$$\begin{gathered} What is the remainder when P\left(\mathcal{x}\right)={\mathcal{x}}^2+5\mathcal{x}+7 is divided by (\mathcal{x}+3)? \\ \left(A\right) \frac{-109}{9} \\ \left(B\right) \frac{19}{9} \\ \left(C\right) 31 \\ \left(D\right) 1 \end{gathered}$$

$$\begin{gathered} The term independent of \mathcal{x} in the expansion of {(\mathcal{x}+\frac{2}{\mathcal{x}})}^6 is: \\ \left(A\right) 160 \\ \left(B\right) 80 \\ \left(C\right) 40 \\ \left(D\right) 20 \end{gathered}$$

$$\begin{gathered} A simplified expression for \left(\begin{array}{c}n+1\\ n-1\end{array}\right) is: \\ \left(A\right) \frac{1}{2}(n^2-n) \\ \left(B\right) \frac{1}{2}(n^2+n) \\ \left(C\right) n^2-n \\ \left(D\right) n^2+n \end{gathered}$$

$Which diagram represents P\left(\mathcal{x}\right)={(\mathcal{x}-a)}^2(b^2-{\mathcal{x}}^2), where a>b?$

When $(\mathcal{x}+3)(\mathcal{x}–2)+2 is divided by \mathcal{x}–k, the remainder is k^2$. Find the value of k

Solve $\frac{\mathcal{x}}{\mathcal{x}-3}>1$

$Solve \frac{\mathcal{x}}{2\mathcal{x}-1}\le 5$

Differentiate 

  $$\begin{gathered} \frac{1}{1+4{\mathcal{x}}^2} \\ {e}^{2\mathcal{x}}{\log }_{e}2\mathcal{x} \end{gathered}$$

$$\begin{gathered} The diagram above shows a circular disc with radius OA. \\ The radius of the disc, OA, is one metre and AB is a rod of length k metres (k > 1). \\ The end of the rod, B, is free to slide along a horizontal axis with origin O. The angle between OA and OB is \theta . \\ Let OB = x metres. \\ \left(i\right) show that x=\cos \theta +\sqrt{k^2-{\sin }^2 \theta }. \\ \left(ii\right) Find \frac{dx}{d\theta } in terms of k and \theta . \\ \left(iii\right) Given that \frac{d\theta }{dt}=4\mathrm{\pi } \mathrm{rad}/\mathrm{s}. \\ \mathrm{find} \frac{\mathrm{dx}}{\mathrm{dt}} \mathrm{in} \mathrm{terms} \mathrm{of} \mathrm{k} \mathrm{when} \mathrm{\theta }=\frac{\mathrm{\pi }}{6} \\ \left(\mathrm{iv}\right) \mathrm{Find} \mathrm{\theta }\le \mathrm{\theta }<2\mathrm{\pi }, \mathrm{when} \mathrm{the} \mathrm{velocity} \mathrm{of} \mathrm{point} \mathrm{B} \mathrm{is} \mathrm{zero}. \end{gathered}$$

$$\begin{gathered} A verticle section of a velly in the form of a parabola x^2=4ay where a is a positive cons\tan t. \\ A gun placed at the origin fires with speed \sqrt{2gh} at an angle of elevation \alpha where \\ 0<\alpha <\frac{\mathrm{\pi }}{2} and h is positive cons\tan t. \\ The equation of the motion of a projectile fired from the origion with initial velocity V m{s}^{-1} \\ at angle \theta to the horizontal are \\ x=Vt\cos \alpha and y=Vt\sin \alpha -\frac{1}{2}g{t}^2 \\ \left(i\right) If the shell strikes the section of the valley at the point P(x, y) show that \\ x=\frac{4ah}{(a+h)cot\alpha +a \tan \alpha } \\ \left(ii\right) Let f\left(\theta \right)=(a+h)cot\alpha +a \tan \theta for 0<\theta <\frac{\mathrm{\pi }}{2} \\ iii) Show that the greatest value of x is given by \\ x=2h\sqrt{\frac{a}{a+h}} \end{gathered}$$

