| Exam Board | CAIE |
|---|---|
| Module | Further Paper 1 (Further Paper 1) |
| Year | 2020 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Solve |f(x)| > k using sketch |
| Difficulty | Standard +0.8 This is a multi-step Further Maths question requiring sketching a rational function with asymptotes, applying modulus transformation, then solving an inequality graphically. It demands understanding of rational function behavior, modulus effects, and careful algebraic manipulation to find critical points where |f(x)| = a/2, which is moderately challenging but follows standard Further Maths techniques. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02l Modulus function: notation, relations, equations and inequalities1.02m Graphs of functions: difference between plotting and sketching1.02n Sketch curves: simple equations including polynomials1.02s Modulus graphs: sketch graph of |ax+b| |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Graph with correct axes and asymptotes (vertical asymptote at \(x = 0\), horizontal asymptote at \(y = 0\)) | B1 | For axes and asymptotes correct |
| Correct branches of the curve (curve in second and fourth quadrants, approaching asymptotes, passing through approximately \((-1, -7)\) region) | B1 | For branches correct |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Sketch showing correct shape with asymptotes | B1 FT | Follow through from sketch in part (a) |
| \(\frac{ax}{x+7} = \frac{a}{2}\) or \(\frac{ax}{x+7} = -\frac{a}{2}\) | M1 | Setting up equations |
| \(x = 7\) and \(x = -\frac{7}{3}\) | A1 | Both values correct |
| \(x < -7\), \(-7 < x < -\frac{7}{3}\), \(x > 7\) | A1 | Correct intervals |
## Question 1:
### Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Graph with correct axes and asymptotes (vertical asymptote at $x = 0$, horizontal asymptote at $y = 0$) | **B1** | For axes and asymptotes correct |
| Correct branches of the curve (curve in second and fourth quadrants, approaching asymptotes, passing through approximately $(-1, -7)$ region) | **B1** | For branches correct |
| | **Total: 2** | |
## Question 1:
### Part 1(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Sketch showing correct shape with asymptotes | B1 FT | Follow through from sketch in part (a) |
| $\frac{ax}{x+7} = \frac{a}{2}$ or $\frac{ax}{x+7} = -\frac{a}{2}$ | M1 | Setting up equations |
| $x = 7$ and $x = -\frac{7}{3}$ | A1 | Both values correct |
| $x < -7$, $-7 < x < -\frac{7}{3}$, $x > 7$ | A1 | Correct intervals |
---1 Let $a$ be a positive constant.
\begin{enumerate}[label=(\alph*)]
\item Sketch the curve with equation $\mathrm { y } = \frac { \mathrm { ax } } { \mathrm { x } + 7 }$.
\item Sketch the curve with equation $y = \left| \frac { a x } { x + 7 } \right|$ and find the set of values of $x$ for which $\left| \frac { a x } { x + 7 } \right| > \frac { a } { 2 }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 1 2020 Q1 [6]}}