Indian Institute of Technology Guwahati (IIT-G) 2005 JAM Geophysics - Question Paper

expiai-¹

25.    An n - type semiconductor has

(A)    more holes than electrons

(B)    equal number of holes and electrons

(C)    boron as impurity

(D)    phosphorous as impurity

26.    A satellite is moving around earth in a circular orbit of radius R . The time period T of the satellite is

(A)    proportional to R

(B)    proportional to R ²

(C)    proportional to R³²

(D)    independent of R

27.    In a p-n junction diode, the current

(A)    gets saturated for small forward bias voltage

(B)    never gets saturated for forward bias voltage

(C)    is strictly zero for any forward bias voltage

(D)    is strictly zero for any reverse bias voltage

28. The n moles of an ideal gas are in volume V/2 of an isolated chamber of total volume V. The other half of the chamber is empty. Now the valve in the wall separating the two halves is opened and the gas fills the whole volume. The change in the entropy of the gas is

(A) nR lnV

(B)    nRV^r

(C)    nR In 2

(D)    zero

29.    In the Fraunhofer diffraction of light of wavelength X at a slit of width a, the angular positions of different diffraction maxima (other than the central maximum) are given by

m 2

(A)    sin 6 = ; m = 0, 1, 2, 3, ...

a

(B)    sin 0 =    j ; m - 0, 1, 2, 3, ...

m 2

(C)    sin6> = ; 771 = 1, 2, 3, ...

a

(D)    sin0 = jm + ; m = 1,2,3,...

30.    The number of lattice points in a unit cell of a FCC lattice is

(A)    1

(B)    2

(C)    4

(D)    8

Space for rough work

31.    Three unit vectors a, b, c , (b and c not parallel) are such that a x

> > angles which a makes with b and c, respectively, are

(A)    30 and 90

(B)    150 and 90

(C)    60 and 90

(D)    90 and 30

32.    The general solution of differential equation 4x²y" - 8x y' + 9y = 0 is

(A)    C,e⁵*^/2 + C₂e^_3/2

(B)    Cje³*^/2 +C₂e~^3xl2

(C)    (Cj+C₂ \ogx)x^m

(D)    C_lX^3/2 +C₂*-^3/2

33.    The Particular integral of the following differential equation

y" + 2y' + 5y = e^x2 +18 cos Ax -11 sin 4x

4

is

(A) e²+5cos4x

4

(B) 5 cos 4x + 2 sin 4x

(C) e^x2 + 2cos4x + 5sin4x

5

34. U and W are subspaces of vector space V. If DimiV) = 12, Dim{U) = 6 and Dim(W) = 8, then

(A)    Dim{U nW)<6 and DimiU uf )>8

(B)    Dim(U nW)>6 and DimiXJ uW)>8

(C)    DimiU nW)<6 and DimiU uW)< 8

(D)    DimiU r\W)<6 and DimiU uf)> 12

35. For linear transformation T{x₁,x₂,x₃) = (x_x + x₂,x₂ + x₃ ,x₃-x₁), the associated matrix {A;B₁,B₂}, where

B_x = {(1, 2, 1), (-1, 1, 0), (5, -1, 2)} and

B₂ ={(1, 0, 0), (0, 1, 0), (0, 0, 1)}

is

36. The set {e, a, a², a³, 6, ab, a²b, a³&}, where the identity element e = a⁴, is

(A)    a cyclic group of order 8 when b² = a³

(B)    a cyclic group of order less than 8 when b² = a³

(C)    a group but not a cyclic group when b² = a³

(D)    not a group when b² = a³

_A_

37.    Let a,(3,Y be the three roots of the equation e^2x sin2x -7 = 0. Then the root of the equation e^2x sin 2x + 7 = 0 lies between C₁ and C₂, where

(A)    both C₁ and C₂G(a,j3)

(B)    both C₁ and C₂e(fl,y)

(C)    Cj g {a,(3) and C₂ g

(D)    Cj g    and C₂

oo X

38.    The value of the integral J J x e~^x dy dx is

o y=0

39.    Choose the correct answer for the function f{z) = e^y e^lx = u + iv

(A)    f{z) is analytic everywhere in complex plane C

(B)    v is Harmonic conjugate of u

(C)    f(z) is nowhere analytic in complex plane C

(D)    v is Harmonic but u is not Harmonic.

