Find e^A, If A =\(\begin{bmatrix} 3/2 & 1/2\\ 1/2& 3/2 \end{bmatrix}\\\)

Consider the following problem:
maximise Z=2x₁-2x₁ - 2x₂ + 4x₃-5x₄
Subject to x₁ + 4x₂-2x₃ + 8x₄ \(\le\) 2
-x₁ +2x₂+3x₃+4x₄\(\le \)1 and x₁,x₂,x₃,x₄ \(\ge\)0.

DETERMINE:

• all basic solution.
• all feasible basic solutions.
• optimal feasible basic solution.

If f(z) = u+iv is analytic and \(u+v =\dfrac{2sin2x}{e^{2y}+e^{-2y}-2cos2x}\\\)find f(z)

Compute A⁹-6A⁸+10A⁷-3A⁶+A+I
where,A =\( \begin{bmatrix} 1 &2 & 3\\ -1& 4 & 1\\ 1&0 & 3 \end{bmatrix}\\\)

\(y\le0\)Solve the following LPP by simplex method-
Minimise  \( Z =x_1-3x_2+3x_3\)
subject to  \(3x_1-x_2+2x_3\le7\)
                  \(2x_1+4x_2\ge-12\)
                     \( -4x_1+3x_2+8x_3\le10\)
                         \( x1,x2,x3 \ge0\)

Show that A =\(\begin{bmatrix} -6& -2 &2 \\ -2& 3 &-1 \\ 2& -1 &3 \end{bmatrix}\\\) is derogatory and find its minimal polynomial.

Slove the following LPP by Big M-method -
Minimize Z =\( 2x_1+x_2\)

subject to
                  \(3x_1+x_2 =3\\[2ex] 4x_1+ 3x_2\ge 6\\[2ex] x_1+2x_2 \le3 \\[2ex]and\space x1,x2 \ge0 \).

Show that \(f(z) =\sqrt{|x y|}\)​ ​​ ​ is not analytic at the origin although Cauchy-Riemann equation are satisfied at that point.

Evaluate \(\int_{c}\dfrac{z+6}{z^{2}-4}dz\\\)

where c is the circle

• |z| =1,
• |z+2|=1.

Show that the matrix A =\(\begin{bmatrix} 1 & -6 &-4 \\ 0 & 4 & 2\\ 0 & -6 & -3 \end{bmatrix}\\\) is similar to a diagonal matrix.also find the transforming matrix and the diagonal matrix.

Using Duality solve the following LPP-
Minimise z =4x₁ +3x₂ +6x₃
Subject to x₁+x₃\(\ge\) 2
x₂ +x₃\(\ge\)5
and x₁,x₂,x₃\(\ge0\)

Use the dual simplex method to solve the following LPP-
Maximize Z =-3x₁-2x₂
Subject to x₁+ x₂ \(\ge\) 1
x₁+ x₂ \(\le\) 7
x₁+ 2x₂ \(\le\)10
x₂ \(\le\)3
and x₁, x₂ \(\ge\)0

Evaluate \(\int_{0}^{2\pi}\dfrac{d\theta}{5+3sin\theta}\\\)

Find the characteristics equation of the matrix\( \begin{bmatrix} 1 &2 & -2\\ -1& 3&0 \\ 0& -2 & 1 \end{bmatrix}\\\)and verify that is satisfied by A and hence ,obtain A^-1

Obtain Taylor's or Laurent's series for the function-
\(f(z) =\dfrac{1}{(1+z^{2})(z+2)} for(i) \space 1<|z|<2 \space and (ii)\space |z| >2.\)

Obtain the relative maximum or minimum (if any) of the function
\(z =x_{1} +2x_{3}+x_{2}x_{3}-x_{1}^{2}-x_{2}^{2}-x_{3}^{2}.\\\)

Evaluate\( \int_{c}\dfrac{z^{2}}{(z-1)^{2}(z-2)}dz\\\) where c is the circle |z| =2.5

Find the bilinear transformation which maps the point 2,;-2 onto the points 1,i,-1.

Using the method of Lagrangian multipliers solve the following problem
Optimise Z =4x₁²+2x₂²+x₃²-4₁x₂
Subjected to x₁+x₂+x₃ =15
2x₁-x₂+2x₃=20
x₁,x_2,x₃\(\ge\)0.

Verify Laplace's equation for \( u =\left(r+\dfrac{a^{2}}{r}\right)cos\theta\).also find v and f(z).