Compute A⁹-6A⁸+10A⁷-3A⁶+A+I
where,A =$ \begin{bmatrix} 1 &2 & 3\\\\ -1& 4 & 1\\\\ 1&0 & 3 \end{bmatrix}\\\\$\lt/span\gt\lt/p\gt \lt/span\gt\ltspan class='paper-ques-marks'\gt(7 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt2(c)\lt/b\gt \ltp\gt\ltspan class="math-tex"\gt$y\le0$\lt/span\gtSolve the following LPP by simplex method-\ltbr\gt Minimise  \ltspan class="math-tex"\gt$ Z =x_1-3x_2+3x_3$\lt/span\gt\ltbr\gt subject to  \ltspan class="math-tex"\gt$3x_1-x_2+2x_3\le7$\lt/span\gt\ltbr\gt \ltstrong\gt                  \ltspan class="math-tex"\gt$2x_1+4x_2\ge-12$\lt/span\gt\lt/strong\gt\ltbr\gt \ltsub\gt                     \lt/sub\gt\ltspan class="math-tex"\gt$ -4x_1+3x_2+8x_3\le10$\lt/span\gt\ltbr\gt \ltsub\gt                         \lt/sub\gt\ltspan class="math-tex"\gt$ x1,x2,x3 \ge0$\lt/span\gt\lt/p\gt \lt/span\gt\ltspan class='paper-ques-marks'\gt(7 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt3(a)\lt/b\gt \ltp\gtShow that A =\ltspan class="math-tex"\gt$\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 $.

(7 marks) 3(c)

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

Evaluate $\int_{c}\dfrac{z+6}{z^{2}-4}dz\\\\$\lt/span\gt\lt/p\gt \ltp\gtwhere c is the circle\lt/p\gt \ltul\gt \ltli\gt|z| =1,\lt/li\gt \ltli\gt|z+2|=1.\lt/li\gt \lt/ul\gt \lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4(b)\lt/b\gt \ltp\gtShow that the matrix A =\ltspan class="math-tex"\gt$\begin{bmatrix} 1 & -6 &-4 \\ 0 & 4 & 2\\ 0 & -6 & -3 \end{bmatrix}\\$ 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 $\int_{0}^{2\pi}\dfrac{d\theta}{5+3sin\theta}\\\\$\lt/span\gt\lt/p\gt \lt/span\gt\ltspan class='paper-ques-marks'\gt(7 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5(c)\lt/b\gt \ltp\gtFind the characteristics equation of the matrix\ltspan class="math-tex"\gt$ \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
$z =x_{1} +2x_{3}+x_{2}x_{3}-x_{1}^{2}-x_{2}^{2}-x_{3}^{2}.\\\\$\lt/span\gt\lt/p\gt \lt/span\gt\ltspan class='paper-ques-marks'\gt(7 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6(c)\lt/b\gt \ltp\gtEvaluate\ltspan class="math-tex"\gt$ \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).