STUPIDSID
1 (a)
Find the extremal of \( \int_{x_4}^{x_1} (2xy - y^{-/2}) dx \)
5 M
1 (b)
Find an orthonormal basis for the subspaces of R3 by applying gram-Schmidt process where S={(1, 2, 0) (0, 3, 1)}.
5 M
1 (c)
Show that Eigen values of unitary matrix are of unit modulus.
5 M
1 (d)
Evaluate \( \int \dfrac {dz}{z^3 (z+4)} \text { where }|z|=4 \)
5 M
2 (a)
Find the complete solution of \( \int^{x_1}_{z_0} (2xy - y^{1/2})dx \)
6 M
2 (b)
Find the Eigen value and Eigen vectors of the matrix A^3 where \( A=\begin{bmatrix}
4 &6 &6 \\1
&3 &2 \\-1
&-5 &-2
\end{bmatrix} \)
6 M
2 (c)
Find expansion of \( f(z) = \dfrac {1} {(1+z^2)(z+2)} \) indicating region of convergence.
8 M
3 (a)
Verify Cayley-Hamilton Theorem and find the value A64 for the matrix \( A= \begin{bmatrix}
1 &2 \\2
&-1
\end{bmatrix}\)
6 M
3 (b)
Using Cauchy's Residue Theorem evaluate \( \int^\infty_{-\alpha} \dfrac {x^2}{x^6 +1}dx \)
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)
State and prove Cauchy-Schwartz inequality. Verify the inequality for vector u=(-4, 2, 1) and v=(8, -4, 2).
6 M
4 (b)
Reduce the quadratic form xy+yz+zx to diagonal form through congruent transformation.
6 M
4 (c)
If \( A= \begin{bmatrix} \frac {3}{2} & \frac {1}{2} \\ \frac {1}{2} & \frac {3}{2} \end{bmatrix} \) then find eA and 4A with the help of Modal Matrix.
8 M
5 (a)
Solve the boundary value problem \( \int^1_0 (2xy+y^2 - y^2) dx, \ 0\le x \le 1, \ y(0)=0, \ y(1)=0 \) by Rayleigh - Ritz Method.
6 M
5 (b)
If W={∝; ∝∈Rn and a1 ≥ 0} a subset of V=Rn with ∝=(a1, a2 ....... an) in Rn (n≥3.). Show that W is not a subspace of V by giving suitable counter example.
6 M
5 (c)
Show that the matrix \( A=\begin{bmatrix}
8 &-8 &-2 \\4
&-3 &-2 \\3
&-4 &1
\end{bmatrix} \) is similar to diagonal matrix. Find the diagonalising matrix and diagonal form.
8 M
6 (a)
State and prove Cauchy's integral Formula for the simply connected region and hence evaluate \[ \int \dfrac {z+6}{z^2-4}dz, |z-2|=5 \]
6 M
6 (b)
Show that \( \int^{2\pi}_0 \dfrac {\sin^2 \theta}{a+ b\cos \theta}d \theta = \dfrac {2\pi}{b^2} (a- \sqrt{a^2 - b^2}), \ 0 < b< a. \)
6 M
6 (c)
Find the Singular value decomposition of the following matrix \( A=\begin{bmatrix} 1 &2 \\1 &2 \end{bmatrix} \)
8 M
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