$$\begin{gathered} Calculate the exact volume generated by the solid formed when y={\cos }^{-1}x is \\ rotated about the y-axis between y=0 and y=\mathrm{\pi } \end{gathered}$$

$$\begin{gathered} A particle is oscilliating in simple harmonic motion about fixed point O on a \\ straight line.At time t seconds its displacement \mathcal{x} meters from O satisfies the \\ equation \frac{d^2\mathcal{x}}{d{t}^2}=-4\mathcal{x}. \\ i) Show that \mathcal{x}=a\cos (2t+\beta ) is a possible equation of motion for this particle \\ where \alpha and \beta are cons\tan ts and \alpha >0.State the period. \\ ii) The particle is observed at time t=0 have a velocity of 2 meters \\ per second and a displacement from the origin of 4 meters. \\ Show that the amplitude of the oscillation is \sqrt{17} meters. \\ iii) Show that \beta is approximately -0.245 correct to 3 decimal places \\ iv) Give the general solutions to the times when the particle is back \\ at its starting point \\ v) Find the time at which the particle first returns to its starting point \\ By u\sin g the substitution u=\sqrt{\mathcal{x}+1} find {\int }_3^8\frac{\mathcal{x}-1}{\sqrt{\mathcal{x}+1}}d\mathcal{x}. \\ It is known that the equation e^{-{\mathcal{x}}^2}-5{\mathcal{x}}^2-0.99=0 has a positive root \\ close to the origin.Attempt to find this root u\sin g Newton\text{'}s method,starting \\ with a first approximation of {\mathcal{x}}_0=0.Explain why this method fails. \end{gathered}$$

$By considering the derivative of \ln (\tan x) , find \int \cos ec 2x dx.$

If    $\frac{d\mathcal{x}}{dt}=\mathcal{x}+6$ $\mathcal{x}=-5 when t=0, find an expression for \mathcal{x} in terms of t.$

The speed v m / s of a particle moving in a straight line is given by $V^2=64-16\mathcal{x}-8{\mathcal{x}}^2$ where the displacement from a fixed point 0 is x metres.

Find an expression for the acceleration and show that the motion is simple harmonic. 

Between which two points is the particle oscillating? 

Find the period and amplitude of the motion. 

Find the maximum speed of the particle.

Aspherical bubble is expanding so that its volume is increasing at $10 c{m}^{3}m/s$. Find the rate increase of its radius when the surface area is $500 c{m}^2$

$$\begin{gathered} The polynomial p\left(\mathcal{x}\right) = 2{\mathcal{x}}^3 – {\mathcal{x}}^2- 6\mathcal{x} +k has a factor (\mathcal{x}+2). \\ What is the value of k ? \\ \left(A\right) 8 \\ \left(B\right) 0 \\ \left(C\right) -24 \\ \left(D\right) 32 \end{gathered}$$

$$\begin{gathered} The diagram shows the graph of a function. Which function does \\ the graph represent? \end{gathered}$$

$$\begin{gathered} \left(A\right)\mathcal{y}= 2 {\cos }^{-1}\left(2\mathcal{x}\right) \\ \left(B\right) \mathcal{y}=2 {\sin }^{-1} \left(2\mathcal{x}\right) \\ \left(C\right)\mathcal{y}=2{\sin }^{-1}\left(\frac{\mathcal{x}}{2}\right) \\ \left(D\right)\mathcal{y}=2{\cos }^{-1}\left(\frac{\mathcal{x}}{2}\right) \end{gathered}$$

$$\begin{gathered} Consider the function f\left(\mathcal{x}\right)={(\mathcal{x}+2)}^2-9 , -2\le \mathcal{x}\le 2 \\ Find the equation of the inverse function f^{-1}\left(\mathcal{x}\right) \\ On rhe same diagram, sketch the graphs of \mathcal{y}=f\mathcal{x}) \\ and \mathcal{y}=f^{-1}(\mathcal{x}), showing clearly the coordinates of the endpoints \\ and the intercepts on the axes. \\ Consider the function f(\mathcal{x})=2{\tan }^{-1}\mathcal{x}+{\sin }^{-1}({\log }_e\mathcal{x}) where \mathcal{x}\ge 0 \\ Find f\text{'}(\mathcal{x}) \end{gathered}$$

