1. State the domain and range of ; hence determine the domain and range of the function .

2. A particle moves along the x-axis X'OX so that its acceleration is given by , where x is the displacement of the particle at time t. If the particle is at rest when x = 5, find the velocity of the particle when x = 4 assuming at that instant the particle is moving in the direction OX'.

3. Prove that

4. P and Q are points on a circle and the tangents to the circle at P and Q meets at S. R is a pont on the circle so that PR // QS. Prove that PQ = QR.

5. A circle has equation

(a) Find the centre and radius of the circle.

(b) The line x + 2y = 0 meets the circle at A and B. Find the coordinates of A and B and hence its length.

6. A wall 27 m high stands on a horizontal plane. From a point O in this plane, a ball is projected in a plane at right angles to the wall at an angle with the horizontal, where tan = 3/4. The ball just clears the top of the wall. If O is 96 m from the wall, find the time taken to reach the wall and show that the speed of projection is 40 m/s.

(a) Find the greatest height to which the ball will rise beyond the horizontal plane.

(b) Find the range.

7. The chord PQ in the parabola subtends a right angle at the vertex of the parabola. The normals to P and Q meets at R.

(a) Prove that pq = -4 where p and q are parameters of the points P and Q.

(b) Show that as P and Q take varied positions on the parabola, the locus of R is .

8. Determine the coordinates of the focus and vertex and find the equation of the directrix of the parabola .

9. Six discs are numbered 1, 3, 5, 7, 9, 11. Two discs are drawn at random from these discs (without replacement). What is the probability that the sum of the numbers on the two discs exceeds 13 ?

10. Find the set of values of x for which the quadratic equation is prositive. Hence, determine the set of values of x for which the curve is concave up.

11. Find the derivative of . Show that there is a stationary point at x = 0, and find the abscissae of the first 3 positive and first 3 negative stationary points.

12. Find the area under the curve of between x = 2 and x = 3.

13. A chord AB of length cm is drawn in a circle centre O with radius 2 cm. Find the exact value of the (a) area of the minot sector AOB; (b) length of the major arc AB.

14. Prove by mathematical induction that is divisible by 9, if n is odd.

15. Write down the expansion of by the Binomial Theorem. Noting that , show by evaluating the coefficient of of both sides of this identity, that:

Hence, prove that

Thus, find the value of n if

16. Two bags, each contains 3 balls: one white and two black. A ball is selected randomly from the first bag and placed into the second bag. A ball is then drawn out from the second bag and placed in the first. Find the probability that each bag still contains 1 white and 2 black balls.

17. The probability that a biased coin will fall tails is 3/4. This coin is tossed 4 times. Determine the successive probability that the coin will fall tails 4, 3, 2, 1, 0 times in the 4 throws.

Hence, determine the expected number of tails in the four throws and compare this with the theoretical expected number.