| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Equations & Modelling |
| Type | Exponential relation to line equation |
| Difficulty | Moderate -0.3 This is a straightforward logarithms question requiring standard techniques: taking logs of both sides to linearize an exponential relation, then solving simultaneous equations. Part (a) is routine manipulation with log laws, and part (b) is simple substitution and algebraic rearrangement. The question is slightly easier than average as it's highly structured with clear guidance and uses well-practiced A-level methods. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution1.06c Logarithm definition: log_a(x) as inverse of a^x1.06f Laws of logarithms: addition, subtraction, power rules |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State or imply \(y\log 2 = \log 3 - 2x\log 3\) | B1 | Accept \(y\ln 2 = (1-2x)\ln 3\) |
| State that the graph of \(y\) against \(x\) has an equation which is linear in \(x\) and \(y\), or is of the form \(ay = bx + c\) | B1 | Correct equation. Need a clear statement/comparison with matching linear form. |
| Clear indication that the gradient is \(-\dfrac{2\ln 3}{\ln 2}\) | B1 | Must be exact. Any equivalent e.g. \(-\dfrac{2\log_k 3}{\log_k 2}\), \(\log_2 \dfrac{1}{9}\) |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Substitute \(y = 3x\) in an equation involving logarithms and solve for \(x\) | M1 | |
| Obtain answer \(x = \dfrac{\ln 3}{\ln 72}\) | A1 | Allow M1A1 for the correct answer following decimals |
| 2 |
## Question 3:
### Part 3(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State or imply $y\log 2 = \log 3 - 2x\log 3$ | B1 | Accept $y\ln 2 = (1-2x)\ln 3$ |
| State that the graph of $y$ against $x$ has an equation which is linear in $x$ and $y$, or is of the form $ay = bx + c$ | B1 | Correct equation. Need a clear statement/comparison with matching linear form. |
| Clear indication that the gradient is $-\dfrac{2\ln 3}{\ln 2}$ | B1 | Must be exact. Any equivalent e.g. $-\dfrac{2\log_k 3}{\log_k 2}$, $\log_2 \dfrac{1}{9}$ |
| | **3** | |
### Part 3(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Substitute $y = 3x$ in an equation involving logarithms and solve for $x$ | M1 | |
| Obtain answer $x = \dfrac{\ln 3}{\ln 72}$ | A1 | Allow M1A1 for the correct answer following decimals |
| | **2** | |
---3 The variables $x$ and $y$ satisfy the relation $2 ^ { y } = 3 ^ { 1 - 2 x }$.
\begin{enumerate}[label=(\alph*)]
\item By taking logarithms, show that the graph of $y$ against $x$ is a straight line. State the exact value of the gradient of this line.
\item Find the exact $x$-coordinate of the point of intersection of this line with the line $y = 3 x$. Give your answer in the form $\frac { \ln a } { \ln b }$, where $a$ and $b$ are integers.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2020 Q3 [5]}}