| Exam Board | Edexcel |
|---|---|
| Module | C34 (Core Mathematics 3 & 4) |
| Year | 2018 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Find range using calculus |
| Difficulty | Standard +0.3 This is a straightforward quotient rule differentiation followed by standard optimization. Part (a) is routine algebraic manipulation to reach a given form. Part (b) requires finding the maximum using the derivative (standard technique). Part (c) tests understanding of inverse functions (one-to-one requirement). All techniques are standard C3/C4 material with no novel insight required, making it slightly easier than average. |
| Spec | 1.02u Functions: definition and vocabulary (domain, range, mapping)1.07n Stationary points: find maxima, minima using derivatives1.07q Product and quotient rules: differentiation |
**Question 14 (c)**
**Mark Scheme:**
M1: $f$ is many to one (or 2 values in domain of $f$ map to one in the range)
M1: $f$ is not one to one
M1: $f^{-1}$ would be one to many
M1: the inverse would be one to many
M1: it would be one to many
M1: it is not one to one
M1: the graph illustrates a many to one function
**Do NOT allow:**
- it is many to one
- You can't reflect in $y = x$
**Guidance:**
Any reference to "it" we must assume refers to the inverse because of the wording in the question14. Given that
$$y = \frac { \left( x ^ { 2 } - 4 \right) ^ { \frac { 1 } { 2 } } } { x ^ { 3 } } \quad x > 2$$
\begin{enumerate}[label=(\alph*)]
\item show that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { A x ^ { 2 } + 12 } { x ^ { 4 } \left( x ^ { 2 } - 4 \right) ^ { \frac { 1 } { 2 } } } \quad x > 2$$
where $A$ is a constant to be found.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{a377da06-a968-438c-bec2-ae55283dae47-48_593_1134_865_395}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{center}
\end{figure}
Figure 4 shows a sketch of part of the curve with equation $y = \mathrm { f } ( x )$ where
$$\mathrm { f } ( x ) = \frac { 24 \left( x ^ { 2 } - 4 \right) ^ { \frac { 1 } { 2 } } } { x ^ { 3 } } \quad x > 2$$
\item Use your answer to part (a) to find the range of f.
\item State a reason why f-1 does not exist.
\begin{center}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel C34 2018 Q14 [12]}}