Applied Mathematics 4 - Dec 2012

Electronics Engineering (Semester 4)

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) Find the analytic u + iv given $$u+v=e^{x} (cos y + sin y)+\dfrac{x-y}{x+y}$$(5 marks) 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 marks) 1(c) Evaluate $$\int_{c}\bar{f}.\bar{dr}$$ along the arc of the curve form the point (1,0)to (e^2π,0)
where $$\bar{f}=\dfrac{xi+yj}{(x^{2}+y^{2})^{3/2}}$$ and curve C is $$\bar{r}=e^{t}i+e^{t}sin t j$$(5 marks) 1(d) Prove that $$\int J_{3}(X)dx=-\dfrac{2}{x}J_{1}(x)-J_{2}(x).$$(5 marks) 2(a) Find the Bilinear transformation which maps 1,-1,&infty; onto 1+i.1-I,1.Find its fixed points.(6 marks) 2(b) Evaluate A¹⁰⁰ for $$A=\begin{bmatrix}1 & 0& 0\\ 1&0 &1 \\ 0&1 &0 \end{bmatrix}$$.(6 marks) 2(c) Verify Green's theorem for $$\bar{f}=(x^{2}-xy)i+(x^{2}-y^{2})j$$ and c in Δ^le with vertices (0,0),(1,1)&(1,-1).(8 marks) 3(a) Show that f(x)=x²,o<x<2, $$f(x)="\sum_{i-1}^{\infty}\dfrac{2(\lambda_{i}^{2}-4)}{\lambda_{i}^{3}J_{1}(\lambda_{i})}Jo(\lambda_{i}x)\$$where$$" \lambda_{i}$$,i="0,1,2�..are" roots="" of="" jo(λ)="0\lt/a"\gt\ltbr\gt\ltbr\gt \lt/x\lt2,\gt\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt3(b)\lt/b\gt Show that $$\dfrac{x}{x^{2}+y^{2}}+2tan^{-1}\left(\dfrac{y}{x}\right)$$is imaginary part of an analytic function,find its real part and hence find the analytic function.\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt3(c)\lt/b\gt Evaluate $$\int_{c} \dfrac{z^{2}}{z^{4}-1}dz$$\ltbr\gtc is \ltbr\gt(i)|z-1|=\dfrac{1}{2} \ltbr\gt(ii)|z-1|=1 \ltbr\gt(iii)|z+i|=1\lt/span\gt\ltspan class='paper-ques-marks'\gt(8 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4(a)\lt/b\gt Evaluate using stokes theorem $$\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$$\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4(b)\lt/b\gt Evaluate $$\int_{0}^{\infty} \dfrac{1}{x^{4}+1}dx$$\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4(c)\lt/b\gt Find an orthogonal transformation which reduces the quardratic form $$2x^{2}+y^{2}-3z^{2}-8xy-4xz+12xy $$to a diagonal form.find the rank, index,signature and class value of the given form.\lt/span\gt\ltspan class='paper-ques-marks'\gt(8 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5(a)\lt/b\gt Prove that $$J_{3/2}{x} = \sqrt{\dfrac{2}{\pi x}}\left(\dfrac{sin x}{x}-cos x\right).$$\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5(b)\lt/b\gt Find a minimal polynomial of A hence find\ltbr\gt$$A^{10}\ where\ A= \begin{bmatrix}5 &-6 &-6 \\ -1&4 &2 \\ 3& -6 &4 \end{bmatrix} $$\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5(c)\lt/b\gt Find all possible Laurent's series expansion of $$\dfrac{4z^{2}+2z-4}{z^{3}-4z}$$about z=2 and specify their domain of converence.\lt/span\gt\ltspan class='paper-ques-marks'\gt(8 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6(a)\lt/b\gt Prove that $$2J_{n}^{1}(X) =J_{n-1}^{x}-J_{n+1}^{x}$$\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6(b)\lt/b\gt Evaluate $$\int_{0}^{2\pi}\dfrac{cos 3\theta }{5-4 cos \theta}d\theta$$\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6(c)\lt/b\gt Verify Gauss divergence theorem for F =xi + yj+z\ltsup\gt2\lt/sup\gtk,s in the surface bounded by the x\ltsup\gt2\lt/sup\gt+y\ltsup\gt2\lt/sup\gt=z\ltsup\gt2\lt/sup\gt and plane z=1.\lt/span\gt\ltspan class='paper-ques-marks'\gt(8 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt7(a)\lt/b\gt Show that under the transmission w =z\ltsup\gt2\lt/sup\gt,the circle |z-1|=1 is mapped onto cardiode ρ=2(1+cosϕ) where w=ρe\ltsup\gtiϕ\lt/sup\gt in w plane.\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt7(b)\lt/b\gt Find the matrix represented by A\ltsup\gt8\lt/sup\gt-5A\ltsup\gt7\lt/sup\gt+7A\ltsup\gt6\lt/sup\gt-3A\ltsup\gt5\lt/sup\gt+A\ltsup\gt4\lt/sup\gt-5A\ltsup\gt3\lt/sup\gt+8A\ltsup\gt2\lt/sup\gt-2A+I \ltbr\gt where $$A=\begin{bmatrix}2 & 1 &1 \\ 0& 1 & 0\\ 1& 1 & 2\end{bmatrix}$$\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt7(c)(i)\lt/b\gt State and prove the Cauchy residue theorem.\lt/span\gt\ltspan class='paper-ques-marks'\gt(4 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt7(c)(ii)\lt/b\gt Evaluate $$\int_{c} z^{6}\ e^{\frac{-1}{x^{2}}}\ dz; c:|z|=1$$</span>(4 marks)