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MU Biomedical Engineering (Semester 4)
Applied Mathematics - 4
May 2016
Total marks: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

1(a) Find the extremal of the functional $\int ^1_0[y'^{2}+12xy]dx\ \text {subject to}\ y(0)=0\ \text{and}\ y(1)=1).$
5 M
1(b) Verify Cauchy - Schwartz inequality for u = (1, 2, 1) and v = (3, 0, 4) also find the angle between u & v.
5 M
1(c) If &lambda & X are eigen values and eigen vectors of A then prove that $$\dfrac{1}{\lambda}$$ and X are eigen values and eigen vectors of A-1, provided A is non singular matrix.
5 M
1(d) Evaluate $$\int _C \dfrac{e^{2z}}{(z+1)^4}dz\ \text{where}\ C:|z|=2$$.
5 M

2(a) Find the extremal that minimises the integral $\int ^{x_1}_{x_0}(16y^2-y''^2)dx$
6 M
2(b) Find eigen values and eigen vectors of A3 $\text{where}\ A=\begin{bmatrix} 2 & 1 & 1\\ 2 & 3 & 2\\ 3 & 3 & 4 \end{bmatrix}$
6 M
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 M

3(a) Verify Cayley-Hamilton Theorem for $A=\begin{bmatrix} 2 & -1 & 1\\ -1 & 2 & -1\\ 1 & -1 & 2 \end{bmatrix}\text{and hence find}A^{-1}$
6 M
3(b) Using Cauchy Residue Theorem, evaluate $\int ^{\infty} _{-\infty }\dfrac{x^2-x+2}{x^4+10x^2+9}dx$
6 M
3(c) Show that a closed curve 'C' of given fixed length (perimeter) which encloses maximum area is a circle.
8 M

4(a) Find an orthonormal basis for the subspace of R3 by appling Gram-Schmidt process where S {(1, 1, 1), (0, 1, 1) (0, 0, 1)}.
6 M
4(b) Find A50, where $A=\begin{bmatrix} 2 & 3\\ -3 & -4 \end{bmatrix}$
6 M
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 M

5(a) Using the Rayleigh-Ritz method, find an approximate solution for the extremal of the functional $$\int ^1_0 \left \{ xy+\dfrac{1}{2}y'^2 \right \}dx$$ subject to y(0)=y(1)=0.
6 M
5(b) Prove that W = {(x, y)| x=3y} subspace of R2. Is W1 = {a, 1, 1)| a in R} subspace of R3?
6 M
5(c) Prove that A is 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 M

6(a) By using Cauchy Residue Theorem, evaluate $$\int ^{2\pi}_0 \dfrac{\cos ^2 \theta}{5+4\cos \theta}d\theta.$$
6 M
6(b) Evaluate $$\int _C \dfrac{z+4}{z^2+2z+5}dz\ \text{where}\ C:|z+1+i|=2.$$
6 M
6(c)(i) Determine the function that gives shortest distance between two given points.
5 M
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 M

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