| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2016 |
| Session | March |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Implicit equations and differentiation |
| Type | Find stationary points |
| Difficulty | Standard +0.3 This is a straightforward implicit differentiation question requiring standard technique to find dy/dx, a simple inequality argument about signs, and solving simultaneous equations. While it involves multiple steps, each component is routine for A-level—slightly above average difficulty only due to the implicit differentiation and algebraic manipulation required. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State \(3y^2\frac{dy}{dx}\) as derivative of \(y^3\) | B1 | |
| Equate derivative of left-hand side to zero and solve for \(\frac{dy}{dx}\) | M1 | |
| Obtain \(\frac{dy}{dx} = \frac{6x^2}{-3y^2}\) or equivalent | A1 | |
| Observe \(x^2\) and \(y^2\) never negative and conclude appropriately | A1 | [4] |
| (ii) Equate first derivative to \(-2\) and rearrange to \(y^2 = x^2\) or equivalent | B1 | |
| Substitute in original equation to obtain at least one equation in \(x^3\) or \(y^3\) | M1 | |
| Obtain \(3x^3 = 24\) or \(x^3 = 24\) or \(3y^3 = 24\) or \(-y^3 = 24\) | A1 | |
| Obtain \((2,2)\) | A1 | |
| Obtain \((\sqrt[3]{24}, -\sqrt[3]{24})\) or \((2.88, -2.88)\) and no others | A1 | [5] |
(i) State $3y^2\frac{dy}{dx}$ as derivative of $y^3$ | B1 |
Equate derivative of left-hand side to zero and solve for $\frac{dy}{dx}$ | M1 |
Obtain $\frac{dy}{dx} = \frac{6x^2}{-3y^2}$ or equivalent | A1 |
Observe $x^2$ and $y^2$ never negative and conclude appropriately | A1 | [4]
(ii) Equate first derivative to $-2$ and rearrange to $y^2 = x^2$ or equivalent | B1 |
Substitute in original equation to obtain at least one equation in $x^3$ or $y^3$ | M1 |
Obtain $3x^3 = 24$ or $x^3 = 24$ or $3y^3 = 24$ or $-y^3 = 24$ | A1 |
Obtain $(2,2)$ | A1 |
Obtain $(\sqrt[3]{24}, -\sqrt[3]{24})$ or $(2.88, -2.88)$ and no others | A1 | [5]7 The equation of a curve is $2 x ^ { 3 } + y ^ { 3 } = 24$.\\
(i) Express $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$, and show that the gradient of the curve is never positive.\\
(ii) Find the coordinates of the two points on the curve at which the gradient is - 2 .
\hfill \mbox{\textit{CAIE P2 2016 Q7 [9]}}