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)

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