| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2018 |
| Session | November |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Implicit equations and differentiation |
| Type | Find second derivative d²y/dx² |
| Difficulty | Challenging +1.8 This is a sophisticated Further Maths question requiring implicit differentiation to find higher derivatives, then integration by parts to establish and apply a recurrence relation. While the individual techniques are standard, the multi-layered structure (finding successive derivatives at a point, proving a recurrence for integrals involving these derivatives, then using it iteratively) requires sustained reasoning across 5 connected parts, making it substantially harder than typical A-level questions but not exceptionally difficult for Further Maths. |
| Spec | 1.07s Parametric and implicit differentiation |
The curve $C$ has equation
$$x ^ { 2 } + 2 x y = y ^ { 3 } - 2$$
(i) Show that $A ( - 1,1 )$ is the only point on $C$ with $x$-coordinate equal to - 1 .\\
For $n \geqslant 1$, let $A _ { n }$ denote the value of $\frac { \mathrm { d } ^ { n } y } { \mathrm {~d} x ^ { n } }$ at the point $A ( - 1,1 )$.\\
(ii) Show that $A _ { 1 } = 0$.\\
(iii) Show that $A _ { 2 } = \frac { 2 } { 5 }$.\\
Let $I _ { n } = \int _ { - 1 } ^ { 0 } x ^ { n } \frac { \mathrm {~d} ^ { n } y } { \mathrm {~d} x ^ { n } } \mathrm {~d} x$.\\
(iv) Show that for $n \geqslant 2$,
$$I _ { n } = ( - 1 ) ^ { n + 1 } A _ { n - 1 } - n I _ { n - 1 } .$$
(v) Deduce the value of $I _ { 3 }$ in terms of $I _ { 1 }$.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.\\
\hfill \mbox{\textit{CAIE FP1 2018 Q11 OR}}