| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2023 |
| Session | November |
| Marks | 12 |
| Topic | Chain Rule |
| Type | Equation of normal line |
| Difficulty | Moderate -0.8 This is a straightforward differentiation and tangent/normal question requiring standard techniques: differentiating powers of x (including fractional and negative powers), verifying a point lies on a curve by substitution, finding the gradient at a point, and using the perpendicular gradient property for normals. All steps are routine A-level procedures with no problem-solving insight required, making it easier than average but not trivial due to the algebraic manipulation involved. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations |
3. The curve $C$ has equation
$$y = \frac { 1 } { 2 } x ^ { 3 } - 9 x ^ { \frac { 3 } { 2 } } + \frac { 8 } { x } + 30 , \quad x > 0$$
\begin{enumerate}[label=(\alph*)]
\item Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
\item Show that the point $P ( 4 , - 8 )$ lies on $C$.
\item Find an equation of the normal to $C$ at the point $P$, giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.\\[0pt]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2023 Q3 [12]}}