| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2014 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Equations & Modelling |
| Type | Logarithmic equation solving |
| Difficulty | Moderate -0.8 Part (a) requires standard logarithm laws (combining logs and exponentiating) to solve a straightforward equation. Part (b) is a routine application of taking logs of both sides of an inequality. Both are textbook exercises testing basic logarithm manipulation with no problem-solving insight required, making this easier than average. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks | Guidance |
|---|---|---|
| Use power law to produce \(\ln(x-4)^2\) | B1 | |
| Apply logarithm laws to produce equation without logarithms | M1 | |
| Obtain \((x-4)^2 = 2x\) or equivalent | A1 | |
| Solve 3-term quadratic equation | DM1 | |
| Obtain (finally) \(x = 8\) only | A1 | [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Apply logarithms and use power law (once) | M1 | |
| Obtain \(\frac{\ln 10^{10}}{\ln 1.4}\) or equivalent as part of inequality or equation | A1 | |
| Conclude with single integer 69 | A1 | [3] |
**(a)**
Use power law to produce $\ln(x-4)^2$ | B1 |
Apply logarithm laws to produce equation without logarithms | M1 |
Obtain $(x-4)^2 = 2x$ or equivalent | A1 |
Solve 3-term quadratic equation | DM1 |
Obtain (finally) $x = 8$ only | A1 | [5]
**(b)**
Apply logarithms and use power law (once) | M1 |
Obtain $\frac{\ln 10^{10}}{\ln 1.4}$ or equivalent as part of inequality or equation | A1 |
Conclude with single integer 69 | A1 | [3]4
\begin{enumerate}[label=(\alph*)]
\item Find the value of $x$ satisfying the equation $2 \ln ( x - 4 ) - \ln x = \ln 2$.
\item Use logarithms to find the smallest integer satisfying the inequality
$$1.4 ^ { y } > 10 ^ { 10 }$$
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2014 Q4 [8]}}