Applied Mathematics - 3 - May 2013

Mechanical 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) Determine whether the function f(z)=cosh z is analytic or not. If so, find the derivative.(5 marks) 1 (b) Obtain the Laurent's expansion for the function $$ f(z)=\dfrac {e^{2z}}{(z-1)^3} about z=1 $$(5 marks) 1 (c) Find the inverse Laplace transform of -
$$ \dfrac {S \ e^{-2S}}{S^2 -6S+25} $$(5 marks) 1 (d) $$ If \ A=\begin{bmatrix}0 &1/\sqrt{2} &1/\sqrt{2} \\ \\ \sqrt{2}/\sqrt{3}&1/\sqrt{6} &1/\sqrt{6} \\ \\ 1/\sqrt{3}&1/\sqrt{3} &1/\sqrt{3} \end{bmatrix} \ find \ A^{-1} $$(5 marks) 2 (a) $$ Evaluate \ \int_{c} (z^2 +3z) dz $$ along the circle |z|=2 from (2,0) to (0,2).(6 marks) 2 (b) $$ Evaluate \ \int_{0}^{\infty}\dfrac {t^2 \sin 3t}{e^{2t}}dt $$(6 marks) 2 (c) Determine the value of λ for which the following system of equations possesses a non-trivial solution and obtain these solutions for each value of λ.
$$ 3x_{1}+x_{2}-\lambda x_{3}=0\\4x_{1}-2x_{2}-3x_{3}=0\\2\lambda x_{1}+4x_{2}+\lambda x_{3}=0 $$(8 marks) 3 (a) $$ Show \ that \ L\left \{erf \ \sqrt{t}\right \}=\dfrac {1}{S\sqrt{S+1}} \ hence \ deduce \ L\left \{t\cdot erf (2\sqrt{t}) \right \} $$(6 marks) 3 (b) Reduce to normal form and find the rank of :
\begin{bmatrix} 1 & 3& 5& 7\ 4& 6& 8& 10\ 15& 27& 39& 51\ 6& 12& 18& 24 \end{bmatrix}(6 marks) 3 (c) $$ Evaluate \ \int_c \dfrac {z^2}{z^4 -1}dz \ and \ \int_{c} \dfrac {dz}{z^{3}(z+4)} \ $$ where C is the circle |z|=2(8 marks) 4 (a) Find the residue of the function$$ f(z)= \dfrac {\sin \pi z^2 + \cos \pi z^2}{(z-1)(z-2)^{2}} $$ at their poles.(6 marks) 4 (b) Show that under the transformer$$ W=\dfrac {3-z}{z-2} $$ transforms the circle with centre $$ \left ( \dfrac {5}{2},0 \right ) and \ radius \dfrac {1}{2} $$ in the z-plane into imaginary axis in the W-plane.(6 marks) 4 (c) $$ Solve \ y^n(t)+9y(t)=18t \ if \ y(0)=1, y\left (\dfrac {\pi}{2} \right )=0 $$(8 marks) 5 (a) Find the orthogonal trajectory of the family of curves given by -
e^x cos y-xy=c(6 marks) 5 (b) Is the system of vectors $$ X_{1}= [2 2 1]^T, X_{2}=[1 3 1]^T, X_{3}=[1 2 2]^T \ linearly \ dependent? $$ (6 marks) 5 (c) $$ Evaluate \ \int^{2\pi}_{0}\dfrac {\sin^{2}\theta}{5-4 \cos \theta}d\theta\ $$(8 marks) 6 (a) Obtain the bilinear transformation that maps the points z=0, -i, I onto w=i, 1, 0(6 marks) 6 (b) Find the Laplace Transformation of the periodic function
$$ f(t)=\left\{\begin{matrix} t & 0<t<\pi\\ \pi -t& \pi<t<2\pi \end{matrix}\right. $$(6 marks) 6 (c) Prove that $$ u(x,y)=x^2 - y^2 \ and \ v(x,y)=\dfrac {-y}{x^{2}+y^{2}} $$ are both harmonic functions, but u+iv is not analytic (8 marks) 7 (a) Find the inverse Laplace Transform of$$ \dfrac {S^{2}+S}{(S^{2}+1)(S^{2}+2S+2)} $$using convolution theroem.(6 marks) 7 (b) Determine the analytic function f(z)=u+iv in terms of z, when it is given that 3u+2v=y²-x²+16xy(6 marks) 7 (c) Find the characteristics equation of the symmetric matrix -
$$ A=\begin{bmatrix}2 &-1 &1 \\ -1&2 &-1 \\ 1&-1 &2 \end{bmatrix} $$
Verify Cayley Hamilton theorem for A and A^-1(8 marks)