| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2002 |
| Session | Specimen |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Topic | Applied differentiation |
| Type | Tangent, normal and triangle area |
| Difficulty | Challenging +1.8 This AEA question requires deriving a general formula for triangle area using tangent/normal geometry, then solving a transcendental equation. Part (a) involves multiple coordinate geometry steps with algebraic manipulation of derivatives, while part (b) requires substituting into the formula and solving e^(5a) = (1/2)e^(10a)(26/5), leading to logarithmic manipulation. The geometric setup and algebraic complexity elevate this above standard A-level, though the techniques themselves are accessible. |
| Spec | 1.06b Gradient of e^(kx): derivative and exponential model1.07j Differentiate exponentials: e^(kx) and a^(kx)1.07m Tangents and normals: gradient and equations |
4.A curve $C$ has equation $y = \mathrm { f } ( x )$ with $\mathrm { f } ^ { \prime } ( x ) > 0$ .The $x$-coordinate of the point $P$ on the curve is $a$ .The tangent and the normal to $C$ are drawn at $P$ .The tangent cuts the $x$-axis at the point $A$ and the normal cuts the $x$-axis at the point $B$ .
\begin{enumerate}[label=(\alph*)]
\item Show that the area of $\triangle A P B$ is
$$\frac { 1 } { 2 } [ \mathrm { f } ( a ) ] ^ { 2 } \left( \frac { \left[ \mathrm { f } ^ { \prime } ( a ) \right] ^ { 2 } + 1 } { \mathrm { f } ^ { \prime } ( a ) } \right)$$
\item Given that $\mathrm { f } ( x ) = \mathrm { e } ^ { 5 x }$ and the area of $\triangle A P B$ is $\mathrm { e } ^ { 5 a }$ ,find and simplify the exact value of $a$ .
\end{enumerate}
\hfill \mbox{\textit{Edexcel AEA 2002 Q4 [12]}}