40.    There are three bags containing 2 red & 3 blue; 3 red & 4 blue and 4 red & 4 blue balls, respectively. First a bag is selected at random and then randomly a ball is drawn from the selected bag. The probability that the drawn ball will be red is

(A)    91/210

(B)    93/210

(C)    3/7

41. Let x_lf x₂, , x_n be a random sample from normal population with mean // and variance

x 1 ⁿ 1 ⁿ <j², then the statistic P = ^X , , where x = 'V' ac- and s² =-'V' (* - x )² has

s/Vtz     ti    ⁿ~¹ti

(A)    t distribution with n - 1 degrees of freedom

(B)    t distribution with n degrees of freedom

(C)    standard normal distribution

(D)    X distribution with n - 1 degrees of freedom

42. Let R[a,6] denote the set of all Reimann integrable functions and feR[a,i)]. With following options

P:f²eR[a,b]

Q:afg R [a,b], a scalar

jR :| /*|e R[a,6]

choose the correct answer.

(A)    The options P and Q are correct but not R

(B)    The options P and R are correct but not Q

(C)    The options Q and R are correct but not P

(D)    All the options P, Q and R are correct.

43. lim

(*,y)-(<), 0) X² -y²

(A)    is equal to 0

(B)    is equal to -

2

(C)    is equal to

2

(D)    does not exist

Then the approximate integral value of e is

(A)    1

(B)    2

(C)    -2

(D)    4

45. Let 2 and - 2 be fixed points of function g (x) = 0.4 + x - O.lx² . The sequence {:x_k } is obtained by using the iterative rule x_k+1 =g(x_k). Then with initial approximation x₀ = -1.9 , the sequence {x,}

(A)    converges to 2

(B)    does not converge to - 2

(C)    converges to - 2

(D)    oscillates between - 2 and 2

Space for rough work

GEOLOGY SECTION

Answer the following :

(a)    What is lagoon?

(b)    What is point bar?

(c)    What is terminal moraine?

(d)    What is dendritic pattern?

(e)    What is mesa?

Along a traverse, a sequence 123123 of simple dipping beds is observed. With the help of a block diagram, show how can such a repetition of beds be explained.

(a)    What is pseudomorphism?

(b)    What is isomorphism?

(c)    What is polymorphism?

(d)    What is acicular form?

(e)    What is pleochroism?

A

Answer the following in relation to the magmatic crystallization of a binary system.

(a)    What is liquidus?

(b)    What is solidus?

(c)    What is eutectic point?

(d)    What is solid solution?

(e)    What is graphic texture?

Name one example of each of the following commercial deposits found in India.

(a)    Mica

(b)    Iron

(c)    Copper

(d)    Oil

(e)    Phosphorite

Arrange the geological units/events in order of their relative ages (oldest at the bottom and

youngest at the top) in each of the following :

(a)    Barakar Formation, Talchir Formation and Umaria Marine Beds

(b)    Aravalli Orogeny, Himalayan Orogeny and Satpura Orogeny

(c)    Deccan Volcanics, Malani Volcanics and Dhanjori Volcanics

(d)    Great Boundary Fault, Main Boundary Fault and Himalayan Frontal Fault

(e)    Cuddapah Supergroup, Marwar Supergroup and Semri Group

Answers

(a)    Youngest    ----------------------

Older    ----------------------

Oldest    ----------------------

(b)    Youngest ----------------------

Older    ----------------------

Oldest ----------------------

(c)    Youngest ----------------------

Older    ----------------------

Oldest ----------------------

(d)    Youngest ----------------------

Older    ----------------------

Oldest ..................

(e) Youngest Older Oldest

Mr. Robinson kept 32 grams of a radioactive substance at 10 AM on January 1, 2004 in his laboratory. On the same day at 10 PM, when he went to his laboratory, he found it reduced to 4 grams through decay process. Find the 'half-life of this substance.

A charge Q is uniformly distributed throughout the volume of a solid sphere of radius R. Find the electric field strength E (x) at a point, a distance x away from the centre of the sphere such that (a) 0 < x < R and (b) x > R . Sketch E (x) as a function of x .

54. The n - moles of an ideal gas are in an initial state (P₀, V₀, T₀). The gas is expanded to

. The pressure is then reduced to P₀,

volume 3Vo along a path given by P = P₀ |

Vo

maintaining the volume constant. Finally, the gas undergoes an isobaric compression to the state (P₀,V₀,T₀). (a) Show these processes on a P - V diagram, (b) Calculate the work done by the gas along different paths on the P - V diagram, (c) What is the total work done by the gas during the complete cycle?

