Applied Mathematics - 3 - May 2013

Electronics & Telecomm. (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 (-?,?).(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 (0,2?)(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/s² - 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+y²) dx+x² by where c is a closed
curve of a region bounded by y=x and y²=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+ a²/r) cos ?. (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-x² 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)