 MORE IN Engineering Maths 3
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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: $\dfrac {d^2y}{dx^2} + 5 \dfrac {dy}{dx} + 6y = e^{-2x}\sin 2x$
4 M
1 (a) (ii) (D2-4D+4)y=e2x x-2 (by variation of parameters).
4 M
1 (a) (iii) $x^3 \dfrac {d^3 y}{dx^3} + 2x^2 \dfrac {d^2y}{dx^2}+2y = x+\dfrac{1}{x}$
4 M
1 (b) Solve:
$f(k)-4 f(k-2) = \left ( \dfrac {1}{2} \right )^k , \ k\ge 0$
4 M

2 (a) The charge Q on the plate of condenser satisfies the differential equation: $\dfrac {d^2Q}{dt^2} + \dfrac {Q}{LC}= \dfrac {E}{L}\sin \dfrac {t}{\sqrt{ LC}}$ Assuming $\dfrac {1} {LC} = \omekga^2$ find the charge Q at any time 'l'.
4 M
2 (b) Find the Fourier sine integral representation for the function: $f(x)= \left\{\begin{matrix} \frac {\pi}{2} ;&0< x<\pi \\ 0; &x>\pi \end{matrix}\right.$
5 M
Attempt any one:
2 (c) (i) Find z-transform of f(k) = ke-3k; k≥ 0
4 M
2 (c) (ii) Find $z^{-1} \left [ \dfrac {z^2}{z^2 + 1} \right ]$
4 M

Answer any one question from Q3 and Q4
3 (a) Given $\dfrac {dy}{dx} = 3x + \dfrac {y}{2}; \ y(0)=1 \ \ \ h=0.1$ Evaluate y(0.1) by using Runge-Kutta method of fourth order.
4 M
3 (b) The distance travelled by a point p in XY-plane in a mechanism is given by y in the following table. Estimate distance travelld by p when x=4.5.
 x y 1 14 2 30 3 62 4 116 5 198
4 M
3 (c) Find the directional derivative of function ϕ=xy2+yz3 at (1, -1, 1) along the direction normal to the surface 2x2+y2+2z2=9 at (1, 2, 1).
4 M

Prove that (any one):
4 (a) (i) $\overline a \cdot \nabla \left [ \overline b \cdot \nabla \dfrac {1}{r} \right ] = - \dfrac {(\overline a \cdot \overline b)}{r^3} + \dfrac {3 ( \overline b \cdot \overline r) (\overline a \cdot \overline r)}{r^5}$
4 M
4 (a) (ii) $\nabla \cdot \left [ r \nabla \dfrac {1}{r^5} \right ] = \dfrac {15}{d^6}$
4 M
4 (b) Use Trapezoidal Rule to estimate the value of: $\int^2_0 \dfrac {x} {\sqrt{2+x^2}} dx$ by taking h=0.5.
4 M
4 (c) Show that the vector field f(r)r is always irrotational and then determine F(r) such that vector field f(r)r is solenoidal.
4 M

Answer any one question from Q5 and Q6
5 (a) Evaluate$\int\limits_{c} \left [ \left ( 2x^2 y + y + z^2 \right )i + 2 (1+yz^3)j + (2z+3y^2 z^2)k \right ]\cdot d\overline r$ along the curve C: y2+z2=a2 x=0
4 M
5 (b) Find $\iint_s \overline F \cdot \widehat n \ ds$ where s is the sphere x2+y2+z2=9 and $\overline F = (4x + 3yz^2) \widehat{i} - ( x^2y^2 +y)\widehat{j}+ (y^3+2z)\widehat{k} 4 M 5 (c) Evaluate: \[ \iint_s \nabla \times \overline F \cdot \widehat n ds$ for the surface of the paraboloid z=4 x2 -y2; (z ≥ 0) and $\overline F y^2\widehat {i} + z\widehat{j} + xy\widehat{k}$
5 M

6 (a) Find the total work done in moving a particle is a force field $\overline F = 3xy\widehat{i} 5z\widehat{j}+10x\widehat{k}$ along the curve x=t2+1, y=2t2, z=t3 from t=1 and t=2.
5 M
6 (b) Using divergence theorem to evaluate the surface integral $\iint_s \overline F \cdot \widehat {n} ds \ where \ \overline F = \sin xi + (2 - \cos x) j$ and S is the total surface area of the parallelepiped bounded by x=0, x=3, y=0, y=2, z=0 and z=1.
4 M
6 (c) Equations of electromagnetic wave theory are given by: $i) \ \ \nabla \cdot \overline D = \rho \\ ii) \ \ \nabla \cdot \overline H = 0 \\ iii) \ \ \nabla \times \overline D = \dfrac {-1}{C}\dfrac { \partial \overline H}{\partial t} \ and \ \\ iv)\ \ \nabla \times \overline H = \dfrac {1}{C} \left [ \dfrac {\partial \overline D}{\partial t} + \rho \overline v \right ] \\ prove \ that \\ \nabla^2 \overline D - \dfrac {1}{C} \dfrac {\partial ^2 \overline D}{\partial t^2} = \nabla \rho + \dfrac {1}{C^2} \dfrac {\partial }{\partial t} (\rho \overline v)$
4 M

Answer any one question from Q7 and Q8
7 (a) Find the analytic function f(z)=u+iv if 2u+v=ex(cos y ? sin y).
5 M
7 (b) Evaluate: $\int_c \dfrac {e^{2z}} {(z-1)(z-2)} dz$ where C is circle |z|=3. \]
4 M
7 (c) Find the bilinear transformation which maps the points z=-1, 0, 1 of z-plane into the point w=0,i,3i of w-plane.
4 M

8 (a) Find the analytic function f(z)=u+iv where u=r3 cos 3θ + r sin θ.
4 M
8 (b) Evaluate: $\int_c \dfrac {1-2z}{z(z-1)(z-2)} dz$ where Cis |z|=1.5.
4 M
8 (c) Find the map of the straight line y=2x under the transformation: $w=\dfrac {z-1}{z+1}$
5 M

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