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SATHYABAMA UNIVERSITY

Title of the Paper :Advanced Mathematics Max. Marks :80Date :25/05/2011 Session :AN

PART - A (6 x 5 = 30)Answer ALL the Questions

1. Find the singular value decomposition of

111

111

2. Write down the properties of harmonic function.

3. Find the extremals of the functional V= [Y(x)] = +1

0

)sin2( 22X

Xdxxyyy

4. Find the plane curve of fixed perimeter and maximum area.5. Write down the dual of the following LPP and solve it.

Max Z = 4x1 + 2x2

Subject to x1 x2 -3-x1 + x2 > -2

And x1, x2 >06. The temperature T any point (X,Y,Z) in the space is T = 400 XYZ2.

Find the highest temperature on the surface of the unit sphere X2 +Y2 + Z2 = 1.

PART B (5 x 10 = 50)Answer ALL the Questions

7. Construct a QR decomposition for the matrix

666

333

224

(or)

8. Find the pseudo inverse of

622222

222

9. A curve c joining the points (X1, Y1) and (X2, Y2) is revolved about

the x axis, find the shape of the curve so that the surface thusgenerated is a maximum.(or)

10. Find the curve on which an extermum of the function

==

0

212 0)0(;)( ydxyyI

Can be achieved, if the second boundary point is permitted to

move along the straight line X=/4

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11. A string is stretched and fastened to two points X =0 and X = Lapart. Motion is started by displacing the string into the form y=k(LX X2) from which it is released at time t = 0. Find thedisplacement of the string of any point by laplace transform on thestring at a distance of x from one end at time t.

(or)

12. A rod 30 cm long has its end A and B kept at 20 and 80respectively until steady state conditions prevail. The temperature

at each end is then suddenly reduced to 0C and kept so. Find the

resulting temperature u (x,t) by fourier transform method taking X =0 at A.

13. Solve: uxx + uyy = 0 in 0 x 4, 0 y 4 in given that u (0, y) = 0, u

(4,y)= 8 + 2y u(x,0) = x2 and u (x,4) = 2 taking h=k=1, obtain theresult correct to one decimal.

(or)14. Solve the poissons equation 2 u = -10(X2 + Y2 + 10) by Fourier

transform method over the square mesh with sides X =0, Y=0,X=3, Y=3 with u=0 on the boundary and mesh length 1 unit.

15. Use two phase simplex method to solveMaximize Z=5x1 + 8x2Subject to 3x1 + 2x2 >, 3

X1, 4x2 < 4

X1 + X25Andx1, x2>, 0

(or)

16. Find the Stationary value ofX2 + Y2 + Z2 subject to X2 +32

22ZY

+ and

3x + 2y + z = 0.