A school soccer team has a probability of 0.7 of losing or drawing any match and a probability of 0.3 of winning any match.

(i) Find the probability of the team winning at least one of three consecutive matches.

(ii) What is the least number of consecutive matches the team must play to be 90% certain it will win at least one match?

(b) A light is to be placed over the centre of a circle, radius a units. The intensity, I, of the light varies as the sine of the angle, a, at which the rays strike the illuminated surface, divided by the square of the distance, d, from the light. i.e. $I=\frac{k\sin \alpha }{d^2}$ where k is a constant.

(i) Show that $I=\frac{k\mathcal{y}}{{\left({\mathcal{y}}^2+a^2\right)}^{{\displaystyle \frac{3}{2}}}}$

(ii) Find the best height for a light to be placed over the centre of a circle so as to provide the maximum illumination to the circumference.

$$\begin{gathered} The graph shows the equation \mathcal{y}=\sin (A\mathcal{x}+B ) over the \\ domain -2\pi \le \mathcal{x}\le 2\pi \end{gathered}$$

$$\begin{gathered} What are the values of A and B ? \\ A=2,B=\frac{\pi }{3} \\ A=\frac{1}{2},B=\frac{\mathrm{\pi }}{3} \\ A=2,B=\frac{\mathrm{\pi }}{6} \\ A=\frac{1}{2},B=\frac{\mathrm{\pi }}{6} \end{gathered}$$

$\mathcal{y}= f\left(\mathcal{x}\right) is shown on the number plane$

$$\begin{gathered} Which of the following statements is true? \\ \mathcal{y}=f\left(\mathcal{x}\right) is decrea\sin g and concave up. \\ \mathcal{y}= f\left(\mathcal{x}\right) is decrea\sin g and concave down. \\ \mathcal{y}=f\left(\mathcal{x}\right) is increa\sin g and concave up. \\ \mathcal{y}=f\left(\mathcal{x}\right) is increa\sin g and concave down \end{gathered}$$

In the diagram AB = BC and CD is perpendicular to AB.

CD intersects the $\mathcal{y}$-axis at P.

Copy the diagram onto your answer sheets.

a) Find the length of AB.

b) Hence show the coordinates of C are (2, 0)

c) Show the equation of CD is $3\mathcal{x}+4\mathcal{y}=6$.

d) Show that the coordinates of P are $\left(0,1\frac{1}{2}\right)$.

e) Use Pythagoras theorem on $△$POC to show the length of CP is $2\frac{1}{2}$units.

f) Prove $△$ADP is congruent to $△$COP.

g) Hence calculate the area of the quadrilateral DPOB.

The figure below shows the graph of $\mathcal{y}=f\text{'}\left(\mathcal{x}\right)$ where $f\text{'}\left(\mathcal{x}\right)$ is the derivative of a function $f\left(\mathcal{x}\right)$.

The domain of $f\left(\mathcal{x}\right)$ and $f\text{'}\left(\mathcal{x}\right)$ is $-3\le \mathcal{x}\le 3$.

Copy the diagram onto your answer sheet.

i) By considering the graph of $\mathcal{y}=f\text{'}\left(\mathcal{x}\right)$, explain why the graph of $\mathcal{y}=f\left(\mathcal{x}\right)$ has two, and only two stationary points.

ii) For what values of $\mathcal{x}$ does $\mathcal{y}=f\left(\mathcal{x}\right)$ has a relative maximum? Justify your answer.

iii) Given that $f(-3)=0$ sketch a possible graph of $\mathcal{y}=f\left(\mathcal{x}\right)$ on the same axes that you drew the graph $\mathcal{y}=f\text{'}\left(\mathcal{x}\right)$.

Find the value of $\mathcal{x}$, giving reasons.

The diagram shows the route MPR, followed by a shuttle bus from the main entrance to the Sydney Olympic site (M), via a roundabout (R), to the entrance to Olympic Park (P), where the principal stadiums are sited. R is 550 meters at a bearing of 325$°$ from M, P is 450 meters at a bearing of 250$°$ from R. It is proposed that an overhead cable car be built directly from M to P.

i) Copy the diagram onto your page and show that $\angle$MRP measures 105$°$.

ii) Calculate the distance MP, covered by the cable car (to the nearest m).

The shaded region shown in the diagram is bounded by the $\mathcal{x} \mathrm{and} \mathcal{y}$ axes, the curve $e^{\mathcal{y}}=\frac{\mathcal{x}}{a}$ and the line $\mathcal{y}=1$.

A solid is formed by rotating the shaded region about the $\mathcal{y}$-axes. Show that the volume of the solid is given by $4\mathrm{\pi }{\int }_0^1{\mathrm{e}}^{\mathcal{y}}\mathrm{d}\mathcal{y}$ and find the volume correct to 2 decimal places.

The graph of $\mathcal{y}=1+4\mathcal{x}-{\mathcal{x}}^3$ over the domain $0\le \mathcal{x}\le 5$ is shown below.

i) What is the range of $\mathcal{y}$ over the domain?

ii) Evaluate ${\int }_3^{5}1+4\mathcal{x}-{\mathcal{x}}^2 d\mathcal{x}$ and hence determine which area $A_1 \mathrm{or} A_2$ is greater? giving reasons.