$$\begin{gathered} The point P(2p,p^2) lies on the parabola {\mathcal{x}}^2=4\mathcal{Y} whose focus is S. \\ \left(i\right) Find the equation of the \tan gent to the parabola at P. \\ \left(ii\right) The segement at P mets the \mathcal{x}-axis at Q. The point R divides the interval joining SQ externally in the ratio 3:2. Show that as P moves on the parabola {\mathcal{x}}^2=4\mathcal{Y} the locus of R is a straight line. \end{gathered}$$

$$\begin{gathered} Which expression is equal to \int {\cos }^2\frac{x}{2}dx \\ \left(A\right) \frac{1}{2}\left(x+\sin x\right)+c \\ \left(B\right) \frac{1}{2}\left(x-\sin x\right)+c \\ \left(C\right) 2{\cos }^3\frac{x}{2}+c \\ \left(D\right) 2{\sin }^3\frac{x}{2}+c \end{gathered}$$

$$\begin{gathered} The displacement, \mathcal{x}, of a particle at time tis given by \mathcal{x} = 3\sin 2t + 4\cos 2t. \\ What is the maximum velocity of the particle? \\ \left(A\right) 2 \\ \left(B\right) 5 \\ \left(C\right) 10 \\ \left(D\right) 14 \end{gathered}$$

$$\begin{gathered} \left(i\right) The equation {\mathcal{x}}^3-k{\mathcal{x}}^2+2=0 has exactly one root between \mathcal{x}=0 and \mathcal{x}=1. Prove that k>3. \\ \left(ii\right) The equation {\mathcal{x}}^3-4{\mathcal{x}}^2+2=0 has a root between \mathcal{x}=0 and \mathcal{x}=1. U\sin g a first approximation of this root as \mathcal{x}=0.7, use one application of Netwon\text{'}s Method to find a better approximation correct to 2 decimal places. \end{gathered}$$

$$\begin{gathered} P(2ap, ap²) lies on the parabola {\mathcal{x}}^2=4a\mathcal{y} the line L is a \tan gent at P. S is the focus. \\ Show that L is equally inclined to SP and the axis of the parabola. \end{gathered}$$

Solve $\frac{1}{\left|3-\mathcal{x}\right|}\ge \frac{1}{2}$

The sketch shows $\mathcal{y}=f^1\left(\mathcal{x}\right)$

$$\begin{gathered} sketch \\ \mathcal{y}=f\left(\mathcal{x}\right) \\ \mathcal{y}=f^1\left(\mathcal{x}\right) \\ \mathcal{y}=f^{\text{'}\text{'}}\left(\mathcal{x}\right) \end{gathered}$$

down the page showing clearly the shape and position of each graph at the points which correspond with A, B and C in the graph of $\mathcal{y}=f^1\left(\mathcal{x}\right)$

The roots of

 $$\begin{gathered} {\mathcal{x}}^2+m\mathcal{x}+n are \alpha , \beta \\ Find the values, in term of m \& n of \\ \alpha +\beta \\ \alpha \beta \\ {\alpha }^2+{\beta }^2 \\ 3{\alpha }^2\beta +3\alpha {\beta }^2 \end{gathered}$$

$$\begin{gathered} What is the acute angle to the nearest degree between the lines \mathcal{y}=1-3\mathcal{x} and 4\mathcal{x}-\mathcal{y}-5=0? \\ \left(A\right) 15° \\ \left(B\right) 38° \\ \left(C\right) 52° \\ \left(D\right) 75° \end{gathered}$$

$$\begin{gathered} i) Prove that \cos \left(\frac{\theta }{2}\right)=\frac{\sin \theta }{1-\cos \theta } \\ ii) Hence find the exact value of cot\left(\frac{\theta }{2}\right) given that \sin \theta =\frac{5}{6} and \\ \frac{\mathrm{\pi }}{2}<\theta <\mathrm{\pi }. \end{gathered}$$

$$\begin{gathered} Which of the following is equivalent to the expression \sqrt{\frac{4+4\cos 2\mathcal{x}}{1-\cos 2\mathcal{x}}}? \\ A. 2co{t}^2\mathcal{x} \\ B. 2\left|cot\mathcal{x}\right| \\ C. 2\left|\tan \mathcal{x}\right| \\ D. 2{\tan }^2\mathcal{x} \end{gathered}$$