A body of mass 9 g is thrown vertically upward with an initial velocity of 100 m/s. After one second, a bullet of mass 1 g hits the body at an angle of 45 with the horizontal with a

velocity of 9oV2 m/s and sticks to it. Calculate the total horizontal distance travelled by the body-bullet system. (Take acceleration due to gravity, g = 10 m/s².)

A solid with FCC structure has lattice constant 4.0 A. Each lattice point has an atom of mass 4.0 x 10~²⁶kg. (a) Calculate the (mass) density of the solid, (b) If each atom of the solid contributes two electrons to the conduction band of the solid, then calculate the number of conduction electrons in 1 m³ volume of the solid.

X-Rays of wavelength A = 0.612 xlO"¹⁰m are scattered by free electrons (Compton collision), (a) Calculate the wavelength of the scattered radiation at an angle of 90 with the direction of incidence, (b) Calculate the corresponding kinetic energy (in Joules) imparted to the electron.

In an experiment on Youngs double-slit with the light of wavelength 600 nm, the screen is placed at a distance of 1 m from the slits having separation of 1 mm. Calculate the value of the fringe-width. Now the screen is moved away farther by 0.5 m. What should be the new wavelength of light so that the fringe-width remains the same?

A radioactive source contains two radioisotopes, each with initial activity 10³ ln 2 Bq (lBq = 1 Becquerel = 1 decay per second). The half-life of one of the radioisotopes is one hour and that of the other is two hours, (a) Calculate the total number of radioactive nuclei present initially in the radioactive source, (b) How many nuclei decay in the period of first four hours?

MATHEMATICS SECTION

60. (a) Find the region of convergence of 'S'--- (z + i)ⁿ

4 (n +1)^7/2

z

(b) Expand f (z) - in Laurant series valid for 0 < I z - 3 I < 1

(z - 2) (3 - z)    ¹¹

61. (a) Solve the differential equation y' + xy = y^1/2e ^x secx .

(b) Calculate the conditional expectation of Y given X = x, for bivariate random variable (X, Y) having joint probability density function

_f( , j² 0 < x < y < 1 f {x,y) = <

|0 elsewhere

62. (a) Is the set S = {(z₁,z₂, - z₂ ,z₁) : z₁ ,z₂ e C } with addition and multiplication defined as

(a,b,c,d) + (e,f,g,h) = (a + e, b + f, c. + g, d + h),

{a,b,c,d)- (e,f,g,h) = (ae + bg, af + bh, ce + dg, cf + dh)

non commutative ring with unity e ? If yes, find e and corresponding inverse of (z_x ,2,-22,) where z₁, z₂ 0 .

(b) Set up an isomorphism between the multiplicative Group X of the 4^th root of unity and the permutation group Y whose elements are I = (1)(2)(3)(4), P₁ = (1, 2, 3, 4), P₂ = (1, 3) (2, 4) and P₃ = (1, 4, 3, 2).

63. (a) Using Caley Hamilton theorem, find the inverse of the matrix A =

(b) For which values of a and J3 the following system of linear equations is consistent? For consistent system find the solution also.

x + 3y + z =3 2x + 3y + 5z =4 4x + 9y + az = j3

64. (a) Using Divergence theorem, evaluate ii- ndS , where F = 4xi - 2y²j + z²k and S

s

is the surface bounded by the region x² + y² = 4, z = 0, z -3.

(b) Show that the series e~^nxxⁿ converges uniformly in the interval [0, 10].

65. (a) Determine the number M and the interval width h , so that the Simpsons 1/3^rd rule

7t/6

for 2M intervals can be used to compute the integral cos x dx with an accuracy of

-njQ

1 O⁷ 7T⁵

5 xlO^-9, where , = (17.186)⁴.

6 x27

(b) Using Guass-Seidal iterative method, solve the following system of linear equations up to 2^nd iteration.

8x + y + z = 8 2x + Ay + z = 4 x + 3y + 5z = 5

66. (a) To test H₀ : p = against the alternative H₁ : p = , for a binomial parameter p,

2    3

2

with n = 5, the critical region is : Reject if₀ i/    < 3, for a sample of size 2.

i=i

Calculate the probabilities of Type I and Type II errors.

(b) Let , x₂, ..., x_n be a simple random sample with replacement from population with

finite mean M and finite variance S². Find the variance of sample mean i ⁿ x =-'Yx_l .

T)

¹¹ = 1