| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2015 |
| Session | June |
| Marks | 8 |
| Topic | Implicit equations and differentiation |
| Type | Find stationary points |
| Difficulty | Standard +0.8 This requires implicit differentiation to find dy/dx, setting it to zero for horizontal tangents, solving a cubic equation, then finding corresponding y-values and writing tangent equations. The cubic factorization and handling multiple solutions elevates this above routine implicit differentiation exercises, but the techniques are all standard A-level methods applied systematically. |
| Spec | 1.07m Tangents and normals: gradient and equations1.07s Parametric and implicit differentiation |
**Question 9:**
$$2y\frac{dy}{dx} = 4x^3-12x^2$$
$$4x^2(x-3)=0$$
$x=0$ or $x=3$
$x=0 \Rightarrow y^2=36 \Rightarrow y=\pm 6$
$x=3 \Rightarrow y^2=9 \Rightarrow y=\pm 3$
Hence equations are $y=3,\ y=-3,\ y=6,\ y=-6$
- **M1**: Differentiate implicitly to get at least LHS
- **A1**: Obtain fully correct expression
- **B1**: Use $\frac{dy}{dx}=0$
- **M1**: Attempt to solve for $x$
- **A1**: Obtain $x=0,3$, www
- **M1**: Attempt to find $y$, must include square rooting
- **A1**: Obtain at least two correct equations, www
- **A1**: Obtain all four correct equations, and no others (A1 A0 if final equations given as $y=\pm 3,\ y=\pm 6$)
Notes: Misreading $y$ for $y^2$ gets M0A0B1M1A1M1A1A0. Using $y=\sqrt{x^4-4x^3+36}$ can get full marks.
**Total: [8]**9 Find the equations of all the horizontal tangents to the curve with equation $y ^ { 2 } = x ^ { 4 } - 4 x ^ { 3 } + 36$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2015 Q9 [8]}}