| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2021 |
| Session | September |
| Marks | 10 |
| Topic | Chain Rule |
| Type | Equation of normal line |
| Difficulty | Moderate -0.3 This is a straightforward multi-part differentiation question requiring algebraic simplification, basic power rule differentiation, and finding a normal line. Part (a) is routine algebra, (b)(i) applies the power rule directly, (b)(ii) is standard tangent/normal work, and (b)(iii) involves solving dy/dx=0 which leads to a simple equation. All techniques are standard with no novel insight required, making it slightly easier than average. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations1.07n Stationary points: find maxima, minima using derivatives |
5. A curve has the equation
$$y = \frac { 12 + x ^ { 2 } \sqrt { x } } { x } , \quad x > 0$$
\begin{enumerate}[label=(\alph*)]
\item Express $\frac { 12 + x ^ { 2 } \sqrt { x } } { x }$ in the form $12 x ^ { p } + x ^ { q }$.
\item \begin{enumerate}[label=(\roman*)]
\item Hence find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
\item Find an equation of the normal to the curve at the point on the curve where $x = 4$.
\item The curve has a stationary point $P$. Show that the $x$-coordinate of $P$ can be written in the form $2 ^ { k }$, where $k$ is a rational number.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2021 Q5 [10]}}