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=egin{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}ar{f}.ar{dr}\] along the arc of the curve form the point (1,0)to (e2π,0)
where [ar{f}=dfrac{xi+yj}{(x^{2}+y^{2})^{3/2}}\] and curve C is [ar{r}=e^{t}i+e^{t}sin t j\]
where [ar{f}=dfrac{xi+yj}{(x^{2}+y^{2})^{3/2}}\] and curve C is [ar{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 thhe 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=egin{bmatrix}
1 & 0& 0\
1&0 &1 \
0&1 &0
end{bmatrix}\].
6 M
2(c)
Verify Green's theorem for [ar{f}=(x^{2}-xy)i+(x^{2}-y^{2})j\] and c in Δlewith 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}
ight)\]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 quadratic 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
ight).
6 M
5(b)
Find a minimal polynomial of A hence find
[A^{10} where A= egin{bmatrix} 5 &-6 &-6 \ -1&4 &2 \ 3& -6 &4 end{bmatrix} \]
[A^{10} where A= egin{bmatrix} 5 &-6 &-6 \ -1&4 &2 \ 3& -6 &4 end{bmatrix} \]
6 M
5(c)
Find all possible Laurents series expansion of [dfrac{4z^{2}+2z-4}{z^{3}-4z}\]about z=2 and specify their domain of convergence.
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}^{2pi}dfrac{cos 3 heta }{5-4 cos heta}d heta\]
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=egin{bmatrix} 2 & 1 &1 \ 0& 1 & 0\ 1& 1 & 2 end{bmatrix}\]
where [A=egin{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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