Standard +0.3 This is a straightforward C4 partial fractions question with clear guidance. Part (i) involves routine factor verification and algebraic division. Part (ii) requires standard partial fractions decomposition and integration of logarithmic forms. The 'show that' structure removes problem-solving demands, making this slightly easier than average for C4.
7. (i) Show that ( \(2 x + 3\) ) is a factor of ( \(\left. 2 x ^ { 3 } - x ^ { 2 } + 4 x + 15 \right)\) and hence, simplify
$$\frac { 2 x ^ { 2 } + x - 3 } { 2 x ^ { 3 } - x ^ { 2 } + 4 x + 15 } .$$
(ii) Show that
$$\int _ { 2 } ^ { 5 } \frac { 2 x ^ { 2 } + x - 3 } { 2 x ^ { 3 } - x ^ { 2 } + 4 x + 15 } \mathrm {~d} x = \ln k$$
where \(k\) is an integer.
7. (i) Show that ( $2 x + 3$ ) is a factor of ( $\left. 2 x ^ { 3 } - x ^ { 2 } + 4 x + 15 \right)$ and hence, simplify
$$\frac { 2 x ^ { 2 } + x - 3 } { 2 x ^ { 3 } - x ^ { 2 } + 4 x + 15 } .$$
(ii) Show that
$$\int _ { 2 } ^ { 5 } \frac { 2 x ^ { 2 } + x - 3 } { 2 x ^ { 3 } - x ^ { 2 } + 4 x + 15 } \mathrm {~d} x = \ln k$$
where $k$ is an integer.\\
\hfill \mbox{\textit{OCR C4 Q7 [9]}}