  rα > rd > rp

(4)  rα = rd > rp

Chemistry

31. Which among the following will be named as dibromidobis- (ethylenediamine) chromium (III) bromide?

(1)  [Cr(en)3]Br3

(2)  [Cr(en)2Br2]Br

(3)  [Cr(en)Br4]

(4)  [Cr(en)Br2]Br

32. Which method of purification is represented by the following equation?

(1)  Zone refining

(2)  Cupellation

(3)  Polling

(4)  van-Arkel

33. Lithium forms body-centred cubic structure. The length of the side of its unit cell is 351 pm. Atomic radius of the lithium will be

(1)  75 pm

(2)  300 pm

(3)  240 pm

(4)  152 pm

34. The molecule having smallest bond angle is

(1)  NCl3

(2)  AsCl3

(3)  SbCl3

(4)  PCl3

35. Which of the following compounds can be detected by Molisch’s test?

(1)  Nitro compounds

(2)  Sugars

(3)  Amines

(4)  Primary alcohols

36. The incorrect expression among the following is

(1)

(2)  In isothermal process,

(3)

(4)  K = e−∆G°/RT

37. The density of a solution prepared by dissolving 120 g of urea (mol. mass = 60 u) in 1000 g of water is 1.15 g/mL. The molarity this solution is

(1)  0.50 M

(2)  1.78 M

(3)  1.02 M

(4)  2.05 M

38. The species which can best serve as an initiator for the cationic polymerization is

(1)  LiAlH4

(2)  HNO3

(3)  AlCl3

(4)  BaLi

39. Which of the following on thermal decomposition yields a basic as well as acidic oxide?

(1)  NaNO3

(2)  KClO3

(3)  CaCO3

(4)  NH4NO3

40. The standard reduction potentials for Zn2+/Zn, Ni2+/Ni and Fe2+/Fe are −0.76, −0.23 and −0.44 V respectively. The reaction X + Y2 → X2 + Y will be spontaneous when

(1)  X = Ni, Y = Fe

(2)  X = Ni, Y = Zn

(3)  X = Fe, Y = Zn

(4)  X = Zn, Y = Ni

41. According to Freundlich adsorption isotherm which of the following is correct?

(1)

(2)

(3)

(4)  All of the above are correct for different range of pressure

42. The equilibrium constant (Kc) for the reaction N2(g) + O2(g) → 2NO(g) at temperature T is 4 × 10−4. The value of Kc for the reaction NO(g) → ½ N2(g) + ½ O2(g) at the same temperature is

(1)  0.02

(2)  2.5 × 102

(3)  4 × 104

(4)  50.0

43. The compressibility factor for real gas at high pressure is

(1)  1 + (RT/pb)

(2)  1

(3)  1+(pb/RT)

(4)  1 – (pb/RT)

44. Which one of the following statements is correct?

(1)  All amino acids except lysine are optically active.

(2)  All amino acids are optically active.

(3)  All amino acids except glycine are optically active.

(4)  All amino acids except glutamic acids are optically active.

45. Aspirin is known as

(1)  acetyl salicylic acid

(2)  phenyl salicylate

(3)  acetyl salicylate

(4)  methyl salicylic acid

46. Ortho-nitrophenol is less soluble in water than p- and m-nitrophenols because

(1)  o-nitrophenol is more volatile steam than those of m- and p-isomers

(2)  o-nitrophenol shows intramolecular H-bonding

(3)  o-nitrophenol shows intermolecular H-bonding

(4)  melting point of o-nitrophenol is lower than those of m- and p –isomers.

47. How many chiral compounds are possible on monochlorination of 2-methyl butane?

(1)  8

(2)  2

(3)  4

(4)  6

48. Very pure hydrogen (99.9) can be made by which of the following processes?

(1)  Reaction of methane with steam

(2)  Mixing natural hydrocarbons of high molecular weight

(3)  Electrolysis of water

(4)  Reaction of slats like hydrides with water

49. The electrons identified by quantum numbers n and l

(1) n = 4, l = 1           (2) n = 4, l = 0

(3) n = 3, l = 2           (4)  n = 3, l = 1

(1)  (3) < (4) < (2) < (1)

(2)  (4) < (2) < (3) < (1)

(3)  (2) < (4) < (1) < (3)

(4)  (1) < (3) < (2) < (4)

50. For a first order reaction (A) → products the concentration of A changes from 0.1 M to 0.025 M in 40 min. The rate of reaction when the concentration of A is 0.01 M is

(1)  1.73 × 105 M/min

(2)  3.47 × 104 M/min

(3)  3.47 × 105 M/min

(4)  1.73 × 104 M/min

51. Iron exhibits +2 and +3 oxidation states. Which of the following statements about iron is incorrect?

(1)  Ferrous oxide is more basic in nature than the ferric oxide

(2)  Ferrous compounds are relatively more ionic than the corresponding ferric compounds

(3)  Ferrous compounds are less volatile than the corresponding ferric compounds

(4)  Ferrous compounds are more easily hydrolyzed than the corresponding ferric compounds

52. The pH of a 0.1 molar solution of the acid HQ is 3. The value of the ionization constant, Ka of the acid is

(1)  3 × 101

(2)  1 × 103

(3)  1 × 105

(4)  1 × 107

53. Which branched chain isomer of the hydrocarbon with molecular mass 72 u gives only one isomer of mono substituted alkyl halide?

