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MU Computer Engineering (Semester 4)
Applied Mathematics 4
December 2012
Total marks: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

1(a)

Find eA, If A =$$\begin{bmatrix} 3/2 & 1/2\\ 1/2& 3/2 \end{bmatrix}\\$$

5 M
1(b) Find the orthogonal trajectory of the family of curves x3y -xy3=c.
5 M
1(c) Integrate the function f(z) =x2 +iXY from A(1,1) to B(2,4) along the curves x=t, y=t2
5 M
1(d)

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.
5 M

2(a)

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

6 M
2(b)

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

7 M
2(c)

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

7 M

3(a)

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

6 M
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 M
3(c)

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

7 M

4(a)

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

where c is the circle

• |z| =1,
• |z+2|=1.
6 M
4(b)

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.

7 M
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 M

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 M
5(b)

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

7 M
5(c)

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

7 M

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 M
6(b)

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

7 M
6(c)

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

7 M

7(a)

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

6 M
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 M
7(c)

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

7 M

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