Applied Mathematics 3 - May 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) Find the Laplace transform of f(t) = e^-4t sinht sint(5 marks) 1(b) Express the matrix A as the sum of a symmetric and a skew symmetric matrix
(5 marks) 1(c) If the functions f₁(x)=1, f₂(x)=x and f₃(x)=-1+ax+bx² are orthogonal in [-1,1] then determine the constants a and b.(5 marks) 1(d) Find the Fourier transform of f(x) = e^-|x|(5 marks) 2(a) Find the Laplace transform of f(t) = sin⁵t(6 marks) 2(b) For the matrix A find the non-singular matrices P and Q such that PAQ is in normal form
(6 marks) 2(c) Find the Fourier series for f(x) = x in (0, 2?)(8 marks) 3(a) Find the Laplace transform of
(6 marks) 3(b) Reduce the following matrix to normal form and find its rank:
(6 marks) 3(c) Find Fourier expansion of f(x) = [(?-x)/2]² in (0,2?) and hence prove that:
(8 marks) 4(a) Find inverse Laplace transform of
(6 marks) 4(b) Is the matrix A unitary. If yes, find A^-1
(6 marks) 4(c) Obtain the half range sine series in (0,2?) for f(x) = x(?-x) and hence find the value of ?(-1)ⁿ/(2n-1)³(8 marks) 5(a) Find inverse Laplace transform of
(6 marks) 5(b) Find the complex form of Fourier series for f(x)=e^x in (-?,?)(6 marks) 5(c) Express the function
f(x) = 1 ... (|x| < 1)
= 0 ... (|x| > 1)
as a Fourier integral. Hence evaluate
(8 marks) 6(a) Using convolution theorem find Laplace inverse of
(6 marks) 6(b) Find the Fourier series for f(x) = 1-x² in (-1,1)(6 marks) 6(c) Solve the following system of equations
x + 2y + 3z = 14
3x + y + 2z = 11
2x + 3y + z = 11(8 marks) 7(a) Evaluate the following:
(6 marks) 7(b) Find the z-transform of
(i) f(k)=1 ... (k?0, |z|>1)
(ii) f(k)=a^k ... (k?0, |z|>a)
(iii) f(k)=1/2^k ... (k?0, |2z|>1) (6 marks) 7(c) Solve the following:
(8 marks)