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(te3t 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 = 4e2t, 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: