Question 1:
(i) Express the exact value of 
as a fraction in tis lowest terms.
(ii) Express 
as a fraction in its lowest term.
(iii) ABCD is a parallelogram (as shown below) in which the lengths of AB and AD are 3 cm and 1 cm, respectively, and the angle ABC is 60o ; also the diagonals cut at K and the angle AKB is denoted by q. Calculate the exact values (giving answers as rational numbers or surds) of:
(a) the length of AC; (b) the length of BD (c) cosq.
Question 2:
(i) (a) Find the equation of the line l through (1,0) which passes through the point of intersection between the line y = 2x + 1 and the y-axis.
(b) Find the exact value of the tangent of the acute angle between the line l and the line y = 2x + 1.
(ii) (a) Differentiate 
(b) Find a primitive for 
(iii) Find all real numbers x for which |x + 1| > |x - 1|
Question 3:
(i) Find (a) 
(b) 
(ii) A plane region is bounded by the curves 
and 
, and by the lines x = 1 and x = 9. Find
(a) the area of the region (b) the volume of the solid obtained by rotating this region about the x-axis.
Question 4:
(i) If 
, prove that 
. Hence show that 
(ii) Find the largest positive value of x for which 
, expressing your answer as a rational number or surds.
Question 5:
(i) Find the minimum value of 
for x>0, giving reasons for your answer.
(ii) A cargo services operates by running a ship between port A and port B at a constant speed of v km/h. For a given v , the cost per hour of running the ship is 9000 + 10v2 dollars. Find the value of v which minimises the cost of the trip.
Question 6:
(i) A parabola in the cartesian (x, y) plane has its vertex at (-1, -2) and its focus at (-1, -3). Derive an inequality in x and y which is satisfies by the coordinates of a point P(x, y) if and only if P is closer to the focus of the parabola than it is to the directrix of the parabola.
(ii) The velocity v(t) of a particle moving along the x-axis is given in terms of time t by 
. If x(t) denotes the position of the particle at time t, show that for every possible choice of time 
and 
, |x() - x()| < 4.
Question 7:
In a raffle there is one first prize of $100, one second prize of $20, and one third prize of $10. There are 100 tickets in the raffle and the prize winning tickets are drawn consecutively without replacement, with the first ticket drawn winning first prize. Find the probability that:
(i) a person buying one ticket in the raffle wins (a) 1st prize (b) at least $10 (c) a prize.
(ii) A person buying 2 tickets in the raffle wins (a) 1st prize (b) at least $20.
Question 8:
(i) Sketch the curve y = cosx for 
.
(ii) Define the function 
, specifying its domain and range. Sketch the curve y =
(iii) Find the range of the function y = cos(sinx)
(iv) Sketch the curve y = 
for
Question 9:
(i) The first three terms of an arithmetic series are 50, 43, 36.
(a) write down a formula for the n-th term.
(b) If the last term of the series is -27, how many terms are there in the series ?
(c) Find the sum of the series.
(ii) A loan of $1000 is to be repaid by equal annual instalments, repayments commencing at the end of the first year of the loan. Interest, at the rate of 10%, is calculated each year on this balance owing at the beginning of that year, and added to that balance. If the annual instalment is $P, prove that
(a) the amount owing at the beginning of the 2nd year of the loan is (1100 - P) $
(b) the amount owing at the beginning of the 3rd year of the loan is (1210 - 2.1P) $
(c) if the loan (including interest charges) is exactly repaid at the end of n years, then
P = 