Question Paper: Applied Mathematics 4 : Question Paper Dec 2012 - Computer Engineering (Semester 4) | Mumbai University (MU)
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## Applied Mathematics 4 - Dec 2012

### Computer Engineering (Semester 4)

TOTAL MARKS: 80
TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Assume data if required.
(4) Figures to the right indicate full marks.
1(a)

Find eA, If A =$\begin{bmatrix} 3/2 & 1/2\\\\ 1/2& 3/2 \end{bmatrix}\\\\$\lt/span\gt\lt/p\gt \lt/span\gt\ltspan class='paper-ques-marks'\gt(5 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt1(b)\lt/b\gt Find the orthogonal trajectory of the family of curves x\ltsup\gt3\lt/sup\gty -xy\ltsup\gt3\lt/sup\gt=c.\lt/span\gt\ltspan class='paper-ques-marks'\gt(5 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt1(c)\lt/b\gt Integrate the function f(z) =x\ltsup\gt2\lt/sup\gt +iXY from A(1,1) to B(2,4) along the curves x=t, y=t\ltsup\gt2\lt/sup\gt\lt/span\gt\ltspan class='paper-ques-marks'\gt(5 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt1(d)\lt/b\gt \ltp\gtConsider the following problem:\ltbr\gt maximise Z=2x\ltsub\gt1\lt/sub\gt-2x\ltsub\gt1\lt/sub\gt - 2x\ltsub\gt2\lt/sub\gt + 4x\ltsub\gt3\lt/sub\gt-5x\ltsub\gt4\lt/sub\gt\ltbr\gt Subject to x\ltsub\gt1\lt/sub\gt + 4x\ltsub\gt2\lt/sub\gt-2x\ltsub\gt3\lt/sub\gt + 8x\ltsub\gt4\lt/sub\gt \ltspan class="math-tex"\gt$\le$\lt/span\gt 2\ltbr\gt -x\ltsub\gt1\lt/sub\gt +2x\ltsub\gt2\lt/sub\gt+3x\ltsub\gt3\lt/sub\gt+4x\ltsub\gt4\lt/sub\gt\ltspan class="math-tex"\gt$\le$\lt/span\gt1 and x\ltsub\gt1\lt/sub\gt,x\ltsub\gt2\lt/sub\gt,x\ltsub\gt3\lt/sub\gt,x\ltsub\gt4\lt/sub\gt \ltspan class="math-tex"\gt$\ge$\lt/span\gt0.\lt/p\gt \ltp\gtDETERMINE:\lt/p\gt \ltul\gt \ltli\gtall basic solution.\lt/li\gt \ltli\gtall feasible basic solutions.\lt/li\gt \ltli\gtoptimal feasible basic solution.\lt/li\gt \lt/ul\gt \lt/span\gt\ltspan class='paper-ques-marks'\gt(5 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt2(a)\lt/b\gt \ltp\gtIf f(z) = u+iv is analytic and \ltspan class="math-tex"\gt$u+v =\dfrac{2sin2x}{e^{2y}+e^{-2y}-2cos2x}\\$find f(z)

(6 marks) 2(b)

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.

(6 marks)
3(b)

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|}$​ ​​ ​ is not analytic at the origin although Cauchy-Riemann equation are satisfied at that point. (7 marks) 4(a) Evaluate$\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. (7 marks) 4(c) 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$(7 marks) 5(a) Use the dual simplex method to solve the following LPP- Maximize Z =-3x1-2x2 Subject to x1+ x2$\ge$1 x1+ x2$\le$7 x1+ 2x2$\le$10 x2$\le$3 and x1, x2$\ge$0 (6 marks) 5(b) Evaluate$\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 (7 marks) 6(a) Obtain Taylor's or Laurent's series for the function-$f(z) =\dfrac{1}{(1+z^{2})(z+2)} for(i) \space 1<|z|<2 \space and (ii)\space |z| >2.$(6 marks) 6(b) 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 (7 marks) 7(a) Find the bilinear transformation which maps the point 2,;-2 onto the points 1,i,-1. (6 marks) 7(b) 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,x2,x3$\ge$0. (7 marks) 7(c) Verify Laplace's equation for$ u =\left(r+\dfrac{a^{2}}{r}\right)cos\theta\$.also find v and f(z).

(7 marks)