SPPU Engineering Mathematics 3 - December 2016 Exam Question Paper

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)(i,ii,ii) Solve any one question from Q.1(a,b) & Q.2(a, b, c)

1(a)

$\begin{align*}i)& \left ( D^2-4D+3 \right )y=x^3e^{2x}\\ ii)&\left ( 1+x \right )^2\frac{d^2y}{dx^2}+\left ( 1+x \right )\frac{dy}{dx}+y=2\sin \left [ \log \left ( 1+x \right ) \right ]\\ iii)&\left ( D^2-1 \right )y=\frac{2}{1+e^x} \end{align*}$ / using method of variation of parameters.

8 M

1(b) Solve using Laplace transforms:

$\frac{d^2y}{dx^2}+y=t,\\ \text{given}y(0)=1, y'(0)=-2.$ /

4 M

2(a) A circuit consists of an induactance L and condenser of capacity C in series. An e.m.f. Σsin n t is applied to it at time t =0, the initial chrage and initial current being zero, find the current flowing in the circuit at any time

$t\ \text{for}\frac{1}{\sqrt{LC}}\neq n.$ /

4 M

Solve any one question form Q.2(b)(i,ii)

2(b)(i) Find:

$L\left [ \frac{e^{-at}-e^{-bt}}{t} \right ].$ /

4 M

2(b)(ii) Find:

$L^{-1}\left [ \frac{s+7}{s^2+2s+2} \right ].$ /

4 M

2(c) Find Laplace transform of:

$L\left [ \sin t\ U\left ( t-4 \right ) \right ].$ /

4 M

Solve any one question from Q.3(a, b,c) & Q.4(a, b,c)

3(a) Solve the integral equation:

$\int_{0}^{\infty }f(x)\cos \lambda x\ dx= e^{-\lambda } \ \text{where}\ \lambda >0$ /

4 M

Solve any one question form Q.3(b)(i,ii)

3(b)(i) Find z-transform of

$f(k)=\frac{2^k}{K}, k\geq 1.$ /

4 M

3(b)(ii) Find inverse z-transform of:

$F(z)=\frac{1}{\left ( z-a \right )^2},\ |z| < a.$ /

4 M

3(c) If directional derivative of:

$\phi =ax^2y+by^2z+cz^2x$ / at (1,1,1) has maximum magnitude 15 in the direction parallel to

$\frac{x-1}{2}=\frac{y-3}{-2}=\frac{z}{1},$ / hence find the values of a, b, c.

4 M

Solve any one question fromQ.4(a)(i, ii)

4(a)(i)

$\nabla.\left [ r^\nabla\left ( \frac{1}{r^n} \right ) \right ]=\frac{n\left ( n-2 \right )}{r^{n+1}}$ /

4 M

4(a)(ii)

$\nabla^2f(r)=f"(r)+\frac{2}{r}f'(r).$ /

4 M

4(b) Find the values of the constant scalars a, b, c if the vector point function:

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

4 M

4(c) Obtain f(k), given that:

$f_{k+2}-4f_k=0, k\geq 0,f(0)=0,f(1)=2.$ /

4 M

Solve any two question from Q.5(a,b,c) Solve any one question from Q.5(a, b, c) &Q.6(a, b,c)

5(a) Using Green's theorem, show that the area bounded by a simple closed curve C is given by:

$\frac{1}{2}\int \left ( xdy-ydx \right) .$ / Hence find the area of the ellipe

$x=a\cos \theta , y= b\sin \theta .$

6 M

5(b) Use the divergence theorem to evaluate:

$\iint_s \left ( y^2z^2i+z^2x^2j+x^2y^2k \right ).\overline{dS}$ /
where S is the upper half of the sphere x²+y²+z²=9 above the xoy plane.

6 M

5(c) Verify Stokes' theorem for:

$\bar{F}=\left ( y-z+2 \right )i+\left ( yz+4 \right )+xzk$ / over the surface x=0, y=0, z=0, x=2, y=2.

7 M

Solve any two question from Q.6(a, b,c)

6(a) Evaluate

$\int_{c}\bar{f}.d\bar{r}\ \text{where}\\ \bar{f}=\left ( 5xy-6x^2 \right )i+\left ( 2y-4x \right )j$ / and c is the arc of the curve in the xoy plane, y=x³ from (1, 1) to (2, 8)

6 M

6(b) Evlauate

$\iint _s\bar{F}.\overline{ds}\ \text{where}\\ \bar{F}=yzi+zxj+xyk$ / and s is the part of the surface of the sphere

$x^2+y^2+z^2=1$ / which lies in the first octant.

6 M

6(c) Use Storke's theorem to evaluate:

$\int _c\left ( 4yi+2zj+6yk \right ).d\bar{r}$ / where c is the curve of intersection of

$x^2+y^2+z^2=2z\ \text{and} x=z-1.$ /

7 M

Solve any one question from Q.7(a, b,c) & Q.8(a, b,c)

7(a) If

$v=\frac{-y}{x^2+y^2},$ / find u such that, u+iv is analytic function.

4 M

7(b) Evlauate:

$\oint _c\frac{z+4}{z^2+2z+5}dz,$ / where c is circle |z-2i| = 3 / 2.

5 M

7(c) Find the bilinear transformation which maps points 0, -1, ∞ of z-plane onto -1, -(2+i) pf W-plane.

4 M

8(a) Find the condition satisfied by a, b, c and d under which,

$u=ax^3+bx^2y+cxy^2+dy^3$ /is harmonic function.

4 M

8(b) Evlauate:

$\int_{0}^{2\pi }\frac{d\theta }{5-3\cos \theta }$ / using Cauchy's theorem.

5 M

8(c) Find the image of st. Line y=x under the transformation:

$W=\frac{z-1}{z+1}.$

4 M

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