| Exam Board | Edexcel |
|---|---|
| Module | Paper 2 (Paper 2) |
| Year | 2022 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Deduce related integral from numerical approximation |
| Difficulty | Standard +0.3 Part (a) is a standard trapezium rule application with given values. Parts (b)(i) and (b)(ii) require recognizing logarithm laws (10log₃(2x) and log₃(18x) = log₃(9) + log₃(2x)) to relate back to part (a), which is straightforward manipulation once the pattern is spotted. This is slightly easier than average as it's a structured multi-part question with clear scaffolding and standard techniques. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules1.09f Trapezium rule: numerical integration |
| \(x\) | 3 | 4.5 | 6 | 7.5 | 9 |
| \(y\) | 1.63 | 2 | 2.26 | 2.46 | 2.63 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| States or uses \(h = 1.5\) | B1 | 1.1a |
| Full attempt at trapezium rule: \(= \frac{h}{2}\{1.63 + 2.63 + 2\times(2 + 2.26 + 2.46)\}\) | M1 | Look for \(\frac{\text{their }h}{2}\{1.63 + 2.63 + 2\times(2 + 2.26 + 2.46)\}\); condone copying slips. Note missing brackets scores M0 unless recovered. |
| \(= \text{awrt } 13.3\) or \(\frac{531}{40}\) | A1 | Full accuracy is 13.275. Calculator answer is 13.324 so correct working must be seen. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\int_3^9 \log_3(2x)^{10}\, dx = 10 \times \text{"13.3"} = \text{awrt } 133\) or e.g. \(\frac{531}{4}\) | B1ft | FT on their 13.3; look for 3sf accuracy. A correct method must be seen (minimum \(10 \times \text{"13.3"} = \text{"133"}\)). Attempts to apply trapezium rule again score M0. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\int_3^9 \log_3 18x\, dx = \int_3^9 \log_3(9\times 2x)\, dx = \int_3^9 2 + \log_3 2x\, dx\) \(= [2x]_3^9 + \int_3^9 \log_3 2x\, dx = 18 - 6 + \int_3^9 \log_3 2x\, dx = \ldots\) | M1 | Must reach \([2x]_3^9 + \int_3^9 \log_3 2x\, dx = \ldots\) with correct use of limits. May be implied by finding rectangle area as \(2\times 6\). Just \(12 + \text{"13.3"}\) scores M0. |
| \(\text{Awrt } 25.3\) or \(\frac{1011}{40}\) | A1ft | FT on their 13.3; allow \(12 + \text{their answer to (a)}\) following correct work. Attempts to apply trapezium rule again score M0. |
## Question 5:
### Part 5(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| States or uses $h = 1.5$ | B1 | 1.1a |
| Full attempt at trapezium rule: $= \frac{h}{2}\{1.63 + 2.63 + 2\times(2 + 2.26 + 2.46)\}$ | M1 | Look for $\frac{\text{their }h}{2}\{1.63 + 2.63 + 2\times(2 + 2.26 + 2.46)\}$; condone copying slips. Note missing brackets scores M0 unless recovered. |
| $= \text{awrt } 13.3$ or $\frac{531}{40}$ | A1 | Full accuracy is 13.275. Calculator answer is 13.324 so correct working must be seen. |
### Part 5(b)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\int_3^9 \log_3(2x)^{10}\, dx = 10 \times \text{"13.3"} = \text{awrt } 133$ or e.g. $\frac{531}{4}$ | B1ft | FT on their 13.3; look for 3sf accuracy. A correct method must be seen (minimum $10 \times \text{"13.3"} = \text{"133"}$). Attempts to apply trapezium rule again score M0. |
### Part 5(b)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\int_3^9 \log_3 18x\, dx = \int_3^9 \log_3(9\times 2x)\, dx = \int_3^9 2 + \log_3 2x\, dx$ $= [2x]_3^9 + \int_3^9 \log_3 2x\, dx = 18 - 6 + \int_3^9 \log_3 2x\, dx = \ldots$ | M1 | Must reach $[2x]_3^9 + \int_3^9 \log_3 2x\, dx = \ldots$ with correct use of limits. May be implied by finding rectangle area as $2\times 6$. Just $12 + \text{"13.3"}$ scores M0. |
| $\text{Awrt } 25.3$ or $\frac{1011}{40}$ | A1ft | FT on their 13.3; allow $12 + \text{their answer to (a)}$ following correct work. Attempts to apply trapezium rule again score M0. |
---\begin{enumerate}
\item The table below shows corresponding values of $x$ and $y$ for $y = \log _ { 3 } 2 x$ 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
$y$ & 1.63 & 2 & 2.26 & 2.46 & 2.63 \\
\hline
\end{tabular}
\end{center}
(a) Using the trapezium rule with all the values of $y$ in the table, find an estimate for
$$\int _ { 3 } ^ { 9 } \log _ { 3 } 2 x \mathrm {~d} x$$
Using your answer to part (a) and making your method clear, estimate\\
(b) (i) $\int _ { 3 } ^ { 9 } \log _ { 3 } ( 2 x ) ^ { 10 } \mathrm {~d} x$\\
(ii) $\int _ { 3 } ^ { 9 } \log _ { 3 } 18 x \mathrm {~d} x$
\hfill \mbox{\textit{Edexcel Paper 2 2022 Q5 [6]}}