$O is the centre of the circle. Find the value of \theta$

$$\begin{gathered} \left(A\right) \theta ={80}^{° } \left(B\right) \theta ={160}^{°} \\ \left(C\right) \theta ={200}^{° } \left(C\right) \theta ={260}^{°} \end{gathered}$$

$Prove that if a and b are both positive , then \frac{a+b}{2}\ge \sqrt{ab}$

$The integral of {\cos }^2 2x is ?$

$$\begin{gathered} Solve \frac{4}{5-\mathcal{x}}\ge 1 \\ A represents the area bounded by the \mathcal{x}-axis, \mathcal{y}= {\sin }^{-1}\mathcal{x}and the line \mathcal{x} = 1. \end{gathered}$$

.

$$\begin{gathered} \left(i\right) Write down two different expressions (without evaluating them) \\ involving integrals that can be used to find the magnitude of A. \\ \left(ii\right) Find the value of A. \\ Let f\left(\mathcal{x}\right)=2{\cos }^{-1}\left(\mathcal{x}-1\right) \\ \left(i\right) State the domain and range of the function f \left(\mathcal{x}\right) \\ \left(ii\right) Find the gradient of the graph of \mathcal{y} =f\left(\mathcal{x}\right) at the point where \mathcal{x}=\frac{1}{2} \\ \left(iii\right) Sketch the graph of \mathcal{y} =f\left(\mathcal{x}\right) \end{gathered}$$

$$\begin{gathered} f\left(\mathcal{x}\right)=3{\sin }^{-1}\left(\frac{\mathcal{x}}{2}\right) \\ State the domain and range \\ Hence, sketch the graph of\mathcal{y}=f\left(\mathcal{x}\right)clearly showing this information \\ Consider the function f\left(\mathcal{x}\right)={(\mathcal{x}-1)}^2+2 Sketch the graph of \mathcal{y}=f\left(\mathcal{x}\right),showing the coordinates of the vertex. \\ Find the largest domain for which f\left(\mathcal{x}\right)has an inverse function f^{-1}\left(\mathcal{x}\right) \\ State the domain of f^{-1}\left(\mathcal{x}\right) \end{gathered}$$

$$\begin{gathered} The function f\left(\mathcal{x}\right) =\ln \left(\mathcal{x}\right)-\cos \mathcal{x} has a zero near \mathcal{x} =1.2 \\ Use one application of Newtons method to find a second approximation for this \\ zero, Write your answer correct to 2 decimal place. \end{gathered}$$

$$\begin{gathered} \left(i\right) If \mathcal{Y}=\ln \left(\sin \mathcal{x}\right), find \frac{d\mathcal{Y}}{d\mathcal{x}} \\ \left(ii\right) Hence or otherwise find {\int }_{\frac{\mathrm{\pi }}{4}}^{\frac{3\mathrm{\pi }}{4}}cot\mathcal{x}d\mathcal{x} \\ Fnd {\int }_0^{\frac{\mathrm{\pi }}{12}}{\cos }^2\left(3\mathcal{x}\right)d\mathcal{x} \end{gathered}$$

$$\begin{gathered} The velocity of a particle is given by the equation v=\frac{1}{e^x} \\ If the initial displacement is x=0, find the equation for the displacement x, \\ in terms of t. \end{gathered}$$

$$\begin{gathered} i) Show that {\int }_0^{\frac{3\mathrm{\pi }}{2}}{\sin }^2 x dx=\frac{3\mathrm{\pi }}{4} \\ ii) The graph of y=1+\sin x for 0\le x\le \frac{3\mathrm{\pi }}{2} is rotated about the x-axis . \\ Find the exact volume of the solid generated. \end{gathered}$$

$$\begin{gathered} A projectile is fired from the ground with an angle of projection given by \alpha ={\tan }^{-1}\frac{3}{4} and initial velocity V. \\ It just clears a wall 10m high 100m away. Let acceleration due to gravity be g=10m{s}^{-2}. \\ \left(i\right) Show that the equations of motion are x=\frac{4vt}{5} and y=-5t^2+\frac{3vt}{5} . \\ \left(ii\right) Find the initial velocity, V of the projectile. \\ \left(iii\right) At what speed is the projectile travelling the ins\tan t it clears the wall? \end{gathered}$$

