STUPIDSID
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:
\[ f(k)-4 f(k-2) = \left ( \dfrac {1}{2} \right )^k , \ k\ge 0 \]
\[ f(k)-4 f(k-2) = \left ( \dfrac {1}{2} \right )^k , \ k\ge 0 \]
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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