Applied Mathematics 4 - Dec 2016

Electronics & Telecomm. (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) if f(x) is an algebraix polynomial in x and λ is an eigen value and X is the the corresponding eigen vector of a square matrix A then f(λ) is an eigen value X is the corresponding eigenvector of f (A).(5 marks) 1(b) Find the extremal of $$\int_{x_0}^{x^1}\left ( x +y' \right )y'dx$$(5 marks) 1(c) Express (6, 11, 6) as linear combination of v₁=(2,1,4), v₂=(1,-1,3), v₃=(3,2,5).(5 marks) 1(d) Evaluate $ \int _C\frac{z}{\left ( z-1 \right )^2\left ( z-2 \right )}dz,$/ where C is the circle |z-2|=0.5(5 marks) 2(a) Find the curve y= f(x) for which$\int_{0}^{\pi }\left ( y'^2 -y^2\right )dx $/ is extremum if $$\int_{0}^{\pi }ydx=1$$(6 marks) 2(b) Evaluate $$\int_{0}^{2\pi }\frac{\cos 3\theta }{5+4\cos \theta }d\theta $$(6 marks) 2(c) Find the singular value decomposition of $$\begin{bmatrix} 2 & 3\\ 0 & 2 \end{bmatrix}$$(6 marks) 3(a) Verify Cayley Hamilton theorem for $ A=\begin{bmatrix} 3 & 10 &5 \\\\ -2& -3&-4 \\\\ 3& 5 & 7 \end{bmatrix} $/ and hence, find the matrix represented by $$A^6-GA^5+9A^4+4A^3-12A^2+2A-1$$(6 marks) 3(b) Construction an orthonormal basis of R³ using Gram Schmidt process to S={(3,0,4),(-1,0,7),(2,9,11)}(6 marks) 3(c) Find all possible Laurent's expansions $ \frac{z}{\left ( z-1 \right )\left ( z-2 \right )} $/ about z=-2 indicating the region of covergence.(8 marks) 4(a) Reduce the quadratic from$ 2x^2-2y^2+2z^2-2xy-8yz+6zx $/ to canonical from and hence find its rank, index and signature and value class.(6 marks) 4(b) If $\phi (\alpha )\int _C\frac{4z^2+z+5}{z-\alpha }dz, $/ where C is the contour of the ellipse $ \frac{x^2}{4}+\frac{y^2}{9}=1, $/ find the values of $$\phi (3.5),\phi (i),\phi (-1),\phi (-i)$$(6 marks) 4(c) Using Rayleigh-Riz method, solve the boundry value problem$$I=\int_{0}^{1}\left ( y'^2-y^2-2xy \right )dx;0\leq x\leq 1,$$ given y(0)=y(1)=0.(8 marks) 5(a) Find the extremal of the function $ \int_{0}^{\pi /2}\left ( 2xy+y^2-y'^2 \right )dx;$/ with y(0)=0, y(π/2)=0(6 marks) 5(b) Find the orthogonal matrix P that diagonalises $$A=\begin{bmatrix} 4 & 2&2 \\ 2 & 4 & 2\\ 2 & 2 & 4 \end{bmatrix}$$(6 marks) 5(c) Using Cauchy's Residue theorem, evaluate $ \oint _C\frac{z^2+3}{z^2-1}dz $/ where C is the circle (i) |z-1|=1
(ii) |z+1|=1.(8 marks) 6(a) Find the sum of the residues points of $$f(z)\frac{z}{\left ( z-1 \right )^2\left ( z^2-1 \right )}$$(6 marks) 6(b) If $ A=\begin{bmatrix} 1 &4 \\\\ 2 & 3 \end{bmatrix}$/, prove that $$A^{50}-5A^{40}=\begin{bmatrix} 4 & -4\\ -2 & 2 \end{bmatrix}$$(6 marks) 6(c)(i) Checy whether W={(x,y,z)|y=x+z,x,y,z are in R} is a subspace R³ with usual addition and usual multiplication.(4 marks) 6(c)(ii) Find the unit vector in R³ orthogonal to both u={1,0,1} and v={0,1,1}.(4 marks)