The diagram below shows the parallelogram OABC with vertices 0(0, 0), A(3,5), B(8,6) and C

$$\begin{gathered} \left(i\right) Write down the coordinates of the mid-point of OB. \\ \left(ii\right) Find the coordinates of C. \\ \left(iii\right) Show that the equation of the line OB is 3\mathcal{x}- 4\mathcal{y}=0 \\ \left(iv\right) Show that the length of the interval OB is 10 units. \\ \left(v\right)Calculate the perpendicular dis\tan ce from A to the line OB. \\ \left(vi\right) Calculate the area of the paralle\log ram OABC. \end{gathered}$$

A 6 metre piece of wire is cut into three pieces to form two congruent squares and a circle of radius r metres. 

(i) Show that the total combined area of these three shapes is given by $A=\frac{{\left(3-\mathrm{\pi r}\right)}^2}{8}+{\mathrm{\pi r}}^2$

(ii) Find the exact value of r which will make this total combined area a minimum.

$$\begin{gathered} ABCDE is a regular pentagon with each side \\ being equal in length. \\ Equal diagonals have been drawn between all vertices to form another smaller regular pentagon PQRST \end{gathered}$$

$$\begin{gathered} \left(i\right)Find the size of <CDE \\ \left(ii\right) Show that triangles ADE and CDE are congruent. \\ \left(iii\right) Find the size of <DAE , giving reasons . \\ \left(iv\right) Hence, or otherwise, find the size of <ATP giving reasons for your answer. \end{gathered}$$                  

$$\begin{gathered} The diagram shows the graph of the gradient function of the curve \mathcal{y}= f\left(\mathcal{x}\right) \\ i)For what values of \mathcal{x} does \mathcal{y}= f(\mathcal{x}) have a local minimum? Justify your answer \\ ii)Draw a possible sketch of the curve \mathcal{y}=f\left(\mathcal{x}\right). \end{gathered}$$

$$\begin{gathered} HKLM is a paralle\log ram. \\ The line through L parallel to MK meets HK produced at Q. \\ i)Prove that Hk=KQ \\ ii)Given that P is the mid-point of LQ, Prove that <PKQ =<LHQ \end{gathered}$$

The graph of $\mathcal{y}=g\left(\mathcal{x}\right)$ and $\mathcal{y}=f\left(\mathcal{x}\right)$ intersect at the point A on the $\mathcal{y}$-axis, as shown in the diagram. If $g\left(\mathcal{x}\right)=3\mathcal{x}+4$ and $f\text{'}\left(\mathcal{x}\right)=2\mathcal{x}-3$, find $f\left(\mathcal{x}\right)$.

A) ${\mathcal{x}}^2-3\mathcal{x}+7$

B) $3{\mathcal{x}}^2+4\mathcal{x}+4$

C) ${\mathcal{x}}^2-3\mathcal{x}+4$

D) ${\mathcal{x}}^2+3\mathcal{x}-7$

$in the first three layers of astack of soup cans there are 20 cans in the first layer, 19 cans in the second layer 18 cans in the third layer.$

$$\begin{gathered} this patteren of stacking cans in layers continues. \\ the maximum number of can be stacked in this way is \\ \left(A\right) 190 \\ \left(B\right) 210 \\ \left(C\right) 220 \\ \left(D\right) 380 \end{gathered}$$

The line $\mathcal{y}=2\mathcal{x}-8$ is a tangent to parabola ${\mathcal{x}}^2=4a\mathcal{y}$ at the point P where $\mathcal{x}=8$

i) Show that $a=2$.

ii) Write down the coordinates of the focus and equation of the directrix.

William starts playing the game Space Cadet Pinball and gets an initial high score of 858.

He then regularly plays this game and keeps a record of the improvement in his high scores

at the end of each week. These are recorded in the table below.

(i) How many weeks will it take for William to reach his overall maximum score?                

(ii) What will be William's overall maximum score?                                                                            

$$\begin{gathered} Evaluate \sum _{n=3}^5\frac{{(-1)}^n}{n} \\ For what values of k is the quadratic expression \\ {x}^2—kx+k+3 positive for all real values of x? \\ Consider the quadratic equation 3x^2+(k+2)x+8k=0. \\ Find the value of k if the product of its roots is equal to twice the sum \\ of its roots \end{gathered}$$

$$\begin{gathered} The graph of \mathcal{y}=\frac{1}{2}{\mathcal{x}}^4-{\mathcal{x}}^3+1 is sketched below \\ The points A, B and C are the stationary points of this curve. \end{gathered}$$

$$\begin{gathered} \left(i\right) Find the coordinates of the points A, B and C. \\ \left(ii\right) For what values of x is this curve concave down? Give reasons for your answer. \\ \left(iii\right) U\sin g Part \left(i\right), draw a rough sketch of the gradient function \frac{d\mathcal{y}}{d\mathcal{x}} \\ Identify any points on this sketch where the concavity of the curve changes \end{gathered}$$