(1)  Tertiary butyl chloride

(2)  Neopentane

(3)  Isohexane

(4)  Neohexane

54. Kf for water is 1.86 K kg mol−1. If your automobile radiator holds 1.0 kg of water, how many grams of ethylene glycol (C2H6O2) must you add to get the freezing point of the solution lowered to −2.8℃?

(1)  72 g

(2)  93 g

(3)  39 g

(4)  27 g

55. What is DDT among the following?

(1)  Green house gas

(2)  A fertilizer

56. The increasing order of the ionic radii of the given isoelectronic species is

(1)  Cl, Ca2+, K+, S2

(2)  S2, Cl, Ca2+, K+

(3)  Ca2+, K+, Cl, S2

(4)  K+, S2, Ca2+, Cl

57. 2-hexyne gives trans-2-hexene on treatment with

(1)  Pt/H2

(2)  Li/NH3

(3)  Pd/BaSO4

(4)  LiAlH4

58. Iodoform can be prepared from all except

(1)  ethyl methyl ketone

(2)  isopropyl alcohol

(3)  3-methyl-2-butanone

(4)  isobutyl alcohol

59. In which of the following pairs the two species are not isostructural?

(1)

(2)

(3)  PF5 and BrF5

(4)

60. In the given transformation, which the following is the most appropriate reagent?

(1)

(2)  Zn – Hg/HCl

(3)  Na, Liq. NH3

(4)  NaBH4

Mathematics

61. The equation esin x − e−sin x – 4 = 0 has

(1)  infinite number of real roots

(2)  no real roots

(3)  exactly one real root

(4)  exactly four real roots

62. Let  be two unit vectors. If the vectors  perpendicular to each other, then the angle between  is

(1)  π/6

(2)  π/2

(3)  π/3

(4)  π/4

63. A spherical balloon is filled with 4500π cu m of helium gas. If a leak in the balloon causes the gas to escape at the rate of 72π cu m/min, then the rate (in m/min) at which the radius of the balloon decreases 49 min after the leakage began is

(1)  9/7

(2)  7/9

(3)  2/9

(4)  9/2

64. Statement 1: The sum of the series 1+(1 + 2+ 4) + (4 + 6 + 9) + (9 + 12 +16)+……….+(361 + 380 + 400) is 8000.

Statement 2 :  for any natural number n.

(1)  Statement 1 is false, statement 2 is true.

(2)  Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1.

(3)  Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1.

(4)  Statement 1 is true, statement 2 is false.

65. The negation of the statement “If I become a teacher, then I will open a school”, is

(1)  I will become a teacher and I will not open a school

(2)  Either I will not become a teacher or I will not open a school

(3)  Neither I will become a teacher nor I will open a school

(4)  I will not become a teacher or I will open a school

66. If the integral  then a is equal to

(1)  −1

(2)  −2

(3)  1

(4)  2

67. Statement 1: An equation of a common tangent to the parabola y2 = 16√3x and the ellipse 2x2 + y2 = 4 is y = 2x + 2√3.

Statement 2: If the line  (m≠0) is a common tangent to the parabola y2 = 16√3x and ellipse 2x2 + y2 = 4, then m satisfies m4 + 2m2= 24.

(1)  Statement 1 is false, statement 2 is true.

(2)  Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1.

(3)  Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1.

(4)  Statement 1 is true, statement 2 is false.

68. Let If u1 and u2 are column matrices such that  then u1 + u2 is equal to

(1)

(2)

(3)

(4)

69. If n is a positive integer, then (√3 + 1)2n – (√3 – 1)2n is

(1)  an irrational number

(2)  an odd positive integer

(3)  an even positive integer

(4)  a rational number other than positive integers

70. If 100 times the 100th term of an AP with non-zero common difference equals the 50 times its 50th term, then the 150th term of this AP is

(1)  −150

(2)  150 times its 50th term

(3)  150

(4)  zero

71. In a ∆PQR , if 3 sin P + 4 cos Q = 6 and 4 sin Q + 3 cos P = 1, then the angle R is equal to

(1)  5π/6

(2)  π/6

(3)  π/4

(4)  3π/4

72. An equation of a plane parallel to the plane x – 2y + 2z – 5 = 0 and at a unit distance from the origin is

(1)  x – 2y + 2z – 3 = 0

(2)  x – 2y + 2z + 1 = 0

(3)  x – 2y + 2z – 1 = 0

(4)  x – 2y + 2z + 5 = 0

73. If the line 2x + y = k passes through the point which divides the line segment joining the points (1, 1) and (2, 4) in the ratio 3 : 2, then k equals

(1)  29/5

(2)  5

(3)  6

(4)  11/5

74. Let x1, x2….., xn be n observations and let  be their arithmetic mean and σ2 be the variance.

Statement 1:  Variance of 2x1, 2x2, . . . . , 2xn is 4σ2.

