$$\begin{gathered} Which of the following is equal to \tan [2{\sin }^{-1}\frac{\mathcal{x}}{\sqrt{1+{\mathcal{x}}^2}}? \\ (A) \mathcal{x} \\ (B) 2\sin \mathcal{x} \cos \mathcal{x} \\ (C)\frac{ 2\mathcal{x}}{1-{\mathcal{x}}^2} \\ (D) \frac{2\mathcal{x}}{\sqrt{1+{\mathcal{x}}^2}} \end{gathered}$$

$$\begin{gathered} Which of the following is equal to \int \frac{3}{25+4{\mathcal{x}}^2}d\mathcal{x}? \\ \left(A\right) \frac{3}{10}{\tan }^{-1}\left(\frac{2\mathcal{x}}{5}\right)+c \\ \left(B\right) 3{\tan }^{-1}(\frac{2\mathcal{x}}{5} )+c \\ \left(C\right) \frac{15}{8}{\tan }^{-1}\left(\frac{5\mathcal{x}}{2}\right)+c \\ \left(D\right) 3{\tan }^{-1}\left(\frac{5\mathcal{x}}{2}\right)+c \end{gathered}$$

$$\begin{gathered} \left(1\right)show that {\sin }^2\mathcal{x}{\cos }^2\mathcal{x}=\frac{t {\sin }^{2}2\mathcal{x}}{4} \\ \left(2\right)hence orotherwise find \int {\sin }^2\mathcal{x}{\cos }^2\mathcal{x} d\mathcal{x} \end{gathered}$$

$$\begin{gathered} For what values of \mathcal{x} is \frac{\mathcal{x} }{\mathcal{x}-2}<2? \\ Let f\left(\mathcal{x}\right)={\mathcal{x}}^3+5{\mathcal{x}}^2+17\mathcal{x}-10 . The equation f\left(\mathcal{x}\right)=0 has only one real root. \\ Show that the root lies between 0 and 2. \\ Use one application of Newtons Method with an initial estimate of {\mathcal{x}}_0=1 to \\ find a better approximation of the root (to 2 decimal places). \end{gathered}$$

$$\begin{gathered} For the function given by f\left(\mathcal{x}\right)=-1+\sqrt{\mathcal{x}+4} \\ State the domain for the function f\left(\mathcal{x}\right). \\ Find the inverse function f^{-1}\left(\mathcal{x}\right) for the given function f\left(\mathcal{x}\right). \\ Find the restrictions on the domain and range for f^{-1} \left(\mathcal{x}\right) to be the inverse \\ function of f\left(\mathcal{x}\right). \end{gathered}$$

$$\begin{gathered} If t=\tan \frac{\theta }{2}, then \tan \theta = \\ \frac{2t}{1-t^2} \\ \frac{1+t^2}{1-t^2} \\ \frac{1-t^2}{1+t^2} \\ \frac{2t+1}{1+t^2} \end{gathered}$$

$$\begin{gathered} \left(i\right) Write down the expansion of \tan (A+B ) . \\ \left(ii\right) Find the value of \tan \left(\frac{7\mathrm{\pi }}{12}\right)in simplest surd form. \\ Show that \underset{\mathcal{x}\to 0}{\lim }\frac{\sin 4\mathcal{x}}{9\mathcal{x}}=\frac{4}{9} \end{gathered}$$

$$\begin{gathered} \mathrm{A} \mathrm{particle} \mathrm{is} \mathrm{moving} \mathrm{in} \mathrm{a} \mathrm{straight} \mathrm{line}. \mathrm{At} \mathrm{time} \mathrm{t} \mathrm{seconds} \mathrm{it} \mathrm{has} \mathrm{displacement} \mathcal{x} \mathrm{meters} \mathrm{from} \mathrm{a} \mathrm{fixed} \mathrm{point} \mathrm{O} \mathrm{in} \mathrm{the} \mathrm{line}, \\ \mathrm{v} {\mathrm{ms}}^{-1} \mathrm{is} \mathrm{given} \mathrm{by} \mathrm{v}=\frac{1}{\mathcal{x}} \mathrm{and} \mathrm{acceleration} \mathrm{a} {\mathrm{ms}}^{-2}. \mathrm{Initially} \mathrm{the} \mathrm{particle} \mathrm{is} \mathrm{at} \mathrm{O}. \\ \left(\mathrm{i}\right) \mathrm{Express} \mathrm{a} \mathrm{as} \mathrm{a} \mathrm{function} \mathrm{of} \mathcal{x}. \\ \left(\mathrm{ii}\right) \mathrm{Express} \mathcal{x} \mathrm{as} \mathrm{a} \mathrm{function} \mathrm{of} \mathrm{t}. \end{gathered}$$

