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SPPU Electrical Engineering (Semester 3)
Engineering 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

Solve any one question form Q.1(a, b) &Q.2(a, b, c)Solve any two question from Q.1(a)(i, ii, iii)
1(a) \begin{align*}i)&4\frac{d^2y}{dx^2}+4\frac{dy}{dx}+y=xe^{-x/2}\cos x \\ ii)&x^2\frac{d^2y}{dx^2}-x\frac{dy}{dx}-3y=x^2\log x\\ iii)&\frac{d^2y}{dx^2}-2\frac{dy}{dx}e^x \sin x \end{align*}/ (use method of variation of parameters).
8 M
1(b) Solve the differential equation by using Laplace transform method: $$\frac{dy}{dt}+3y(t)+2\int_{0}^{t}y(t)dt=t\\ \text{given}y(0)=0$$/
4 M

2(a) An inductor of 0.25 henry is connected in series with a capacitor of 0.04 farads and a generator having alternative voltage given by 12 sin 10t. Find the charge and current at any time t.
4 M
Solve any one question from Q.2(b) (i, ii)
2(b) Find the Laplace transform of: $$\frac{d}{dt}\left ( \frac{1-\cos t}{t} \right ).$$/
ii) Find the inverse Laplace transform of: $$\frac{S}{S^4+S^2+1}.$$/
4 M
2(c) Find the Laplace transform of:$f(t)=\left ( 1+2t+3t^2 \right )u\left ( t-2 \right )+sin 2t\delta \left ( t-\frac{\pi }{4} \right ).$
4 M

Solve any one question from Q.3(a,b,c) & Q.4(a,b,c)
3(a) Find the Fourier transform of: $f(x)=\left\{\begin{matrix} 1-x^2,|x| &\leq 1 \\ 0,|x| &>1. \end{matrix}\right.$
4 M
3(b) Find: $$z^{-1}\begin{Bmatrix} \frac{1}{\left ( z-\frac{1}{2} \right )\left ( z-\frac{1}{3} \right )} & \\ For |z|>$\frac{1}{2}.$ \end{Bmatrix}$$/
4 M
3(c) Find the constants a and b, so that the surface $$\ax^2-byz=\left ( a+2 \right )x)/ will be orthogonal to the surface \( 4x^2y+z^3=4$$/
at the point (1, -1, 2).
4 M

4(a) Show that the vector field $$f(r)\underset{r}{\rightarrow}$$/ is always irrotational and determine f(r) such that the field is solential also.
4 M
Solve any one question from Q.4(b)(i, ii)
4(b)(i) Prove the following:\nabla^2\left ( \nabla.\frac{\underset{r}{\rightarrow}}{r^2} \right )=\frac{2}{r^4}
4 M
4(b)(ii) $$\nabla\times\left ( \underset{a}{\rightarrow} \nabla\frac{1}{r}\right )=\frac{\underset{a}{\rightarrow}}{r^3}-\frac{\left ( \underset{a}{\rightarrow}.\underset{r}{\rightarrow} \right )\underset{r}{\rightarrow}}{r^5}.$$/
4 M
4(c) Solve the difference equation: $$f\left ( k+2 \right )-3f\left ( k+1 \right )+2f(k)=1\\ \text{where}f(0)= 0, f(1)=3.$$/
4 M

Solve any one question from Q.5(a,b,c) & Q.6(a,b,c)
5(a) Evaluate: $$\int _C\bar{F}.d\bar{r}\\\text{for}\\\bar{F}=\left ( 2y+3 \right )\bar{i}+xz\bar{j}+\left ( yz-x \right )\bar{k}$$/ along a straight line joining (0, 0, 0) to (3, 1, 1).
4 M
5(b) Evaluate: $$\iint _S\left ( \nabla\times \bar{F} \right ).\hat{n}dS$$/ where S is the curved surface of the paraboloid $$x^2+y^2=2z$$/ bounded by plane z=2 where $$\bar{F}=3\left ( x-y \right )\bar{i}+2xz\bar{j}+xy\bar{k}.$$/
5 M
5(c) Evaluate: $$\iint \bar{r}.\hat{n}dS$$/ over the surface of sphere of radius 2 with centre at origin.
4 M

6(a) Using Green's theorem evaulate: $$\int _C\left [ \cos y\bar{i} +x\left ( 1-\sin y \right )\right\bar{j} ].d\bar{r}$$/ where C is the closed curve $$x^2+y^2=1, z=0.$$/
4 M
6(b) Prove that: $$\int _C\left ( \bar{a}\times \bar{r} \right ).d\bar{r}=2\bar{a}.\iint _Sd\bar{S}$$/ where C is open surface bouded by closed curve C.
4 M
6(c) Evaluate: $$\iint _S\bar{F}.\hat{n}dS\\\text{where} \\\bar{F}=\left ( 2x+3z \right )\bar{i}-\left ( xz+y \right )\bar{j} \left ( y^2+2z \right )\bar{K}$$/ and S is surface of sphere with radius 3.
5 M

Solve any one question from Q.7(a,b,c) & Q.8(a,b,c)
7(a) If $$u=\log \left ( x^2+y^2 \right ),$$/ find v such that $f(z) = u + iv$ is analytic. Determine f(z) in terms of z.
5 M
7(b) Evaluate: $$\oint _C\frac{dz}{z^2}$$/ where C is the circle |z| = 1.
4 M
7(c) Find the bilinear transformation which maps the points 0, -1, i of z-plane on to the points 2, &infty;,$$\frac{1}{2}\left ( 5+i \right )$$/ of the w-plane.
4 M

8(a) Show that the map of straight line parallel to x-axis is family of ellipses under the transformation w = sinh (z).
4 M
8(b) Evaluate: $$\oint _C\frac{z+2}{z^2+1}dz$$/ where C is the circle | z -i| = $\frac{1}{2}$.
4 M
8(c) Find analytic function $$f(z) = u + iv \\\text{where} u= r^3 \cos 3\theta +r\sin \theta .$$/
5 M

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