$$\begin{gathered} In the diagram above, BC and DE are chords of a circle. CB and ED are produced to meet at A. \\ What is the value of \mathcal{x}? \\ A. \frac{11}{28} \\ B. \frac{28}{11} \\ C. 5 \\ D. 12 \end{gathered}$$

$$\begin{gathered} When the interval joining the points (-5,6) and (-2,3) is trisected \\ find the points of trisected. \end{gathered}$$

$$\begin{gathered} Two straight roads intersect at an angle of 120°.A horse and cart starts \\ from the intersection and travel along one road at 20 km/hr. \\ One hour later a cyclist starts from the intersection and travel along \\ the other road at 50/km/hr.At what rate is the dis\tan ce between the \\ horse and cart and the cyclist changing three hours after the cyclist start? \\ A pendulum bob is oscillating in a straight line according to the equation x=-4(x-8).Initially \\ the bob is 3cm to the left of the orgin O and is travelling at 4cm/s to the right \\ Show that the velocity v,of the pendelum bo is given by v=2\sqrt{125-{(x-8)}^2} \\ Determine the end points of the motion in irrational form. \\ Determine an equation for the displacement ,x of the pendelum bob in term of time,t,in \\ exact form. \end{gathered}$$

$$\begin{gathered} Find the co-ordinates of the point C(p,q) below,given that \\ AC:CB=8:3 \\ Find the value value of k if \mathcal{x}+4 is a factor of P\left(\mathcal{x}\right)=2{\mathcal{x}}^3+3{\mathcal{x}}^2+k\mathcal{x}-12 \end{gathered}$$

$$\begin{gathered} A teacher has 9 girls and 16 boys in her class.The teacher randomly chooses 4 \\ student,one after the other,to form a clean up team to pick up papers in \\ the playground .assume that all student are present when the teacher \\ makes her section and that a student,selectionn in a team on one occasion \\ is independent of their selection and that a studenton another occasion. \\ \left(i\right)find the probability that on one occasion thw teacher choose a team \\ which contained only boys. \\ \left(ii\right)find the probability that the youngest boy in the class is \\ includedd in a team one day but tyhe oldest girl is excluded. \\ \left(iii\right)During the course of the term,the teacher has to choose a \\ clean up team on 9 separate occasion.what is the probability \\ that on 3 of these occasions the team contain only bous? \\ Express your answer correct answer correct to 4 decimal place. \end{gathered}$$

$$\begin{gathered} A resting adult’s breathing cycle is 5 seconds long. For time t seconds,0\le t\le 2\frac{1}{2}, air is taken into the \\ lungs. For 2\frac{1}{2}\le t\le 5 air is expelled from the lungs. The rate, R litres/second, at which air is taken in \\ or expelled from the lungs can be modelled on the equation R=\frac{1}{2}\sin \left(\frac{2\mathrm{\pi }}{5}t\right) \\ How many litres of air does a resting adult take into their lungs during one breathing cycle? \end{gathered}$$

$$\begin{gathered} Ten ki\log rams of chlorine is placed in water and begins to dissolve. After t hours the amount A kg of \\ undissolved chlorine is given by A=10e^{-kt}. What is the value of k given that A=3.6 and t=5 ? \\ \left(A\right) -0.717 \\ \left(B\right) -0.204 \\ \left(C\right) 0.204 \\ \left(D\right) 0.717 \end{gathered}$$

$$\begin{gathered} A flat circular disc is being heated so that the rate of increase of the area (A in m^2), after t hours, is given by: \\ \frac{dA}{dt}=\frac{1}{8}\mathrm{\pi t}. Initially the disc has a radius of 2 metres. \\ Which of the following is the correct expression for the area after t hours? \\ \left(A\right) A=\frac{1}{8}{\mathrm{\pi t}}^2 \\ \left(B\right) A=\frac{1}{16}{\mathrm{\pi t}}^2 \\ \left(C\right) A=\frac{1}{8}{\mathrm{\pi t}}^2+4\mathrm{\pi } \\ \left(D\right) A=\frac{1}{16}{\mathrm{\pi t}}^2+4\mathrm{\pi } \end{gathered}$$

$$\begin{gathered} The displacement, x metres, from the origin of a particle moving in a straight line at any time (t seconds) \\ is shown in the graph. When was the particle moving with greatest speed? \\ \left(A\right) t=0 \\ \left(B\right) t=4.5 \\ \left(C\right) t=8 \\ \left(D\right) t=11.5 \end{gathered}$$