| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2022 |
| Session | March |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Equations & Modelling |
| Type | ln(y) vs x: find constants from two points |
| Difficulty | Moderate -0.8 This is a straightforward application of logarithmic transformation to linearize an exponential relationship. Part (a) requires taking ln of both sides and identifying the gradient as ln(a), then exponentiating. Part (b) is simple substitution. Both parts are routine bookwork with no problem-solving insight required, making this easier than average. |
| Spec | 1.06g Equations with exponentials: solve a^x = b1.06h Logarithmic graphs: reduce y=ax^n and y=kb^x to linear form |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply equation is \(\ln y = \ln 3^{2a} + x\ln a\) | B1 | |
| Equate gradient of line involving \(a\) to 0.239 | M1 | |
| Obtain \(\ln a = 0.239\) and hence \(a = 1.27\) | A1 | |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Substitute \(y = 36\) in \(\ln y = ...\) equation and solve for \(x\) | M1 | Or substitute in original equation with necessary manipulation |
| Obtain 3.32 | A1 | |
| Total | 2 |
## Question 3(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply equation is $\ln y = \ln 3^{2a} + x\ln a$ | B1 | |
| Equate gradient of line involving $a$ to 0.239 | M1 | |
| Obtain $\ln a = 0.239$ and hence $a = 1.27$ | A1 | |
| **Total** | **3** | |
## Question 3(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Substitute $y = 36$ in $\ln y = ...$ equation and solve for $x$ | M1 | Or substitute in original equation with necessary manipulation |
| Obtain 3.32 | A1 | |
| **Total** | **2** | |3 The variables $x$ and $y$ satisfy the equation $y = 3 ^ { 2 a } a ^ { x }$, where $a$ is a constant. The graph of $\ln y$ against $x$ is a straight line with gradient 0.239 .
\begin{enumerate}[label=(\alph*)]
\item Find the value of $a$ correct to 3 significant figures.
\item Hence find the value of $x$ when $y = 36$. Give your answer correct to 3 significant figures.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2022 Q3 [5]}}