| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2018 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differential equations |
| Type | Implicit differentiation for d²y/dx² |
| Difficulty | Standard +0.8 This Further Maths question requires implicit differentiation to find dy/dx from a cubic equation, then a second implicit differentiation to find d²y/dx². The algebraic manipulation is non-trivial, especially differentiating the cubic term and solving for d²y/dx², but follows standard techniques without requiring novel insight. |
| Spec | 1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| \(\left(0+\frac{dy}{dx}\right)^3 = (-1)^2 + 0 \Rightarrow \frac{dy}{dx} = 1\) | B1 | — |
| Answer | Marks | Guidance |
|---|---|---|
| \(3\left(x+\frac{dy}{dx}\right)^2\left(1+\frac{d^2y}{dx^2}\right)\) | M1 A1 | Method mark for good attempt at implicit differentiation of LHS |
| \(= 2y\frac{dy}{dx}+1\) | B1 | Note: may expand bracket before differentiation but M1 is still for implicit differentiation |
| \(\Rightarrow 3\left(1+\frac{d^2y}{dx^2}\right) = -2+1 \Rightarrow \frac{d^2y}{dx^2} = -\frac{4}{3}\) | A1 | — |
**Question 1:**
**Part (i):**
$\left(0+\frac{dy}{dx}\right)^3 = (-1)^2 + 0 \Rightarrow \frac{dy}{dx} = 1$ | **B1** | —
**Part (ii):**
$3\left(x+\frac{dy}{dx}\right)^2\left(1+\frac{d^2y}{dx^2}\right)$ | **M1 A1** | Method mark for good attempt at implicit differentiation of LHS
$= 2y\frac{dy}{dx}+1$ | **B1** | Note: may expand bracket before differentiation but M1 is still for implicit differentiation
$\Rightarrow 3\left(1+\frac{d^2y}{dx^2}\right) = -2+1 \Rightarrow \frac{d^2y}{dx^2} = -\frac{4}{3}$ | **A1** | —
**Total: 5**
---1 The variables $x$ and $y$ are such that $y = - 1$ when $x = 0$ and
$$\left( x + \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 3 } = y ^ { 2 } + x$$
(i) Find the value of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ when $x = 0$.\\
(ii) Find also the value of $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ when $x = 0$.\\
\hfill \mbox{\textit{CAIE FP1 2018 Q1 [5]}}