Question Paper: Applied Mathematics - 3 : Question Paper Dec 2015 - Chemical Engineering (Semester 3) | Mumbai University (MU)
0

## Applied Mathematics - 3 - Dec 2015

### Chemical Engineering (Semester 3)

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 the Laplace Transform of cos t cos 2t cos 3t.(5 marks) 1(b) If $A=\begin{bmatrix} 1 & 4\\\\ 1 & 1 \end{bmatrix}$, find A7-9A2+I.(5 marks) 1(c) Show that $\oint _c \log z dz=2\pi i$ , where C is the unit circle |z|=1.(5 marks) 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 marks) 2(a) Find an analytic function whose real part is e-x (y cos y-x sin y).(6 marks) 2(b) Evaluate $\int ^{\infty}_0e^{-1}\int ^t_0\dfrac{\sin u}{u}du\ dt$(6 marks) 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 marks) 3(a) Find the inverse Laplace Transform of $\dfrac{s+4}{(s^2-1)(s+1)}$(6 marks) 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 marks) 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 marks)
4(a) Find the bilinear transformation which maps the points z=2, i, -2 onto the points w=1, i, -1.(6 marks) 4(b) Find the orthogonal trajectory of the family of curves given by ex cos y-xy=c.(6 marks) 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 marks)
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 marks) 5(b) Evaluate $\oint _{c} \dfrac{x+2}{x^3-2x^2}dz$ where C is the circle |z-2-i|=2.(6 marks) 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 marks)
6(a) Find the poles and calculate the residues at them for $f(z)=\dfrac{z}{(z-1)(z+2)^2}$.(6 marks) 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 marks)
6(c) Reduce the following quadratic form to canonical form. Also find it's rank and signature x2 +2y2 +2z2 -2xy -2yz+zx.(8 marks)