## 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) = r^{2} cos 2? + ir^{2} sin p? is analytic(5 marks)
**1(d)** Find the Fourier Series for f(x)=x^{2} 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)=x^{2} --- 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:

(D^{2} - 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)