STUPIDSID
1(a)
prove that Eigen values of a hermitian matrix are real.
5 M
1(b)
Evaluate\[ \displaystyle \int_{c}\dfrac{e^{kx}}{z}dz \] over the circle |z|=1 and k is real.hence prove that\[ \displaystyle\int_{0}^{\pi}c^{k cos \theta}cos(k sin \theta)d \theta =\pi \]
5 M
1(c)
Find the extremal of \[ \displaystyle \int_{x_{2}}^{x_{1}}(16y^{2}-(y^n)^{2}+x^{2})dx \]
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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