| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2017 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear regression |
| Type | Linearize non-linear relationships |
| Difficulty | Standard +0.3 This is a standard linearization problem requiring students to recognize that ln y = ln C + x ln a gives a linear relationship, plot given points, draw a line of best fit, and extract constants from gradient and intercept. It's slightly easier than average as the data is already transformed and the algebraic manipulation is straightforward, though it does require understanding of logarithms and linear regression concepts. |
| 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 |
|---|---|---|
| Answer | Mark | Guidance |
| Plot the four points and draw straight line | B1 | |
| State or imply that \(\ln y = \ln C + x \ln a\) | B1 | |
| Carry out a completely correct method for finding \(\ln C\) or \(\ln a\) | M1 | |
| Obtain answer \(C = 3.7\) | A1 | |
| Obtain answer \(a = 1.5\) | A1 |
## Question 2:
| Answer | Mark | Guidance |
|--------|------|----------|
| Plot the four points and draw straight line | B1 | |
| State or imply that $\ln y = \ln C + x \ln a$ | B1 | |
| Carry out a completely correct method for finding $\ln C$ or $\ln a$ | M1 | |
| Obtain answer $C = 3.7$ | A1 | |
| Obtain answer $a = 1.5$ | A1 | |
---2 Two variable quantities $x$ and $y$ are believed to satisfy an equation of the form $y = C \left( a ^ { x } \right)$, 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.\\
\includegraphics[max width=\textwidth, alt={}, center]{21878d10-7f16-4dbb-86ef-65a7ba5eeafb-03_759_944_749_596}\\
\hfill \mbox{\textit{CAIE P3 2017 Q2 [5]}}