| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2015 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modulus function |
| Type | Sketch y=|f(x)| for non-linear f(x) |
| Difficulty | Moderate -0.8 This is a straightforward modulus function question requiring standard techniques: sketching an exponential function, reflecting negative parts for the modulus, finding intercepts (ln(5/2)), and solving |f(x)|=2 by cases. All steps are routine C3 material with no novel problem-solving required, making it easier than average. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.02n Sketch curves: simple equations including polynomials1.06a Exponential function: a^x and e^x graphs and properties1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Correct shape of \(y = 2e^x - 5\) | B1 | Exponential curve shape |
| \(\left(\ln\left(\frac{5}{2}\right), 0\right)\) and \((0,-3)\) marked | B1 | Both intercepts required |
| Asymptote \(y=-5\) shown | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Correct shape of \(y = | 2e^x-5 | \) including cusp |
| \(\left(\ln\left(\frac{5}{2}\right), 0\right)\) and \((0,3)\) marked | B1ft | Both intercepts, follow through |
| Asymptote \(y=5\) shown | B1ft | Follow through |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(x \ldots \ln\left(\frac{5}{2}\right)\) | B1ft | Follow through |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(2e^x - 5 = -2 \Rightarrow x = \ln\left(\frac{3}{2}\right)\) | M1A1 | |
| \(x = \ln\left(\frac{7}{2}\right)\) | B1 |
## Question 2:
### Part (ai)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Correct shape of $y = 2e^x - 5$ | B1 | Exponential curve shape |
| $\left(\ln\left(\frac{5}{2}\right), 0\right)$ and $(0,-3)$ marked | B1 | Both intercepts required |
| Asymptote $y=-5$ shown | B1 | |
### Part (aii)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Correct shape of $y = |2e^x-5|$ including cusp | B1ft | Shape including cusp, follow through |
| $\left(\ln\left(\frac{5}{2}\right), 0\right)$ and $(0,3)$ marked | B1ft | Both intercepts, follow through |
| Asymptote $y=5$ shown | B1ft | Follow through |
### Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $x \ldots \ln\left(\frac{5}{2}\right)$ | B1ft | Follow through |
### Part (c)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $2e^x - 5 = -2 \Rightarrow x = \ln\left(\frac{3}{2}\right)$ | M1A1 | |
| $x = \ln\left(\frac{7}{2}\right)$ | B1 | |2. Given that
$$\mathrm { f } ( x ) = 2 \mathrm { e } ^ { x } - 5 , \quad x \in \mathbb { R }$$
\begin{enumerate}[label=(\alph*)]
\item sketch, on separate diagrams, the curve with equation
\begin{enumerate}[label=(\roman*)]
\item $y = \mathrm { f } ( x )$
\item $y = | \mathrm { f } ( x ) |$
On each diagram, show the coordinates of each point at which the curve meets or cuts the axes.
On each diagram state the equation of the asymptote.
\end{enumerate}\item Deduce the set of values of $x$ for which $\mathrm { f } ( x ) = | \mathrm { f } ( x ) |$
\item Find the exact solutions of the equation $| \mathrm { f } ( x ) | = 2$
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 2015 Q2 [10]}}