| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2021 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Laws of Logarithms |
| Type | Solve log equation then substitute trig/exponential expression |
| Difficulty | Moderate -0.3 Part (a) requires standard logarithm laws (quotient and power rules) to reach a simple linear equation. Part (b) is a direct substitution x = cot y, then solving for y using inverse trigonometry. Both parts are routine applications of A-level techniques with no problem-solving insight required, making this slightly easier than average. |
| Spec | 1.05o Trigonometric equations: solve in given intervals1.06f Laws of logarithms: addition, subtraction, power rules |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use at least one relevant logarithm property correctly | M1 | |
| Obtain correct equation \(\frac{2+x}{x} = 9\) or equivalent with no logarithms | A1 | |
| Obtain \(x = \frac{1}{4}\) | A1 | |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Attempt value of \(y\) from \(\tan y = 1 \div (\text{their } \textbf{(a)})\) | M1 | May be implied by an answer in degrees (76.0) |
| Obtain \(1.326\) | A1 | AWRT; and no other answers in the range |
| Total: 2 |
## Question 1:
### Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use at least one relevant logarithm property correctly | M1 | |
| Obtain correct equation $\frac{2+x}{x} = 9$ or equivalent with no logarithms | A1 | |
| Obtain $x = \frac{1}{4}$ | A1 | |
| **Total: 3** | | |
### Part (b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Attempt value of $y$ from $\tan y = 1 \div (\text{their } \textbf{(a)})$ | M1 | May be implied by an answer in degrees (76.0) |
| Obtain $1.326$ | A1 | AWRT; and no other answers in the range |
| **Total: 2** | | |1
\begin{enumerate}[label=(\alph*)]
\item Solve the equation $\ln ( 2 + x ) - \ln x = 2 \ln 3$.
\item Hence solve the equation $\ln ( 2 + \cot y ) - \ln ( \cot y ) = 2 \ln 3$ for $0 < y < \frac { 1 } { 2 } \pi$. Give your answer correct to 4 significant figures.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2021 Q1 [5]}}