MU Applied Mathematics - 3 - December 2011 Exam Question Paper

MU Mechanical Engineering (Semester 3)
Applied Mathematics - 3
December 2011

Total marks: --
Total time: --

INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

1 (a) Find Laplace transform of sin √t

5 M

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 M

1 (c) Evaluate

$\int ^{1+2i}_{0}z^{2}dz$ along the curve 2x²=y

5 M

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 M

2 (a) Evaluate

$\int^{\infty}_{0}t\left (\dfrac {\sin t}{e^{t}} \right )^{2}dt$

7 M

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 M

2 (c) Find the imaginary part of the analytic function whose real part is e^2x(x cos 2y-y sin 2y)

6 M

3 (a) Find inverse Laplace transform of

$\dfrac {5s^{2}-15s-11}{(s+1)(s-2)^{2}}$

7 M

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 M

3 (c) Find the bilinear transformation which maps the points 0, 1 ∞ onto the points w=-5, -1, 3.

6 M

4 (a) Evaluate

$\int^{\pi}_{0}\dfrac {d\theta}{3+2\cos \theta}$

7 M

4 (b) Find inverse Laplace transform by convolution thm, of

$\dfrac {1}{(s+3)(s^{2}+2s+2)}.$

7 M

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 M

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 M

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 M

5 (c) Obtain Taylor's and Laurent's expansion of

$f(z)= \dfrac {z-1}{z^{2}-2z-3}$ indicating regions of convergece.

6 M

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 M

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 M

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 M

7 (a) Solve the foll: equation using Laplace transform,

$\dfrac {dy}{dt}+2y+\int^{t}_{0}ydt=\sin t \ given \ y(0)=1$

7 M

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 M

7 (c) Find sum of residue at singular points of

$f(z)=\dfrac {z}{az^2+bz+c}$

6 M

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