| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2022 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Equations & Modelling |
| Type | ln(y) vs x: find constants from two points |
| Difficulty | Moderate -0.3 This is a straightforward logarithmic transformation question requiring students to take ln of both sides of an exponential equation and identify gradient/intercept. Part (a) is routine algebraic manipulation (ln 4^(2x-a) = (2x-a)ln 4 = 2ln 4 · x - a ln 4, so gradient is 2ln 4 = ln 16). Part (b) requires using the y-intercept to find a. While it involves multiple steps, each step follows standard A-level techniques with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.06h Logarithmic graphs: reduce y=ax^n and y=kb^x to linear form |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State or imply equation is \(\ln y = (2x - a)\ln 4\) | B1 | OE. Do not condone poor use of brackets |
| State gradient is \(2\ln 4\) and confirm \(\ln 16\) | B1 | AG – necessary detail needed |
| Total | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Substitute for \(\ln y\) and attempt value of \(a\) | M1 | Allow if \(\ln y = 2x - a\ln 4\) |
| Obtain \(a = 15\) | A1 | Integer answer required, but condone 15.0 |
| Total | 2 |
## Question 1:
### Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| State or imply equation is $\ln y = (2x - a)\ln 4$ | B1 | OE. Do not condone poor use of brackets |
| State gradient is $2\ln 4$ and confirm $\ln 16$ | B1 | AG – necessary detail needed |
| **Total** | **2** | |
### Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Substitute for $\ln y$ and attempt value of $a$ | M1 | Allow if $\ln y = 2x - a\ln 4$ |
| Obtain $a = 15$ | A1 | Integer answer required, but condone 15.0 |
| **Total** | **2** | |
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\includegraphics[max width=\textwidth, alt={}, center]{ed12a4fb-e3bf-4d00-ad09-9ba5be941dd5-02_654_396_258_872}
The variables $x$ and $y$ satisfy the equation $y = 4 ^ { 2 x - a }$, where $a$ is an integer. As shown in the diagram, the graph of $\ln y$ against $x$ is a straight line passing through the point $( 0 , - 20.8 )$, where the second coordinate is given correct to 3 significant figures.
\begin{enumerate}[label=(\alph*)]
\item Show that the gradient of the straight line is $\ln 16$.
\item Determine the value of $a$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2022 Q1 [4]}}