Applied Mathematics 3 - Dec 2012

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) State Dirichlet conditions for the expansion of f(x) as Fourier series. Examine whether f(x)=sin(1/x) can be expanded in Fourier series in [-?,?](5 marks) 1(b) Find Laplace transform of:
(5 marks) 1(c) Find Z {f(k)} where f(k) is given by:
(5 marks) 1(d) Express the function f(x) as a Fourier integral hence evaluate the integral that follows:
(5 marks) 2(a) Define linear dependence and independence of vectors. If the vectors (0,1,a),(1,a,1) and (a,1,0) are linearly dependent then find the value of 'a'(6 marks) 2(b) Find Laplace transform of:
(6 marks) 2(c) Find {f(k)} if F(z) is as given below and if ROC of F(Z) is:
(i)|z|<2 (ii)2<|z|<3 (iii) |z|>3
(8 marks) 3(a) Determine the value of a and b for which the system:
x + 2y + z = 6
x + 3y + 5z = 9
2x + 5y + az = b
has (i)no solution (ii)unique solution (iii)infinite solutions.
Find the solutions in case of (ii) and (iii)(6 marks) 3(b) Evaluate the following:
(6 marks) 3(c) Find the Fourier series for f(x) in (0,2?) -
f(x) = x ... (0 < x ? ?)
= 2? - x ... (? ? x < 2?)
Hence deduce that
(8 marks) 4(a) Find two non singular matrices P and Q such that PAQ is in the normal form where

(6 marks) 4(b) Find L(|cost|)(6 marks) 4(c) (i) If A, B are Hermitian prove that AB-BA is skew Hermitian
(ii) Show that A is Hermitian and iA is skew Hermitian if:
(8 marks) 5(a) Solve y'' + 2y = r(t); y(0) = 0, y'(0) = 0
Using Laplace Transform where
r(t) = 1 ... (0 ? t ? 1)
= 0 ... (t > 1) (6 marks) 5(b) Find the complex form of the Fourier series of the function
f(x) = x² + x ... (-? < x < ?)(6 marks) 5(c) Find z(aⁿ), z(cos n?), z(sin n?)(8 marks) 6(a) Show that e^x is equal to:

(6 marks) 6(b) Find the inverse Laplace Transform of
(6 marks) 6(c) Obtain the Fourier series for the function
f(x) = x ... (-? < x < 0)
= -x ... (0 < x < ?)
Hence deduce that:
(8 marks) 7(a) Find the Fourier Transform of:
(6 marks) 7(b) Using Laplace Transform evaluate

(6 marks) 7(c) Show that the system of equations
ax + by + cz = 0
bx + cy + az = 0
cx + ay + bz = 0
has a non Trivial solution if a+b+c=0 or if a=b=c.
Find the non Trivial solution when the condition is satisfied(8 marks)