## Applied Mathematics - 3 - May 2014

### Electronics & Telecomm. (Semester 3)

TOTAL MARKS: 80

TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.

(2) Attempt any **three** from the remaining questions.

(3) Assume data if required.

(4) Figures to the right indicate full marks.
**1 (a)** Evaluate $$\int_0^{\infty{}}\frac{\left(\cos{6t-cos4t}\right)}{t}\ dt$$(5 marks)
**1 (b)** Obtain complex form of fourier series for f(x)= e^{ax} in (-1, 1)(5 marks)
**1 (c) ** Find the work done in moving a particle in a force field given by $$\bar{F}=3xy\ \hat{i}-5z\hat{j}+10x\hat{k}$$ along the curve x=t^{2}+1, y=2t^{2}, z=t^{3} from t=1 to t=2(5 marks)
**1 (d)** Find the orthogonal trajectory of the curves 3x^{2}y+2x^{3}-y^{3}-2y^{2} = ?, where &lpha; is a constant(5 marks)
**2 (a)** Evaluate $$\frac{d^2y}{dt^2}+2\frac{dy}{dt}-3y=sint,$$ y(0)=0, y'(0)=0, by Laplace transform(6 marks)
**2 (b)** Show that $$ J_{\frac{5}{2}}=\ \sqrt{\frac{2}{\pi{}x}} \left[\frac{3-x^2}{x^2}\sin{x-\frac{3}{x}\cos{x\ }}\right] $$(6 marks)
**2 (c) (i)** Find the constant a,b,c so that $$\bar{F}=\left(x+2y+az\right)\hat{i}+\left(bx-3y-z\right)\hat{j}+(4x+\left(y+2z\right)\hat{k}$$(4 marks)
**2 (c) (ii)** Prove that the angle between two surface x^{2}+y^{2}+z^{2}=9 and x^{2}+y^{2}-z=3 at the point (2,-1,2) is $${\cos}^{-1}{\left(\frac{8}{3\sqrt{21}}\right)}$$(4 marks)
**3 (a)** Obtain the fourier series of f(x) given by

$$f\left(x\right)=\left\{\begin{array}{l}0,\ \ \&-\pi{}\leq{}x\leq{}0 \\x^2,\ \ \&0\leq{}x\leq{}\pi{}\end{array}\right.$$(6 marks)
**3 (b)** Find the analytic function f(z)= u+iv where u=r^{2} cos2θ-r cosθ+2(6 marks)
**3 (c) ** Find Laplace transform of

(i) te^{-3t} cos2t.cos3t

(ii) $$\frac{d}{dt}\left[\frac{\sin{3t}}{t}\right]$$(8 marks)
**4 (a) ** Evaluate ∫ J_{3}(x) dx and Express the result in terms of J_{0} and J_{1}(6 marks)
**4 (b)** Find half range sine series for f(x)= πx-x^{2} in (0, π) Hence deduce that $$\frac{{\pi{}}^3}{32}=\frac{1}{12}-\frac{1}{3^2}+\frac{1}{5^2}-\frac{1}{7^2}+\pi $$(6 marks)
**4 (c) ** Find inverse Laplace transform of :-

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

$$\left(ii\right)\ \frac{se^{-2s}}{\left(s^2+2s+2\right)}$$(8 marks)
**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 marks)
**5 (b)** Find half range cosine series of f(x)= sinx in 0 ≤ x ≤ π Hence deduce that

$$\frac{1}{1.3}+\frac{1}{3.5}+\frac{1}{5.7}+?=\frac{1}{2}$$(6 marks)
**5 (c) ** Verify Green's theorem, for $$\oint_C\left(3x^2-8y^2\right)dx+\left(4y-6xy\right)$$ by where c is boundary of the region defined by x=0, y=0, and x+y=1(8 marks)
**6 (a)** Using convolution theorem; evaluate

$$L^{-1}\left\{\frac{1}{\left(S-1\right)\left(s^24\right)}\right\}$$(6 marks)
**6 (b)** Find the bilinear transformation which maps the points z=1, I, -1 onto w=0, 1, ?(6 marks)
**6 (c) ** By using the appropriate theorem, evaluate the following :-

$$\left(i\right)\ \int\bar{F}c \dot{}d\bar{r}\ where\\bar{F}=\left(2x-y\right)\hat{i}-\left(yz^2\right)\hat{j}-\left(y^2z\right)\hat{k}$$

and c is the boundary of the upper half of the sphere x^{2}+y^{2}+z^{2}=4

$$\left(ii\right)\ \iint_s\ (9x\hat{i}+6y\hat{j}-10z\hat{k})\ c \dot{}d\bar{s} $$

where s is the surface of sphere with radius 2 uints.(8 marks)