| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2018 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Laws of Logarithms |
| Type | Solve log equation then substitute trig/exponential expression |
| Difficulty | Standard +0.3 Part (i) is a standard logarithm equation requiring application of log laws (power rule, subtraction rule) to form a quadratic, which is routine A-level material. Part (ii) adds a substitution step (recognizing 2^u = x) but is straightforward once part (i) is solved. The question requires multiple techniques but follows predictable patterns with no novel insight needed, making it slightly easier than average. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use \(2\ln(2x) = \ln(4x^2)\) | B1 | |
| Use law for addition or subtraction of logarithms | M1 | |
| Obtain correct equation \(\frac{4x^2}{x+3} = 16\) or equivalent | A1 | With no logarithms involved |
| Solve 3-term quadratic equation | M1 | Dependent on previous M1 |
| Conclude with \(x = 6\) and, finally, no other solutions | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Apply logarithms and use power law for \(2^a = k\) or \(2^{a+1} = 2k\) where \(k > 0\) | M1 | |
| Obtain 2.585 | A1 |
## Question 4(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use $2\ln(2x) = \ln(4x^2)$ | B1 | |
| Use law for addition or subtraction of logarithms | M1 | |
| Obtain correct equation $\frac{4x^2}{x+3} = 16$ or equivalent | A1 | With no logarithms involved |
| Solve 3-term quadratic equation | M1 | Dependent on previous M1 |
| Conclude with $x = 6$ and, finally, no other solutions | A1 | |
## Question 4(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Apply logarithms and use power law for $2^a = k$ or $2^{a+1} = 2k$ where $k > 0$ | M1 | |
| Obtain 2.585 | A1 | |4 (i) Solve the equation $2 \ln ( 2 x ) - \ln ( x + 3 ) = 4 \ln 2$.\\
(ii) Hence solve the equation
$$2 \ln \left( 2 ^ { u + 1 } \right) - \ln \left( 2 ^ { u } + 3 \right) = 4 \ln 2$$
giving the value of $u$ correct to 4 significant figures.\\
\hfill \mbox{\textit{CAIE P2 2018 Q4 [7]}}