$$\begin{gathered} An isosceles trapezium ABCD is drawn with its vertices on a semi-circle centre O and diameter 20 cm (see diagram). \\ OE is the altitude of ABCD. \\ i. Prove that ∆BOE \equiv ∆COE \\ ii. Hence or otherwise, show that the area of the trapezium ABCD is given by: \\ A=\frac{1}{4}(\mathcal{x}+20)\sqrt{400-{\mathcal{x}}^2} \\ where \mathcal{x} is the length of BC. \\ iii. Hence find the length of BC so that the area of the trapezium ABCD is a maximum. \end{gathered}$$

$$\begin{gathered} Given {\log }_m p = 1.75 and {\log }_m q = 2.25. \\ Find the following: \\ {\log }_{m}pq \\ {\log }_m\frac{q}{p} \\ \sqrt[5]{p{q}^2 }in terms of m. \end{gathered}$$

$$\begin{gathered} The shaded region below is between the curve \mathcal{y}={\mathcal{x}}^6+2 , the y-axis \\ and the line \mathcal{y}=8. \end{gathered}$$

$$\begin{gathered} Find the volume of the solid of revolution when the shaded region is \\ rotated about the \mathcal{y}-axis \end{gathered}$$

The velocity v (kms/min) of a train travelling from Epping to Eastwood, is given by $v=\raisebox{1ex}{$t^2\left(3-t\right)$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$ , where t is the time in minutes since leaving Epping. 

i) If the first stop is at Eastwood, how long does the journey take? 

ii) Find an expression for the distance $\mathcal{x} km$ travelled from Epping after time t $\left(where 0\le t\le 3\right)$

iii) Hence find the distance from Epping to Eastwood.  

iv) Where and when, between the two stations, was the train travelling the fastest?

$$\begin{gathered} A \tan k initially containing 18 000 litres of water is to be drained. After \\ t minutes, the rate at which the volume of water is decrea\sin g is given \\ by: \frac{dV}{dt}=-40(30-t) \\ Derive a formula for the volume of water remaining after s minutes. \\ How long will it take the \tan k to empty? \\ By u\sin g the expression for \frac{dV}{dt} or otherwise, sketch the volume – time graph. \end{gathered}$$

The points A, B and C have coordinates (2,0), (1,8) and (8,4) respectively. The angle between the AC and the $\mathcal{x}$-axis is $\theta$.

Copy this diagram

(i) Find the gradient of the line AC.

(ii) Calculate the size of angle $\theta$ to the nearest minute.

(iii) Find the equation of the line AC.

(iv) Find the coordinates of D, the midpoint of AC.

(v0 Show that AC is perpendicular to BD.

$Use simpson\text{'}s rule with 3 functions values to find the shaded area.$

$The curves \mathcal{y}={\mathcal{x}}^2 and \mathcal{y}=8-{\mathcal{x}}^2 are sketched below.$

$$\begin{gathered} i) find the points of intersection of two curvess. \\ ii)The shaded are between the curves and the \mathcal{y}-axis is rotated about \mathcal{y}-axis \\ by the splitting th shaded area into 2 parts or otherwise find the volume of soild \\ formed. \end{gathered}$$

The diagram shows a farmhouse F that is located 240m from a straight section of road, at the end of which is the bus depot D. The front gate G of the farmhouse is 3000m from the bus depot. The school bus leaves the depot at 8am and travels along the road at $15m{s}^{-1}$. Peter lives in the farmhouse and can run across the open paddock at a speed of $4m{s}^{-1}$. The bus will stop for Peter anywhere on the road but will not wait. Assume that Peter catches the bus at the point P where $\angle GFP=\theta$. 

i) Show that the time, in seconds, taken for the bus to go from D to P is given by 

$200+16\tan \theta$

ii) Find an expression, in terms of θ, for the time taken by Peter to run from F to P.  

iii) If Peter leaves home T seconds after 8am and he and the bus arrive at P at the same time, show that $T=200+16\tan \theta -60sec\theta$ 

iv) What is the latest time, to the nearest second, that Peter can leave home and still catch the bus?

In the diagram AB = BC and CD is perpendicular to AB. CD intersects the $\mathcal{y}$-axis at P. 

i) Find the length of AB.

ii) Hence show that the co-ordinates of point C are ( 2, 0 ).

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

iv) Show that the co-ordinates of P are $\left(0,1,\frac{1}{2}\right)$.

v) Show that the length of CP is $2\frac{1}{2}$.

vi) Prove that ∆ADP is congruent to ∆COP 

vii) Hence calculate the area of the quadrilateral DPOB.

$$\begin{gathered} A region in the first quadrant is bounded by the line \mathcal{y}=3\mathcal{x}+1, the \mathcal{x}-axis, the \mathcal{y}- axis, \\ and the line \mathcal{x}=2 . \end{gathered}$$

$$\begin{gathered} What is the volume of the solid of revolution formed when this region is rotated about the x-axis? \\ \left(A\right) 8 unit{s}^2 \\ \left(B\right) 38 unit{s}^2 \\ \left(C\right) 8\mathrm{\pi } unit{s}^2 \\ \left(D\right) 38\mathrm{\pi } unit{s}^2 \end{gathered}$$