Applied Mathematics 3 - Dec 2011

Computer Engineering (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 the Z-transform of f(k)=(3^k)/k where k?1.(5 marks) 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 marks) 1(c) Find the complex form of Fourier series of f(x) = cosh 2x + sinh 2x in (-5, 5)(5 marks) 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)=e^3t.g(t) where
g(t) = (t-4)² ... (t > 4)
= 0 ... (t < 4)(5 marks) 2(a) Find the Fourier series of f(x) where
(6 marks) 2(b) Find all the possible values of k for which rank of A is 1,2,3 where
(6 marks) 2(c) (i) Find L{J₀(t)} where

(ii) Find L{(1+te^-t)³}(8 marks) 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 marks) 3(b) Find Fourier expansion of f(x) = [(?-x)/2]² in (0,2?) and hence prove that:
(6 marks) 3(c) Find the Laplace inverse of
(i) cot^-1s
(ii) (s+1)e^-s/(s²+s+1)(8 marks) 4(a) Find the Z-inverse transform of z/(z-a) for |z| < a and |z| > a
Given a>0(6 marks) 4(b) Using Convolution theorem find and verify
(6 marks) 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 marks) 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 marks) 5(b) Find the Laplace transformation of
(6 marks) 5(c) Express the function
f(x) = 1 ... (|x| < 1)
= 0 ... (|x| > 1)
as a Fourier integral and hence evaluate:
(8 marks) 6(a) Show that the fourier transform of f(x)=e^-x2/2 is given by F(s)=e^-s2/2(6 marks) 6(b) Find Z{f(k)} where

(6 marks) 6(c) Find the fourier series of
f(x) = 0 ... (-? < x < 0)
= sinx ... (0 < x < ?)
Hence deduce that
(8 marks) 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 marks) 7(b) Solve the equation if y=1 at t=0
(6 marks) 7(c) Show that the set of functions:
1, sinx, cosx, sin2x, cos2x,...is orthogonal on (0,2?) but not on (0,?) (8 marks)