Smso 2003 Paper 1

PAPER 1

2.

Find the equation of the normal to the curve y = x2 – 4x + 5 at (3, 2).

3.

Express f(x) = 9x2 – 36x + 52 in the form (Ax – B)2 + C, where A, B and C are integers. (3)

(4)

Hence, or otherwise, find the range of f(x) for x > 2 and its inverse.

6.

(5)

The diagram shows a sector OAB of a circle, with centre O, in which angle AOB = 120°.

Area of sector OAB 4 ,  Area of traingle OAB 3 3 Length of arc AB (b) Find the exact value of . Length of chord AB

(a) Show that

(4) (3)

7.

8.

The tennis ball is dropped vertically from a height of 1 m onto a flat surface. After each bounce the maximum height reached by the ball is 90% of the maximum height before the bounce. Given that after the nth bounce, the maximum height reached is the first below 0.1m, find n. [6]

2

10. Points A and B have position vectors 2i – 4j – 2k and -3i + 2j + k respectively, relative to the origin O. The points P and Q are such that OP = 3 OA and OQ = 2OB. Show that PQ is parallel to 3i – 4j – 2k, and find the length of PQ. [4]

9

11. Find the coefficient of x3 in the expansion

2  x  . x 

[4]

12. A flask has a volume of V cm3 and is made up of two parts: a right circular cylinder of radius r cm and height h cm and a hemisphere at the bottom of the cylinder of radius r cm, as shown in the diagram. (i)

Show that h 

V 2  r. 2 3  r

[4]

(ii) The flask is made of some thin metal sheets and the cost of the cylindrical surface and the base is $3 per cm2 and that of the hemispheric surface is $6 per cm2. Let $C be the total cost of making the flask. Show that

C  8 r 2  If V 

6V r

9 , find the value of r which gives the minimum value of C. [6] 8

3