## 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 f_{1}(x)=1, f_{2}(x)=x and f_{3}(x)=-1+ax+bx^{2} 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) = sin^{5}t(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-x^{2} 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)