MORE IN Applied Mathematics - 3
MU Chemical Engineering (Semester 3)
Applied Mathematics - 3
December 2015
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 the Laplace Transform of cos t cos 2t cos 3t.
5 M
1(b) If $$A=\begin{bmatrix} 1 & 4\\ 1 & 1 \end{bmatrix}$$, find A7-9A2+I.
5 M
1(c) Show that $$\oint _c \log z dz=2\pi i$$ , where C is the unit circle |z|=1.
5 M
1(d) The incidence of an occupational disease in an industry is such that workers have 20% chance of suffering from it. What is the probability that out of 6 workers 4 or more will catch the disease?
5 M

2(a) Find an analytic function whose real part is e-x (y cos y-x sin y).
6 M
2(b) Evaluate $$\int ^{\infty}_0e^{-1}\int ^t_0\dfrac{\sin u}{u}du\ dt$$
6 M
2(c) Find the orthogonal matrix that will diagonalise the matrix $$A=\begin{bmatrix} 7 & 0 & -2\\ 0 & 5 & -2\\ -2 & -2 & 6 \end{bmatrix}$$
8 M

3(a) Find the inverse Laplace Transform of $$\dfrac{s+4}{(s^2-1)(s+1)}$$
6 M
3(b) Check whether the matrix $$A=\begin{bmatrix} 5 & -6 & -6\\ -1 & 4 & 2\\ 3 & -6 & -4 \end{bmatrix}$$ is derogatory or not.
6 M
3(c) Using Kuhn-Tucker conditions solve the following NLPP
Maximize Z=2x1 +x2 -$$x_{1}^{2}$$ Subject to 2x1+3x2≤6; 2x1+x2≤4; x1, x2≥0.
8 M

4(a) Find the bilinear transformation which maps the points z=2, i, -2 onto the points w=1, i, -1.
6 M
4(b) Find the orthogonal trajectory of the family of curves given by ex cos y-xy=c.
6 M
4(c) Solve the following NLPP using Lagrange's multipliers method
Optimise Z=4x1+8x2-$$x_{1}^{2}$$ -$$x_{2}^{2}$$
Subject to x1+x2=4; x1, x2 ≥0.
8 M

5(a) Find the Eigen values and Eigen vectors of $$A=\begin{bmatrix} 2 & -1 & 1\\ 1 & 2 & -1\\ 1 & -1 & 2 \end{bmatrix}$$.
6 M
5(b) Evaluate $$\oint _{c} \dfrac{x+2}{x^3-2x^2}dz$$ where C is the circle |z-2-i|=2.
6 M
5(c) Find the inverse Laplace Transform of (i) $$\log \left ( \dfrac{s^2+a^2}{s^2+b^2} \right )$$
(ii) $$\dfrac{(s+1)e^{-s}}{(s^2+s+1)}$$
8 M

6(a) Find the poles and calculate the residues at them for $$f(z)=\dfrac{z}{(z-1)(z+2)^2}$$.
6 M
6(b) From the following data calculate Spearman's rank correlation coefficient between X and Y
X: 36  56  20  42  33  44  50  15  60
Y:50  35  70  58  75  60  45  80  38 .
6 M
6(c) Reduce the following quadratic form to canonical form. Also find it's rank and signature x2 +2y2 +2z2 -2xy -2yz+zx.
8 M

More question papers from Applied Mathematics - 3