| Exam Board | OCR MEI |
|---|---|
| Module | Paper 2 (Paper 2) |
| Year | 2020 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find composite function expression |
| Difficulty | Moderate -0.3 This is a straightforward composite function question requiring standard techniques: finding fg(x) by substitution, evaluating at a point, and using the inverse function derivative formula. The domain consideration is simple since g(x) > 0 for x > 2. Part (c) uses the standard result that the gradient of the inverse at a point equals the reciprocal of the original function's gradient, which is a bookwork application. All parts are routine A-level procedures with no novel problem-solving required. |
| Spec | 1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \([\text{h}(x)\) or \(\text{fg}(x) =]\ \sqrt{x^3 - x - 6}\) oe | B1 (1.1) | expression; mark the final answer |
| \(x > 2\) | B1 (1.1) | domain |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\sqrt{18}\) oe isw FT *their* \(h(x)\) | B1 (1.1) | allow \(4.2426406872\ldots\) rounded to 2 or more sf |
| [1] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{1}{2} \times \frac{3x^2-1}{\sqrt{(x^3-x-6)}}\) or \(\frac{3x^2-1}{2h(x)}\) oe | M1 (3.1a) | chain rule used; allow one slip in differentiation, e.g. sign error |
| (all correct) | A1 (1.1) | \(h(x)\) must be correct for first M1 |
| *their* \(\frac{dh}{dx}\) evaluated at \(x = 3\) | M1 (1.1) | |
| \(\frac{3\sqrt{2}}{13}\) or \(0.326356975932\) rounded to 2 sf or better | A1 (3.2a) | |
| [4] | ||
| *OR* | ||
| \(x^2 = y^3 - y - 6 \Rightarrow 2x\frac{dx}{dy} = 3y^2 - 1\) oe | M1 | allow one slip e.g. sign error |
| \(\frac{dx}{dy} = \frac{3y^2-1}{2x}\) or \(\frac{dy}{dx} = \frac{2x}{3y^2-1}\) | A1 | rearrangement to find \(h^{-1}(x)\) explicitly in terms of \(x\) followed by differentiation does not score |
| substitution of \(y = 3\) and \(x = \text{their}\ \sqrt{18}\) | M1 | |
| \(\frac{3\sqrt{2}}{13}\) or \(0.326356975932\) rounded to 2 sf or better | A1 | |
| [4] |
# Question 15:
## Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $[\text{h}(x)$ or $\text{fg}(x) =]\ \sqrt{x^3 - x - 6}$ oe | B1 (1.1) | expression; mark the final answer |
| $x > 2$ | B1 (1.1) | domain |
| **[2]** | | |
## Part (b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\sqrt{18}$ oe isw FT *their* $h(x)$ | B1 (1.1) | allow $4.2426406872\ldots$ rounded to 2 or more sf |
| **[1]** | | |
## Part (c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{1}{2} \times \frac{3x^2-1}{\sqrt{(x^3-x-6)}}$ or $\frac{3x^2-1}{2h(x)}$ oe | M1 (3.1a) | chain rule used; allow one slip in differentiation, e.g. sign error |
| (all correct) | A1 (1.1) | $h(x)$ must be correct for first M1 |
| *their* $\frac{dh}{dx}$ evaluated at $x = 3$ | M1 (1.1) | |
| $\frac{3\sqrt{2}}{13}$ or $0.326356975932$ rounded to 2 sf or better | A1 (3.2a) | |
| **[4]** | | |
| *OR* | | |
| $x^2 = y^3 - y - 6 \Rightarrow 2x\frac{dx}{dy} = 3y^2 - 1$ oe | M1 | allow one slip e.g. sign error |
| $\frac{dx}{dy} = \frac{3y^2-1}{2x}$ or $\frac{dy}{dx} = \frac{2x}{3y^2-1}$ | A1 | rearrangement to find $h^{-1}(x)$ explicitly in terms of $x$ followed by differentiation does not score |
| substitution of $y = 3$ and $x = \text{their}\ \sqrt{18}$ | M1 | |
| $\frac{3\sqrt{2}}{13}$ or $0.326356975932$ rounded to 2 sf or better | A1 | |
| **[4]** | | |15 Functions $\mathrm { f } ( x )$ and $\mathrm { g } ( x )$ are defined as follows.\\
$\mathrm { f } ( x ) = \sqrt { x }$ for $x > 0$ and $\mathrm { g } ( x ) = x ^ { 3 } - x - 6$ for $x > 2$.
The function $\mathrm { h } ( x )$ is defined as\\
$\mathrm { h } ( x ) = \mathrm { fg } ( x )$.
\begin{enumerate}[label=(\alph*)]
\item Find $\mathrm { h } ( x )$ in terms of $x$ and state its domain.
\item Find $\mathrm { h } ( 3 )$.
Fig. 15 shows $\mathrm { h } ( x )$ and $\mathrm { h } ^ { - 1 } ( x )$, together with the straight line $y = x$.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{cea67565-8074-4703-8e1a-09b98e380baf-17_780_796_895_242}
\captionsetup{labelformat=empty}
\caption{Fig. 15}
\end{center}
\end{figure}
\item Determine the gradient of $\mathrm { y } = \mathrm { h } ^ { - 1 } ( \mathrm { x } )$ at the point where $y = 3$.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Paper 2 2020 Q15 [7]}}