Applied Mathematics - 3 - Dec 2012

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) Discuss the analyticity of the function $$f(z)=\dfrac {1}{z-1}$$(5 marks) 1 (b) Obtain Laurent's expansion for the function $$f(z)=\dfrac {1}{z^2 \sin h \ z}$$(5 marks) 1 (c) $$ Prove \ that \ L\left \{ \sqrt[2]{\dfrac {t}{P}}\right \}=\dfrac {1}{s^{2/3}}\ and \ hence \ show \ that \ L \left \{ \dfrac {1}{\sqrt{\pi t}} \right \}=\dfrac {1}{\sqrt{s}} $$(5 marks) 1 (d) The matrix A is given by $$ A=\begin{bmatrix}1 &2 &-3 \\ 0&3 &2 \\ 0&0 &-2 \end{bmatrix}$$ Find the eigen values of $$3A^3+5A^2-6A+2I-6SA^{-1} $$(5 marks) 2 (a) Evaluate $$\int^{4+2i}_{0} \bar{z}\ dz $$along the path of the line from 0 to I and then to 4+2i.(6 marks) 2 (b) Evaluate the integral $$ \int^{\infty}_{t=0}\int^{t}_{u=0}\dfrac {e^t \sin u}{u}du \ dt $$(6 marks) 2 (c) Discuss for the values of k, the following system of equations possesses trivial and non-trivial solutions -
$$ 2x+3ky+(3k+4)z=0 \\\ltbr\gt x+(k+4)y+(4k+2)z=0\\ \ltbr\gt x+2(k+1)y+(3k+4)z=0 $$(8 marks) 3 (a) $$ Show \ that \ L^{-1}\left \{ \dfrac {1}{s} \cos \left ( \dfrac {1}{s} \right )\right \}=1 \dfrac {t^2}{(2!)^2}+\dfrac {t^4}{(4!)^2}-\dfrac {t^6}{(6!)^2}+.....$$(6 marks) 3 (b) $$ if \ A(\alpha)= \begin{bmatrix}\cos \alpha &-\sin \alpha &0 \\ \sin \alpha&\cos \alpha & 0\\ 0&0 &1 \end{bmatrix} \ prove \ that [A(\alpha)]^{-1}= A(-\alpha)$$ (6 marks) 3 (c) Evaluate $$ \int_{c}\dfrac {\cos \pi z}{z^2-1}dz $$ where c is
(i) a rectangle with vertices at 2 ± i and -2 ± i.
(ii) a square with vertices at ± i and 2± i.(8 marks) 4 (a) Prove that the sum of the residues of the function $$ f(z)=\dfrac {e^{z}}{z^2+a^2}\ is \ \dfrac {\sin a}{a} $$(6 marks) 4 (b) Prove that is circle |z|=1 in the z-plane mapped onto the cardiode in the w-plane under the transformation w=z²+2z(6 marks) 4 (c) Obtain the Laplace transformation of $$ \left \{ t\cdot erf \left ( 3\sqrt{t} \right ) \right \} $$(8 marks) 5 (a) Find the orthogonal trajectories of u=constant where
$$ u=x^2-y^2 +5x+y-\dfrac {y}{x^2 +y^2} $$(6 marks) 5 (b) Examine the linear depedence of the vector [1 0 2 1], [3 1 2 1], [4 6 2 -4] and [-6 0 -3 -4] and find the relation between them if possible.(6 marks) 5 (c) $$ Evaluate \ \int^{2 \pi}_{0}\dfrac {d\theta}{3-2 \cos \theta + \sin \theta }$$ (8 marks) 6 (a) Find the bilinear transformation which maps the points z=2, 1, 0 onto w=1, 0, i.(6 marks) 6 (b) $$ If \ f(t)=\left\{\begin{matrix}3t &&0<t<2 \\ 6 &&2<t<4 \end{matrix}\right.$$ where f(t) has period 4.
(i) Draw graph of f(t)
(ii) Find L { f(t) }(6 marks) 6 (c) Show that $$ u=\left ( \gamma + \dfrac {a^2}{\gamma} \right )\cos \theta $$ is harmonic, find v(γ, θ) so that u+iv is analytic.(8 marks) 7 (a) Obtain -
$$ (i) \ L^{-1}\left \{ \dfrac {3s-8}{s^{2}+4}- \dfrac {4s-24}{s^{2}-16}\right \}\$$ii) \ L^{-1} \left \{ \dfrac {3s-2}{s^{5/2}}- \dfrac {7}{3s+2} \right \} $$(6 marks) 7 (b) If f(z)=u+iv is an analytic function of z=x+iy and $$ u-v=\dfrac {e^y -\cos x + \sin x}{\cos hy-\cos x}, $$ find f(z) subject to the condition $$ f\left (\dfrac {\pi}{2} \right )=\dfrac {3-i}{2} $$(6 marks) 7 (c) Show that the matrix $$ A=\begin{bmatrix}9 &-1 &9 \\ 3&-1 &3 \\ -7&1 &-7 \end{bmatrix} $$ is diagonalizable. Find the transforming matrix and the diagonal form.(8 marks)