## 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) = x^{2} + x ... (-? < x < ?)(6 marks)
**5(c)** Find z(a^{n}), 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)