$$\begin{gathered} Find \int \frac{1}{9+25{\mathcal{x}}^2}d\mathcal{x} \\ A) \frac{1}{15}{\tan }^{-1}\frac{5\mathcal{x}}{3}+C \\ B) \frac{1}{25}{\tan }^{-1}\frac{5\mathcal{x}}{3}+C \\ C) \frac{1}{25}{\tan }^{-1}\frac{3\mathcal{x}}{5}+C \\ D) \frac{1}{15}{\tan }^{-1}\frac{3\mathcal{x}}{5}+C \end{gathered}$$

$$\begin{gathered} The solution to \left|2\mathcal{x}-1\right|\le \left|\mathcal{x}-2\right| is \\ \left(A\right) \mathcal{x}\le 1 \\ \left(B\right) \mathcal{x}\ge 1 \\ \left(C\right) -1\le \mathcal{x}\le 1 \\ \left(D\right) \mathcal{x}\le -1 or \mathcal{x}\ge 1 \end{gathered}$$

$$\begin{gathered} The three numbers a, b, c are consecutive tenns in an arithmetic progression. Show that the three numbers e^a, e^b, e^c are consecutive terms in a geometric progression. \\ Find the exact area between the curve \mathcal{y} = {\sin }^{-1} \mathcal{x}, the \mathcal{x}-axis and the lines \mathcal{x}=\frac{1}{2} and \mathcal{x}=1. \\ Find the equation of the vertical and horizontal asymptotes of the curve \mathcal{y}=\frac{2{\mathcal{x}}^2+1}{{\mathcal{x}}^2-4\mathcal{x}}. \\ For what values of \mathcal{x} will 1-{\tan }^2\mathcal{x}+{\tan }^4\mathcal{x}-{\tan }^6\mathcal{x}+... have a limiting sum for 0\le \mathcal{x}\le 2\mathrm{\pi } ? \end{gathered}$$

$Use the substitution u=x+3 to find \int \frac{x}{\sqrt{x+3}}dx.$

$d) Sketch the graph of y=f(x), where f(x)=\frac{1}{2}{\cos }^{-1}(1-3x).$

$$\begin{gathered} \left(i\right) Show that f \left(x\right) = e^x - x^3+1 has a zero between x = 4.4 and x = 4.6 \\ \left(ii\right) Starting at x = 4.5, find an approximation for the zero in part \left(i\right) \\ u\sin g Newton’s method. \\ Express this approximation correct to 2 decimal places \end{gathered}$$

$$\begin{gathered} Finding \underset{x\to 0}{\lim }\frac{\sin 4x}{x}gives \\ A. \frac{1}{4} \\ B. 4 \\ C. 2\mathrm{\pi } \\ \mathrm{D}. \frac{\mathrm{\pi }}{8} \end{gathered}$$

$What is a possible equation of this function?$

$$\begin{gathered} The points A, B and C lie on the circle with centre O. \\ OA is parallel to CB. AC intersects OB at D and \angle ODC = \alpha \\ What is the size of \angle OAD in terms of \alpha ? \end{gathered}$$

$$\begin{gathered} \frac{\alpha }{2} \\ \frac{\alpha }{3} \\ \frac{2\alpha }{3} \\ 3\alpha \end{gathered}$$

$Find the exact value of \sin \frac{\mathrm{\pi }}{12} waith a rational denominator.$

$$\begin{gathered} The point P divides the interval AB externally in the ratio 3:2, whereA= (-5,6) \\ and B= (-2,3).The \mathcal{x}-value of P is: \\ \left(A\right) 1 \\ \left(B\right) 2 \\ \left(C\right) 4 \\ \left(D\right) 5 \end{gathered}$$

$$\begin{gathered} The line AT is a \tan gent to the circle at A and BT is a secant meeting the \\ circle at B and C. Given that AT=10cm and BC=6, the value of \mathcal{x} Is \\ closest to? \end{gathered}$$

$$\begin{gathered} \left(A\right) 1.67 \\ \left(B\right) 4 \\ \left(C\right) 7.44 \\ \left(D\right) 16.67 \end{gathered}$$

$$\begin{gathered} Research into Alzheimer\text{'}s disease suggests that the rate of loss of percentage brain function is proportional to \\ the percentage brain function already lost. A particular Alzheimer\text{'}s disease patient was initially diagnosed with \\ a 20\% loss of brain function. If L is the percentage brain function lost and k is a cons\tan t, which of the \\ following equations represents the loss of percentage brain function for this paiiicular patient? \\ A: L=k{e}^{0.2t} \\ B: L=k{e}^{20t} \\ C: L=20e^{kt} \\ D: L=80e^{kt} \end{gathered}$$

$$\begin{gathered} A bowl of soup is cooling in a room that has a cons\tan t temperature of 20 °. At time t, measured in minutes, the \\ temperature, T, of the soup is decrea\sin g according to the differential equation:\frac{dT }{dt}=-k\left(T-20\right) \\ where k is a positive cons\tan t. The initial temperature of the soup is 100° and it cools to 70° C after 5 minutes. \\ \left(i\right) Verify that T=20+A{e}^{-kt} is a solution to the differential equation, where A is a cons\tan t. \\ \left(ii\right) Find the values of A and k. \\ \left(iii\right) Find the temperature of the soup after 15 minutes, Give your answer to the nearest degree. \end{gathered}$$

$$\begin{gathered} The rate of descent of a submarine, from the surface, into the ocean is given as: \frac{dh}{dt}=1-{\left(1+t\right)}^{-2} \\ where h is the depth of the submarine in metres and t the time in seconds. \\ Find the depth of the submarine after 1 minute, correct to one decimal place. \end{gathered}$$

$$\begin{gathered} The graph above shows the average weight W of a herd of beef cattle over a period of time t, \\ where t is in months. After a period of drought, the average weight of the herd stabilised. \\ Sketch the graph of the rate, at which \frac{dW}{dt} was changing over this period. \end{gathered}$$