$$\begin{gathered} A football, lying at point F on level ground is 4 metres away from 1 metre \\ below the top of a flat-roofed long narrow green house. The football is kicked \\ with an initial velocity of 12 m/s at an angle of projection \theta . \\ \left(i\right) U\sin g g=-10m{s}^{-2}, show that the football\text{'}s trajectory at time \\ t seconds after being kicked may be defined by the equations \\ \mathcal{x}=12t \cos \theta and \mathcal{y}=-5t^2+12t \sin \theta -1 where \mathcal{x} and \mathcal{y} are horizontal \\ and vertical displacements in metres , of the football from the origin O shown \\ in the diagram . (Neglect air resis\tan ce). \\ \left(ii\right) Given that \theta =30°, how far from D will the football land on top of the green house? \\ \left(iii\right) Find the range of values of \theta , to the nearest degree, at which the football must \\ be kicked so that it will land to the right of D. \end{gathered}$$

$$\begin{gathered} A particle moves on a line so that its distance from the origin at time t is x and \\ its velocity is v. \\ \left(i\right) Prove \frac{d^{2}x}{d{t}^2}=\frac{d}{dx}\left(\frac{1}{2}{v}^2\right). \\ \left(ii\right) If \frac{d^{2}x}{d{t}^2}=n^2(3-x) where n is a constant \\ and if the particle is released from rest at x=0, \\ show \frac{1}{2}{v}^2-n^2(3x-\frac{1}{2}{x}^2)=0 \\ iii) Hence show that the particle never moves outside a certain interval. \end{gathered}$$

$$\begin{gathered} i) Express \sqrt{3} \sin \theta -\cos \theta in the form R \sin (\theta -\alpha ) where \alpha is acute. \\ ii) Hence solve \sqrt{3} \sin \theta -\cos \theta =0, for 0\le \theta \le 2\mathrm{\pi }, \mathrm{leaving} \mathrm{your} \mathrm{answer} \\ \mathrm{in} \mathrm{exact} \mathrm{form}. \end{gathered}$$

$$\begin{gathered} a) The acute angle between the \tan gents to the curve y = \sin x and y = \cos x \\ at their point 4 of intersection( where 0<x<\frac{\mathrm{\pi }}{2}) is \theta . \\ Find the value of \theta correct to the nearest degree \end{gathered}$$

$Sketch \mathcal{y}=\frac{\mathcal{x}}{{\mathcal{x}}^2-4}, clearly indicating its asymptotes$

$Which of the following could be the polynomial y=P\left(x\right)$

$$\begin{gathered} \left(A\right) P\left(x\right)=x^3\left(2-x\right) \\ \left(B\right) P\left(x\right)=x^2{\left(2-x\right)}^2 \\ \left(C\right) P\left(x\right)=x^3\left(x-2\right) \\ \left(D\right) P\left(x\right)=-x^3\left(x+2\right) \end{gathered}$$

$$\begin{gathered} What is the derivative of {\cos }^{-1} \left(\frac{\mathcal{x}}{3}\right)? \\ A) \frac{-1}{3\sqrt{9-{\mathcal{x}}^2}} \\ B) \frac{1}{3\sqrt{9-{\mathcal{x}}^2}} \\ C) \frac{-1}{\sqrt{9-{\mathcal{x}}^2}} \\ D)\frac{1}{\sqrt{9-{\mathcal{x}}^2}} \end{gathered}$$

$$\begin{gathered} Let f\left(x\right)=5-\sqrt{x}. \\ \left(i\right) Sketch the inverse function y=f^{\text{'}}\left(x\right). \\ \left(ii\right) Find the equation of the inverse function y=f^{\text{'}}\left(x\right). \end{gathered}$$

$$\begin{gathered} A partical move in a straight line so that its velocity \mathcal{y} at a position \mathcal{x} is \\ given by v=9+4{\mathcal{x}}^2.it start from a fixed point where \mathcal{x}=0 at a \\ velocity of 9m{s}^{-1} \\ finnd its accelaration , \mathcal{x} as a function of the displacement , \mathcal{x} \\ express its displacement , \mathcal{x}as a function of time t. \\ find the displacement after \frac{\mathrm{\pi }}{24} seconds from its starting points. \end{gathered}$$

