STUPIDSID
1(a)
Find the extremal of the functional
\( \displaystyle \int ^1_0[y'^2+12xy]dx \) subject to y(0) = 0 and y(1) = 1.
\( \displaystyle \int ^1_0[y'^2+12xy]dx \) subject to y(0) = 0 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 the 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 \( \int _C \dfrac{e^{2x}}{(z+1)^4}dz \) where C : |z| = 2
5 M
2(a)
Find the extremal that minimise the integral \[\displaystyle \int ^{x_1}_{x_0}(16y^2-y^{''2})dx\]
6 M
2(b)
Find eigrn 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
\( A=\begin{bmatrix} 2 & -1 & 1\\ -1 & 2 & -1\\ 1 & -1 & 2 \end{bmatrix} \) and hence find A-1
\( A=\begin{bmatrix} 2 & -1 & 1\\ -1 & 2 & -1\\ 1 & -1 & 2 \end{bmatrix} \) 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 orthonomal 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 \( \displaystyle \int ^1_0 \left \{ xy +\dfrac{1}{2}y'^2\right \}dx \) 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 us 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 \( \displaystyle \int ^{2\pi}_0 \dfrac{\cos^2 \theta}{5+4\cos \theta}d\theta. \)
6 M
6(b)
Evaluate \( \int _C \dfrac{z+4}{z^2+2z+5} dz \) 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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