$$\begin{gathered} Consider the function f\left(\mathcal{x}\right)= \frac{\mathcal{x}}{4-{\mathcal{x}}^2} \\ \left(i\right) Find the domain of the function. \\ \left(ii\right) Show that the function is increa\sin g throughout its domain. \\ \left(iii\right) Sketch the graph of the function showing clearly the coordinates of any points of intersection with \\ the \mathcal{x}-axis and the \mathcal{y}-axis and the equations of any aslymptotes. \end{gathered}$$

$$\begin{gathered} The diagram below shows a sketch of the graph of \mathcal{y}=f\left(\mathcal{x}\right) where f\left(\mathcal{x}\right)=\sqrt{{\mathcal{x}}^2}-4 for all \mathcal{x}\ge 2. \\ \left(i\right) Copy this diagram onto your answer sheet. On the same set of axes, sketch the graph of the inverse function \mathcal{y}=f^{-1}\left(\mathcal{x}\right) \\ \left(ii\right) State the domain of f^{-1}\left(\mathcal{x}\right) \\ \left(iii\right) Find an expression for \mathcal{y}=f^{-1}\left(\mathcal{x}\right) in terms of \mathcal{x}. \end{gathered}$$

$$\begin{gathered} The equation f\left(\mathcal{x}\right)=\sqrt{\mathcal{x}}+\sqrt{\mathcal{x}+1}\_+\sqrt{\mathcal{x}+2}-5 has a root \alpha betweem \\ \mathcal{x}=1.5 and \mathcal{x}=2 \\ Find the interval in which \alpha lies by applying halving the interval twice \end{gathered}$$

$$\begin{gathered} The derivative of 3{\sin }^{-1}\frac{\mathcal{x}}{2}is, \\ \left(A\right)\frac{3}{\sqrt{4-{\mathcal{x}}^2}} \\ \left(B\right)\frac{3}{\sqrt{1-4{\mathcal{x}}^2}} \\ \left(C\right)\frac{3}{\sqrt{{\displaystyle \frac{1}{4}}-{\mathcal{x}}^2}} \\ \left(D\right)\frac{3}{\sqrt{1-{\displaystyle \frac{{\mathcal{x}}^2}{4}}}} \end{gathered}$$

$$\begin{gathered} Two circles centres P and Q respectively have radii 3 cm. They intersect each other at A and B. AB=PQ. \\ Find the area of the shaded region. \end{gathered}$$

$$\begin{gathered} What is another expression for \cos \left(\mathcal{x}+\mathcal{y}\right)? \\ \left(A\right) \cos \mathcal{x}\cos \mathcal{y}-\sin \mathcal{x}\sin \mathcal{y} \\ \left(B\right) \sin \mathcal{x}\cos \mathcal{y}+\cos \mathcal{x}\sin \mathcal{y} \\ \left(C\right) \cos \mathcal{x}\cos \mathcal{y}+\sin \mathcal{x}\sin \mathcal{y} \\ \left(D\right) \sin \mathcal{x}\cos \mathcal{y}-\cos \mathcal{x}\sin \mathcal{y} \end{gathered}$$

$$\begin{gathered} Line TA is a \tan gent to the circle at A and TB is a secant meeting the circle at B and C. \\ Given that TA=6 , CB=9 and TC=\mathcal{x}, what is the value of \mathcal{x}? \\ \left(A\right) -12 \\ \left(B\right) 2 \\ \left(C\right) 3 \\ \left(D\right) 4 \end{gathered}$$

$Find the general solutions of 2{\sin }^3\mathcal{x}-\sin \mathcal{x}-2{\sin }^2\mathcal{x}+1=0$

$Show that \sin A\cos B=\frac{1}{2}\left[\sin \left(A+B\right)+\sin \left(A-B\right)\right]$

$Show that \frac{1+\cos \mathcal{x}}{\sin \mathcal{x}}=\frac{1}{t} for t=\tan \frac{\mathcal{x}}{2}$

$$\begin{gathered} O is the centre of the circle ABP. MO\perp AB. M, P and B are collinear. MO intersects AP at L. \\ \left(i\right) Prove that A, O, P and M are concyclic. \\ \left(ii\right) Prove that \angle OPA=\angle OMB. \end{gathered}$$