$$\begin{gathered} The diagram shows the points P (0, 2) and Q (-4, 0). The point M is the midpoint of PQ. \\ The line MN is perpendicular to PQ and meets the y axis at N. \end{gathered}$$

$\left(i\right) Show that the gradient of PQ is \frac{1}{2}.$

$\left(ii\right) Find the coordinates of M.$

$\left(iii\right) Find the equation of the line MN.$

$$\begin{gathered} \left(vi\right) The point R lies in the second quadrant, and PNQR is a rhombus. \\ Find the coordinates of R. \end{gathered}$$

$Find the equation of the \tan gent to the curve y = e^{\mathcal{x}}at the point where x = 1 .$

$$\begin{gathered} A particle moves on a straight line so that its velocity, mv/s , at any time t seconds is given by v={(t-1)}^4+\frac{t}{2} , t \ge 0 \\ Find the initial velocity and show that the particle never stops. \\ Find the initial acceleration of the particle. \\ Find the least value of the velocity. \\ Sketch the velocity-time graph. \\ Find the dis\tan ce travelled by the particle in the first 2 seconds. \end{gathered}$$

$$\begin{gathered} In the diagram, the shaded region is bounded by \\ \mathcal{y}={\log }_e(\mathcal{x}-2) the \mathcal{x} axis and the line \mathcal{x}= 7. \\ Find the exact value of the area of the shaded region. \end{gathered}$$

$$\begin{gathered} A cone is inscribed in a sphere of radius a, centred at O. \\ The height of the cone is \mathcal{x} and the radius of the base is r, as shown in the diagram. \\ \left(i\right) Show that the volume, V,of the cone is given by \\ V=\frac{1}{3}\mathrm{\pi }(2\mathrm{a}{\mathcal{x}}^2-{\mathcal{x}}^3) \\ \left(ii\right) Find the value of x for which the volume of the cone is a maximum. 3 \\ You must give reasons why your value of x gives the maximum volume. \end{gathered}$$

$$\begin{gathered} In the diagram, O is the centre of a circle of radius \sqrt{6} cm and PQ is a chord of length 4 cm. ABCD is a rec\tan gle constructed in the minor segment cut off by chord PQ. OM is drawn perpendicular to PQ so that M is the mid-point of both chord PQ and side AB. N is the mid-point of side CD. \\ Let \angle CON=\theta , \theta in radians. \\ Show that OM = \sqrt{2} cm \\ Show that 0<\theta <1 \\ Show that the area, a , of rec\tan gle ABCD is given by \\ a=4\sqrt{3}\sin \theta \left(\sqrt{3}\cos \theta -1\right) \\ Show that \frac{da}{d\theta }=4\sqrt{3}\left(2\sqrt{3}{\cos }^2\theta -\cos \theta -\sqrt{3}\right) \\ Find the maximum area of the rec\tan gle. \end{gathered}$$

$$\begin{gathered} Use Simpson\text{'}s Rule with three function values to approximate \\ {\int }_0^1\sqrt{18{\mathcal{x}}^2-3\mathcal{x}+1}d\mathcal{x} \end{gathered}$$

$$\begin{gathered} O is the centre of the circle containing the arc AB. P is the midpoint of OA and Q is the \\ midpoint of OB. \\ \angle AOB = 120 ° OQ = OP = 5 cm \\ \left(i\right) Find the exact length of the arc AB. \\ \left(ii\right) Find the shaded area PQBA in exact form. \\ Find, justifying your answer, any point\left(s\right) of inflexion on the curve \\ \mathcal{y}=\frac{1}{{\mathcal{x}}^3}-{\mathcal{x}}^2+1 \end{gathered}$$

$$\begin{gathered} The diagram shows the points A(-2,2) , B(4,6) , O(0,0) and C for the paralle\log ram OABC. \\ The equation of line AB is 2\mathcal{x} - 3\mathcal{y} +10 = 0 \\ Do NOT prove this \\ \left(i\right) Show that the length of AB is 2\sqrt{13} \\ \left(ii\right) Calculate the perpendicular dis\tan ce from O to the line AB. \\ \left(iii\right) Calculate the area of the paralle\log ram OABC. \end{gathered}$$

$Find the obtuse angle \theta correct to the nearest minute$

$Without u\sin g calculus, sketch the curve y = e^x – 2$

$$\begin{gathered} By letting u = {\mathcal{x}}^{\frac{1}{3}} , or otherwise, solve {\mathcal{x}}^{\frac{2}{3}}+ {\mathcal{x}}^{\frac{1}{3}} - 6 = 0 \\ Give your answer as an exact value. \end{gathered}$$$$\begin{gathered} The graph below shows the function f\text{'}\text{'}\left(\mathcal{x}\right) \\ Copy the diagram into your writing books and draw the primitive graph on the same axes. \end{gathered}$$