MU Applied Mathematics 4 - May 2015 Exam Question Paper

MU Electronics and Telecom Engineering (Semester 4)
Applied Mathematics 4
May 2015

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) Evaluate

i n t_{c} | z | d z

$int_c |z| dz$ where c is the left half of unit circle |z|=1 from z=-i to z=i

5 M

1 (b) If λ is an Eigen value of the matrix A with corresponding Eigen vector X. Prove that λⁿ is an Eigen value of Aⁿ with corresponding Eigen vector X.

5 M

1 (c) Find the external of

$\int^{x_2}_{x_1} \dfrac {\sqrt{1+y'^2}}{x} dx$

5 M

1 (d) Find the unit vector orthogonal to both [1, 1, 0] & [0, 1, 1]

5 M

2 (a) Find the curve on which the functional

$\int^1_0 [y'^2 +12 x y$ dx \] with y(0)=0 & y(1)=1 can be Extremised.

6 M

2 (b) Find the Eigen values and Eigen vectors for the matrix

$\begin{bmatrix} 2 &2 &1 \\1 &3 &1 \\1 &2 &2 \end{bmatrix}$

6 M

2 (c) Obtain two distinct Laurent's series expansions of

$f(z) = \dfrac {2z-3} {z^2 -4z +3}$ in power of (z-4) indicating the region of convergence in each case.

8 M

3 (a)

$if \ A= \begin{bmatrix} 2 &1 \\1 &2 \end{bmatrix} \ find \ A^{20}$

6 M

3 (b) Evaluate

$\int_c \dfrac {\sin Π z^2 - \cos \pi z^2}{(z-1)(z-2)}dz,$ where c is the circle |z|=3.

6 M

3 (c) Using Reyleigh-Ritz method, find an approximate solution for the external of the functional

$I(y) = \int^1_0 (y'^2-2y-2xy) dx$ subject to y(0)=2, y(1)=1 \]

8 M

4 (a) Find the vector orthogonal to both [-6, 4, 2] & [3, 1, 5]

6 M

4 (b) Show that the matrix

$A= \begin{bmatrix} 7 &4 &-1 \\4 &7 &-1 \\-4 &-4 &4 \end{bmatrix}$ is derogatory and find is minimal polynomial.

6 M

4 (c) Reduce the matrix of the quadratic form [ 6x^2_1 + 3x^2_2 + 3x^2_3 - 4x_1x_2-2x_2x_3 ] to canonical form through congruent transformation and find its rank, signature, and value class.

8 M

5 (a) Find the external of

$\int^{x_1}_{x_0} (2xy-y''^2) dx$

6 M

5 (b) Show that the set W={[x,y,z] | y=x+z} is a subspace of Rⁿ under the usual addition and scalar multiplication.

6 M

5 (c) Show that the following matrix

$A= \begin{bmatrix} 6 &-2 &2 \\-2 &3 &-1 \\2 &-1 &3 \end{bmatrix}$ is diagonalisable. Also find the diagonal form and a diagonalising matrix.

8 M

6 (a)

$If f(a) = \int_c \dfrac {3z^2 = 7z +1}{z-a} dz$ where c is a circle |z|=2, find the values of i) f(-3), ii) f(i), iii) f'(1-i)

6 M

6 (b) Evaluate

$\int^{2\pi}_0 \dfrac {d\theta}{13+5 \sin \theta }$

6 M

6 (c) Verify Caylex-Hamilton theorem for the matrix A and hence find A^-1 and A⁴ where

$A= \begin{bmatrix} 1 &2 &-2 \\-1 &3 &0 \\0 &-2 &1 \end{bmatrix}$

8 M

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