MU Applied Mathematics 3 - December 2015 Exam Question Paper

MU Computer Engineering (Semester 3)
Applied Mathematics 3
December 2015

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 of {t⁵ cosht}

5 M

1 (b) Find Fourier series for f(x)=1-x² in (-1, 1)

5 M

1 (c) Find a, b, c, d, e if,

f (z) = (a x^{4} + b x^{2} y^{2} + c y^{4} + d x^{2} - 2 y^{2}) + (4 x^{3} y - e x y^{3} + 4 x y)

$f(z)= (ax^4 + bx^2y^2+ cy^4+dx^2 - 2y^2)+ (4x^3y - exy^3 + 4xy)$ is analytic.

5 M

1 (d) Prove that

\nabla (\frac{1}{r}) = \frac{f}{r^{3}}

$\nabla \left ( \dfrac {1}{r} \right ) = \dfrac {f}{r^3}$

5 M

2 (a) If f(z)=u+iv is analytic and

u + v = \frac{2 \sin 2 x}{e^{2 y} + e^{- 2 y} - 2 \cos 2 x}, find f(z)

$u+v = \dfrac {2 \sin 2x}{e^{2y}+e^{-2y}-2\cos 2x}, \text {find f(z)}$

6 M

2 (b) Find inverse Z-transform of

f (z) = \frac{z + 2}{z^{2} - 27 + 1} for | z | > 1

$f(z)= \dfrac {z+2}{z^2 -27 +1 } \ \text {for} \ |z|>1$

6 M

2 (c) Find Fourier series for

f (x) = \sqrt{1 ⊖ \cos x} in (0, 2 π)

$f(x)= \sqrt{1 \ominus \cos x} \ \text {in} \ (0,2\pi)$ Hence, deduce that

\frac{1}{2} = \sum_{1}^{\infty} \frac{1}{- 4 n^{2} - 1}

$\displaystyle \dfrac {1}{2} = \sum^{\infty}_{1} \dfrac {1}{-4n^2 - 1}$

8 M

3 (a) Find

L^{- 1} {\frac{1}{(s - 2) (s + 3)}}

$L^{-1} \left \{ \dfrac {1}{(s-2)(s+3)} \right \}$ using Convolution theorem.

6 M

3 (b) prove that f₁(x)=1, f₂(x)=x, f₃(x)=(3x²-1)/2 are orthogonal over (-1,1)

6 M

3 (c) Verify Green's theorem for

\int_{c} \bar{F} \cdot \bar{d r} where \bar{F} = (x^{2} - y^{2}) i + (x + y) j

$\displaystyle \int_c \overline {F}\cdot \overline {dr}\ \text{where} \ \overline {F}= (x^2 - y^2)i + (x+y)j$ and c is the triangle with vertices (0,0), (1,1), (2,1).

8 M

4 (a) Find Laplace Transform of f(t)=|sinpt|, t≥0.

6 M

4 (b) Show that F= (ysinz-sinx)i + (xsinz+2yz)j+(xycosz+y²) k is irrotational. Hence, find its scalar potential.

6 M

4 (c) Obtain Fourier expansion of \[ \begin {align*} f(x) &=x+\dfrac {\pi}{2} \ \text {where} \ -\pi

8 M

5 (a) Using Gauss Divergence theorem to evaluate

\iint_{s} \bar{N} \cdot \bar{F} d s where \bar{F} = 4 x i - 2 y^{2} j + z^{2} k

$\iint_s \overline{N}\cdot \overline{F}ds \text{where} \overline {F}=4xi-2y^2j+z^2k$ and S is the region bounded by x²+y²=4, z=0, z=3

6 M

5 (b) Find Z{2^k cos (3k+2)}, k≥0

6 M

5 (c) Solve (D²+2D+5)y=e^-t sint, with y(0) and y'(0)=1.

8 M

6 (a) Find

L^{- 1} {\tan^{- 1} (\frac{2}{s^{2}})}

$L^{-1}\left \{ \tan ^{-1} \left ( \dfrac {2}{s^2} \right ) \right \}$

6 M

6 (b) Find the bilinear transformation which maps the points 2, i, -2 onto point 1, j, -1 by using cross-ratio property.

6 M

6 (c) Find Fourier Sine integral representation for

f (x) = \frac{e^{- a x}}{x}

$f(x) = \dfrac {e^{-ax}}{x}$

8 M

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