SPPU Engineering Maths 3 - December 2014 Exam Question Paper

SPPU Electronics and Telecom Engineering (Semester 4)
Engineering Maths 3
December 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

Answer any one question from Q1 and Q2

1 (a) (i) Solve any two: -
(D² -2D) y=e^x sin x by method of variation of parameters.

4 M

1 (a) (ii)

x^{2} \frac{d^{2} y}{d x^{2}} - x \frac{d y}{D x} + 4 y = cos (log x) + x sin (log x)

$x^2 \dfrac {d^2 y}{dx^2} - x \dfrac {dy}{Dx}+4y = \cos (\log x) + x \sin (\log x)$

4 M

1 (a) (iii) (D²-2D+D) y =x e^x sin x.

4 M

1 (b) Find Fourier sine transform of:

$\begin {align*} f(x)&=x^2, \ \ 0\le x \le 1 \\ &=0 \ \ , \ \ x>1 \end{align*}$

4 M

2 (a) An electric current consists of an inductance 0.1 Henry, a resistance R of 20 Ω and a condenser of capacitance C of 25 μfarad. If the differential equation of electric circuit is:

$L \dfrac {d^2 q}{dt^2}+ R \dfrac {dq}{dt}+ \dfrac {q}{C}=0$ then find the at time t, given that at t=0, q=0.05 coulombs

$\dfrac {dq}{dt} =0$

4 M

Solve any one:

2 (b) (i) Find z transform of:

$f(k) = \dfrac {2^k}{k}, \ k\ge 1$

4 M

2 (b) (ii) Find inverse z transform:

$F(z) = \dfrac {1}{(z-3)(z-2)}, |z|<2$

4 M

2 (c) Solve:
12f (k+2)-7f(k+1)+f(k)=0, k≥0,
F(0)=0, F(1)=3.

4 M

Answer any one question from Q3 and Q4

3 (a) Solve the following differential equation to get y(0.2):

$\dfrac {dy}{dx} = \dfrac {1}{x+y}, \ y(0)=1, \ h=0.2$ by using Runge-Kutta fourth order method.

4 M

3 (b) Find Lagrange's interpolating polynomial passing through set of points:

x	y
0	4
1	3
2	6

Use it to find y at x=2,

$\dfrac {dy}{dx} \ at \ x=0.5 \ and \ \int^3_0 \ y \ dx$

4 M

3 (c) Find the directional derivative of:
ϕ=5x²y -5y²z+2z²x
at the point (1,1,1) in the direction of the line:

$\dfrac {x-1}{2} = \dfrac {y-3}{-2}=\dfrac {z}{1}$

4 M

Show that (any one):

4 (a) (i)

$\nabla \left ( \dfrac {\overline a \cdot \overline r }{r^n}\right )= \dfrac {\overline a}{r^n} - \dfrac {n(\overline a \cdot \overline r)}{r^{n+2}}\overline r$

4 M

4 (a) (ii)

$\nabla^2 f(r) = \dfrac {d^2 f}{dr^2}+ \dfrac {2}{r} \dfrac {df}{dr}$

4 M

4 (b) Find the function f(r) so that f(r)r is solenoidal.

4 M

4 (c) Evaluate:

$\int_0^1 \dfrac {dx} {1+x^2} \ using \ Simpson's \ \dfrac {3}{8} \ rule \ taking \ h=\dfrac {1} {6}$

4 M

Answer any one question from Q5 and Q6

5 (a) Find the work done by the force: (2xy+3z²)i+(x² + 4yz)j+ (2y²+6xz)k
it taking a particle from (0,0,0) to (1,1,1).

4 M

5 (b) Apply Stoke's theorem to calculate:

$\int_c (4y \ dx + 2z \ dy + 6y \ dz)$ where c is the curve of intersection of x²+y²+z²=6z, z=x+3.

5 M

5 (c) Evaluate:

$\iint_s (xz^2 \ dydz + (x^2y-z^2) dzdx + (2xy+y^z) dxdy)$ where s is the surface enclosing a region bounded by hemisphere x²+y²+z²=4 above xoy plane.

4 M

6 (a)

$If \ \overline F = \dfrac {1} {x^2+y^2} (-y\overline i + x \overline j)$ then show that:

$\oint_c \overline F\cdot d\overline r =2 \pi$ where c is circle x²+ y²=1.

4 M

6 (b) Evaluate:

$\iint_s (4xz \overline i - y^2 \overline j + yz \overline k) \cdot d\overline s$ over the cube bounded by the planes:
x=0, x=2, y=0, y=2, z=0, z=2.

5 M

6 (c) Maxwell's electromagnetic equations are:

$\nabla \cdot \overline B =0, \ \nabla \times \overline E = \dfrac {\partial \overline B} {\partial t}$ Given B=curl A then deduce that:

$\overline E + \dfrac {\partial \overline A} {\partial t} = - grad \ V$ where V is the scalar point function.

4 M

Answer any one question from Q7 and Q8

7 (a) Show that: u=e^-x(x sin y ? y cos y) is harmonic and determine an analytic function f(z)=u+iv.

5 M

7 (b) Evaluate:

$\int_c (z-z^2) dz$ where c is the upper half circle |z|=1.

4 M

7 (c) Find the Bilinear transformation which maps the points z=0, -1, ∞ in the z-plane onto the point w=-1, -(2+i), i in the w-plane.

4 M

8 (a) Find the analytic function f(z)=u+iv if:
v=(r-1/r) sin θ, r≠0.

4 M

8 (b) Using Cauchy's integral formula, evaluate the integral:

$\int_c \dfrac {(z+4)} {(z^2+2z +5)} dz$ where c is the curve |z+1-i |=2.

5 M

8 (c) Find the image in the w-plane of the circle |z-3|=2 in the z-plane under the inverse mapping

$w = \dfrac {1} {z}.$

4 M

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