## 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{J_{0}(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^{-1}s

(ii) (s+1)e^{-s}/(s^{2}+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 = k^{2}

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)