MU Applied Mathematics 4 - December 2015 Exam Question Paper

MU Electronics and Telecom Engineering (Semester 4)
Applied Mathematics 4
December 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) Find the extremal of

\int_{x_{4}}^{x_{1}} (2 x y - y^{- / 2}) d x

$\int_{x_4}^{x_1} (2xy - y^{-/2}) dx$

5 M

1 (b) Find an orthonormal basis for the subspaces of R³ by applying gram-Schmidt process where S={(1, 2, 0) (0, 3, 1)}.

5 M

1 (c) Show that Eigen values of unitary matrix are of unit modulus.

5 M

1 (d) Evaluate

\int \frac{d z}{z^{3} (z + 4)} where | z | = 4

$\int \dfrac {dz}{z^3 (z+4)} \text { where }|z|=4$

5 M

2 (a) Find the complete solution of

\int_{z_{0}}^{x_{1}} (2 x y - y^{1 / 2}) d x

$\int^{x_1}_{z_0} (2xy - y^{1/2})dx$

6 M

2 (b) Find the Eigen value and Eigen vectors of the matrix A^3 where

$A=\begin{bmatrix} 4 &6 &6 \\1 &3 &2 \\-1 &-5 &-2 \end{bmatrix}$

6 M

2 (c) Find expansion of

$f(z) = \dfrac {1} {(1+z^2)(z+2)}$ indicating region of convergence.

8 M

3 (a) Verify Cayley-Hamilton Theorem and find the value A⁶⁴ for the matrix

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

6 M

3 (b) Using Cauchy's Residue Theorem evaluate

$\int^\infty_{-\alpha} \dfrac {x^2}{x^6 +1}dx$

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) State and prove Cauchy-Schwartz inequality. Verify the inequality for vector u=(-4, 2, 1) and v=(8, -4, 2).

6 M

4 (b) Reduce the quadratic form xy+yz+zx to diagonal form through congruent transformation.

6 M

4 (c) If

$A= \begin{bmatrix} \frac {3}{2} & \frac {1}{2} \\ \frac {1}{2} & \frac {3}{2} \end{bmatrix}$ then find e^A and 4^A with the help of Modal Matrix.

8 M

5 (a) Solve the boundary value problem

$\int^1_0 (2xy+y^2 - y^2) dx, \ 0\le x \le 1, \ y(0)=0, \ y(1)=0$ by Rayleigh - Ritz Method.

6 M

5 (b) If W={∝; ∝∈Rⁿ and a₁ ≥ 0} a subset of V=Rⁿ with ∝=(a₁, a₂ ....... a_n) in Rⁿ (n≥3.). Show that W is not a subspace of V by giving suitable counter example.

6 M

5 (c) Show that the matrix

$A=\begin{bmatrix} 8 &-8 &-2 \\4 &-3 &-2 \\3 &-4 &1 \end{bmatrix}$ is similar to diagonal matrix. Find the diagonalising matrix and diagonal form.

8 M

6 (a) State and prove Cauchy's integral Formula for the simply connected region and hence evaluate

$\int \dfrac {z+6}{z^2-4}dz, |z-2|=5$

6 M

6 (b) Show that

$\int^{2\pi}_0 \dfrac {\sin^2 \theta}{a+ b\cos \theta}d \theta = \dfrac {2\pi}{b^2} (a- \sqrt{a^2 - b^2}), \ 0 < b< a.$

6 M

6 (c) Find the Singular value decomposition of the following matrix

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

8 M

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