| Exam Board | Edexcel |
|---|---|
| Module | PMT Mocks (PMT Mocks) |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Deduce related integral from numerical approximation |
| Difficulty | Standard +0.3 This question requires applying the trapezium rule (routine numerical integration) in part (a), then using logarithm laws to relate the given integral to new ones in part (b). The manipulations are straightforward: log₃(√x) = ½log₃(x) and log₃(9x³) = 2 + 3log₃(x). While multi-step, each component is standard A-level technique with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02b Surds: manipulation and rationalising denominators1.06f Laws of logarithms: addition, subtraction, power rules1.09f Trapezium rule: numerical integration |
| \(x\) | 3 | 4.5 | 6 | 7.5 | 9 |
| \(y\) | 1 | 1.37 | 1.63 | 1.83 | 2 |
| Answer | Marks |
|---|---|
| Part a. | (3 marks) |
| B1 | States and uses \(h = 1.5\) |
| M1 | A full attempt at the trapezium rule. Look for \(\frac{\text{their } h}{2}\{1 + 2 + 2 \times (1.37 + 1.63 + 1.83)\}\); but condone copying slips. Allow this if they add the areas of individual trapezoid. Example: \(\frac{\text{their } h}{2}\{1 + 1.37\} + \frac{\text{their } h}{2}\{1.37 + 1.63\} + \frac{\text{their } h}{2}\{1.63 + 1.83\} + \frac{\text{their } h}{2}\{1.83 + 2\}\) |
| A1 | 9.50 |
| Answer | Marks | Guidance |
|---|---|---|
| Part b.i. | (1 mark) | |
| B1 | awrt 4.75 | Example: \(\frac{1}{2} \times \log_3 x = \frac{1}{2} \times 9.50 = 4.75\) |
| Answer | Marks |
|---|---|
| Part b.ii. | (2 marks) |
| M1 | States and implies that \(\log_3 9x^3 = 2 + 3\log_3 x\) and \(\int_3^{18} \log_3(9x^3) \, dx = [2x]_3^{18} + 3 \times 9.50 = \cdots\) or equivalent work for finding the area of the rectangle as \(2 \times 15\) |
| A1 | Correct working followed by awrt 58.5. Example: \([2x]_3^{18} + 3 \times 9.50 = 2 \times 18 - 2 \times 3 + 3 \times 9.50 = 58.5\) |
(6 marks)
**Part a.** | (3 marks)
**B1** | States and uses $h = 1.5$
**M1** | A full attempt at the trapezium rule. Look for $\frac{\text{their } h}{2}\{1 + 2 + 2 \times (1.37 + 1.63 + 1.83)\}$; but condone copying slips. Allow this if they add the areas of individual trapezoid. Example: $\frac{\text{their } h}{2}\{1 + 1.37\} + \frac{\text{their } h}{2}\{1.37 + 1.63\} + \frac{\text{their } h}{2}\{1.63 + 1.83\} + \frac{\text{their } h}{2}\{1.83 + 2\}$
**A1** | 9.50
---
**Part b.i.** | (1 mark)
**B1** | awrt 4.75 | Example: $\frac{1}{2} \times \log_3 x = \frac{1}{2} \times 9.50 = 4.75$
---
**Part b.ii.** | (2 marks)
**M1** | States and implies that $\log_3 9x^3 = 2 + 3\log_3 x$ and $\int_3^{18} \log_3(9x^3) \, dx = [2x]_3^{18} + 3 \times 9.50 = \cdots$ or equivalent work for finding the area of the rectangle as $2 \times 15$
**A1** | Correct working followed by awrt 58.5. Example: $[2x]_3^{18} + 3 \times 9.50 = 2 \times 18 - 2 \times 3 + 3 \times 9.50 = 58.5$
---\begin{enumerate}
\item The table below shows corresponding values of $x$ and $y$ for $y = \log _ { 3 } ( x )$ The values of $y$ are given to 2 decimal places as appropriate.
\end{enumerate}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$x$ & 3 & 4.5 & 6 & 7.5 & 9 \\
\hline
$y$ & 1 & 1.37 & 1.63 & 1.83 & 2 \\
\hline
\end{tabular}
\end{center}
a. Obtain an estimate for $\int _ { 3 } ^ { 9 } \log _ { 3 } ( x ) \mathrm { d } x$, giving your answer to two decimal places.
Use your answer to part (a) and making your method clear, estimate\\
b. i) $\int _ { 3 } ^ { 9 } \log _ { 3 } \sqrt { x } \mathrm {~d} x$\\
ii) $\int _ { 3 } ^ { 18 } \log _ { 3 } \left( 9 x ^ { 3 } \right) \mathrm { d } x$\\
\hfill \mbox{\textit{Edexcel PMT Mocks Q5 [6]}}