Applied Mathematics - 3 - Dec 2011

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) Find Laplace transform of sin √t(5 marks) 1 (b) If $$ A=\begin{bmatrix}123 &231 &312 \\ 231&312 &123 \\ 312&123 &231 \end{bmatrix} $$ P.T. (i) one of the characteristics roots of is 666
(ii) given A is non-singular, one of the characteristics roots A is negative.(5 marks) 1 (c) Evaluate $$ \int ^{1+2i}_{0}z^{2}dz $$ along the curve 2x²=y(5 marks) 1 (d) Find the image of the circle with centre at (0,3) and radius 3 in the z-plane into the w-plane inder the transformation $$ w=\dfrac {1}{2} $$ (5 marks) 2 (a) Evaluate $$ \int^{\infty}_{0}t\left (\dfrac {\sin t}{e^{t}} \right )^{2}dt $$(7 marks) 2 (b) Find non-singular matrices P and Q such that PAQ is in normal form. Also find of A and A^-1 where
\begin{bmatrix}1 &2 &-2 \ -1&3 &0 \ 0&-2 &1 \end{bmatrix}(7 marks) 2 (c) Find the imaginary part of the analytic function whose real part is e^2x(x cos 2y-y sin 2y) (6 marks) 3 (a) Find inverse Laplace transform of $$ \dfrac {5s^{2}-15s-11}{(s+1)(s-2)^{2}} $$(7 marks) 3 (b) Find the characteristics equation of the matrix A and hence find the matrix represented by
$$ A^{8}-5A^{7}+7A^{6}-3A^{5}+A^{4}-5A^{3}+8A^{2}-2A+1 \\where \ A=\begin{bmatrix}2 &1 &1 \\ 0&1 &0 \\ 1&1 &2 \end{bmatrix}$$(7 marks) 3 (c) Find the bilinear transformation which maps the points 0, 1 ∞ onto the points w=-5, -1, 3. (6 marks) 4 (a) Evaluate $$ \int^{\pi}_{0}\dfrac {d\theta}{3+2\cos \theta} $$(7 marks) 4 (b) Find inverse Laplace transform by convolution thm, of $$ \dfrac {1}{(s+3)(s^{2}+2s+2)}. $$(7 marks) 4 (c) Investigation for what values of λ and μ, the equation 2x+3y+5z=9, 7x+3y-2z=8, 2x+3y+λz=μ have (i) no solution, (ii) unique solution (iii) infinite no. of solutions.(6 marks) 5 (a) Find the inverse of the matrix
$$ S= \begin{bmatrix}0 &1 &1 \\ 1&0 &1 \\ 1&1 &0 \end{bmatrix} \ and \ if \ A=\dfrac {1}{2}\begin{bmatrix}4 &-1 &1 \\ -2&3 &-1 \\ 2&1 &5 \end{bmatrix} $$ S.T. SAS^-1 is a diagonal matrix.(7 marks) 5 (b) Evaluate $$ \int_{c}\dfrac {z^{2}+4}{(z-2)(z+3i)}dz $$ where C is
(i) |z+1|=2
(ii) |z-2|=2 (7 marks) 5 (c) Obtain Taylor's and Laurent's expansion of $$ f(z)= \dfrac {z-1}{z^{2}-2z-3} $$ indicating regions of convergece.(6 marks) 6 (a) Find Laplace transform of $$ \dfrac {e^{-at}-\cos at}{t} \ hence \ evaluate \ \int^{\infty}_{0}\dfrac {e^{-1}-\cos t}{te^{4t}}dt $$(7 marks) 6 (b) Find Eigen value Eigen vectors of $$ A^3 +1 \ where \ A=\begin{bmatrix}2 &2 & 1\\ 1&3 &1 \\ 1&2 &2 \end{bmatrix} $$(7 marks) 6 (c) Evaluate using residue theorem $$ \int_c \dfrac {(z+4)^{2}}{z^4+5z^3+6z^2}dz $$ where C is is |z|=1(6 marks) 7 (a) Solve the foll: equation using Laplace transform,
$$ \dfrac {dy}{dt}+2y+\int^{t}_{0}ydt=\sin t \ given \ y(0)=1 $$(7 marks) 7 (b) Express the foll: matrix A as P+iQ where P and Q are both hermitian given: $$ A= \begin{bmatrix}2 &3-i &2+i \\ i&0 &1-i \\ 1+2i&1 &3i \end{bmatrix} $$(7 marks) 7 (c) Find sum of residue at singular points of
$$ f(z)=\dfrac {z}{az^2+bz+c} $$(6 marks)