1(a)
Evaluate the following:

3 M

1(b)
Solve (D

^{4}+4)y=0
3 M

1(c)
Prove that E∇ = Δ =∇E

3 M

1(d)
Solve the following:

3 M

1(e)
Evaluate ∫∫r

^{3}drdθ over the regions between the circles r=2sinθ and r=4sinθ
4 M

1(f)
Evaluate the following:

4 M

2(a)
Solve (x

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

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

6 M

2(c)
Prove that:

8 M

3(a)
Evaluate the following:

6 M

3(b)
Find the area of one loop of the lemniscate r

^{2}=a^{2}cos2θ
6 M

3(c)
Solve (D

^{3}+2D^{2}+D)y = x^{2}e^{3x}+sin^{2}x+2^{x}
8 M

4(a)
Show that the length of the arc of the parabola y

^{2}=4ax cut off by the line 3y=8x is alog2+15/16
6 M

4(b)
Using the method of variation of parameters, solve:

6 M

4(c)
Compute y(0.2) given (dy/dx)+y+xy

^{2}=0, y(0)=0, By taking h=0.1 using Runge-Kutta method of fourth order correct to 4 decimals.
8 M

5(a)
Solve the following:

6 M

5(b)
Solve (dy/dx)-2y=3e

^{x},y(0)=0 using Taylor Series method. Find approximate value of y for x=0.1 and 0.2
6 M

5(c)
Evaluate

using Trapezoidal rule, Simpson's 1/3rd rule and Simpson's 3/8th rule. Compare the result with exact values.

using Trapezoidal rule, Simpson's 1/3rd rule and Simpson's 3/8th rule. Compare the result with exact values.

8 M

6(a)
The current in a circuit containing an inductance L, resistance R and a voltage Esinωt is given by L (di/dt)+Ri=E sinωt. If i=0 at t=0, find i.

6 M

6(b)
Evaluate ∫∫e

^{(2x-3y)}dxdy over the triangle bounded by x+y=1, x=1 and y=1
6 M

6(c)(1)
Find the volume of solid bounded by the surfaces y

^{2}=4ax, x^{2}=4ay and the planes z=0, z=3.
4 M

6(c)(2)
Change to polar co-ordinates and evaluate:

4 M

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