1-The Surface:F(x, y, z) =x2 + y2 + z2 =1, g(x, y, z) =x + y + z + 5 intersected in a curve C .Find the line tangent to C at the Po= (1, 2, 2)

2-Given Z=U(x, y) e(ax +by)

Where U is a function of X, Y such that Uxy =0, a, b are constants .Find the Value of a & b that will make the expression zxy-zx-zy identically zero

3-Find the points on the sphere: x2+y2+z2=9 such that their Distance from the point (4,-8, 8) are extremism

4-Find the envelope of the family of ellipses of constant area and whose axes of symmetry coincide

5-Evaluate dx, P -1, then Prove that dx= ,,m=1,2.3…..

6-Determind whether the following series is convergent or divergent:

(a) (b) (c) (d)

7-Find the interval of convergent of the following series:

(a) (b)

8-expand in Fourier series the function:

F(x) = , 0<x<2

9- Expand in Fourier series the function

F(x) =x ( -x), 0<x<

10-Find the complex Fourier series of function:

F(x) =

11-(tany-2x) dx + x(1+xtany)dy=0

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15-Find the orthogonal trajectories of the family of curves: 4(x+1)2 +(y-2)2=CWhere C is the parameter.

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18-Given y1=x+1 is a solution of the homogenous eqn

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20-Given Y=sinx is solution of