Applied Mathematics 3 - Dec 2011

Information Technology (Semester 3)

TOTAL MARKS: 80
TOTAL TIME: 3 HOURS (1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Assume data if required.
(4) Figures to the right indicate full marks. 1(a) Express the following matrix as the sum of symmetric and skew symmetric matrices where
(5 marks) 1(b) Obtain the Fourier series for the function f(x) = 2x - 1 . . . 0 < x < 3 (5 marks) 1(c) Evaluate the following:
(5 marks) 1(d) If f(z) = u + iv is an analytic function of z = x + iy and u + v = cosx.coshy - sinx.sinhy. Find f(z) in terms of z.(5 marks) 2(a) Find the Laplace transform of the following:

(6 marks) 2(b) Reduce to normal form and find the rank of the matrix:
(6 marks) 2(c) Find the Fourier series of the function
f(x)= x --- 0 ? x ? ?
= 2? - x --- ? ? x ? 2?
Hence deduce that
(8 marks) 3(a) Construct an analytic function f(z) if its real part is:
(6 marks) 3(b) Find adj A, A^-1 and also find B such that:
(6 marks) 3(c) Find inverse Laplace transform of the following:
(8 marks) 4(a) Obtain Taylor's and Laurent's expansion of f(z) indicating regions of convergence

(6 marks) 4(b) Find the half range sine series for the function

(6 marks) 4(c) Find the Laplace transform of the following functions:
(8 marks) 5(a) Evaluate the expression that follows. Take C as (i) |z| = 1 (ii) |z + 1 -i| = 2 (iii) |z + 1 + i| = 2

(6 marks) 5(b) Find non singular matrices P and Q such that PAQ is in the normal form. Hence find the rank of A where:

(6 marks) 5(c) Solve y'' + 2y' + 5y = e^-tsint
where y(0) = 0, y'(0) = 1 (8 marks) 6(a) Evaluate the following along the path (i)y = x (ii) y= x²:
(6 marks) 6(b) Use residue theorem to evaluate
where C is |z| = 3 (6 marks) 6(c) Investigate for what values of a, b the following equations
x + 2y + 3z = 4
x + 3y + 4z = 5
x + 3y + az = b
have (i)no solution (ii)a unique solution (iii) an infinite no. of solutions(8 marks) 7(a) Show that the set S={sinx,sin3x,sin5x,...} is orthogonal over [0, ?/2]. Find the corresponding orthonormal set.(6 marks) 7(b) Find the Fourier series of the function
(6 marks) 7(c) (i) If u, v are harmonic conjugate functions, show that uv is a harmonic function
(ii) Find the Laplace transform
(8 marks)