| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2019 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Topic | First order differential equations (integrating factor) |
| Type | Separable variables |
| Difficulty | Challenging +1.8 This AEA question requires students to (a) verify algebraic properties of the golden ratio through manipulation, then (b) find where two curves touch by solving simultaneous equations with the tangency condition (equal derivatives). The golden ratio algebra is routine verification, but part (b) requires recognizing that 'touch' means both equal function values AND equal derivatives, leading to a system that yields x = φ. The connection between parts and the non-standard answer form elevates this above typical A-level but doesn't require deep novel insight. |
| Spec | 1.02b Surds: manipulation and rationalising denominators1.06b Gradient of e^(kx): derivative and exponential model1.07m Tangents and normals: gradient and equations1.07n Stationary points: find maxima, minima using derivatives |
3.Given that $\phi = \frac { 1 } { 2 } ( \sqrt { 5 } + 1 )$ ,
\begin{enumerate}[label=(\alph*)]
\item show that
\begin{enumerate}[label=(\roman*)]
\item $\phi ^ { 2 } = \phi + 1$
\item $\frac { 1 } { \phi } = \phi - 1$
\end{enumerate}\item The equations of two curves are
$$\begin{array} { r l r l }
y & = \frac { 1 } { x } & x > 0 \\
\text { and } & y & = \ln x - x + k & x > 0
\end{array}$$
where $k$ is a positive constant.\\
The curves touch at the point $P$ .\\
Find in terms of $\phi$
\begin{enumerate}[label=(\roman*)]
\item the coordinates of $P$ ,
\item the value of $k$ .
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{Edexcel AEA 2019 Q3 [11]}}