Applied Mathematics 4 - Dec 2014

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 value of μ which satisfy the equation A¹⁰⁰x=μ X, where $$ A= \begin{bmatrix}2 &1 &-1 \\0 &-2 &-2 \\ 1 &1 &0 \end{bmatrix} $$(5 marks) 1 (b) $$ Evaluate \ \int^{1+i}_{0} (x^2+iy)dz \ along \ y=x \ and \ y=x^2 $$(5 marks) 1 (c) Find the external of the function $$ \int^{x_2}_{x_1} [y^2-y^2 -2y\cosh x]dx $$(5 marks) 1 (d) Verify Cauchy-Schwartz inequality for the vectors.
u=(-4, 2, 1) & V=(8, -4, -2)(5 marks) 2 (a) Determine the function that gives the shortest distance between two given points(6 marks) 2 (b) Find Eigen values and Eigen vectors of :- $$ A= \begin{bmatrix} 2 & 1 &1 \\2 &3 &2 \\3 &3 &4 \end{bmatrix} $$(6 marks) 2 (c) Obtain Taylor's and two distinct Laurent's series expansion of $$ f(z) = \dfrac {z-1}{z^2-2z-3} $$ about z=0 indicating the region of convergence.(8 marks) 3 (a) Verify Caley-Hamilton theorem for $$ A= \begin{bmatrix}1 &2 &0 \\2 &-1 &0 \\0 &0 &-1 \end{bmatrix} \ hence \ find \ A^{-2} $$(6 marks) 3 (b) Evaluate by using Residue theorem. $$ \int^{2\pi}_0 \dfrac {d\theta}{(2+\cos \theta )^2} $$(6 marks) 3 (c) Solve the boundary value problem. $$ I= \int^1_0 \left ( 2xy -y^2 -y^{1^{2 }} \right )dx $$ given y(0)=y(1)=0 by Rayleigh-Ritz method.(8 marks) 4 (a) Reduce the following Quadratic form $$ Q=3x^2_1 + 5x_2^2 + 3x^2_3 -2x_1x_2-2x_2x_3 +2x_3x_1 $$ into canonical form. Hence find its rank index and signature.(6 marks) 4 (b) Show that the matrix $$ A= \begin{bmatrix} 7 &4 &-1 \\4 &7 &-1 \\-4 &-4 &4 \end{bmatrix} \ is \ derogatory $$(6 marks) 4 (c) (i) Show that the set W={(1,x)|x∈R} is a subspace of R² under operations [1,x]+[1,y]=[1, x+y]; k[1,x]=[1,kx]; k is any scalar:(4 marks) 4 (c) (ii) Is the set W={[a,1,1]|a∈R} a subspace of R³ under the usual addition and scalar multiplication?(4 marks) 5 (a) Find the plane curve of fixed perimeter and maximum area.(6 marks) 5 (b) Construct an orthonormal basis of R² by applying Gram Schmidt orthogonalization to S={[3,1],[2,2]}(6 marks) 5 (c) Show that the matrix $$ A= \begin{bmatrix} -9 &4 &4 \\-8 &3 &4 \\-16 &8 &7 \end{bmatrix} $$ is diagonable. Also find diagonal form and diagonalising matrix.(8 marks) 6 (a) Evaluate $$ \int^{\infty}_{-\infty} \dfrac {\cos 3x}{(x^2+1)(x^2+4)} dx $$ using Cauchy Residue Theorem.(6 marks) 6 (b) $$ If \ \phi(\alpha)= \oint_{c}\dfrac {ze^z}{z-\alpha}dz \ where \ c \ is \ |z-2i|=3 \\ find \phi (1), \ \phi '(2), \ \phi ' (3), \ \phi ' (4) $$(6 marks) 6 (c) Show that the set V of position real numbers with operations Addition : x+y=xy, Scalar multiplication: kx=x^k is a vector space where x, y are any two real numbers and k is any scalar.(8 marks)