At Luna Park, a chair is released from a height of 50 metres and falls vertically. Magnetic brakes are applied to stop the fall.

$$\begin{gathered} The height of the chair at time t seconds is x metres. The acceleration of the chair is given \\ by \ddot{\mathcal{x}} =-9.8 At the release point, t=0, \mathcal{x}=50 and \dot{\mathcal{x}}=0 \\ \left(i\right) Show that the chair’s displacement x at time t is given by\mathcal{x}=-4.9t^2+50 \\ \left(ii\right) If it takes half a second for the brakes to stop the fall, find the latest time the brakes could be applied? \\ (Correct your answer to two decimal places) \\ \left(iii\right) How far has the chair fallen and what is its speed when the brakes are applied? \\ (correct your answer to two decimal places) \end{gathered}$$

A rectangular piece of paper is 30 centimetres high and 15 centimetres wide. The lower right-hand corner is folded over so as to reach the leftmost edge of the paper. Let x be the horizontal distance folded and y be the vertical distance folded as shown in the diagram.

$$\begin{gathered} \left(i\right) By considering the areas of three triangles and trapezium that make up the total area of the paper, show \mathcal{y}=\frac{\mathcal{x}\sqrt{15\left(2\mathcal{x}-15\right)}}{2\mathcal{x}-15} \\ \left(ii\right) Show that the crease, C, is found by the expression C=\sqrt{\frac{2{\mathcal{x}}^3}{2\mathcal{x}-15}} \\ \left(iii\right) Hence, find the minimum length of C. \end{gathered}$$

i) Use the trapezoidal Rule with 2 strips (i.e. 3 functions value), to find an approximate value for ${\int }_0^4{3}^{\mathcal{x}}d\mathcal{x}$.

ii) Hence find an approximate value for the shaded area in the diagram below.

The diagram shows the graphs $\mathcal{y}={\mathcal{x}}^2-2\mathcal{x}-3$ and $\mathcal{y}=\mathcal{x}+7$. The graphs intersect at the point A and B.

i) Find the coordinates of A and B.

ii) Find the area enclosed by $\mathcal{y}={\mathcal{x}}^2-2\mathcal{x}-3$ and $\mathcal{y}=\mathcal{x}+7$.

$$\begin{gathered} Find integers a and b such that \\ \frac{1}{\sqrt{3}+2}=a\sqrt{3}+b \end{gathered}$$

Prove that ∆BUT is similar to ∆SAT. 

Hence, or otherwise, find the length of BU

$simplify \frac{2^{n+1}-2^n}{2^{2n+1}-2^{2n}}$

$$\begin{gathered} In the diagram above <ABC=<AED, AB=2, BD=3, CE=5 \\ and AC=\mathcal{x}. Copy the diagram in your book \\ i) Prove that triangle ABC is similar to triangle AED \\ ii) Hence find thr value of \mathcal{x} \end{gathered}$$

The diagram shows the origin O and the points A(—3,6), B(5,2) and                                      

The lines AB and BC are perpendicular.                                                                                                                    

Copy or trace this diagram onto your writing sheet.

a) Show that A and B lie on the line $\mathcal{x}+2y=9$

b) Show that the length of AB is $4\sqrt{5}$ units.

c) Find the perpendicular distance from 0 to AB .

d) Find the area of triangle AOB.

e) Show that C has coordinates (2,—4).       

f) Does the line AC pass through the origin? Explain. 

g) The point D is not shown on the diagram. The point D lies on the $\mathcal{x}$

axis and ABCD is a rectangle. Find the coordinates of D. Note: D is not the point Of intersection Of line AB extended to meet the $\mathcal{x}$ axis, 

h) On your diagram, shade the region satisfying the inequality$\mathcal{x}+2y\ge 9$

$$\begin{gathered} The geomatric derise a+ar+a{r}^2+..... has the secound term of \\ \frac{1}{4} and has limiting sum of 1 \\ i) Show that a=1-r \\ ii) Solve the pair of simul\tan eous equation to find r \end{gathered}$$