Applied Mathematics 3 - Dec 2012

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) Find L(te^3t sin t)(5 marks) 1(b) Use the adjoint method to find the inverse of
(5 marks) 1(c) Find p if f(z) = r² cos 2? + ir² sin p? is analytic(5 marks) 1(d) Find the Fourier Series for f(x)=x² in (-1,1) (5 marks) 2(a) Show that u=cosx.coshy is a harmonic function. Find its harmonic conjugate and corresponding analytic function.(8 marks) 2(b) Show that the set of functions cosx, cos2x, cos3x,... form a orthonormal set in the interval (?, -?)(6 marks) 2(c) For a matrix A verify that A(adj.A)=|A|I
(6 marks) 3(a) Find the Laplace Transform of each of the following:
(6 marks) 3(b) Find half-range series for the function:
(6 marks) 3(c) Find non singular matrics P and Q such that PAQ is normal form. Hence find its rank where A is given by
(8 marks) 4(a) Solve the system of equations x-y+2z=9, 2x-5y+3z=18, 6x+7y+10z=35(6 marks) 4(b) Find the inverse Laplace Transform of the following:
(6 marks) 4(c) Expand the function f(x) with the period 'a' into a fourier series f(x)=x² --- 0 ? x ? a(8 marks) 5(a) Using Convolution Theorem, find the inverse Laplace transform of the following:
(8 marks) 5(b) Find the analytic function and its imaginary part if real part is:
(6 marks) 5(c) Find the Fourier series of the function
(6 marks) 6(a) Using Laplace Transformation, solve the following equation:
(D² - 3D + 2)y = 4e^2t, with y(0) = -3 & y'(0) = 5(8 marks) 6(b) Find the Fourier series of the function
f(x) = ?x ... 0 < x < 1
= 0 ... 1 < x < 2(6 marks) 6(c) Determine l, m, n and find A^-1 if A is orthogonal
(6 marks) 7(a) Evaluate the following integral by using Laplace Transform
(6 marks) 7(b) Find the values of f(1), f(i), f'(-1), f''(-i) if:
(8 marks) 7(c) Reduce the following matrix to normal form and find its rank:
(6 marks)