STUPIDSID
1(a)
Find the extremal of the functional \[\int ^1_0[y'^{2}+12xy]dx\ \text {subject to}\ y(0)=0\ \text{and}\ y(1)=1).\]
5 M
1(b)
Verify Cauchy - Schwartz inequality for u = (1, 2, 1) and v = (3, 0, 4) also find the angle between u & v.
5 M
1(c)
If &lambda & X are eigen values and eigen vectors of A then prove that \( \dfrac{1}{\lambda} \) and X are eigen values and eigen vectors of A-1, provided A is non singular matrix.
5 M
1(d)
Evaluate \( \int _C \dfrac{e^{2z}}{(z+1)^4}dz\ \text{where}\ C:|z|=2 \).
5 M
2(a)
Find the extremal that minimises the integral \[\int ^{x_1}_{x_0}(16y^2-y''^2)dx\]
6 M
2(b)
Find eigen values and eigen vectors of A3 \[\text{where}\ A=\begin{bmatrix}
2 & 1 & 1\\
2 & 3 & 2\\
3 & 3 & 4
\end{bmatrix}\]
6 M
2(c)
Obtain Taylor's and two distinct laurent's expansion of \( f(z)=\dfrac{z-1}{z^2-2z-3} \) indicating the region of convergence.
8 M
3(a)
Verify Cayley-Hamilton Theorem for \[A=\begin{bmatrix}
2 & -1 & 1\\
-1 & 2 & -1\\
1 & -1 & 2
\end{bmatrix}\text{and hence find}A^{-1}\]
6 M
3(b)
Using Cauchy Residue Theorem, evaluate \[\int ^{\infty} _{-\infty }\dfrac{x^2-x+2}{x^4+10x^2+9}dx\]
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)
Find an orthonormal basis for the subspace of R3 by appling Gram-Schmidt process where S {(1, 1, 1), (0, 1, 1) (0, 0, 1)}.
6 M
4(b)
Find A50, where \[A=\begin{bmatrix}
2 & 3\\
-3 & -4
\end{bmatrix}\]
6 M
4(c)
Reduce the following Quadratic form into canonical form & hence find its rank, index, signature and value class where,
Q = 3x12+5x22+3x32-2x1x2-2x2x3+2x3x1
Q = 3x12+5x22+3x32-2x1x2-2x2x3+2x3x1
8 M
5(a)
Using the Rayleigh-Ritz method, find an approximate solution for the extremal of the functional \( \int ^1_0 \left \{ xy+\dfrac{1}{2}y'^2 \right \}dx \) subject to y(0)=y(1)=0.
6 M
5(b)
Prove that W = {(x, y)| x=3y} subspace of R2. Is W1 = {a, 1, 1)| a in R} subspace of R3?
6 M
5(c)
Prove that A is diagonizable matrix. Also find diagonal form and transforming matrix where \( A=\begin{bmatrix}
1 & -6 & -4\\
0 & 4 & 2\\
0 & -6 & -3
\end{bmatrix} \)
8 M
6(a)
By using Cauchy Residue Theorem, evaluate \( \int ^{2\pi}_0 \dfrac{\cos ^2 \theta}{5+4\cos \theta}d\theta. \)
6 M
6(b)
Evaluate \( \int _C \dfrac{z+4}{z^2+2z+5}dz\ \text{where}\ C:|z+1+i|=2. \)
6 M
6(c)(i)
Determine the function that gives shortest distance between two given points.
5 M
6(c)(ii)
Express any vector (a, b, c) in R3 as a linear combination of v1, v2, v3 where v1, v2, v3 are in R3.
3 M
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