(a) Find the equation of the tangent to the curve y=2x$e^x$ at the point (1,e)

(b) Consider the parabola $y=x^2+12$

(i) Find the coordinates of the vertex and focus of the parabola.

(ii) the area between parabola and the line y=16 is rotated about the y-axis. calculate the volume of the solid formed by this rotation leaving your answer in terms of pi.

(C) Calculate the approximate area (to two decimal places) between the curve y=ln2x, the x-axis and the line x=2, using the trapezoidal rule with four function values.

(d) 

(i) calculate ${\int }_0^{4}f\left(x\right)dx$

(ii) explain why $$\begin{gathered} {\int }_4^{8}f\left(x\right)dx=0 \\ \left(iii\right) what is the value of a if {\int }_1^{a}f\left(x\right)dx=-6 \end{gathered}$$

$A regular octagon is drawn inside a circle with centre O so that its vertices lie on the circumference. The circle has radius 1 cm.$

$$\begin{gathered} \left(i\right) Find the area of △AOB and hence find the area of this octagon. \\ Leave your answer in surd form.) \\ Another regular octagon is drawn outside the circle. The altitude OM of △OPS is 1 cm. \end{gathered}$$

$$\begin{gathered} \left(ii\right) Find the area of △OPS and hence find the area of this outer octagon. \\ \left(iii\right) By considering the results in parts \left(i\right) and \left(ii\right), show that: \\ \sqrt{2}<\frac{\mathrm{\pi }}{2}<4\tan \frac{\mathrm{\pi }}{8} \end{gathered}$$

(i) Solve $2{\mathcal{x}}^2-\mathcal{x}-1=0$ 

(ii) Hence or otherwise, solve $2\cos {\mathcal{x}}^2-\cos \mathcal{x}-1=0 for 0\le \mathcal{x}\le 2\mathrm{\pi }$

Dan is doing the following question for homework

" The area enclosed by $\mathcal{Y}=f\left(\mathcal{x}\right)$ and the axis is rotated about the x-axis. Find its volume."

Dan write down the correct statement

$Volume=2\mathrm{\pi }{\int }_0^{3}36{\mathcal{x}}^2-12{\mathcal{x}}^3+{\mathcal{x}}^{4}d\mathcal{x}$

(i) Write the equation of f(x)

(ii) Continue with Dan's question and find the volume in the question. Answer in terms of $\mathrm{\pi }$

$$\begin{gathered} In the diagram above, ABCD is a paralle\log ram with the corner points A(2,0), B(7 , 2), C(1,10) and D(- 4,8) . \\ The point M is the midp oint of AD. \\ \left(i\right) Show that the gradient of AD is -\frac{4}{3} \\ \left(ii\right) Find the length of AD. \\ \left(iii\right) Show that the coordinat es of M, the mid point of AD, are (-1,4). \\ \left(iv\right) Show that the equation of BC is 4\mathcal{x}+3\mathcal{y} -34 - 0. Find the perpendicular dis\tan ce from M to BC. \\ Find the area of the paralle\log ram ABCD. \end{gathered}$$

The 2-dimensional shape shown has the are BC formed from centre A and the are CA formed from centre D. AC, BC and AB are all equal. A, C and B are all equidistant from D. AB is 2√3 cm.Calculate the exact area of this shape.

A 12m long piece of string is cut into two pieces to form a circle of radius r metres and a square of side x metres.

Show that the total area of the circle and square is given by

$A={\mathrm{\pi r}}^2 +9-3\mathrm{\pi r}+\frac{{\mathrm{\pi }}^2{\mathrm{r}}^2}{4}$

Find the exact radius r of the circle such that the total area will be a minimum.

$$\begin{gathered} ABC is a right-angled triangle in which \angle ABC = 90° . Points D and E lie on AB and AC respectively such that AC \\ is perpendicular to DE. AD = 8 cm, EC = 11 cm and DB = 2 cm \end{gathered}$$

$$\begin{gathered} Prove that \Delta ABC is similar to \Delta AED . \\ Find the length of AE. \end{gathered}$$

$$\begin{gathered} In the diagram, ACD is a triangle where AB =2 cm, BC=4 cm, CD=9 cm and \angle CDE: 30\text{'} . Also, BE rs parallel to CD. \\ \left(i\right) Find the size of \angle BED. Give a reason for your answer. \\ \left(ii\right) Find the length of BE. \end{gathered}$$

$$\begin{gathered} The diagram represents an archway of a building that is 5m high and 6m wide. \\ The curved part is in the shape of a parabola with vertex 3m above the ground. \end{gathered}$$

$$\begin{gathered} Use the axis shown in the diagram to: \\ show that the equation of the parabola is \mathcal{y}=-\frac{1}{3}{\mathcal{x}}^2+3 \\ find the shaded area. \end{gathered}$$

$The line I has intercepts at A (4, 0) and B(O, 3). D is a point on line I and C has co-ordinates (0, 8) .$

$$\begin{gathered} Show that the equation of line I is 3\mathcal{x} — 4y + 12 = 0 \\ Find the lengths of AB and BC \\ What type of triangle is △ABC? Give a clear reason. \\ If B is the midpoint ofAD, find the co-ordinates of D. \\ Write down the equation of the circle through A, C and D with centre B \\ Find the shortest dis\tan ce from point C to the line l. \\ Hence, or otherwise, find the area of △ABC . \end{gathered}$$