1(a)

prove that:**\(\Gamma({\dfrac {3}{4}-x}).\Gamma({\dfrac {3}{4}+x})=(\dfrac{1}{4}-x^2)\pi\sec x.\)** Provided -1<2x<1

5 M

1(b)
Solve (D

^{4}- 4D^{3}+ 8D^{2}- 8D + 4)y = e^{x}+ 1.
5 M

1(c)
Find the length of the curve y = log(e

^{x}+ 1) - log(e^{x}- 1) from x=1 to x=2.
5 M

1(d)
Find the area bounded by the curves : xy =2, 4y = x

^{2}, y=4
5 M

2(a)
Change the order of integration:

6 M

2(b)
Solve by the method of variation of parameters:

6 M

2(c)
Solve dy/dx = 2 + (xy)

^{xy}with x_{0}= 1.2 and y_{0}=1.6403 by Euler's modifies formula for x = 1.6. Correct the four places of decimal. Take h = 0.2
8 M

3(a)
Evaluate:

6 M

3(b)
Change to polar co-ordinates:

6 M

3(c)
Solve the differential equation dy/dx = x + y

^{2}, y(0) = 1 by Runge-Kutta method of fourth order, for the interval (0,0.2) in steps of h =0.1
8 M

4(a)
Solve (D

^{2}- 2D + 1)y = xe^{x}sin(x)
6 M

4(b)
Evaluate:

6 M

4(c)
Solve:

8 M

5(a)
Solve dy/dx = e

^{x-y}(e^{x}-e^{y})
6 M

5(b)
Solve:

6 M

5(c)
Find the volume of the tetrahedron bounded by the planes x = 0, y = 0, z=a,x + y + z = a

8 M

6(a)
Find the mass of the lamina bounded by the curves ay

^{2}= x^{3}and the line by=x, if the density at a point varies as the distance of the point from the x-axis.
6 M

6(b)
Using Duplication Formula prove:

6 M

6(c)
Solve: (D

^{2}- 1)y = x^{2}sin(3x).
8 M

7(a)
Evaluate ∫ ∫ (x

^{2}+ y^{2})dxdy over the area of a triangle whose vertices are (0,1) (1,1) and (1,2).
6 M

7(b)
Solve the following:

6 M

7(c)
Evaluate the following:

and hence deduce that:

and hence deduce that:

8 M

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