1(a)
Evaluate the following:

3 M

1(b)
Solve (D

^{2}-1)(D-1)^{2}y=0
3 M

1(c)
Prove that E=1+Δ=e

^{hD}
3 M

1(d)
Solve the following:

3 M

1(e)
Change into polar co-ordinates and evaluate:

4 M

1(f)
Evaluate the following:

4 M

2(a)
Solve (x

^{3}y^{3}-xy)dy=dx
6 M

2(b)
Change the order of Integration and evaluate:

6 M

2(c)(1)
Prove that:

4 M

2(c)(2)
Evaluate , where a>0

4 M

3(a)
Evaluate the following:

6 M

3(b)
Find the area bounded by 9xy=4 and 2x+y=2

6 M

3(c)(1)
Solve the following:

4 M

3(c)(2)
Solve the equation by variation of parameters:

4 M

4(a)

Show that for the parabola

from θ=0 to θ=π/2, length of the arc is

6 M

4(b)
Solve the following:

6 M

4(c)
Apply Runge-Kutta method of fourth order to find an approximation value of y at x=0.2, if dy/dx=x+y

^{2}, given y=1 when x=0, in steps of h=0.1
8 M

5(a)
Solve: (2xy

^{4}e^{y}+2xy^{3}+y)dx+(x^{2}y^{4}e^{y}-x^{2}y^{2}-3x)dy=0
6 M

5(b)
Solve dy/dx=2x+y with x

_{0}=0,y_{0}=0 by Taylor’s method. Obtain y as a series in power of x. Find approximation value of y for x=0.2,0.4. Compare your result with exact values.
6 M

5(c)
Evaluate the following equation

by Trapezoidal method, Simpson's 1/3rd and 3/8th methods. Compare result with exact values.

by Trapezoidal method, Simpson's 1/3rd and 3/8th methods. Compare result with exact values.

8 M

6(a)
In a circuit containing inductance L, resistance R, voltage E, the current i is given by L(di/dt)+Ri=E. Find i at a time t if at t=0,i=0, and if L, R, E are constants.

6 M

6(b)
Evaluate ∫∫xy dxdy bounded by y=x, x

^{2}+y^{2}-2x=0, and y^{2}=2x
6 M

6(c)(1)
Find the volume of a tetrahedron bounded by the plane x=0,y=0,z=0 and x+y+z=a

4 M

6(c)(2)
Find the volume bounded by the cone z

^{2}=x^{2}+y^{2}and paraboloid z=x^{2}+y^{2}.
4 M

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