Statement 2:  Arithmetic mean 2x1, 2x2 …….., 2xn is

(1)  Statement 1 is false, statement 2 is true.

(2)  Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1.

(3)  Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1.

(4)  Statement 1 is true, statement 2 is false.

75. The population p(t) at time t of a certain mouse species satisfies the differential equation  If p(0) = 850, then the time at which the population becomes zero is

(1)  2 log 18

(2)  log 9

(3)  1/2 log 18

(4)  log 18

76. Let a, b ∈R be such that the function f given by f(x) = log|x| + bx2 + ax, x ≠ 0 has extreme values x = −1 and x = 2.

Statement 1: f has local maximum at x = −1 and at x = 2

Statement 2: a = ½ and b = −1/4

(1)  Statement 1 is false, statement 2 is true.

(2)  Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1.

(3)  Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1.

(4)  Statement 1 is true, statement 2 is false.

77. The area bounded between the parabolas x2 = y/4 and x2= 9y and the straight line y = 2 is

(1)  20√2

(2)  (10√2)/3

(3)  (20√2)/3

(4)  10√2

78. Assuming the balls to be identical except for difference in colours, the number of ways in which one or more balls can be selected from 10 white, 9 green and 7 black balls is

(1)  880

(2)  629

(3)  630

(4)  879

79. If f : R → R is a function defined by  where [x] denotes the greatest integer function, then f is

(1)  continuous for every real x

(2)  discontinuous only at x = 0

(3)  discontinuous only at non-zero integral vales of x

(4)  continuous only at x = 0

80. If the line  interest, then k is equal to

(1)  −1

(2)  2/9

(3)  9/2

(4)  0

81. Three numbers are chosen at random without replacement from {1, 2, 3, ….8}. The probability that their minimum is 3, given that their maximum is 6, is

(1)  3/8

(2)  1/5

(3)  1/4

(4)  2/5

82. If z ≠ 1 and  is real, then the point represented by the complex number z lies

(1)  either on the real axis or on a circle passing through the origin

(2)  on a circle with centre at the origin

(3)  either on the real axis or on a circle not passing through the origin

(4)  on the imaginary axis

83. Let P and Q be 3 × 3 matrices P ≠ Q. If P3 = Q3 and P2Q = Q2P, then determinant of (P2 + Q2) is equal to

(1)  −2

(2)  1

(3)  0

(4)  −1

84. If  then g(x + π) equals

(1)  g(x)/g(π)

(2)  g(x) + g(π)

(3)  (g(x) − g(π))1/2

(4)  g(x) ∙g(π)

85. The length of the diameter of the circle which touches the x-axis at the point (1, 0) and passes through the point (2, 3) is

(1)  10/3

(2)  3/5

(3)  6/5

(4)  5/3

86. Let X = {1, 2, 3, 4, 5}. The number of different ordered pairs (Y, Z) that can formed such that Y ⊆ X, Z ⊆ X and Y ∩ Z is empty, is

(1)  52

(2)  35

(3)  25

(4)  53

87. An ellipse is drawn by taking a diameter of the circle (x – 1)2 + y2 = 1 as its semi-minor axis and a diameter of the circle x2 + (y – 2)2 = 4 is semi-major axis. If the centre of the ellipse is at the origin and its axes are the coordinate axes, then the equation of the ellipse is

(1)  4x2 + y2 = 4

(2)  x2 + 4y2 = 8

(3)  4x2 + y2 = 8

(4)  x2 + 4y2 = 16

88. Consider the function f(x) = |x – 2| |x – 5|, x ∈

Statement 1: f'(4) = 0

Statement 2: f is continuous in [2, 5], differentiable in (2, 5) and f(2) = f(5).

(1)  Statement 1 is false, statement 2 is true.

(2)  Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1.

(3)  Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1.

(4)  Statement 1 is true, statement 2 is false.

89. A line is drawn through the point (1, 2) to meet the coordinate axes at P and Q such that it forms a ∆PQR, where O is the origin, if the area of the ∆PQR is least, then the slope of the line PQ is

(1)  −1/4

(2)  −4

(3)  −2

(4)  −1/2

90. Let ABCD be a parallelogram such that AB = q, AD = p and ∠BAD be an acute angle. If r is the vector that coincides with the altitude directed from the vertex B to the side AD, then r is given by

(1)

(2)

(3)

(4)