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MU Electronics 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

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1(a) prove that Eigen values of a hermitian matrix are real.
5 M
1(b) Evaluate \[\oint_{c} \dfrac{e^{kx}}{z}dz\]over the circle |z|=1 and k is real. Hence prove that \[\int_{0}^{\pi}c^{k \cos \theta}\cos(k \sin \theta)d \theta =\pi\]
5 M
1(c) Find the extremal of \[\int_{x_{2}}^{x_{1}}(16y^{2}-(y)^{2}+x^{2})dx\]
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 \[\int_{x_{1}}^{x_{2}}y\sqrt{1+(y)^{2}}dx\]is minimum subject to the constraint \[\int_{x_{1}}^{x_{2}}\sqrt{1+(y)^{2}}dx=l\]
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 \[\int_{0}^{2\pi}e^{\cos \theta}\cos(\sin \theta -n \theta)d \theta=\dfrac{2\pi}{n!}\]
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 f(a) =\(\int _{c}\dfrac{3x^{2}+11z+7}{z-a}dz \),c:|z|=2
6 M
6(b)

Evaluate\( \int_{0}^{\infty}\dfrac{x^{3}sin x}{(x^{2}+z^{2})^{2}}\)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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