STUPIDSID
1(a)
if f(x) is an algebraix polynomial in x and λ is an eigen value and X is the the corresponding eigen vector of a square matrix A then f(λ) is an eigen value X is the corresponding eigenvector of f (A).
5 M
1(b)
Find the extremal of \[\int_{x_0}^{x^1}\left ( x +y' \right )y'dx\]
5 M
1(c)
Express (6, 11, 6) as linear combination of v1=(2,1,4), v2=(1,-1,3), v3=(3,2,5).
5 M
1(d)
Evaluate \( \int _C\frac{z}{\left ( z-1 \right )^2\left ( z-2 \right )}dz,\)/ where C is the circle |z-2|=0.5
5 M
2(a)
Find the curve y= f(x) for which\(\int_{0}^{\pi }\left ( y'^2 -y^2\right )dx \)/ is extremum if \[\int_{0}^{\pi }ydx=1\]
6 M
2(b)
Evaluate \[\int_{0}^{2\pi }\frac{\cos 3\theta }{5+4\cos \theta }d\theta \]
6 M
2(c)
Find the singular value decomposition of \[\begin{bmatrix}
2 & 3\\
0 & 2
\end{bmatrix}\]
6 M
3(a)
Verify Cayley Hamilton theorem for \( A=\begin{bmatrix}
3 & 10 &5 \\
-2& -3&-4 \\
3& 5 & 7
\end{bmatrix} \)/ and hence, find the matrix represented by \[A^6-GA^5+9A^4+4A^3-12A^2+2A-1\]
6 M
3(b)
Construction an orthonormal basis of R3 using Gram Schmidt process to S={(3,0,4),(-1,0,7),(2,9,11)}
6 M
3(c)
Find all possible Laurent's expansions \( \frac{z}{\left ( z-1 \right )\left ( z-2 \right )} \)/ about z=-2 indicating the region of covergence.
8 M
4(a)
Reduce the quadratic from\( 2x^2-2y^2+2z^2-2xy-8yz+6zx \)/ to canonical from and hence find its rank, index and signature and value class.
6 M
4(b)
If \(\phi (\alpha )\int _C\frac{4z^2+z+5}{z-\alpha }dz, \)/ where C is the contour of the ellipse \( \frac{x^2}{4}+\frac{y^2}{9}=1, \)/ find the values of \[\phi (3.5),\phi (i),\phi (-1),\phi (-i)\]
6 M
4(c)
Using Rayleigh-Riz method, solve the boundry value problem\[I=\int_{0}^{1}\left ( y'^2-y^2-2xy \right )dx;0\leq x\leq 1,\] given y(0)=y(1)=0.
8 M
5(a)
Find the extremal of the function \( \int_{0}^{\pi /2}\left ( 2xy+y^2-y'^2 \right )dx;\)/ with y(0)=0, y(π/2)=0
6 M
5(b)
Find the orthogonal matrix P that diagonalises \[A=\begin{bmatrix}
4 & 2&2 \\
2 & 4 & 2\\
2 & 2 & 4
\end{bmatrix}\]
6 M
5(c)
Using Cauchy's Residue theorem, evaluate \( \oint _C\frac{z^2+3}{z^2-1}dz \)/ where C is the circle (i) |z-1|=1
(ii) |z+1|=1.
(ii) |z+1|=1.
8 M
6(a)
Find the sum of the residues points of \[f(z)\frac{z}{\left ( z-1 \right )^2\left ( z^2-1 \right )}\]
6 M
6(b)
If \( A=\begin{bmatrix}
1 &4 \\
2 & 3
\end{bmatrix}\)/, prove that \[A^{50}-5A^{40}=\begin{bmatrix}
4 & -4\\
-2 & 2
\end{bmatrix}\]
6 M
6(c)(i)
Checy whether W={(x,y,z)|y=x+z,x,y,z are in R} is a subspace R3 with usual addition and usual multiplication.
4 M
6(c)(ii)
Find the unit vector in R3 orthogonal to both u={1,0,1} and v={0,1,1}.
4 M
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