Applied Mathematics - 3 - May 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) Show that $$ L\left [ \dfrac {\cos \sqrt{t}}{\sqrt{t}} \right ]= \sqrt {\dfrac {\pi}{s}} \ e^{-\frac {1}{4s}} $$(5 marks) 1 (b) Use adjoint method to find inverse of -
$$ A= \begin{bmatrix}7 &-3 &-3 \\ -1&1 &0 \\ -1&0 &1 \end{bmatrix} $$(5 marks) 1 (c) Evaluate $$ \int^{1+i}_{0} (x^{2}-iy) dz $$ along the path
(i) y=x
(ii) y=x²(5 marks) 1 (d) IF f(z)=u+iv is analytic and u-v=e^x (cos y -sin y) then find f(z) in terms of z.(5 marks) 2 (a) Evaluate $$ \int^{\infty}_{0} \dfrac {t^2 \sin 3t}{e^{2t}} dt $$(6 marks) 2 (b) Find non-singular matrices P and Q such that PAQ is in normal form. Also find rank of A where
$$ A \begin{bmatrix}1 &2 &3 &2 \\ 2&3 &5 &1 \\ 1&3 &4 &5 \end{bmatrix} $$(6 marks) 2 (c) Find the bilinear transformation which maps the points 2, I, -2 onto points 1, I, -1. Also the fixed points of this bilinear transformation.(8 marks) 3 (a) $$ \begin {align*} Find \ &(i) \ L^{-1} \left [ \dfrac {s+2}{s^{2}+4s+7} \right ] \\ \\ &(ii) \ L^{-1}\left [log \dfrac {s^{2}+a^{2}}{\sqrt{s+b}} \right ]\\ \end{align*} $$(6 marks) 3 (b) $$ Compute \ A^{7}-4A^6-20A^5-34A^4-4A^3-20A^2-33A+I \\ where \ A =\begin{bmatrix}1 &3 &7 \\ 4&2 &3 \\ 1&2 &1 \end{bmatrix} $$(6 marks) 3 (c) Prove that the circle |z-3|=5 is mapped onto the circle $$ \left |w+\dfrac {3}{16} \right | =\dfrac {5}{16} $$ under the transformation $$ w=\dfrac {1}{z} $$(8 marks) 4 (a) Evaluate $$ \int^{2\pi}_{0}\dfrac {\cos 3 \theta}{5+4 \cos \theta} d \theta $$(6 marks) 4 (b) Test for consistency and solve -
$$ 2x_{1}-3x_{2}+5x_{3}=1 \\3x_{1}+x_{2}-x_{3}=2\\x_{1}+4x_{2}-6x_{3}=1$$(6 marks) 4 (c) Find $$ L^{-1}\left [ \dfrac {s}{(s^{2}+a^{2})(s^{2}+b^{2})} \right ] $$ by using convolution theorem.(8 marks) 5 (a) Express the Hermitian matrix $$ A= \begin{bmatrix}3 &2-i &1+2i \\ 2+i&2 &3-2i \\ 1-2i&3+2i &0 \end{bmatrix}$$ as P+iQ where P is real symmetric and Q is real skew-symmetric.(6 marks) 5 (b) Evaluate $$ \int_{C} \dfrac {\sin \pi z^2+\cos \pi z^{2}}{z^{2}+3z+2}dz $$ where C is (i) |z|=1 (ii) |z|=2(6 marks) 5 (c) Find all possible Laurent's expansion of the function $$ f(z)=\dfrac {7z-2}{z(z-2)(z+1)} $$ about z=-1(8 marks) 6 (a) Find $$ L\left [ \dfrac {d}{dt}\left ( \dfrac {1-\cos 2t}{t} \right ) \right ] $$(6 marks) 6 (b) Find Eigen value and Eigen vector for -
$$ A= \begin{bmatrix}4 &6 &6 \\ 1&3 &2 \\ -1&-5 &-2 \end{bmatrix}$$(6 marks) 6 (c) Evaluate using residue theorem $$ \int_c \dfrac {4z^2 +1}{(2z-3)(z+2)^2}dz $$ where C is |z|=4(8 marks) 7 (a) State and prove Cauchy's integral theorem.(6 marks) 7 (b) $$ if \ A=\begin{bmatrix}1/3 &2/3 &a \\ 2/3&1/3 &b \\ 2/3&-2/3 &c \end{bmatrix} $$ is orthogonal matrix then find a,b,c. Also find A^-1.(6 marks) 7 (c) Solve (D²-D-2) y=20 sin 2 t with y(0)=1 and y'(0)=2 by using Laplace transform.(8 marks)