$$\begin{gathered} \mathrm{A} \mathrm{particle} \mathrm{is} \mathrm{moving} \mathrm{in} \mathrm{a} \mathrm{straight} \mathrm{line} \mathrm{with} \mathrm{Simple} \mathrm{Harmonic} \mathrm{Motion}. \mathrm{At} \mathrm{time} \mathrm{t} \mathrm{seconds} \\ \mathrm{it} \mathrm{has} \mathrm{displacement} \mathcal{x} \mathrm{meters} \mathrm{from} \mathrm{a} \mathrm{fixed} \mathrm{point} \mathrm{O} \mathrm{on} \mathrm{the} \mathrm{line}, \mathrm{given} \mathrm{by} \mathcal{x}=1+3\cos \frac{\mathrm{t}}{2}, \\ \mathrm{velocity} \mathrm{v} {\mathrm{ms}}^{-1} \mathrm{and} \mathrm{acceleration} \mathrm{a} {\mathrm{ms}}^{-2}. \\ \left(\mathrm{i}\right) \mathrm{Show} \mathrm{that} \mathrm{a}=-\frac{1}{4}\left(\mathcal{x}-1\right). \\ \left(\mathrm{ii}\right) \mathrm{Find} \mathrm{the} \mathrm{distance} \mathrm{travelled} \mathrm{and} \mathrm{the} \mathrm{time} \mathrm{taken} \mathrm{by} \mathrm{the} \mathrm{particle} \mathrm{over} \mathrm{one} \mathrm{complete} \mathrm{oscillation} \\ \mathrm{of} \mathrm{its} \mathrm{motion}. \end{gathered}$$

$$\begin{gathered} A particle exhibits simple harmonic motion according to the equation v^2=(\mathcal{x}-1)(5-\mathcal{x}). \\ The amplitude is: \\ \left(A\right) 1 \\ \left(B\right) 2 \\ \left(C\right) 3 \\ \left(D\right) 4 \end{gathered}$$

$$\begin{gathered} In a kitchen, a leg of ham is removed from a fridge and its temperature H degrees Celsius 1 is monitored. It is found that after t hours the temperature is given by \\ H=16(1-\frac{3}{4}{e}^{-4t}) \\ The temperature of the kitchen is: \\ \left(A\right) 3^{°}C \\ \left(B\right) 4^{°}C \\ \left(C\right) {10}^{°}C \\ \left(D\right) {16}^{°}C \end{gathered}$$

$$\begin{gathered} A, B and C are points defined by the position vectors \\ \underset{~}{a}=\underset{~}{i}+3\underset{~}{j} , \underset{~}{b}=2\underset{~}{i}+\underset{~}{j} , and \underset{~}{c}=\underset{~}{i}-2\underset{~}{j} respectively \\ Find \overrightarrow{BA} and \overrightarrow{BC} \\ Find \left|\overrightarrow{BA}\right| and \left|\overrightarrow{BC}\right| \\ Find \angle ABC. \end{gathered}$$

$In the diagram below \angle BAC = 34° and \angle ADE = 85° . What is the size of angle \angle ACB ?$

$Sketch the curve y=3{\sin }^{-1}2x$

$$\begin{gathered} The roots of x^3-3x^2=6x-5=0 are \alpha ,\beta and \gamma . find the value of \\ i) (\alpha +1)+(\beta +1)+\left(\gamma +1\right) \\ ii) (\alpha +1)(\beta +1)(\gamma +1) \end{gathered}$$

$$\begin{gathered} Use the identity {(1+\mathcal{x})}^8{(1+\mathcal{x})}^8={(1+\mathcal{x})}^{16} to show \\ {\left(\begin{array}{c}8\\ 0\end{array}\right)}^2+{\left(\begin{array}{c}8\\ 1\end{array}\right)}^2+{\left(\begin{array}{c}8\\ 2\end{array}\right)}^2+......+{\left(\begin{array}{c}8\\ 8\end{array}\right)}^2=\left(\begin{array}{c}16\\ 8\end{array}\right) \end{gathered}$$

$$\begin{gathered} Prove, by Mathematical induction, that all positive integer values of n, \\ 2\times 1!+5\times 2!+10\times 3!+......+(n^2+1)n!=n(n+1) \end{gathered}$$

$$\begin{gathered} Expand \cos (\alpha +\beta ) \\ Show that \cos 2\alpha =1-2{\sin }^2\alpha \\ Evaluate \underset{\mathcal{x}\to 0}{\lim }\frac{1-\cos 2\mathcal{x}}{{\mathcal{x}}^2} \end{gathered}$$

$$\begin{gathered} At a dinner party, the host, hostess and their six guests sit at a round table. In how \\ many ways can they be arranged if the host and hostess are separated? \\ \left(A\right) 720 \\ \left(B\right) 1440 \\ \left(C\right) 3600 \\ \left(D\right) 5040 \end{gathered}$$

Three Mathematics study guides, four Mathematics textbooks and five exercise books are randomly placed along a bookshelf. What is the probability that the Mathematics textbooks are all next to each other?