| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2010 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Implicit equations and differentiation |
| Type | Find tangent equation at point |
| Difficulty | Standard +0.3 This is a straightforward implicit differentiation question requiring application of the product rule and chain rule, followed by algebraic rearrangement and substitution into the tangent equation formula. While it involves multiple steps, each is standard technique with no novel insight required, making it slightly easier than average. |
| Spec | 1.07m Tangents and normals: gradient and equations1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State \(2xy + x^2 \frac{dy}{dx}\) as derivative of \(x^2y\) | B1 | |
| State \(2y \frac{dy}{dx}\) as derivative of \(y^2\) | B1 | |
| Equate derivatives of LHS and RHS, and solve for \(\frac{dy}{dx}\) | M1 | |
| Obtain given answer | A1 | [4] |
| (ii) Substitute and obtain gradient \(\frac{2}{3}\), or equivalent | B1 | |
| Form equation of tangent at the given point \((1, 2)\) | M1 | |
| Obtain answer \(2x - 5y + 8 = 0\), or equivalent | A1 | [3] |
**(i)** State $2xy + x^2 \frac{dy}{dx}$ as derivative of $x^2y$ | B1 |
State $2y \frac{dy}{dx}$ as derivative of $y^2$ | B1 |
Equate derivatives of LHS and RHS, and solve for $\frac{dy}{dx}$ | M1 |
Obtain given answer | A1 | [4]
**(ii)** Substitute and obtain gradient $\frac{2}{3}$, or equivalent | B1 |
Form equation of tangent at the given point $(1, 2)$ | M1 |
Obtain answer $2x - 5y + 8 = 0$, or equivalent | A1 | [3]
[The M1 is dependent on at least one of the B marks being obtained.]6 The equation of a curve is
$$x ^ { 2 } y + y ^ { 2 } = 6 x$$
(i) Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 - 2 x y } { x ^ { 2 } + 2 y }$.\\
(ii) Find the equation of the tangent to the curve at the point with coordinates ( 1,2 ), giving your answer in the form $a x + b y + c = 0$.
\hfill \mbox{\textit{CAIE P2 2010 Q6 [7]}}