Michael is in training for a local marathon. He is training by completing practice runs over the marathon course. So far he has completed three practice runs with times shown below. 

(i) Show that these times form a geometric series with a common ratio r = 0.95.

(ii) If this series continues, what would be his expected time in Week 5, to the nearest second? 

(iii) How many hours, minutes and seconds (to the nearest second) will he have run in total in his practice runs in these 5 weeks? 

(iv) If the previous winning time for the marathon was 2 hours and 6 minutes, how many weeks must he keep practising to be able to run the marathon in less than the previous winning time?

Grandma wants to invest a certain amount of money for her grandchild so that she will have $5 000 in ten years time. If the bank account pays 5% p.a. interest, compounding quarterly, how much will she need to invest now?

$$\begin{gathered} find the area of shaded section below that is bounded by the \mathcal{x} axis \\ the curve \mathcal{y}= e^{\mathcal{x}}-1 and the line \mathcal{x}=-1 \left(in exact form\right) \end{gathered}$$

Tiarn borrows $500 000 to buy a house. An interest rate of 9% p.a. compounded monthly is charged on the outstanding balance. The loan is to be repaid in equal monthly installments (R) over a 25 year period. 

(i) Show the amount owing after 3 months is: $A_3=\$500000{\left(1.0075\right)}^3-R\left[1.1.0075+1.{0075}^2\right]$

ii) Assuming this pattern continues the monthly repayment can be calculated using: $A_n=\$500000{\left(1.0075\right)}^n-R\left[1.1.0075+1.{0075}^2+...+1.{0075}^{n-1}\right]$ How much should Tiarn be paying each month? 

(iii) How much interest does Tiarn pay over the 25 years? 

(iv) What is the equivalent simple interest rate of this loan?

$find \underset{\mathcal{x}\to 0}{\lim }\frac{\sin \mathcal{x}}{2\mathcal{x}}$

$$\begin{gathered} Find the values of the cons\tan ts A, B and C such that \\ 3{\mathcal{x}}^2-8\mathcal{x}+6=A{\left(\mathcal{x}-1\right)}^2+B\left(\mathcal{x}-1\right)+C \end{gathered}$$

$The value of a certain function g\left(x\right) for values of x from 1 to 5 is given in the table below:$

$U\sin g Simpson\text{'}s Rule with 5 function values, find an approximation for the area enclosed by the curve y=g\left(x\right), the x-axis and the lines x= 1 and x=5.$

$Consider ∆ABC in the diagram below. AB is extended to point D forming triangle ACD. \angle ABC = \angle ACD, AB = 6 cm, AC = 8 cm and BD = w.$

$$\begin{gathered} Prove that triangle ABC is similar to triangle ACD. \\ Find the value of w, giving reasons. \end{gathered}$$

$$\begin{gathered} \text{ A box contains six yellow marbles and four green marbles. Two marbles are drawn from the box, without replacement.} \\ \text{ The probability that they are the same colour is:} \\ \begin{array}{r}\left(A\right) \frac{1}{3}\\ \left(B\right) \frac{28}{45}\\ \left(C\right) \frac{7}{15}\\ \left(D\right) \frac{13}{25}\end{array} \end{gathered}$$

There are two groups of people at a party and Minh is blind folded. 

In the first group there are 4 men, 3 women and 2 children. In the second group there are 7 men and 5 women. 

Minh is spun around and asked to select one person at random. 

Find the probability that Minh approaches the first group and 

then selects a woman. 

Find the probability that a woman from either group is selected.