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MU Information Technology (Semester 3)
Applied Mathematics 3
May 2012
Total marks: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

1(a) Find adj A, A-1 if A is a matrix as given below. Also find B.
5 M
1(b) Find Laplace transform of
5 M
1(c) A regular function of constant magnitude is constant.
5 M
1(d) Find the Fourier series f(x) =1-x2 in (-1,1)
5 M

2(a) Expand f(x) with period 2 into a Fourier series.
6 M
2(b) Find the orthogonal trajectories of the family of curves e-x (x siny - y cosy) = c
7 M
2(c) Using convolution theorem, prove that,
7 M

3(a) Show that every square matrix A can be uniquely expressed as P+iQ where P and Q are Hermitian matrices.
6 M
3(b) Using Cauchy's residue theorem evaluate the following where C is the circle (i) |z| = 1/2 (ii) |z + i| = 3:
7 M
3(c) Solve the following equation by using Laplace transform. Given that y(0) = 1
7 M

4(a) State Laplace equation in polar form and verify it for u = r2cos 2? and also find V and f(z).
6 M
4(b) Find the Fourier expansion for
f(x)= ?(1-cosx) ... 0 < x < 2? and hence show that
7 M
4(c) Evaluate the following:
7 M

5(a) Using residue theorem evaluate:
6 M
5(b) Reduce the following matrix to normal form and find its rank
7 M
5(c) (i) Express the function as Heaviside's unit step function and find their Laplace Transforms
f(t) = 0 ... 0 < t < 1
= t2 ... 1 < t < 3
= 0 ... t > 3

(ii) Find L {f(t)} where
f(t) = t ... 0 < t < 1
= 0 ... 1 < t < 2
7 M

6(a) Investigate for what values of ? and ? the equations:
x + 2y + 3z = 4
x + 3y +4z = 5
x + 3y + ?z = ?
have (i) no solution (ii) a unique solution (iii) an infinite number of solutions.
6 M
6(b) Show that the set of functions sin(2n + 1)x where n=0, 1, 2, ... is orthogonal over [0, ?/2]. Hence construct the orthogonal set of functions.
7 M
6(c) Find all Laurent's expansion of the function f(z)
7 M

7(a) Find L{cost cos2t cos3t}
6 M
7(b) Show that the vectors [1, 0, 2, 1], [3, 1, 2, 1], [4, 6, 2, -4], [-6, 0, -3, -4] are linearly dependent and find the relation between them
7 M
7(c) Obtain half range sine series for f(x) where
7 M

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