## Applied Mathematics - 3 - May 2013

### Electronics 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)** Show that $$ f(z)=\frac{\bar{z}}{{\left\vert{}z\right\vert{}}^2} \ \ \ \ \ \ , |z|≠0 $$ is analytic function.Hence find f'(z)(5 marks)
**1 (b)** Find the Fourier Expansion of f(x)=sinx in (-π,π).(5 marks)
**1 (c) ** Find the Laplace Transform of $$t\sqrt{1+sint}$$(5 marks)
**1 (d)** Find z transformation of { a^{k} sin ak}, k ≥ 0. where a is constant(5 marks)
**2 (a)** Using Laplace Transform evaluate $$\int_0^{\infty{}}e^{-t}\frac{\sin{3t}}{t}dt.\ \ \ \ \ \ \ \ \ \ \ \ \ \ $$(6 marks)
**2 (b)** Find the Fourier expansion of f(x)=cospx where p is not an integer in (0,2π)(7 marks)
**2 (c) ** Find the matrix A, if adj A = $$\ \left[\begin{array}{ccc}-2 & 1 & 3 \\-2 & -3 & 11 \\2 & 1 & -5\end{array}\right]$$(7 marks)
**3 (a)** Find inverse Laplace transform of

(I) log (s-2/s-3)

(II) s+1/s^{2} - 4(6 marks)
**3 (b)** Find non Singular matrices P and Q such that PAQ is in normal form. Also find rank of a matrix A where

A =$$\begin{array}{cccc}2 & -4 & 3 & 1 & 0 \\1 & -2 & 1 & -4 & 2 \\0 & 1 & -1 & 3 & 1 \\4 & -7 & 4 & -4 & 5\end{array}$$

(7 marks)
**3 (c) ** Verify Green's Theorem in the plane for ∮_{c} (xy+y^{2}) dx+x^{2} by where c is a closed

curve of a region bounded by y=x and y^{2}=x (7 marks)
**4 (a)** Obtain Complex form of Fourier series for f(x)=e^{-ax} in (-2,2)where a is not an integer.(6 marks)
**4 (b)** if A= $$\left[\begin{array}{ccc}1 & 1 & 1 \\2 & 5 & 7 \\2 & 1 & -1\end{array}\right]$$ compute A ^{l} and hence, Solve the system of equation

x + y + z = 9, 2x + 5y + 7z =52, 2x + y - z = 0. (7 marks)
**4 (c) ** Find the Laplace transform of

f(t) = 1, 0 ≤ t ≤a

f(t) =1, a < t ≤ 2a &

f(t + 2a)= f(t)(7 marks)
**5 (a)** Find the analytic function f(z)=u+iv if u=(r+ a^{2}/r) cos θ. (6 marks)
**5 (b)** Show that the equations.

ax + by +cz =0

bx + cy + az =0

cx + ay + bz = 0

has non trivial solution if a+b+c=0 or if a=b=c. Find the non trivial solution when the condition is satisfied.(7 marks)
**5 (c) ** Find Fourier integral representing f(x) = $$\left\{\begin{array}{l}1-x^2\ \ \ \ \ \ \ \ \left\vert{}x\right\vert{}\leq{}1 \\0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left\vert{}x\right\vert{}>1\end{array}.\right.$$(7 marks)
**6 (a)** Find the half range cosine series for f(x)=2x-x^{2} in (0,2)(6 marks)
**6 (b)** Find the Bilinear Transformation which maps the points 2,i,-2 on the points 1,i,-1(7 marks)
**6 (c) ** Using Lapace transform solve the differential equation

$$\frac{d^2y}{{dt}^2}-2\frac{dy}{dt}-8y=4\ ,\ y\left(0\right)=0,\ y^{'}\left(0\right)=1.$$(7 marks)
**7 (a)** Find inverse z-transform of F(z)= 1/(z-2)(z-3) if ROC is 2 < |z| < 3.(6 marks)
**7 (b)** Verify Stoke's theorem for $$\bar{F}=x^2\bar{i}+xy\bar{j}$$ & C is the boundary of the rectangle x=0, x=2, y=3.(7 marks)
**7 (c) ** Using Gauss Divergence Theorem evaluate $$\iint_S\vec{F}\bullet{}\hat{n}ds\ where\ \vec{F}=4x\bar{i}+3y\bar{j}-4z^2\vec{k}$$

and S is the closed surface bounded the planes x=0,y=0,z=0 and 2x+2y+z=4 (7 marks)