| Exam Board | Edexcel |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2020 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Deduce related integral from numerical approximation |
| Difficulty | Standard +0.3 This is a straightforward trapezium rule application followed by algebraic manipulation of logarithms. Part (a) is routine numerical integration. Parts (b)(i) and (b)(ii) require recognizing log properties (log(4x²) = 2log(2x) and log(2/x) = log(2x) - 2log(x)) to relate back to part (a), but these are standard A-level techniques with clear signposting. The question is slightly easier than average due to its structured, step-by-step nature. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules1.09f Trapezium rule: numerical integration |
| \(x\) | 2 | 5 | 8 | 11 | 14 |
| \(y\) | 2 | 3.32 | 4 | 4.46 | 4.81 |
\begin{enumerate}
\item The table below shows corresponding values of $x$ and $y$ for $y = \log _ { 2 } ( 2 x )$
\end{enumerate}
The values of $y$ are given to 2 decimal places as appropriate.
Using the trapezium rule with all the values of $y$ in the given table,\\
(a) obtain an estimate for $\int _ { 2 } ^ { 14 } \log _ { 2 } ( 2 x ) \mathrm { d } x$, giving your answer to one decimal place.
Using your answer to part (a) and making your method clear, estimate\\
(b) (i) $\quad \int _ { 2 } ^ { 14 } \frac { \log _ { 2 } \left( 4 x ^ { 2 } \right) } { 5 } \mathrm {~d} x$\\
(ii) $\int _ { 2 } ^ { 14 } \log _ { 2 } \left( \frac { 2 } { x } \right) \mathrm { d } x$
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$x$ & 2 & 5 & 8 & 11 & 14 \\
\hline
$y$ & 2 & 3.32 & 4 & 4.46 & 4.81 \\
\hline
\end{tabular}
\end{center}
\hfill \mbox{\textit{Edexcel P2 2020 Q1 [7]}}