| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2021 |
| Session | May |
| Marks | 8 |
| Topic | Implicit equations and differentiation |
| Type | Find normal equation at point |
| Difficulty | Standard +0.3 This is a standard implicit differentiation question with straightforward application of techniques. Part (a) requires routine use of the product rule and chain rule to differentiate implicitly, then algebraic rearrangement. Part (b) involves substituting a point to find the gradient, then finding the normal (negative reciprocal). While it requires careful algebra, it follows a well-practiced procedure with no novel insight needed, making it slightly easier than average. |
| Spec | 1.07m Tangents and normals: gradient and equations1.07s Parametric and implicit differentiation |
A curve has equation $x^3 - 3x^2y + y^2 + 1 = 0$.
\begin{enumerate}[label=(\alph*)]
\item Show that $\frac{dy}{dx} = \frac{6xy - 3x^2}{2y - 3x^2}$. [4]
\item Find the equation of the normal to the curve at the point $(1, 2)$. [4]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2021 Q5 [8]}}