2. Fig. 5 shows the curve with polar equation \(r = a ( 3 + 2 \cos \theta )\) for \(- \pi \leqslant \theta \leqslant \pi\), where \(a\) is a constant.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c545da50-9478-47e9-a6ff-4ec69bd00fc7-06_620_734_269_262}
\captionsetup{labelformat=empty}
\caption{Fig. 5}
\end{figure}
- Write down the polar coordinates of the points A and B .
- Explain why the curve is symmetrical about the initial line.
- In this question you must show detailed reasoning.
Find in terms of \(a\) the exact area of the region enclosed by the curve.
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\section*{3. In this question you must show detailed reasoning.}
The roots of the equation \(2 x ^ { 3 } - 5 x + 7 = 0\) are \(\alpha , \beta\) and \(\gamma\). - Find \(\frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma }\).
- Find an equation with integer coefficients whose roots are \(2 \alpha - 1,2 \beta - 1\) and \(2 \gamma - 1\).
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\section*{4. In this question you must show detailed reasoning.} - Given that
$$\frac { 1 } { r ( r + 1 ) ( r + 2 ) } = \frac { A } { r ( r + 1 ) } + \frac { B } { ( r + 1 ) ( r + 2 ) }$$
show that \(A = \frac { 1 } { 2 }\) and find the value of \(B\).
- Use the method of differences to find
$$\sum _ { r = 10 } ^ { 98 } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }$$
giving your answer as a rational number.
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