[br]\begin{enumerate}[br]\item[br]\begin{enumerate}[br]\item A = 1, B = -1 \hfill \textbf{\underline{[3]}} \\[br]\item \dfrac{1}{2} - \dfrac{1}{(n+2)!} \hfill \textbf{\underline{[2]}} \\[br]\end{enumerate}[br]\end{enumeate}[br]
[br]\begin{enumerate}[br]\setcounter{enumi}{1}[br]\item [br]\begin{enumerate}[br]\item \text{Sketch} y = tanhx \text{asymptotes are} y = \pm 1 \hfill \textbf{\underline{[3]}} \\[br]\item \text{Show} sech^{2}x + tanh^{2}x = 1 \hfill \textbf{\underline{[3]}} \\[br]\item x = \dfrac{ln 3}{2}, \dfrac{-ln 5}{2} \hfill \textbf{\underline{[5]}} \\[br]\end{enumerate}[br]\end{enumerate}[br]
[br]\begin{enumerate}[br]\setcounter{enumi}{2}[br]\item[br]\begin{enumerate}[br]\item \text{Show something} \hfill \textbf{\underline{[4]}} \\[br]\item S_{x} = \pi (6ln 2 - 2) \hfill \textbf{\underline{[5]}} \\[br]\end{enumerate}[br]\end{enumerate}[br]
[br]\begin{enumerate}[br]\setcounter{enumi}{3}[br]\item[br]\begin{enumerate}[br]\item f(k+1) - 16f(k) = 33(3^{3k}) \hfill \textbf{\underline{[3]}} \\[br]\item \text{Inductive proof.} \hfill \textbf{\underline{[4]}} \\[br]\end{enumerate}[br]\end{enumerate}[br]
[br]\begin{enumerate}[br]\setcounter{enumi}{4}[br]\item[br]\begin{enumerate}[br]\item \text{Sketch} \left|z - 2 + 4i\right| = \left|z\right| \hfill \textbf{\underline{[3]}} \\[br]\item[br]\begin{enumerate}[br]\item \text{Show midpoint} = \dfrac{5}{2} - \dfrac{5i}{4} \hfill \textbf{\underline{[4]}} \\[br]\item \text{Equation of circle} is \left|z-\left(\dfrac{5}{2} - \dfrac{5i}{4}\right)\right| = \dfrac{5\sqrt{5}}{4} \hfill \textbf{\underline{[2]}} \\[br]\end{enumerate}[br]\end{enumerate}[br]\end{enumerate}[br]
[br]\begin{enumerate}[br]\setcounter{enumi}{5}[br]\item[br]\begin{enumerate}[br]\item \dfrac{dy}{dx} = 2\sqrt{5 + 4x - x^{2}} \hfill \textbf{\underline{[5]}} \\[br]\item I = \dfrac{9 \sqrt{3} }{8} + \dfrac{3\pi}{4} \hfill \textbf{\underline{[3]}} \\[br]\end{enumerate}[br]\end{enumerate}[br]
[br]\begin{enumerate}[br]\setcounter{enumi}{6}[br]\item [br]\begin{enumerate}[br]\item[br]\begin{enumerate}[br]\item \alpha = \dfrac{-1}{3}, \beta = \gamma = \dfrac{2}{3} \hfill \textbf{\underline{[5]}} \\[br]\item k = 1 \hfill \textbf{\underline{[1]}} \\[br]\end{enumerate}[br]\item[br]\begin{enumerate}[br]\item \alpha^{2} = -2i, \alpha^{3} = -2 - 2i \hfill \textbf{\underline{[2]}} \\[br]\item k = 25 - 27i \hfill \textbf{\underline{[2]}} \\[br]\end{enumerate}[br]\item 4x^{3} - 12x^{2} + 35 = 0 \hfill \textbf{\underline{[7]}} \\[br]\end{enumerate}[br]\end{enumerate}[br]
[br]\begin{enumerate}[br]\setcounter{enumi}{7}[br]\item [br]\begin{enumerate}[br]\item[br]\begin{enumerate}[br]\item \text{Show} \omega \text{ is a root of } z^{5} = 1 \hfill \textbf{\underline{[1]}} \\[br]\item \text{Show} 1 + \omega + \omega^{2} + \omega^{3} + \omega^{4} = 0 \hfill \textbf{\underline{[1]}} \\[br]\item \texts{Roots in form of} \omega^{a} = cos\dfrac{k\pi}{5} + isin\dfrac{k\pi}{5} \text{ for } a = 2,3,4; k = 4,-1,-3 \hfill \textbf{\underline{[1]}} \\[br]\end{enumerate}[br]\item[br]\begin{enumerate}[br]\item \text{Show that } \left(\omega + \dfrac{1}{\omega}\right)^{2} + \left(\omega + \dfrac{1}{\omega}\right) + 1 = 0 \hfill \textbf{\underline{[2]}} \\[br]\item \text{Show} cos \dfrac{2\pi}{5} = \dfrac{\sqrt{5} - 1}{4} \hfill \textbf{\underline{[4]}} \\[br]\end{enumerate}[br]\end{enumerate}[br]\end{enumerate}[br]
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