Triangle ABC is a right angled triangle with $\angle ABC=90°$. D is a point on AC such that AB = BD = DC. E lies on BC such that AE bisects $\angle BAD$. Let $\angle ADB=\mathcal{x}°$

Copy the diagram into your booklets showing this information.

i) Show that $\angle DBC=(2\mathcal{x}-90)°$

ii) Hence find the value of $\mathcal{x}$.

iii) Show that triangle AEC is isosceles.

In the diagram above, $\angle ABC=\angle AED, AB=2,BD=3,CE=5 \& AC=\mathcal{x}$. Copy the diagram into your booklet

i) Prove the triangle ABC is similar to triangle AED.

ii) Hence find the value of $\mathcal{x}$.

In the diagram above, PQ = 18.6 kilometers, PR = 12.5 kilometers and $\angle PQR=37°$. $\angle PRQ$ is obtuse. Find the size of $\angle PRQ$ correct to the nearest minute.

The diagram shows the origin 0 and the points A(-3, 6), B(5, 2) and C(2, $\mathcal{y}$)

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 $\mathcal{x}+2\mathcal{y}=9$

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 A0B.

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 $\mathcal{x}+2\mathcal{y}\ge 9$.

The spinner shown above is used in a game. Once spun, it is equally likely to stop at any one of the letters A, E, I, O or U.

i) If the spinner is spun twice, find the probability that it stops on the same letter twice.

ii) How many times must the spinner be spun  for it to be 99% certain that it will stop on the letter E at least once?

$$\begin{gathered} A scientist grows the number of bacteria according to the equation \\ N\left(t\right)=A{e}^{0.15t} where t is measured in days and A is a cons\tan t. \\ \left(i\right) Show that the number of bacteria increases at a rate proportional to the number present. \\ \left(ii\right) When t=3 the number of bacteria was estimated at 1.5\times {10}^8 . Evaluate A. Answer correct to 2 significant figures. \\ \left(iii\right) The number of bacteria doubles every \mathcal{x} days. Find \mathcal{x}. Answer correct to 1 decimal place. \end{gathered}$$

$$\begin{gathered} The velocity of an object moving along the x-axis is given by v=2sint+1 for 0\le t\le 2\mathrm{\pi } \\ where v is measured in metres per second and t in seconds. \\ \left(i\right) When is the object at rest? \\ \left(ii\right) Sketch the graph of v as a function of t for 0\le t\le 2\mathrm{\pi } \\ \left(iii\right) Find the maximum velocity of the object for this period. \\ \left(iv\right) When is the object travelling in the negative direction during this period? \\ \left(v\right) Calculate the total dis\tan ce travelled by the object in the period \mathrm{\pi }\le t\le 2\mathrm{\pi } \end{gathered}$$

i) The curve $\mathcal{y}={\mathcal{x}}^2$ and the line $\mathcal{y}=3\mathcal{x}+4$, intersect at the points A and B as shown in the diagram above.

Find the $\mathcal{x}$ coordinates of the point A and B.

ii) Find the area bounded by the curve $\mathcal{y}={\mathcal{x}}^2$ and the line $\mathcal{y}=3\mathcal{x}+4$

$$\begin{gathered} For the points A\left(3,2\right) and B\left(-5,5\right) \\ i) Find gradiant between A and B \\ ii) Find midpoint of A and B \\ iii) Find dis\tan ce between A and B \\ iv) Show that equation of line L through A and B is 7\mathcal{x}-8y-5=0 \\ v) Show that the point C\left(-3,4\right) does not lies on line L \\ vi) Find the perpendicular dis\tan ce From the line L to \left(-3,4\right) \end{gathered}$$

$$\begin{gathered} Paint at the local hardware store is sold at a profit of 30\% on the \cos t price. \\ If a drum of paint is sold for \$67\bullet 50, find the \cos t price. \end{gathered}$$