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MU First Year Engineering (Semester 2)
Applied Mathematics 2
December 2011
Total marks: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

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 (D4 - 4D3 + 8D2 - 8D + 4)y = ex + 1.
5 M
1(c) Find the length of the curve y = log(ex + 1) - log(ex - 1) from x=1 to x=2.
5 M
1(d) Find the area bounded by the curves : xy =2, 4y = x2, 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 x0 = 1.2 and y0 =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 + y2, 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 (D2 - 2D + 1)y = xexsin(x)
6 M
4(b) Evaluate:

6 M
4(c) Solve:

8 M

5(a) Solve dy/dx = ex-y(ex-ey)
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 ay2 = x3 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: (D2 - 1)y = x2sin(3x).
8 M

7(a) Evaluate ∫ ∫ (x2 + y2)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:

8 M

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