(i) Show that hence evaluate .

(ii) Solve the equation , . Use this result to give the solution of the equation for .

(iii) Use the Newton's method to find an improved solution to the equation if the initial solution is x = 0.

Question 2:

(i) The sum of the roots of the cubic equation is equal to twice their product. Find the value of k.

(ii) The respective probabilities that each of the three golfers qualifies for a certain turnament are 1/7, 1/10, and 1/4. What is the probability that none of them qualifies for the tournament.

(iii) State the domain and range of the function and sketch this graph,

Question 3:

(i) The points and lie on the parabola such that the tangent at P is parallel to the normal at Q. Find the relation expressing in terms of .

(ii) Find the following limits (a) (b)

(iii) Find and hence show that .

Question 4:

(i) Use the method of mathematical induction to prove that .

(ii) Show that if where V(x) is a polynomial, then is a polynomial with (x - a) as a factor. Hence or otherwisse find the values of the constant k, l, for which has a factor .

Question 5:

(i) The acceleration of a particle moving in a straight line is given by , where x is the displacement in metres from the origin O and t is the time in seconds. Initially the particle is at rest at x = 4.

(a) If the velocity of the particle is v m/s show that .

(b) Show that the particle does not pass through the origin.

(c) Determine the position of the particle when v = 10. (1991/QQ4).

(ii) Containers are coded by different arrangements of coloured dots in a row. The colours used are red, white and blue. In an arrangement at most three of the dots are red, at most two of the dots are white and at most one is blue.

(a) Find the number of different codes possible if six dots are used.

(b) On some containers only five dots are used. Find the number of different codes possible in this case. (1991/Q4)

Question 6:

(i) The volume V of a sphere of radius r cm is increasing at a constant rate of 200 per second. Find the rate of increase of the surface area (S) of the sphere when the radius is 50 cm . (1991/Q3).

(ii) If , find dy/dx when x = 1. (1991/Q3)

(iii) Evaluate (a) (b) , using . (1991/Q1)

Question 7:

(i) Consider the function .

(a) Evaluate f(2).

(b) Draw the graph of this function

(c) State the domain and range,

(ii) Sketch the parabola whose parametric equations are . On the diagram, mark the points P and Q which correspond to t = -1 and t = 2, respectively.