Applied Mathematics 4 - May 2016

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 extremal of the functional
$ \displaystyle \int ^1_0[y'^2+12xy]dx $ subject to y(0) = 0 and y(1) = 1.(5 marks) 1(b) Verify Cauchy ' Schwartz inequality for u = (1, 2, 1) and v = (3, 0, 4) also find the angle between u & v.(5 marks) 1(c) If &lambda & X are eigen values and eigen vectors of A the prove that $ \dfrac{1}{\lambda} $ and X are eigen values and eigen vectors of A^-1, provided A is non singular matrix.(5 marks) 1(d) Evaluate $ \int _C \dfrac{e^{2x}}{(z+1)^4}dz $ where C : |z| = 2(5 marks) 2(a) Find the extremal that minimise the integral $$\displaystyle \int ^{x_1}_{x_0}(16y^2-y^{''2})dx$$(6 marks) 2(b) Find eigrn values and eigen vectors of A³ $$\text {where} A=\begin{bmatrix} 2 & 1 & 1\\ 2 & 3 & 2\\ 3 & 3 & 4 \end{bmatrix}$$(6 marks) 2(c) Obtain Taylor's and two distinct Laurent's expansion of $ f(z)=\dfrac{z-1}{z^2-2z-3} $ indicating the region of convergence.(8 marks) 3(a) Verify Cayley-Hamilton Theorem for
$ A=\begin{bmatrix} 2 & -1 & 1\\\\ -1 & 2 & -1\\\\ 1 & -1 & 2 \end{bmatrix} $ and hence find A^-1(6 marks) 3(b) Using Cauchy Residue Theorem, evaluate $$\int ^{\infty}_{-\infty}\dfrac{x^2-x+2}{x^4+10x^2+9}dx$$(6 marks) 3(c) Show that a closed curve 'C' of given fixed length (perimeter) which encloses maximum area is a circle.(8 marks) 4(a) Find an orthonomal basis for the subspace of R³ by appling Gram-Schmidt process where S {(1, 1, 1), (0, 1, 1) (0, 0, 1)}(6 marks) 4(b) Find A⁵⁰, where $$A=\begin{bmatrix} 2 & 3\\ -3 & -4 \end{bmatrix}$$(6 marks) 4(c) Reduce the following Quadratic form into canonical form & hence find its rank, index, signature and value class where,
Q = 3x₁² + 5x₂² + 3x₃² - 2x₁x₂ - 2x₂x₃ + 2x₃x₁(8 marks) 5(a) Using the Rayleigh- Ritz method, find an approximate solution for the extremal of the functional $ \displaystyle \int ^1_0 \left \{ xy +\dfrac{1}{2}y'^2\right \}dx $ subject to y(0) = y(1) = 0.(6 marks) 5(b) Prove that W = {(x, y)| x = 3y} subspace of R². Is W₁ = {(a, 1, 1)| a in R} subspace of R³?(6 marks) 5(c) Prove that A us diagonizable matrix. Also find diagonal form and transforming matrix where $ A=\begin{bmatrix} 1 & -6 & -4\\\\ 0 & 4 & 2\\\\ 0 & -6 & -3 \end{bmatrix} $(8 marks) 6(a) By using Cauchy residue Theorem, evaluate $ \displaystyle \int ^{2\pi}_0 \dfrac{\cos^2 \theta}{5+4\cos \theta}d\theta. $(6 marks) 6(b) Evaluate $ \int _C \dfrac{z+4}{z^2+2z+5} dz $ where C : |z+1+i| = 2.(6 marks) 6(c)(i) Determine the function that gives shortest distance between two given points.(5 marks) 6(c)(ii) Express any vector (a,b,c) in R³ as a linear combination of v₁, v₂, v₃ where v₁, v₂, v₃ are in R³.(3 marks)