Standard +0.3 This is a standard C4 partial fractions question with routine algebraic division followed by integration. Part (a)(i) is textbook partial fractions decomposition, (a)(ii) is straightforward integration of logarithmic forms, and part (b) connects the pieces together but requires no novel insight. The multi-part structure and 'show that' element add slight complexity, but all techniques are standard and well-practiced at this level.
Express \(\frac { 5 - 8 x } { ( 2 + x ) ( 1 - 3 x ) }\) in the form \(\frac { A } { 2 + x } + \frac { B } { 1 - 3 x }\), where \(A\) and \(B\) are integers.
(3 marks)
Hence show that \(\int _ { - 1 } ^ { 0 } \frac { 5 - 8 x } { ( 2 + x ) ( 1 - 3 x ) } \mathrm { d } x = p \ln 2\), where \(p\) is rational.
(4 marks)
Given that \(\frac { 9 - 18 x - 6 x ^ { 2 } } { 2 - 5 x - 3 x ^ { 2 } }\) can be written as \(C + \frac { 5 - 8 x } { 2 - 5 x - 3 x ^ { 2 } }\), find the value of \(C\).
(1 mark)
Hence find the exact value of the area of the region bounded by the curve \(y = \frac { 9 - 18 x - 6 x ^ { 2 } } { 2 - 5 x - 3 x ^ { 2 } }\), the \(x\)-axis and the lines \(x = - 1\) and \(x = 0\).
You may assume that \(y > 0\) when \(- 1 \leqslant x \leqslant 0\).
1
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Express $\frac { 5 - 8 x } { ( 2 + x ) ( 1 - 3 x ) }$ in the form $\frac { A } { 2 + x } + \frac { B } { 1 - 3 x }$, where $A$ and $B$ are integers.\\
(3 marks)
\item Hence show that $\int _ { - 1 } ^ { 0 } \frac { 5 - 8 x } { ( 2 + x ) ( 1 - 3 x ) } \mathrm { d } x = p \ln 2$, where $p$ is rational.\\
(4 marks)
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Given that $\frac { 9 - 18 x - 6 x ^ { 2 } } { 2 - 5 x - 3 x ^ { 2 } }$ can be written as $C + \frac { 5 - 8 x } { 2 - 5 x - 3 x ^ { 2 } }$, find the value of $C$.\\
(1 mark)
\item Hence find the exact value of the area of the region bounded by the curve $y = \frac { 9 - 18 x - 6 x ^ { 2 } } { 2 - 5 x - 3 x ^ { 2 } }$, the $x$-axis and the lines $x = - 1$ and $x = 0$.
You may assume that $y > 0$ when $- 1 \leqslant x \leqslant 0$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C4 2013 Q1 [10]}}