Find the value of μ which satisfy the equation A¹⁰⁰x=μ X, where \(A= \begin{bmatrix}2 &1 &-1 \\0 &-2 &-2 \\ 1 &1 &0 \end{bmatrix}\) 

Evaluate \[\int^{1+i}_{0} (x^2+iy)dz\ along\ y=x\ and\ y=x^2\]

Find the external of the function  \(\int^{x_2}_{x_1} [y^2-y^{'2} -2ycosh x]dx \)

Find eigen values and eigen vectors of :- \(A= \begin{bmatrix} 2 & 1 &1 \\2 &3 &2 \\3 &3 &4 \end{bmatrix}\)

Obtain Taylor's and two distinct Laurent's series expansion of \(f(z) = \dfrac {z-1}{z^2-2z-3} \) about z=0 indicating the region of convergence.

Verify Caley-Hamilton theorem for \(A= \begin{bmatrix}1 &2 &0 \\2 &-1 &0 \\0 &0 &-1 \end{bmatrix}\)  hence find \(A^{-2}\) 

Evaluate by using Residue theorem. \(\int^{2\pi}_0 \dfrac {d\theta}{(2+cos \theta )^2}\)

Solve the boundary value problem: \(I=\int_0^1(2xy-y^{2}-y'^{2})dx\)

given y(0)=y(1)=0 by Rayleigh Ritz method.

Reduce the following Quadrature form  \(Q=3x^2_1 + 5x_2^2 + 3x^2_3 -2x_1x_2-2x_2x_3 +2x_3x_1\)  into canonical form. Hence find its rank index and signature.

Show that the matrix  \(A= \begin{bmatrix} 7 &4 &-1 \\4 &7 &-1 \\-4 &-4 &4 \end{bmatrix}\) is derogatory .

Show that the matrix \(A= \begin{bmatrix} -9 &4 &4 \\-8 &3 &4 \\-16 &8 &7 \end{bmatrix}\) is diagnosable. Also find diagonal form and diagonalising matrix.

Evaluate  \(\int^{\infty}_{-\infty} \dfrac {cos 3x}{(x^2+1)(x^2+4)} dx\) using Cauchy Residue Theorem.

[ If \(\phi(\alpha)= \oint_{c}\dfrac {ze^z}{z-\alpha}dz\) where \(c \space is \space|z-2i|=3\)  find \(\phi (1), \phi '(2), \phi ' (3), \phi ' (4)\)