$If a focal chord of the parabola x^2=4ay cuts the parabola at two distinct points (x_1, y_1) and (x_2, y_2) , then;$

$$\begin{gathered} A, B, C and D are concyclic points on a circle centre O. E, F and G are points on the chords AD, CD and AC respectively, \\ such that AF, CE and DG are concurrent at O, and D, O, G and B are collinear. AE = DE and CF = DF. \\ Which of the following statements is NOT true? \end{gathered}$$

$$\begin{gathered} Find the nearest degree the size of the acute angle between the line \\ \mathcal{y}+2\mathcal{y}-1=0 \\ 3\mathcal{y}-2\mathcal{y}+4=0 \\ For the point A(-5,2) and B(2,0) \\ Write down the coordinate of P,the point that divides AB \\ internally in the ratio k:1 \\ If P ies on \mathcal{xy}=1, show that k^2-2k+11=0 \end{gathered}$$

$$\begin{gathered} Find all value of \mathcal{x} which satisfy the inequaliy \\ \frac{\mathcal{x}+1}{\mathcal{x}-1}>2 \end{gathered}$$

$$\begin{gathered} prove by mathamatical induction \\ {1}^3+2^3+3^3+....+n^3=\frac{1}{4}{n}^2{(n+1)}^2 for n \ge 1 \\ hence evaluate :2^3+4^3+6^3+...+{20}^3 \end{gathered}$$

$$\begin{gathered} Prove by mathematical induction that 9^{n+2}-4^n is divisible by 5 \\ for integers n \ge 1. \end{gathered}$$

$$\begin{gathered} Let \alpha ,\beta ,\gamma be the roots of the equation {\mathcal{x}}^3-3{\mathcal{x}}^2-6\mathcal{x}-1=0. \\ \left(i\right) Find 2\alpha +2\beta +2\gamma \\ \left(ii\right) Find {\alpha }^2+{\beta }^2+{\gamma }^2 \\ A particle moves in a straight line and its position in metres at anytime seconds is given by \\ \mathcal{x}=3\cos 2t-4\sin 2t \\ \left(i\right) Express the motion in terms of A\cos (nt+a). \\ \left(ii\right) Find the particle’s greatest speed. (Answer to the nearest whole number). \end{gathered}$$

$$\begin{gathered} A coffee maker has the shape of a double cone 60cm high. The radii at both ends are 4cm. Coffee is flowing from the top cone at the rate of 5c{m}^3/s. . \\ \left(i\right) Show that radius in the bottom cone is \frac{2(30-H)}{15} \\ ii) How fast is the level of coffee in the bottom cone ri\sin g at the ins\tan t when the coffee in this cone is 6 cm deep? \end{gathered}$$

$$\begin{gathered} In the diagram ST is \tan gent to both the circles at A. \\ The points B and Care on the larger cicles, and the line BC is a \tan gent to the smaller \\ circle at D .The line ܤܣ intersects the smaller circle at X ܺ. \\ Copy or trace the diagram into your answer booklet. \\ i) Explain why\angle AXD =\angle ABD+\angle XDB \\ ii) Explain why \angle AXD= \angle TAC+\angle CAD iii) Hence show that AD bisects \angle BAC . \end{gathered}$$

$$\begin{gathered} At the Aquarium in the middle of the pier there is a \tan k of \\ 8 Clownfish, and another \tan k of 7 Blue \tan g. Captain Jack Sparrow \\ wants his fish \tan k to contain 6 fish. Fish are selected at random from \\ both \tan ks. \\ What is the probability Jack’s \tan k will contain at least 4 clownfish? \end{gathered}$$

5. $The co-efficient of {\mathcal{x}}^2 in the expansion of {\left[2\mathcal{x}+\frac{3}{{\mathcal{x}}^2}\right]}^{11} is$