STUPIDSID
1(a)
Find the analytic u + iv given \[u+v=e^{x} (cos y + sin y)+\dfrac{x-y}{x+y}\]
5 M
1(b)
The matrix A is given by \[A=\begin{bmatrix}1 & 0 &-3 \\ 0& 3 &2 \\ 0& 0&-2 \end{bmatrix}\].Find the Eigen values and Eigen vectors of B where \[B=I-6A^{-1}\]
5 M
1(c)
Evaluate \[\int_{c}\bar{f}.\bar{dr}\] along the arc of the curve form the point (1,0)to (e2π,0)
where \[\bar{f}=\dfrac{xi+yj}{(x^{2}+y^{2})^{3/2}}\] and curve C is \[\bar{r}=e^{t}i+e^{t}sin t j\]
where \[\bar{f}=\dfrac{xi+yj}{(x^{2}+y^{2})^{3/2}}\] and curve C is \[\bar{r}=e^{t}i+e^{t}sin t j\]
5 M
1(d)
Prove that \[\int J_{3}(X)dx=-\dfrac{2}{x}J_{1}(x)-J_{2}(x).\]
5 M
2(a)
Find the Bilinear transformation which maps 1,-1,&infty; onto 1+i.1-I,1.Find its fixed points.
6 M
2(b)
Evaluate A100 for \[A=\begin{bmatrix}1 & 0& 0\\ 1&0 &1 \\ 0&1 &0 \end{bmatrix}\].
6 M
2(c)
Verify Green's theorem for \[\bar{f}=(x^{2}-xy)i+(x^{2}-y^{2})j\] and c in Δle with vertices (0,0),(1,1)&(1,-1).
8 M
3(a)
Show that f(x)=x2,o
6 M
3(b)
Show that \[\dfrac{x}{x^{2}+y^{2}}+2tan^{-1}\left(\dfrac{y}{x}\right)\]is imaginary part of an analytic function,find its real part and hence find the analytic function.
6 M
3(c)
Evaluate \[\int_{c} \dfrac{z^{2}}{z^{4}-1}dz\]
c is
(i)|z-1|=\dfrac{1}{2}
(ii)|z-1|=1
(iii)|z+i|=1
c is
(i)|z-1|=\dfrac{1}{2}
(ii)|z-1|=1
(iii)|z+i|=1
8 M
4(a)
Evaluate using stokes theorem \[\int_{c}y dx +zdy+xdz\],where c is the curve of intersection of surfaces \[x^{2}+y^{2}+z^{2}=a^{2} and x+z=a\]
6 M
4(b)
Evaluate \[\int_{0}^{\infty} \dfrac{1}{x^{4}+1}dx\]
6 M
4(c)
Find an orthogonal transformation which reduces the quardratic form \[2x^{2}+y^{2}-3z^{2}-8xy-4xz+12xy \]to a diagonal form.find the rank, index,signature and class value of the given form.
8 M
5(a)
Prove that \[J_{3/2}{x} = \sqrt{\dfrac{2}{\pi x}}\left(\dfrac{sin x}{x}-cos x\right).\]
6 M
5(b)
Find a minimal polynomial of A hence find
\[A^{10}\ where\ A= \begin{bmatrix}5 &-6 &-6 \\ -1&4 &2 \\ 3& -6 &4 \end{bmatrix} \]
\[A^{10}\ where\ A= \begin{bmatrix}5 &-6 &-6 \\ -1&4 &2 \\ 3& -6 &4 \end{bmatrix} \]
6 M
5(c)
Find all possible Laurent's series expansion of \[\dfrac{4z^{2}+2z-4}{z^{3}-4z}\]about z=2 and specify their domain of converence.
8 M
6(a)
Prove that \[2J_{n}^{1}(X) =J_{n-1}^{x}-J_{n+1}^{x}\]
6 M
6(b)
Evaluate \[\int_{0}^{2\pi}\dfrac{cos 3\theta }{5-4 cos \theta}d\theta\]
6 M
6(c)
Verify Gauss divergence theorem for F =xi + yj+z2k,s in the surface bounded by the x2+y2=z2 and plane z=1.
8 M
7(a)
Show that under the transmission w =z2,the circle |z-1|=1 is mapped onto cardiode ρ=2(1+cosϕ) where w=ρeiϕ in w plane.
6 M
7(b)
Find the matrix represented by A8-5A7+7A6-3A5+A4-5A3+8A2-2A+I
where \[A=\begin{bmatrix}2 & 1 &1 \\ 0& 1 & 0\\ 1& 1 & 2\end{bmatrix}\]
where \[A=\begin{bmatrix}2 & 1 &1 \\ 0& 1 & 0\\ 1& 1 & 2\end{bmatrix}\]
6 M
7(c)(i)
State and prove the Cauchy residue theorem.
4 M
7(c)(ii)
Evaluate \[\int_{c} z^{6}\ e^{\frac{-1}{x^{2}}}\ dz; c:|z|=1\]
4 M
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