| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2020 |
| Session | Specimen |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Equations & Modelling |
| Type | Quadratic in exponential form |
| Difficulty | Moderate -0.3 Part (a) is a standard quadratic-in-exponential substitution (let u=5^x, solve u²+u=13/8) requiring routine algebraic manipulation and logarithms. Part (b) involves straightforward logarithm laws to rearrange and isolate y. Both parts are textbook exercises with no novel insight required, making this slightly easier than average A-level difficulty. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Recognise quadratic equation in \(5^x\) and attempt solution for \(5^x\) | M1 | |
| Obtain \(5^x = 3\) only | A1 | |
| Use logarithms to solve equation of the form \(5^x = k\) where \(k > 0\) | M1 | |
| Obtain 0.683 | A1 | |
| Total | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use \(2\ln x = \ln x^2\) | B1 | |
| Use law for addition or subtraction of logarithms | M1* | |
| Arrange equation not involving logarithms to the form \(y = \ldots\) | DM1 | |
| Obtain \(y = \dfrac{5}{x^2 - 1}\) | A1 | Or equivalent explicit \(y = \ldots\) form |
| Total | 4 |
## Question 4:
**Part 4(a):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Recognise quadratic equation in $5^x$ and attempt solution for $5^x$ | M1 | |
| Obtain $5^x = 3$ only | A1 | |
| Use logarithms to solve equation of the form $5^x = k$ where $k > 0$ | M1 | |
| Obtain 0.683 | A1 | |
| **Total** | **4** | |
**Part 4(b):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use $2\ln x = \ln x^2$ | B1 | |
| Use law for addition or subtraction of logarithms | M1* | |
| Arrange equation not involving logarithms to the form $y = \ldots$ | DM1 | |
| Obtain $y = \dfrac{5}{x^2 - 1}$ | A1 | Or equivalent explicit $y = \ldots$ form |
| **Total** | **4** | |4 (a) Solve the equation $5 ^ { 2 x } + 5 ^ { x } = 12$, giving your answer correct to 3 significant figures.\\
(b) It is given that $\ln ( y + 5 ) - \ln y = 2 \ln x$.
Express $y$ in terms of $x$, in a form not involving logarithms.\\
\hfill \mbox{\textit{CAIE P2 2020 Q4 [8]}}