$$\begin{gathered} ) A freshly caught fish, initially at 18°C, is placed in a freezer that has a cons\tan t \\ unknown temperature of \mathcal{x}°C.The cooling rate of the fish is proportional to the difference \\ between the temperature of the freezer and the temperature T°C ,of the fish. \\ It is known that ܶT satisfies the equation \frac{dT}{dt}=-k(t-\mathcal{x}), where t is the number of minutes after the fish is placed in the freezer. \\ (i) Show that T=\mathcal{x}+A{e}^{-kt} satisfies this equation. \\ (ii) If the temperature of the fish is 10° c after 7 \frac{1}{2} minutes, \\ Show that the fish’s temperature after t minutes is given by \\ T=\mathcal{x}+(18-\mathcal{x})e^{\frac{2}{15}kg\left[\frac{10-\mathcal{x}}{18-\mathcal{x}}\right]t} \\ (iii) Find the temperature of the fish after 15 minutes when the initial freezer temperature is \\ 5°c .Answer to the nearest degree. \end{gathered}$$

$$\begin{gathered} Consider the function f\left(\mathcal{x}\right)={\cos }^{-1}\left(\mathcal{x}\right)+{\cos }^{-1}(-\mathcal{x}) \\ \left(i\right) Show that f\left(\mathcal{x}\right) is cons\tan t by finding f\text{'}\left(\mathcal{x}\right). \\ \left(ii\right) Find the value of the cons\tan t. \end{gathered}$$

$Given \mathcal{Y}={\tan }^{-1}\left(\frac{a}{\mathcal{x}}\right), show that \frac{d\mathcal{Y}}{d\mathcal{x}}=\frac{-a}{{\mathcal{x}}^2+a^2}$

$$\begin{gathered} A particle moves in two dimensional space where \stackrel{\rightharpoonup }{i} and \stackrel{\rightharpoonup }{j} are unit vectors \\ in the \mathcal{x} and \mathcal{y} directions respectively. At time t seconds its displacement from \\ the origin is given by \stackrel{\rightharpoonup }{r}= (6t-4t^2)\stackrel{\rightharpoonup }{i}+ 2t\stackrel{\rightharpoonup }{j} where all lengths are measured \\ in metres. \\ Write down the particle’s velocity vector in component form. \\ Find the speed of the particle when t = 2. \\ Show that the equation of the path of the particle is \mathcal{x}= 3\mathcal{y}-{\mathcal{y}}^2 . \end{gathered}$$

$$\begin{gathered} Points P(4, -6), Q(1, 2), R(7, 5) form a triangle P QR in the Cartesian Plane. \\ Find the vectors \stackrel{\rightharpoonup }{PQ} and \stackrel{\rightharpoonup }{QR}, representing two sides of this triangle. Give your \\ answer in component form. \\ Use the dot product to find angle P QR. Give your answer correct to the nearest degree. \end{gathered}$$

$$\begin{gathered} In the diagram above, 𝐴𝐵𝐶𝐷 is a cyclic quadrilateral and 𝐾 is the intersection of the diagonals 𝐴𝐶 and 𝐵𝐷. \\ 𝑀 is the point on 𝐵𝐷 such that \angle 𝐴𝐶𝐵 = \angle 𝐷𝐶𝑀. \\ \left(i\right) Prove that \frac{AC}{CD}=\frac{AB}{MD} \\ \left(ii\right) Ptolemy’s Theorem states that in a cyclic quadrilateral the product of the diagonals \\ is equal to the sum of the products of the pairs of opposite sides, that is: AC\times BD=AB\times CD+BC\times AD . \\ Prove Ptolemy’s theorem. \end{gathered}$$

$Use the principle of mathematical induction to show that n^3+2n is divisible by 3 for all positive integers n.$

$$\begin{gathered} Ansett Airlines offer two options on all flights for their meal service – chicken or beef (vegetarians choose not to fly with Ansett). \\ If 60\% of the time Ansett passengers select the chicken dish \\ what is the probability that out of 7 randomly selected passengers at least 2 will select chicken for their meal? \end{gathered}$$

$$\begin{gathered} How many numbers greater than 6000 can be formed with the digits \\ 1, 4, 5, 7, 8 if no digit is repeated? \end{gathered}$$

$$\begin{gathered} Use one application of Newton\text{'}s methode to find an approximation to the \\ root of the equation \cos \mathcal{x}=\mathcal{x} near \mathcal{x}=0.5. Give your answer correct to \\ two decimal places. \end{gathered}$$

By considering the sum of the terms of an arithmetic series show that

$(1+2+3+...+n)² = \frac{1}{4}n²(n+1)²$

By using the Principle of Mathematical induction prove that

$1^3 +2^3+...+n^3 = {(1+2+...+n)}^2 for all n\ge 1.$