| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2010 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Laws of Logarithms |
| Type | Two unrelated log parts: one non-log algebraic part |
| Difficulty | Moderate -0.8 This question tests basic logarithm laws and solving a simple quadratic equation. Part (a) requires only direct recall of logarithm definitions (log_9(1)=0 and log_9(9^(1/2))=1/2). Part (b) applies standard log laws to get n²=18(n-4), yielding a straightforward quadratic. All steps are routine C2-level techniques with no problem-solving insight required. |
| Spec | 1.06c Logarithm definition: log_a(x) as inverse of a^x1.06f Laws of logarithms: addition, subtraction, power rules |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(x = 1\) | B1 | CAO |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(x = 3\) | B1 | CAO |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\log_a n^2 = \log_a 18(n-4)\) | M1 | A valid law of logs applied to correct logs |
| M1 | A second valid law of logs applied to correct logs | |
| \(n^2 - 18n + 72 = 0\) | A1 | ACF of these terms eg \(n^2 - 18n = -72\) |
| \((n-6)(n-12) = 0\) | m1 | Valid method to solve quadratic, dep on both previous Ms |
| \(n = 6,\ n = 12\) | A1 | Both values required. SC NMS max (out of 5) B3 for both 6 and 12 without uniqueness considered; max B1 for either 6 or 12 only |
## Question 3:
**Part (a)(i)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $x = 1$ | B1 | CAO |
**Part (a)(ii)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $x = 3$ | B1 | CAO |
**Part (b)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\log_a n^2 = \log_a 18(n-4)$ | M1 | A valid law of logs applied to correct logs |
| | M1 | A second valid law of logs applied to correct logs |
| $n^2 - 18n + 72 = 0$ | A1 | ACF of these terms eg $n^2 - 18n = -72$ |
| $(n-6)(n-12) = 0$ | m1 | Valid method to solve quadratic, dep on both previous Ms |
| $n = 6,\ n = 12$ | A1 | Both values required. **SC** NMS max (out of 5) B3 for both 6 and 12 without uniqueness considered; max B1 for either 6 or 12 only |
---3
\begin{enumerate}[label=(\alph*)]
\item Find the value of $x$ in each of the following:
\begin{enumerate}[label=(\roman*)]
\item $\quad \log _ { 9 } x = 0$;
\item $\quad \log _ { 9 } x = \frac { 1 } { 2 }$.
\end{enumerate}\item Given that
$$2 \log _ { a } n = \log _ { a } 18 + \log _ { a } ( n - 4 )$$
find the possible values of $n$.
\end{enumerate}
\hfill \mbox{\textit{AQA C2 2010 Q3 [7]}}