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MU Computer Engineering (Semester 3)
Applied Mathematics 3
December 2011
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 the Z-transform of f(k)=(3k)/k where k?1.
5 M
1(b) Prove that every skew Hermitian Matrix A can be expressed as B+iC where B is real skew symmetric and C is real skew symmetric
5 M
1(c) Find the complex form of Fourier series of f(x) = cosh 2x + sinh 2x in (-5, 5)
5 M
1(d) Show that L{f(t) }=e-as g(s) where
f(t) = g(t-a) ... (t > a)
= 0 ... (t < a)
And hence find L{f(t)} for f(t)=e3t.g(t) where
g(t) = (t-4)2 ... (t > 4)
= 0 ... (t < 4)
5 M

2(a) Find the Fourier series of f(x) where
6 M
2(b) Find all the possible values of k for which rank of A is 1,2,3 where
6 M
2(c) (i) Find L{J0(t)} where

(ii) Find L{(1+te-t)3}
8 M

3(a) Define orthogonal matrix. If A is an orthogonal matrix prove that |A| = +/- 1.
Also find whether A is an orthogonal matrix or not where
6 M
3(b) Find Fourier expansion of f(x) = [(?-x)/2]2 in (0,2?) and hence prove that:
6 M
3(c) Find the Laplace inverse of
(i) cot-1s
(ii) (s+1)e-s/(s2+s+1)
8 M

4(a) Find the Z-inverse transform of z/(z-a) for |z| < a and |z| > a
Given a>0
6 M
4(b) Using Convolution theorem find and verify
6 M
4(c) Find the values of k for which the following equations have a solution:
x + y + z = 1
x + 2y + 3z = k
x + 5y + 9z = k2
Also find the solutions for these values of k.
8 M

5(a) Examine whether the vectors [1,0,2,1], [3,1,2,1], [4,6,2,-4], [-6,0,-3,-4] are linearly independent or dependent.
6 M
5(b) Find the Laplace transformation of
6 M
5(c) Express the function
f(x) = 1 ... (|x| < 1)
= 0 ... (|x| > 1)
as a Fourier integral and hence evaluate:
8 M

6(a) Show that the fourier transform of f(x)=e-x2/2 is given by F(s)=e-s2/2
6 M
6(b) Find Z{f(k)} where
6 M
6(c) Find the fourier series of
f(x) = 0 ... (-? < x < 0)
= sinx ... (0 < x < ?)
Hence deduce that
8 M

7(a) Test for the consistency of the following equations and solve them if possible
x + 2y -z = 1
x + y + 2z = 9
2x + y - z = 2
6 M
7(b) Solve the equation if y=1 at t=0
6 M
7(c) Show that the set of functions:
1, sinx, cosx, sin2x, cos2x,...is orthogonal on (0,2?) but not on (0,?)
8 M

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