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 e^x 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)