| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2002 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | Implicit equations and differentiation |
| Type | Find second derivative d²y/dx² |
| Difficulty | Standard +0.8 This is a multi-part implicit differentiation question requiring finding dy/dx from a cubic implicit equation, analyzing where the derivative equals zero, and computing the second derivative at a specific point. While implicit differentiation is standard Further Maths content, the second derivative calculation involves substantial algebraic manipulation and the proof in part (i) requires analyzing when 3x² + y² = 0 (which has no real solutions). This is moderately challenging but still a fairly standard FP1 exercise. |
| Spec | 1.07s Parametric and implicit differentiation |
6 A curve has equation $x ^ { 3 } + x y ^ { 2 } - y ^ { 3 } = 3$.\\
(i) Show that there is no point of the curve at which $\frac { d y } { d x } = 0$.\\
(ii) Find the values of $\frac { d y } { d x }$ and $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ at the point $( 1 , - 1 )$.
\hfill \mbox{\textit{CAIE FP1 2002 Q6 [9]}}