MU Applied Mathematics 4 - May 2014 Exam Question Paper

MU Electronics and Telecom Engineering (Semester 4)
Applied Mathematics 4
May 2014

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) prove that Eigen values of a hermitian matrix are real.

5 M

1(b) Evaluate

\int_{c} \frac{e^{k x}}{z} d z

$\displaystyle \int_{c}\dfrac{e^{kx}}{z}dz$ over the circle |z|=1 and k is real.hence prove that

\int_{0}^{π} c^{k c o s θ} c o s (k s i n θ) d θ = π

$\displaystyle\int_{0}^{\pi}c^{k cos \theta}cos(k sin \theta)d \theta =\pi$

5 M

1(c) Find the extremal of

\int_{x_{2}}^{x_{1}} (16 y^{2} - (y^{n})^{2} + x^{2}) d x

$\displaystyle \int_{x_{2}}^{x_{1}}(16y^{2}-(y^n)^{2}+x^{2})dx$

5 M

1(d) Find a vector orthogonal to both u =(-6,4,2)and v=(3,1,5).

5 M

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

$\displaystyle \int_{x_{1}}^{x_{2}}y\sqrt{1+(y')^{2}}dx$ is minimum subject to the constraint

$\displaystyle \int_{x_{1}}^{x_{2}}y\sqrt{1+(y')^{2}}dx=l.$

6 M

2(b)

Find eigen values and eigen vectors of the matrix $A=\begin{bmatrix} -2 & 5 & 4\\ 5& 7 &5 \\ 4& 5& -2 \end{bmatrix}$

6 M

2(c)

Obtain Taylors series and two distinct Laurents series expansion of $f(z)=\dfrac{z^{2}-1}{z^{2}+5z+6}$ about z=0, indicating region of convergence.

8 M

3(a)

State Cayley-Hamilton Theorem,hence deduce that A⁸=6251,where $A=\begin{bmatrix} 1&2 \\ 2 &-1 \end{bmatrix}$

6 M

3(b) Using calculus of Residues,prove that

$\displaystyle \int_{0}^{2pi}e^{cos \theta}\cos(sin \theta -n \theta)d \theta=\dfrac{2\pi}{n!}$

6 M

3(c) Find the plane curve of fixed perimeter and maximum area.

8 M

4(a) State Cauchy-Schwartz inequality and hence show that

$(x^{2}+y^{2}+z^{2})^{1/2} \ge\dfrac{1}{13}(3x+4y+12z),x,y,z$ are positive.

6 M

4(b) Reduce the quadratic form Q =x²+y²-2z²-4xy-2yz+10xz to to Canonical form using congruent transformation.

6 M

4(c) (ii)

Show that the matrix $A=\begin{bmatrix} 5 &-6 &-6 \\ -1&4 &2 \\ 3 &-6 &-4 \end{bmatrix}$ is Derogatory.

4 M

4(c)(i) If

$A=\begin{bmatrix} \pi/2 &3\pi/2 \\ \pi&\pi \end{bmatrix}$ , find Sin A

4 M

5(a) Using Rayleigh-Ritz method,find an appropriate solution for the extremal of the functional

$I \left [ y(x) \right ]=\int_{0}^{1}\left [ xy+\dfrac{1}{2}(y^{'})^{2}\right]dx$ subject to y(0)=y(1)=0.

6 M

5(b) Find an orthonormal basis of the following subspace of R³,S ={[1,2,0][0,3,1]}.

6 M

5(c)

Is the matrix $A=\begin{bmatrix} 2 & 1 &1 \\ 1& 2 &1 \\ 0 & 0 & 1 \end{bmatrix}$ diagonalizable.If so find diagonal form and transforming matrix.

8 M

6(a) Find f(3),f'(1+i),f"(1-i),if

$\displaystyle f(a) = (\int _{c}\dfrac{3x^{2}+11z+7}{z-a}dz$ c:|z|=2

6 M

6(b) Evaluate

$\displaystyle \int_{0}^{\infty}\dfrac{x^{3}sin x}{(x^{2}+z^{2})^{2}}$ using contour integration.

6 M

6(c) Find the singular value decomposition of the matrix

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

8 M

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