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=t²+1, y=2t², z=t³ from t=1 to t=2(5 marks) 1 (d) Find the orthogonal trajectory of the curves 3x²y+2x³-y³-2y² = ?, 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 x²+y²+z²=9 and x²+y²-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=r² 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 ∫ J₃(x) dx and Express the result in terms of J₀ and J₁(6 marks) 4 (b) Find half range sine series for f(x)= πx-x² 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 x²+y²+z²=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)