OCTOBER 2003
U/IC 4718/PAP

Time: Three hours
Maximum: 100 marks

SECTION A - (10 x 2 = 20 marks)
Answer ALL questions.

1.	If f(X,Y) = X3 + Y3 - 3axy, find dy/dx.

2.	Write the conditions for a point x =a to be a maximum point or a minimum point.

3. If a curve is defined by the parametric equation x = f(O), y = 0(0), write the formula to find the radius of curvature.

4. From the polar equation of the curve 2a = 1 - cos 0 , show that p 2 = ar. r

5.	Find the envelope of the family of lines

a Y = Mx + - where a is constant.

M

6.	Define an asymptote of a curve.

7.	Evaluate f logxdx.

Ir /2

8.	Evaluate f sin 6 X COS8 xdx.

0

9.	Find the circumference of the circle x 2 +Y2 =r 2 .

2 	2

10. Find the area of the ellipse 2 +- 2 =1. a b

SECTION B - (5 x 16 = 80 marks)


Answer ALL questions.

Each question carries 16 marks.


11. (a) Find the nth derivative ofy = (X X
	_1)2(X +2)

	(b)	Determine the maximum and minimum of
X 3 -18X2 + 96x + 4. 	(8+8)

Or

(c) Find the nth differential coefficient of cos x cos 2x cos 3x .

	(d) If U=Iog(X3+y3+Z3 - 3xyz),	show that
	au + au + au =	3	(8+8)

ax O'Y az X +Y +z

12. (a) If a curve is defined by the parametric equation x = f(O) and y = 0(0), then derive the formula to find the radius of curvature.


(b) Find p at the point 't' of the curve

x=a(cost+tsint), Y=a(sint-tcost). 	(8+8)

Or

14.	(a)	Evaluate	2 +3	dx.
			V;-i-+ X + 1


	(b)	Establish	a	reduction formulafor

	7r12

in	f Cos n xdx.	(8+8)

0

Or

7r

(c) Evaluate	r x sinx dx.
	j j+ -CoS2 X
	0


(d) Prove that F(n +1) =n[(n).	(8+8)

15.	(a) Find the arc length of the curve YI =X3 from

the origin to the point (4, 8).


	b
(b) Evaluate f x A by expressing it as the limit

a

of a sum.	(8+8)

Or


		n a cosO

(c)	Evaluate	f f rsin0drdO.

		0 0






		a x y

(d)	Evaluate	f f f xyz dz dy dx .	(8+8)

0 0 0