| Exam Board | Edexcel |
|---|---|
| Module | C34 (Core Mathematics 3 & 4) |
| Session | Specimen |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Find range using calculus |
| Difficulty | Standard +0.3 Part (a) is algebraic manipulation of fractions with a common denominator (routine but slightly tedious). Part (b) is a standard quotient rule application. Part (c) requires finding the maximum using calculus (setting h'(x)=0, solving a quadratic, evaluating h at critical point and boundary). This is a multi-step question but uses entirely standard C3/C4 techniques with no novel insight required, making it slightly easier than average. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.07i Differentiate x^n: for rational n and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{2}{x+2}+\frac{4}{x^2+5}-\frac{18}{(x+2)(x^2+5)}=\frac{2(x^2+5)+4(x+2)-18}{(x+2)(x^2+5)}\) | M1 A1 | |
| \(=\frac{2x(x+2)}{(x+2)(x^2+5)}\) | M1 | |
| \(=\frac{2x}{(x^2+5)}\) | A1* |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(h'(x)=\frac{(x^2+5)\times2-2x\times2x}{(x^2+5)^2}\) | M1 A1 | |
| \(h'(x)=\frac{10-2x^2}{(x^2+5)^2}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Maximum when \(h'(x)=0 \Rightarrow 10-2x^2=0 \Rightarrow x=\ldots\) | M1 | |
| \(\Rightarrow x=\sqrt{5}\) | A1 | |
| When \(x=\sqrt{5} \Rightarrow h(x)=\frac{\sqrt{5}}{5}\) | M1 A1 | |
| Range of \(h(x)\) is \(0\leq h(x)\leq\frac{\sqrt{5}}{5}\) | A1ft |
# Question 8:
## Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{2}{x+2}+\frac{4}{x^2+5}-\frac{18}{(x+2)(x^2+5)}=\frac{2(x^2+5)+4(x+2)-18}{(x+2)(x^2+5)}$ | M1 A1 | |
| $=\frac{2x(x+2)}{(x+2)(x^2+5)}$ | M1 | |
| $=\frac{2x}{(x^2+5)}$ | A1* | |
## Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $h'(x)=\frac{(x^2+5)\times2-2x\times2x}{(x^2+5)^2}$ | M1 A1 | |
| $h'(x)=\frac{10-2x^2}{(x^2+5)^2}$ | A1 | |
## Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Maximum when $h'(x)=0 \Rightarrow 10-2x^2=0 \Rightarrow x=\ldots$ | M1 | |
| $\Rightarrow x=\sqrt{5}$ | A1 | |
| When $x=\sqrt{5} \Rightarrow h(x)=\frac{\sqrt{5}}{5}$ | M1 A1 | |
| Range of $h(x)$ is $0\leq h(x)\leq\frac{\sqrt{5}}{5}$ | A1ft | |8.
$$\mathrm { h } ( x ) = \frac { 2 } { x + 2 } + \frac { 4 } { x ^ { 2 } + 5 } - \frac { 18 } { \left( x ^ { 2 } + 5 \right) ( x + 2 ) } , \quad x \geqslant 0$$
\begin{enumerate}[label=(\alph*)]
\item Show that $\mathrm { h } ( x ) = \frac { 2 x } { x ^ { 2 } + 5 }$
\item Hence, or otherwise, find $\mathrm { h } ^ { \prime } ( x )$ in its simplest form.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{e375f6ad-4a76-42a0-b7bf-ae47e5cbdaeb-26_679_1168_733_390}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
Figure 2 shows a graph of the curve with equation $y = \mathrm { h } ( x )$.
\item Calculate the range of $\mathrm { h } ( x )$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C34 Q8 [12]}}