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3 C 20647

يسمه

2 3
21. Evaluate ர்‌ e* (y+ 2z) dz dy dx.
0 0

22. Determine whether the vector field F = e* (cosy i —siny ¢ ) is conservative. If so, find a potential

function for the vector field.

23. Evaluate (९ + ೨3) dx+ (x? + 3xy) dy , where Cis the positively oriented closed curve lying on the
0

boundary of the semi annular region R bounded by the upper semicircles x? + y? = 1 814 2 + y? =9
and the x-axis.
(5 x 6 = 30 marks)
Section C (Essay Questions)

Answer any two questions.
Each question carries 10 marks.

24. (a) Sketch the graph of f(x, y)=9- 2 - 02.
It xy .
(b) Show that (೬.೫)3(00 3. 2 does not exist.

25. (a) Find the relative extrema of f(x,y) = x3 + y? 2), + 7 -8) + 2.

(b) Find the minimum value of f (x,y,z) = 2x 2 + y? 4+ 327> subject to the constraint

Qn — By —4z = 49.

26. Let R be the region bounded by the square with vertices (0,1),(1,2),(2,1) and (1,0). Evaluate

1 (x +y) sin? (x—y)dA.
R

27. Let F =(x,y,z)=2xyz” 1+ ७222 + Qn? y2 kh.
(a) Show that # is conservative and find a scalar function f such that F = Vf.

(b) If # is a force field, find the work done by # in moving a particle along any path from
(0, 1, 0) to (1, 2, -1).
(2 x 10 = 20 marks)

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