MU Applied Mathematics - 3 - May 2012 Exam Question Paper

MU Mechanical Engineering (Semester 3)
Applied Mathematics - 3
May 2012

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) Show that

L [\frac{\cos \sqrt{t}}{\sqrt{t}}] = \sqrt{\frac{π}{s}} e^{- \frac{1}{4 s}}

$L\left [ \dfrac {\cos \sqrt{t}}{\sqrt{t}} \right ]= \sqrt {\dfrac {\pi}{s}} \ e^{-\frac {1}{4s}}$

5 M

1 (b) Use adjoint method to find inverse of -

A = [\begin{matrix} 7 & - 3 & - 3 \\ - 1 & 1 & 0 \\ - 1 & 0 & 1 \end{matrix}]

$A= \begin{bmatrix}7 &-3 &-3 \\ -1&1 &0 \\ -1&0 &1 \end{bmatrix}$

5 M

1 (c) Evaluate

\int_{0}^{1 + i} (x^{2} - i y) d z

$\int^{1+i}_{0} (x^{2}-iy) dz$ along the path
(i) y=x
(ii) y=x²

5 M

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 M

2 (a) Evaluate

\int_{0}^{\infty} \frac{t^{2} \sin 3 t}{e^{2 t}} d t

$\int^{\infty}_{0} \dfrac {t^2 \sin 3t}{e^{2t}} dt$

6 M

2 (b) Find non-singular matrices P and Q such that PAQ is in normal form. Also find rank of A where

A [\begin{matrix} 1 & 2 & 3 & 2 \\ 2 & 3 & 5 & 1 \\ 1 & 3 & 4 & 5 \end{matrix}]

$A \begin{bmatrix}1 &2 &3 &2 \\ 2&3 &5 &1 \\ 1&3 &4 &5 \end{bmatrix}$

6 M

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 M

3 (a)

\begin{aligned} F i n d & (i) L^{- 1} [\frac{s + 2}{s^{2} + 4 s + 7}] \\ (i i) L^{- 1} [l o g \frac{s^{2} + a^{2}}{\sqrt{s + b}}] \end{aligned}

$\begin {align*} Find \ &(i) \ L^{-1} \left [ \dfrac {s+2}{s^{2}+4s+7} \right ] \\ \\ &(ii) \ L^{-1}\left [log \dfrac {s^{2}+a^{2}}{\sqrt{s+b}} \right ]\\ \end{align*}$

6 M

3 (b)

C o m p u t e A^{7} - 4 A^{6} - 20 A^{5} - 34 A^{4} - 4 A^{3} - 20 A^{2} - 33 A + I w h e r e A = [\begin{matrix} 1 & 3 & 7 \\ 4 & 2 & 3 \\ 1 & 2 & 1 \end{matrix}]

$Compute \ A^{7}-4A^6-20A^5-34A^4-4A^3-20A^2-33A+I \\ where \ A =\begin{bmatrix}1 &3 &7 \\ 4&2 &3 \\ 1&2 &1 \end{bmatrix}$

6 M

3 (c) Prove that the circle |z-3|=5 is mapped onto the circle

| w + \frac{3}{16} | = \frac{5}{16}

$\left |w+\dfrac {3}{16} \right | =\dfrac {5}{16}$ under the transformation

w = \frac{1}{z}

$w=\dfrac {1}{z}$

8 M

4 (a) Evaluate

\int_{0}^{2 π} \frac{\cos 3 θ}{5 + 4 \cos θ} d θ

$\int^{2\pi}_{0}\dfrac {\cos 3 \theta}{5+4 \cos \theta} d \theta$

6 M

4 (b) Test for consistency and solve -

2 x_{1} - 3 x_{2} + 5 x_{3} = 1 3 x_{1} + x_{2} - x_{3} = 2 x_{1} + 4 x_{2} - 6 x_{3} = 1

$2x_{1}-3x_{2}+5x_{3}=1 \\3x_{1}+x_{2}-x_{3}=2\\x_{1}+4x_{2}-6x_{3}=1$

6 M

4 (c) Find

L^{- 1} [\frac{s}{(s^{2} + a^{2}) (s^{2} + b^{2})}]

$L^{-1}\left [ \dfrac {s}{(s^{2}+a^{2})(s^{2}+b^{2})} \right ]$ by using convolution theorem.

8 M

5 (a) Express the Hermitian matrix

A = [\begin{matrix} 3 & 2 - i & 1 + 2 i \\ 2 + i & 2 & 3 - 2 i \\ 1 - 2 i & 3 + 2 i & 0 \end{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 M

5 (b) Evaluate

\int_{C} \frac{\sin π z^{2} + \cos π z^{2}}{z^{2} + 3 z + 2} d z

$\int_{C} \dfrac {\sin \pi z^2+\cos \pi z^{2}}{z^{2}+3z+2}dz$ where C is (i) |z|=1 (ii) |z|=2

6 M

5 (c) Find all possible Laurent's expansion of the function

f (z) = \frac{7 z - 2}{z (z - 2) (z + 1)}

$f(z)=\dfrac {7z-2}{z(z-2)(z+1)}$ about z=-1

8 M

6 (a) Find

L [\frac{d}{d t} (\frac{1 - \cos 2 t}{t})]

$L\left [ \dfrac {d}{dt}\left ( \dfrac {1-\cos 2t}{t} \right ) \right ]$

6 M

6 (b) Find Eigen value and Eigen vector for -

A = [\begin{matrix} 4 & 6 & 6 \\ 1 & 3 & 2 \\ - 1 & - 5 & - 2 \end{matrix}]

$A= \begin{bmatrix}4 &6 &6 \\ 1&3 &2 \\ -1&-5 &-2 \end{bmatrix}$

6 M

6 (c) Evaluate using residue theorem

\int_{c} \frac{4 z^{2} + 1}{(2 z - 3) (z + 2)^{2}} d z

$\int_c \dfrac {4z^2 +1}{(2z-3)(z+2)^2}dz$ where C is |z|=4

8 M

7 (a) State and prove Cauchy's integral theorem.

6 M

7 (b)

i f A = [\begin{matrix} 1 / 3 & 2 / 3 & a \\ 2 / 3 & 1 / 3 & b \\ 2 / 3 & - 2 / 3 & c \end{matrix}]

$if \ A=\begin{bmatrix}1/3 &2/3 &a \\ 2/3&1/3 &b \\ 2/3&-2/3 &c \end{bmatrix}$ is orthogonal matrix then find a,b,c. Also find A^-1.

6 M

7 (c) Solve (D²-D-2) y=20 sin 2 t with y(0)=1 and y'(0)=2 by using Laplace transform.

8 M

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