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MU Information Technology (Semester 3)
Applied Mathematics 3
May 2013
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 L{t e31sint}
5 M
1 (b) Show that every square matrix can be uniquely expressed as the sum of a Hermitian and skew-Hermitian matrix.
5 M
1 (c) Find Z-transform and region of convergence of f(k)=3k, k≥0.
5 M
1 (d) Find the Fourier expansion of f(x)= x2 where -π ≤x ≤π.
5 M

2 (a) Prove that following matrix is orthogonal and hence find its inverse.
A=19[841148474]
6 M
2 (b) Find L1{s+2(s2+4s+5)2}
6 M
2 (c) Obtain the Fourier expansion of f(x)=(πx2)2 in the internal and 0≤x≤2π and f(x+2π)=f(x). Also deduce that,
(i) π26=112+122+132+142+
(ii)π490=114+124+134+144+
8 M

3 (a) Investing for what values of λ and μ the equations.
x+y+z=6
x+2y+3z=10
x+2y+λz=μ have,
(i) No Solution
(ii) a unique solution
(iii) Infinite no. of solutions.
6 M
3 (b) Obtain complex form of Fourier series for f(x)=eax (-π,π) where is not integer.
6 M
3 (c) Solve (D2 - D - 2) y=20 sin 2t with y(0)=1, y'(0)=2.
8 M

4 (a) Find Laplace transform of
f(t)=asinpt 0<tπpf(t)=0πP<t2πPand  f(t)=(t+2πP)
6 M
4 (b) Find the inverse Z-transform for
f(z)=1(z3)(z2)
for 2<|z|<3.
6 M
4 (c) Find inverse Laplace transform of
(i) e43s(s+4)52(ii)tan12s
8 M

5 (a) Examine whether the following vectors are linearly independent or dependent [2,1,1], [1,3,1], [1,2,-1]
6 M
5 (b) Using Convolution theorem prove that
l1[1s ln (s+1s+2)]= t0(e2ueuu)du 
6 M
5 (c) Using Fourier cosine Integral prove that
excosx=1π0w2+2w4+4 coswx dw
8 M

6 (a) Find the Fourier Transform 0+f(x)=e-1x1
6 M
6 (b) Find z[f(x)] where f(k)=cos(kπu+a) where k≥0.
6 M
6 (c) Find Fourier expansion of f(x)=2x - x2 where 0 ≤ x ≤ 3 and period is 3.
8 M

7 (a) Reduce the following matrix to noraml form and find its rank.
A=[1136133453311]
6 M
7 (b) Evalute 0cos6tcos4ttdt
6 M
7 (c) Show that the set of function
sin(πx2L),sin(3πx2L),sin(5πx2L),
is orthogonal over..(0,L) Hence construct corresponding orthonormal set.
8 M



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