MU Applied Mathematics - 3 - December 2016 Exam Question Paper

MU Electronics and Telecom Engineering (Semester 3)
Applied Mathematics - 3
December 2016

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) Determine the constants 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)

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

5 M

1(b) Find half range Fourier sine series for f(x)=x², 0

5 M

1(c) Find the directional derivative of

φ (x, y, z) = x y^{2} + y z^{3}

$\varphi \left ( x,y,z \right )=xy^2+yz^3$ / at the point (2,-1,1) in the direction of the vector i + 2j + 2k.

5 M

1(d) Evaluate

\int_{0}^{\infty} e^{- 2 t} t^{5} \cosh t d t .

$\int_{0}^{\infty }e^{-2t}t^{5}\cosh t\ dt.$

5 M

2(a) Prove that

ȷ_{\frac{3}{2}} (x) = \sqrt{\frac{2}{π x}} (\frac{\sin x}{x} - \cos x)

$\jmath_ \frac{3}{2}(x)=\sqrt{\frac{2}{\pi x}}\left ( \frac{\sin x}{x}-\cos x \right )$

6 M

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

u - v = e^{x} (\cos y - \sin y)

$u-v=e^x\left ( \cos y-\sin y \right )$ /, fin f(z) in terms of z.

6 M

2(c) Obtain Fourier series for \(\begin{align*} \begin{matrix} f(x)&= x+\frac{\pi }{2} &-\pi / Hence deduce that

\frac{π^{2}}{8} = \frac{1}{1^{2}} + \frac{1}{3^{2}} + \frac{1}{s^{2}} + . . . . .

$\frac{\pi ^2}{8}=\frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{s^2}+.....$

8 M

3(a) Show that

F = (2 x y + z^{3}) i + x^{2} j + 3 x z^{2} k

$F=\left ( 2xy+z^3 \right )i+x^2j+3xz^2k$ /, is a conservative field. Find its scalar potential and also find the work done by the force F in moving a praticle from (1,-2,1) to (3, 1, 4).

6 M

3(b) Show that the set of functions

{\sin (2 n + 1) x}, n = 0, 1, 2, . . .

$\left \{ \sin \left ( 2n+1 \right )x \right \},n=0, 1, 2,...$ / is orthogonal over [0,π/2}. Hence consturct orthonormal set of fucntions.

6 M

3(c) i)

L^{- 1} {\cot^{- 1} (s + 1)}

$L^{-1}\left \{ \cot ^{-1}\left ( s+1 \right ) \right \}$
ii)

L^{- 1} (\frac{e^{- 2 s}}{s^{2} + 8 s + 25})

$L^{-1}\left ( \frac{e^{-2s}}{s^2+8s+25} \right )$

8 M

4(a) Prove that

\int ȷ_{3} (x) d x = \frac{2 ȷ_{1} (x)}{x} - ȷ_{2} (x)

$\int \jmath _3(x)dx=\frac{2\jmath _1(x)}{x}-\jmath _2(x)$

6 M

4(b) Find inverse Laplace of

\frac{s}{(s^{2} + a^{2}) (s^{2} + b^{2})} (a \neq b)

$\frac{s}{\left ( s^2+a^2 \right )\left ( s^2+b^2 \right )}\left ( a\neq b \right )$ / using Convolution theorem.

6 M

4(c) Expand f(x) = xsinx in the interval 0≤x≤2π as a Fourier series. Hence, deduce that

\sum_{n = 2}^{\infty} \frac{1}{n^{2} - 1} = \frac{3}{4}

$\sum_{{n=2}}^{\infty }\ \ \frac{1}{n^2-1}=\frac{3}{4}$

8 M

5(a) Using Gauss Diveragence theorem evaluate

\int \int_{s} \bar{N .} \bar{F} d s where \bar{F} = x^{2} i + z j + y z k

$\int \int _s\bar{N.}\bar{F}ds\ \ \text{where} \bar{F}=x^2i+zj+yzk$ / and S is the cube bounded by x=0, x=1, y=0, y=1, z=0, z-1

6 M

5(b) Prove that

\jmath ^'_2(x)=\left ( 1-\frac{4}{x^2} \right )\jmath _1(x)+\frac{2}{x}\jmath 0(x)

$\jmath ^'_2(x)=\left ( 1-\frac{4}{x^2} \right )\jmath _1(x)+\frac{2}{x}\jmath 0(x)$

6 M

5(c) Solve

(D^{2} 3 D + 2) y = 2 (t^{2} + t + 1)

$\left ( D^23D+2 \right )y=2\left ( t^2+t+1 \right )$ /, with y(0)=2 and y(0)=0 by using Laplace transform

8 M

6(a) Evaluate by Green's theorem for

\int_{c} (e^{- x} \sin d x + e^{- x} \cos y d y)

$\int _c\left ( e^{-x}\sin dx+e^{-x} \cos y dy\right )$ / where C is the rectangle whose vertices are (0,0), (π, 0), (π, π/2)

6 M

6(b) Show that under the transformation

w = \frac{z - i}{z + i}

$w=\frac{z-i}{z+i}$ / real axis in the z-plane is mapped onto the circle |w|=1

6 M

6(c) Find Fourier integral representation for

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

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

8 M

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