MS – 297

*MS297*

VI Semester B.A./B.Sc. Examination, May/June 2013 (Semester Scheme) Paper – VIII : MATHEMATICS Time : 3 Hours

Max. Marks : 90

Instruction : Answer all questions. I. Answer any fifteen questions.

(15×2=30)

1) Find the locus of the point z, satisfying z  4 – 5 . Ì z 2  1Ü 2) Evaluate lim Í Ý. z ‘1 i z 2  1 Î Þ

3) Write the polar form of the C-R (Cauchy-Riemann) equations. 4) Show that f(z) = cos z is analytic. 5) Show that u = excos y is harmonic. 6) Find the invariant points of the bilinear transformation w

7) Evaluate

1 i

Õ (y  x  3 x

z2 . z3

2

i) dz along the line y = 2x.

0

8) State Fundamental theorem of algebra. 9) Evaluate

Õ c

10) Evaluate

e z dz where c is |z| = 1. z2 1

Õc z2  1 dz where c is |z| = 2.

11) Prove that F[f(at)] =

1ˆ ÉsÙ f Ê Ú. a ËaÛ P.T.O.

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12) Define the Fourier cosine and sine transform of f(x). 13) Find the Fourier transform of f( x )

ˆ Ì 1 sin t (x ! 0 ) Ígiven Õ dt x 0 t ÎÍ

14) Find the Fourier sine transform of f( x )

SÜ Ý. 2 ÞÝ

2

e x .

15) Show that Fs >f …( x )@ D Fc >f (x )@. 16) Using bisection method find a real root of cos x – xex = 0 between 0 and 1 in two steps. 17) Using the method of false-position find a real root of x3 – 5x + 1 = 0 in (0, 1). (Do two steps only). 18) Write the formula for finding a root of f(x) = 0 by Newton-Raphson method. 19) Write the Jacobi’s iteration formula for solving a system of three equations. 20) Write the formula for Euler’s modified method for y1(1) and y1(2 ) to solve dy dx

f( x, y ) with initial conditions x = x0 and y = y0.

II. Answer any four of the following. É z  1 i Ù 1) Show that arg Ê Ú Ë zi Û

(4×5=20)

S represents a circle. 4

2) Prove that u(x, y) and v(x, y) are harmonic conjugates of each other if and only if they are constants. 3) If f(z) = u + iv is an analytic function and u  v

x 2

x  y2

( x œ 0, y œ 0 ) , find f(z).

4) Discuss the transformation w = sinz. 5) Prove that bilinear transformation transforms circles into circles or straight lines. 6) Find the bilinear transformation which maps z = ˆ , i, 0 into w = 0, i, ˆ .

*MS297*

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III. Answer any two of the following.

(2×5=10)

( 2,5 )

Õ

1) Evaluate

(3 x  y)dx  (2 y  x ) dy along the curve y = x2 + 1.

( 0,1)

2) State and prove Cauchy’s integral theorem. 3) Evaluate Õ

z 1

2 c ( z  1) ( z  2 )

dz where C : z  i

2.

4) State and prove Cauchy’s inequality. IV. Answer any three of the following.

(3×5=15)

1) By using Fourier integral formula, show that

f( x )

1 S

ˆ

Õ

0

cos sx  cos s(S  x ) ds , where f( x ) 1  s2

0†x†S Ïsin x, . Ð Ñ 0 , x † 0, x ! S

2) Find the Fourier transform of f( x )

ÏÒx , x † 1 Ð . ÒÑ 0, x ! 1

3) Find the Fourier Sine transform of xe–ax(a > 0). 4) Find the inverse Fourier cosine transform of sin a D . D 5) Prove that Fc [ f ……(x )]



2 f …(0)  D 2 Fc [ f (x )] . S

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V. Answer any three of the following.

(3×5=15)

1) Use bisection method, upto 4 stages, to find a real root of x3 – 4x – 9 = 0 2) Using Newton-Raphson method find the root near 2.9, of the equation x + log10x = 3.375 3) Solve the following system of equations by Gauss-Seidel method 20x + y – 2z = 17 3x + 20y – z = 18

(Do three iterations only)

2x – 3y + 20z = 25 4) Applying power method find the largest eigen value of the matrix

A

Ì4 1 Í0 20 Í ÍÎ0 1

0Ü 1Ý Ý 4 ÝÞ

(Do three iterations only)

5) Using Runge-Kutta method solve

dy dx

1  y 2 with y(0) = 0. Compute y(0.2)

by taking h = 0.2.

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