| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2006 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Verify composite identity |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question testing basic composite function mechanics, routine differentiation, and simple algebraic manipulation. Part (a) requires substituting one function into another with basic exponential/logarithm properties (e^ln2 = 2). Parts (b)-(d) are standard bookwork: sketching an exponential, stating its range, and solving a simple equation involving the derivative. All steps are routine C3 techniques with no problem-solving insight required. |
| Spec | 1.02v Inverse and composite functions: graphs and conditions for existence1.06a Exponential function: a^x and e^x graphs and properties1.07j Differentiate exponentials: e^(kx) and a^(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(gf(x) = e^{2(2x+\ln 2)}\) | M1 | |
| \(= e^{4x}e^{2\ln 2}\) | M1 | |
| \(= e^{4x}e^{\ln 4}\) | M1 | |
| \(= 4e^{4x}\) | A1 | Give mark at this point, cso |
| (Hence \(gf: x \mapsto 4e^{4x}, \quad x \in \mathbb{R}\)) | (4) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Correct exponential shape passing through \((0, 4)\) | B1 | Shape and point (1) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Range is \(\mathbb{R}_+\) | B1 | Accept \(gf(x) > 0\), \(y > 0\) (1) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{d}{dx}[gf(x)] = 16e^{4x}\) | M1 A1 | |
| \(e^{4x} = \frac{3}{16}\) | M1 | |
| \(4x = \ln\frac{3}{16}\) | ||
| \(x \approx -0.418\) | A1 | (4) |
| [10] |
## Question 8:
**(a)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $gf(x) = e^{2(2x+\ln 2)}$ | M1 | |
| $= e^{4x}e^{2\ln 2}$ | M1 | |
| $= e^{4x}e^{\ln 4}$ | M1 | |
| $= 4e^{4x}$ | A1 | Give mark at this point, cso |
| (Hence $gf: x \mapsto 4e^{4x}, \quad x \in \mathbb{R}$) | | **(4)** |
**(b)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| Correct exponential shape passing through $(0, 4)$ | B1 | Shape and point **(1)** |
**(c)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| Range is $\mathbb{R}_+$ | B1 | Accept $gf(x) > 0$, $y > 0$ **(1)** |
**(d)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{d}{dx}[gf(x)] = 16e^{4x}$ | M1 A1 | |
| $e^{4x} = \frac{3}{16}$ | M1 | |
| $4x = \ln\frac{3}{16}$ | | |
| $x \approx -0.418$ | A1 | **(4)** |
| | | **[10]** |8. The functions $f$ and $g$ are defined by
$$\begin{array} { l l }
\mathrm { f } : x \rightarrow 2 x + \ln 2 , & x \in \mathbb { R } , \\
\mathrm {~g} : x \rightarrow \mathrm { e } ^ { 2 x } , & x \in \mathbb { R } .
\end{array}$$
\begin{enumerate}[label=(\alph*)]
\item Prove that the composite function gf is
$$\operatorname { gf } : x \rightarrow 4 \mathrm { e } ^ { 4 x } , \quad x \in \mathbb { R }$$
\item In the space provided on page 19, sketch the curve with equation $y = \operatorname { gf } ( x )$, and show the coordinates of the point where the curve cuts the $y$-axis.
\item Write down the range of gf.
\item Find the value of $x$ for which $\frac { \mathrm { d } } { \mathrm { d } x } [ \operatorname { gf } ( x ) ] = 3$, giving your answer to 3 significant figures.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 2006 Q8 [10]}}