1. Factorise

2. Evaluate (i) (ii) using u=1+x as a subst.

3. Find the coordinates of the point which divides the interval AB with A(1,4) and B(5,2) externally in the ratio 1:3.

4. Solve the inequation

5. PQ is a tangent to a circle QRS, while PRS is a secant intersecting the circle in R and S, as in the diagram. Given that PQ=6, RS=5, PR = x, find x.

6. Find all angles q with 0 < q < 2p for which sin 2q = sinq.

7. (a) Show that

(b) The acceleration of a particle moving in a straight line is given by where x is the displacement from O. Initially, the particle at the origin with velocity 2 m/s. Prove that

(c) What happens to v as x increases without bound.

8. A committee of 3 is to be selected from a club of 8 members.

(a) How many different committees can be formed.

(b) If there are 4 Queenslanders in the club, what is the probability that a randomly selected committee of 3 contains only Queenslanders ?

9. Find the constant term in the expansion of

10. The angle of elevation of a tower PQ of height h metres at a point A due east of it is 12o . From another point B, the bearing of the tower is O51oT and the angle of elevation is 11o. The points A and B are 1000 m apart and on the same level as the base Q of the tower.

(a) Show that the angle AQB = 141o.

(b) Consider the triangle APQ and show that AQ = h.tan78o.

(c) Find a similar expression for BQ.

(d) Use the cosine rule to calculate h to the nearest metre.

11. A circular plate of radius r is heated so that the area of the plate expands at a constant rate of 3.2 cm2/min . At what rate does x increase when r=10 cm.

12. (a) The polynomial equation P(x) = 0 has a double root at x = a. By writing where Q(x) is a polynomial, show that P'(a) = 0.

(b) Hence or otherwise find the values of a and b if x = 1 is a double root of

13. Let each different arrangement of all letters of DELETED be called a word.

(a) How many words are possible.

(b) In how many of these words will D's be separated.

14. AB and CD are two intersecting chords of a circle and CD is parallel to the tangent to the circle at B.

(a) Draw a neat sketch of the above data

(b) Prove that AB bisects the angle CAD

15. (a) By considering the sum of the terms of an AP serioes show that

(1+2+ ....+ n)2 =

(b) By using the principle of math induction prove that for all n > 1

16. Sketch the parabola y =

(a) Find the largest positive domain such that the graph define a function f(x) which has an inverse.

(b) Find its inverse function and state its domain

(c) State the domain for which the function does not have an inverse

17. Two points and lie on the parabola .

(a) Derive the equation of the tangent to the parabola at P.

(b) Find the coordinate of the point of intersection T of the tangent to the parabola at P and Q.

(c) You are given that the tangents at P and Q in (ii) intersect at an angle of 45o. Show that p - q = 1 + pq.

(d) By evaluating the expression at T, or otherwise, find the locus of T when the tangents at P and Q intersect as given in (c).

18. A "six" is scored in a cricket game when the ball is hit over the boundary fence on the full as in the diagram. A ball is hit from O with velocity V 32 m/s at an angle q to the horizontal and towards the 1 metre high boundary fence 100 m away.

(a) Derive the equations of motion for the ball in flight using axes as in the diagram. (g=10)

(b) Show that the ball just clears the boundary fence when

50000 tan2q=102400tanq+51024 = 0

(c) In what range must q lies for a "six" to be scored.

(d) If during the flight of the ball, its velocity is reduced by piercing an extremely thin "board", show by a sketch how the path is altered.

Without further calculation, discuss qualitatively the effect of air resistance on your answer

19. (a) Differentiate and hence show that

(b) Sketch the curve .