$$\begin{gathered} A cable car is travelling at a cons\tan t height of 45m above the ground. An observer on the \\ ground at point O sees the cable car on the bearing of 335°T from O with an angle of \\ elevation of 28°. \\ After 1 minute the cable car has a bearing of 025°T from O and a new angle elevation is 53°. \end{gathered}$$

$$\begin{gathered} For the points A\left(3,-5\right) and B\left(-4,2\right), find the coordinates of the point P which divides \\ the interval AB externally in the ratio 2:1. \end{gathered}$$

$$\begin{gathered} Evaluate lim\underset{\mathcal{x}\to \infty }{\lim }\left(\frac{1+\mathcal{x}}{2-\mathcal{x}}\right). \\ \left(A\right) -2 \\ \left(B\right) –1 \\ \left(C\right) 1 \\ \left(D\right) 2 \end{gathered}$$

$$\begin{gathered} . What is the correct expression for \int \frac{d\mathcal{x}}{9+4{\mathcal{x}}^2}? \\ \left(A\right)\frac{1}{4}{\tan }^{-1}\left(\frac{2\mathcal{x}}{3}\right) \\ \left(B\right)\frac{1}{3}{\tan }^{-1}\left(\frac{2\mathcal{x}}{3}\right) \\ \left(C\right)\frac{1}{6}{\tan }^{-1}\left(\frac{2\mathcal{x}}{3}\right) \\ \left(D\right)\frac{2}{3}{\tan }^{-1}\left(\frac{2\mathcal{x}}{3}\right) \end{gathered}$$

$$\begin{gathered} Segment AD lies on a \tan gent to the \\ circle centre, O, radius 5 cm. \\ BC is 6 cm and CD is 9 cm. \\ Find the exact length of AD. \\ \left(A\right)\sqrt{14} \\ B) 3\sqrt{6} \\ (C) 3\sqrt{10} \\ (D) 3\sqrt{15} \end{gathered}$$

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

$$\begin{gathered} Calculate the acute angle between the lines \mathcal{x}– 5\mathcal{y}– 2 = 0 \\ and \mathcal{x} – 2\mathcal{y}=0 to the nearest degree. \\ i)Express \cos 2\mathcal{x} in terms of {\sin }^2\mathcal{x}. \\ ii) Hence evaluate \underset{\mathcal{x}\to \infty }{\lim } \frac{\cos 2\mathcal{x}-1}{\mathcal{x} \sin \mathcal{x}} \\ Evaluate {\int }_0^3\mathcal{x}\sqrt{9-{\mathcal{x}}^2 d\mathcal{x}} u\sin g the substitution u=9-{\mathcal{x}}^2. \\ Ms Namvar bought a slurpy in Port Douglas which had a temperature \\ of 5°C. The temperature in Port Douglas was 35°C.The slurpy warms \\ at a rate proportional to the difference between the air temperature \\ and the temperature\left(T\right) of the slurpy. \\ That is, T satisfies the equation \frac{dT}{dt}=k(T-35). \\ i) Show that T=35+A{e}^{kt} satisfies this equation. \\ ii) If the temperature of the slurpy after ten minutes is 10° C, find its \\ temperature, to the nearest whole degree, after 20 minutes. \end{gathered}$$

$$\begin{gathered} The height of a giraffe has been modeled u\sin g the equation \\ H= 5.40 -4.80e^{-kt} \\ where H is the height in metres, t is the age in years and k is a positive cons\tan t \\ If a 6 year old giraffe has a height of 5.16 metres, find the value of k. \\ Find the limiting height of the giraffe. \end{gathered}$$

$$\begin{gathered} If A is the point (–2, –1) and B is the point (1, 5), find the coordinates of the \\ point P which divides the interval AB externally in the ratio 2:5. \end{gathered}$$

$$\begin{gathered} Two circles C_1 and C_2 centred at P and Q with equal radii r \\ intersect at A and B respectively. AC is a diameter in circle C_1 and \\ AD is a diameter in C_2. \end{gathered}$$

$$\begin{gathered} Redraw the diagram in your answer booklet. \\ Show that ∆ABC is congruent to ∆ABD. \\ Show that PB\parallel AD. \\ Show that PQDB is a paralle\log ram. \end{gathered}$$