MU Applied Mathematics 4 - December 2016 Exam Question Paper

MU Electronics and Telecom Engineering (Semester 4)
Applied Mathematics 4
December 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) 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 M

1(b) Find the extremal of

\int_{x_{0}}^{x^{1}} (x + y^{'}) y^{'} d x

$\int_{x_0}^{x^1}\left ( x +y' \right )y'dx$

5 M

1(c) Express (6, 11, 6) as linear combination of v₁=(2,1,4), v₂=(1,-1,3), v₃=(3,2,5).

5 M

1(d) Evaluate

\int_{C} \frac{z}{{(z - 1)}^{2} (z - 2)} d z,

$\int _C\frac{z}{\left ( z-1 \right )^2\left ( z-2 \right )}dz,$ / where C is the circle |z-2|=0.5

5 M

2(a) Find the curve y= f(x) for which

\int_{0}^{π} (y^{' 2} - y^{2}) d x

$\int_{0}^{\pi }\left ( y'^2 -y^2\right )dx$ / is extremum if

\int_{0}^{π} y d x = 1

$\int_{0}^{\pi }ydx=1$

6 M

2(b) Evaluate

\int_{0}^{2 π} \frac{\cos 3 θ}{5 + 4 \cos θ} d θ

$\int_{0}^{2\pi }\frac{\cos 3\theta }{5+4\cos \theta }d\theta$

6 M

2(c) Find the singular value decomposition of

[\begin{matrix} 2 & 3 \\ 0 & 2 \end{matrix}]

$\begin{bmatrix} 2 & 3\\ 0 & 2 \end{bmatrix}$

6 M

3(a) Verify Cayley Hamilton theorem for

A = [\begin{matrix} 3 & 10 & 5 \\ - 2 & - 3 & - 4 \\ 3 & 5 & 7 \end{matrix}]

$A=\begin{bmatrix} 3 & 10 &5 \\ -2& -3&-4 \\ 3& 5 & 7 \end{bmatrix}$ / and hence, find the matrix represented by

A^{6} - G A^{5} + 9 A^{4} + 4 A^{3} - 12 A^{2} + 2 A - 1

$A^6-GA^5+9A^4+4A^3-12A^2+2A-1$

6 M

3(b) Construction an orthonormal basis of R³ using Gram Schmidt process to S={(3,0,4),(-1,0,7),(2,9,11)}

6 M

3(c) Find all possible Laurent's expansions

\frac{z}{(z - 1) (z - 2)}

$\frac{z}{\left ( z-1 \right )\left ( z-2 \right )}$ / about z=-2 indicating the region of covergence.

8 M

4(a) Reduce the quadratic from

2 x^{2} - 2 y^{2} + 2 z^{2} - 2 x y - 8 y z + 6 z x

$2x^2-2y^2+2z^2-2xy-8yz+6zx$ / to canonical from and hence find its rank, index and signature and value class.

6 M

4(b) If

ϕ (α) \int_{C} \frac{4 z^{2} + z + 5}{z - α} d z,

$\phi (\alpha )\int _C\frac{4z^2+z+5}{z-\alpha }dz,$ / where C is the contour of the ellipse

\frac{x^{2}}{4} + \frac{y^{2}}{9} = 1,

$\frac{x^2}{4}+\frac{y^2}{9}=1,$ / find the values of

ϕ (3.5), ϕ (i), ϕ (- 1), ϕ (- i)

$\phi (3.5),\phi (i),\phi (-1),\phi (-i)$

6 M

4(c) Using Rayleigh-Riz method, solve the boundry value problem

I = \int_{0}^{1} (y^{' 2} - y^{2} - 2 x y) d x; 0 \leq x \leq 1,

$I=\int_{0}^{1}\left ( y'^2-y^2-2xy \right )dx;0\leq x\leq 1,$ given y(0)=y(1)=0.

8 M

5(a) Find the extremal of the function

\int_{0}^{π / 2} (2 x y + y^{2} - y^{' 2}) d x;

$\int_{0}^{\pi /2}\left ( 2xy+y^2-y'^2 \right )dx;$ / with y(0)=0, y(π/2)=0

6 M

5(b) Find the orthogonal matrix P that diagonalises

A = [\begin{matrix} 4 & 2 & 2 \\ 2 & 4 & 2 \\ 2 & 2 & 4 \end{matrix}]

$A=\begin{bmatrix} 4 & 2&2 \\ 2 & 4 & 2\\ 2 & 2 & 4 \end{bmatrix}$

6 M

5(c) Using Cauchy's Residue theorem, evaluate

\oint_{C} \frac{z^{2} + 3}{z^{2} - 1} d z

$\oint _C\frac{z^2+3}{z^2-1}dz$ / where C is the circle (i) |z-1|=1
(ii) |z+1|=1.

8 M

6(a) Find the sum of the residues points of

f (z) \frac{z}{{(z - 1)}^{2} (z^{2} - 1)}

$f(z)\frac{z}{\left ( z-1 \right )^2\left ( z^2-1 \right )}$

6 M

6(b) If

A = [\begin{matrix} 1 & 4 \\ 2 & 3 \end{matrix}]

$A=\begin{bmatrix} 1 &4 \\ 2 & 3 \end{bmatrix}$ /, prove that

A^{50} - 5 A^{40} = [\begin{matrix} 4 & - 4 \\ - 2 & 2 \end{matrix}]

$A^{50}-5A^{40}=\begin{bmatrix} 4 & -4\\ -2 & 2 \end{bmatrix}$

6 M

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 M

6(c)(ii) Find the unit vector in R³ orthogonal to both u={1,0,1} and v={0,1,1}.

4 M

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