MU Applied Mathematics - 3 - May 2014 Exam Question Paper

MU Electronics and Telecom Engineering (Semester 3)
Applied Mathematics - 3
May 2014

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) Evaluate

\int_{0}^{\infty} \frac{(cos 6 t - c o s 4 t)}{t} d t

$\int_0^{\infty{}}\frac{\left(\cos{6t-cos4t}\right)}{t}\ dt$

5 M

1 (b) Obtain complex form of fourier series for f(x)= e^ax in (-1, 1)

5 M

1 (c) Find the work done in moving a particle in a force field given by

¯ F = 3 x y^i - 5 z^j + 10 x^k

$\bar{F}=3xy\ \hat{i}-5z\hat{j}+10x\hat{k}$ along the curve x=t²+1, y=2t², z=t³ from t=1 to t=2

5 M

1 (d) Find the orthogonal trajectory of the curves 3x²y+2x³-y³-2y² = ?, where &lpha; is a constant

5 M

2 (a) Evaluate

\frac{d^{2} y}{d t^{2}} + 2 \frac{d y}{d t} - 3 y = s i n t,

$\frac{d^2y}{dt^2}+2\frac{dy}{dt}-3y=sint,$ y(0)=0, y'(0)=0, by Laplace transform

6 M

2 (b) Show that

J_{\frac{5}{2}} = \sqrt{\frac{2}{π x}} ⎡ ⎣ \frac{3 - x^{2}}{x^{2}} sin x - \frac{3}{x} cos x ⎤ ⎦

$J_{\frac{5}{2}}=\ \sqrt{\frac{2}{\pi{}x}} \left[\frac{3-x^2}{x^2}\sin{x-\frac{3}{x}\cos{x\ }}\right]$

6 M

2 (c) (i) Find the constant a,b,c so that

¯ F = (x + 2 y + a z)^i + (b x - 3 y - z)^j + (4 x + (y + 2 z)^k

$\bar{F}=\left(x+2y+az\right)\hat{i}+\left(bx-3y-z\right)\hat{j}+(4x+\left(y+2z\right)\hat{k}$

4 M

2 (c) (ii) Prove that the angle between two surface x²+y²+z²=9 and x²+y²-z=3 at the point (2,-1,2) is

{cos}^{- 1} (\frac{8}{3 \sqrt{21}})

${\cos}^{-1}{\left(\frac{8}{3\sqrt{21}}\right)}$

4 M

3 (a) Obtain the fourier series of f(x) given by

f (x) = {\begin{matrix} 0, & - π \leq x \leq 0 x^{2}, & 0 \leq x \leq π \end{matrix}

$f\left(x\right)=\left\{\begin{array}{l}0,\ \ \&-\pi{}\leq{}x\leq{}0 \\x^2,\ \ \&0\leq{}x\leq{}\pi{}\end{array}\right.$

6 M

3 (b) Find the analytic function f(z)= u+iv where u=r² cos2θ-r cosθ+2

6 M

3 (c) Find Laplace transform of
(i) te^-3t cos2t.cos3t
(ii)

\frac{d}{d t} [\frac{sin 3 t}{t}]

$\frac{d}{dt}\left[\frac{\sin{3t}}{t}\right]$

8 M

4 (a) Evaluate ∫ J₃(x) dx and Express the result in terms of J₀ and J₁

6 M

4 (b) Find half range sine series for f(x)= πx-x² in (0, π) Hence deduce that

\frac{π^{3}}{32} = \frac{1}{12} - \frac{1}{3^{2}} + \frac{1}{5^{2}} - \frac{1}{7^{2}} + π

$\frac{{\pi{}}^3}{32}=\frac{1}{12}-\frac{1}{3^2}+\frac{1}{5^2}-\frac{1}{7^2}+\pi$

6 M

4 (c) Find inverse Laplace transform of :-

(i) \frac{1}{s} {tan h}^{- 1} (s)

$\left(i\right)\frac{1}{s}{\tan h}^{-1}{\left(s\right)}$

(i i) \frac{s e^{- 2 s}}{(s^{2} + 2 s + 2)}

$\left(ii\right)\ \frac{se^{-2s}}{\left(s^2+2s+2\right)}$

8 M

5 (a) Under the transformation w+2i=z 1/z, show that the map of the circle |z|=2 is an ellipse in w-plane

6 M

5 (b) Find half range cosine series of f(x)= sinx in 0 ≤ x ≤ π Hence deduce that

\frac{1}{1.3} + \frac{1}{3.5} + \frac{1}{5.7} + ? = \frac{1}{2}

$\frac{1}{1.3}+\frac{1}{3.5}+\frac{1}{5.7}+?=\frac{1}{2}$

6 M

5 (c) Verify Green's theorem, for

\oint_{C} (3 x^{2} - 8 y^{2}) d x + (4 y - 6 x y)

$\oint_C\left(3x^2-8y^2\right)dx+\left(4y-6xy\right)$ by where c is boundary of the region defined by x=0, y=0, and x+y=1

8 M

6 (a) Using convolution theorem; evaluate

L^{- 1} ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ \frac{1}{(S - 1) (s^{2} 4)} ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

$L^{-1}\left\{\frac{1}{\left(S-1\right)\left(s^24\right)}\right\}$

6 M

6 (b) Find the bilinear transformation which maps the points z=1, I, -1 onto w=0, 1, ?

6 M

6 (c) By using the appropriate theorem, evaluate the following :-

(i) \int ¯ F c ˙ d ¯ r w h e r e b a r F = (2 x - y)^i - (y z^{2})^j - (y^{2} z)^k

$\left(i\right)\ \int\bar{F}c \dot{}d\bar{r}\ where\\bar{F}=\left(2x-y\right)\hat{i}-\left(yz^2\right)\hat{j}-\left(y^2z\right)\hat{k}$
and c is the boundary of the upper half of the sphere x²+y²+z²=4

(i i) \iint_{s} (9 x^i + 6 y^j - 10 z^k) c ˙ d ¯ s

$\left(ii\right)\ \iint_s\ (9x\hat{i}+6y\hat{j}-10z\hat{k})\ c \dot{}d\bar{s}$
where s is the surface of sphere with radius 2 uints.

8 M

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