STUPIDSID
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 M
1 (b)
Find the Fourier Expansion of f(x)=sinx in (-?,?).
5 M
1 (c)
Find the Laplace Transform of t? \[\sqrt{1+sint}\]
5 M
1 (d)
Find z transformation of { ak sin ak}, k ? 0. where a is constant
5 M
2 (a)
Using Laplace Transform evaluate \[\int_0^{\infty{}}e^{-t}\frac{\sin{3t}}{t}dt.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \]
6 M
2 (b)
Find the Fourier expansion of f(x)=cospx where p is not an integer in (0,2?)
7 M
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 M
3 (a)
Find inverse Laplace transform of
(I) log (s-2/s-3)
(II) s+1/s2 - 4
(I) log (s-2/s-3)
(II) s+1/s2 - 4
6 M
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 =\[\begin{array}{cccc}2 & -4 & 3 & 1 & 0 \\1 & -2 & 1 & -4 & 2 \\0 & 1 & -1 & 3 & 1 \\4 & -7 & 4 & -4 & 5\end{array}\]
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 M
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
curve of a region bounded by y=x and y2=x
7 M
4 (a)
Obtain Complex form of Fourier series for f(x)=e-ax in (-2,2)where a is not an integer.
6 M
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.
x + y + z = 9, 2x + 5y + 7z =52, 2x + y - z = 0.
7 M
4 (c)
Find the Laplace transform of
f(t) = 1, 0 ? t ?a
f(t) =1, a < t ? 2a &
f(t + 2a)= f(t)
f(t) = 1, 0 ? t ?a
f(t) =1, a < t ? 2a &
f(t + 2a)= f(t)
7 M
5 (a)
Find the analytic function f(z)=u+iv if u=(r+ a2/r) cos ?.
6 M
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.
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 M
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 M
6 (a)
Find the half range cosine series for f(x)=2x-x2 in (0,2)
6 M
6 (b)
Find the Bilinear Transformation which maps the points 2,i,-2 on the points 1,i,-1
7 M
6 (c)
Using Lapace transform solve the differential equation
\[\frac{d^2y}{{dt}^2}-2\frac{dy}{dt}-8y=4\ ,\ y\left(0\right)=0,\ y^{'}\left(0\right)=1.\]
\[\frac{d^2y}{{dt}^2}-2\frac{dy}{dt}-8y=4\ ,\ y\left(0\right)=0,\ y^{'}\left(0\right)=1.\]
7 M
7 (a)
Find inverse z-transform of F(z)= 1/(z-2)(z-3) if ROC is 2 < |z| < 3.
6 M
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 M
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
and S is the closed surface bounded the planes x=0,y=0,z=0 and 2x+2y+z=4
7 M
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