STUPIDSID
Answer any one question from Q1 and Q2
1 (a) (i)
Solve any two: -
(D2 -2D) y=ex sin x by method of variation of parameters.
(D2 -2D) y=ex sin x by method of variation of parameters.
4 M
1 (a) (ii)
\[ x^2 \dfrac {d^2 y}{dx^2} - x \dfrac {dy}{Dx}+4y = \cos (\log x) + x \sin (\log x) \]
4 M
1 (a) (iii)
(D2-2D+D) y =x ex 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.
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:
Use it to find y at x=2, \[ \dfrac {dy}{dx} \ at \ x=0.5 \ and \ \int^3_0 \ y \ dx \]
| 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: \[ \dfrac {x-1}{2} = \dfrac {y-3}{-2}=\dfrac {z}{1} \]
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).
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.
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.
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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