| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2017 |
| Session | November |
| 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 standard linearization problem requiring students to plot given points, draw a line of best fit, and extract constants from gradient and intercept. The transformation ln y = ln C + x ln a is routine A-level material, and all computational steps (reading gradient/intercept from graph, exponentiating) are straightforward with no conceptual challenges beyond recognizing the linear relationship. |
| Spec | 1.06h Logarithmic graphs: reduce y=ax^n and y=kb^x to linear form |
| \(x\) | 0.9 | 1.6 | 2.4 | 3.2 |
| \(\ln y\) | 1.7 | 1.9 | 2.3 | 2.6 |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | Plot the four points and draw straight line | B1 |
| State or imply that lny=lnC+xlna | B1 | |
| Carry out a completely correct method for finding lnCor lna | M1 | |
| Obtain answer C = 3.7 | A1 | |
| Obtain answer a = 1.5 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 2:
2 | Plot the four points and draw straight line | B1
State or imply that lny=lnC+xlna | B1
Carry out a completely correct method for finding lnCor lna | M1
Obtain answer C = 3.7 | A1
Obtain answer a = 1.5 | A1
5
Question | Answer | MarksTwo variable quantities $x$ and $y$ are believed to satisfy an equation of the form $y = C(a^x)$, where $C$ and $a$ are constants. An experiment produced four pairs of values of $x$ and $y$. The table below gives the corresponding values of $x$ and $\ln y$.
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
$x$ & 0.9 & 1.6 & 2.4 & 3.2 \\
\hline
$\ln y$ & 1.7 & 1.9 & 2.3 & 2.6 \\
\hline
\end{tabular}
\end{center}
By plotting $\ln y$ against $x$ for these four pairs of values and drawing a suitable straight line, estimate the values of $C$ and $a$. Give your answers correct to 2 significant figures. [5]
\includegraphics{figure_2}
\hfill \mbox{\textit{CAIE P3 2017 Q2 [5]}}