STUPIDSID
1 (a)
Evaluate \[ int_c |z| dz \] 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 \[ \int^1_0 [y'^2 +12 x y \mathrm {dx} \] 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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