| Exam Board | CAIE |
|---|---|
| Module | Further Paper 1 (Further Paper 1) |
| Year | 2020 |
| Session | November |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Multiple transformations including squared |
| Difficulty | Standard +0.8 This Further Maths question requires sketching a rational function and its transformations (squared and absolute value), finding stationary points using the quotient rule, analyzing intersections, and solving an inequality graphically. While systematic, it demands careful handling of multiple transformations, sign analysis, and integration of algebraic and graphical reasoning across several connected parts. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.02n Sketch curves: simple equations including polynomials1.02s Modulus graphs: sketch graph of |ax+b|1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| [Graph with correct asymptotes labelled] | B1 | Asymptotes labelled |
| [Correct curve through \(x = 2a\)] | B1 | Correct position and shape. Not too truncated |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{dy}{dx} = -\frac{2a(x-a)}{(x-2a)^3} = 0\) | M1 | Sets \(\frac{dy}{dx} = 0\) |
| Solves for \(x\) | M1 | |
| Since \(x \neq 2a\), stationary point is \((a, 0)\) | A1 | |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((x-a)^2 = (x-a)(x-2a)\) or \((x-a)^2 = -(x-a)(x-2a)\) | M1 | Finds a critical point |
| \(x = a\) and \(\frac{3}{2}a\) or \((a, 0)\) or \((\frac{3}{2}a, 1)\) | A1 | Both \(x\) values or one correct point |
| \((a, 0)\) and \((\frac{3}{2}a, 1)\) | A1 | |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| [Axes and asymptotes shown] | B1 | Axes and asymptotes |
| [\(C_3\) curve shown] | B1 FT | \(C_3\) FT from sketch in part (a). Clearly identified |
| [Correct shape of \(C_2\)] | B1 | Correct shape of \(C_2\). Clearly identified |
| [Relative positions and intersections] | B1 | Relative positions and intersections correct |
| \(x \leqslant \frac{3}{2}a\) | B1 | Accept algebraic method |
| Total: 5 |
## Question 6(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| [Graph with correct asymptotes labelled] | **B1** | Asymptotes labelled |
| [Correct curve through $x = 2a$] | **B1** | Correct position and shape. Not too truncated |
| **Total: 2** | | |
---
## Question 6(b)(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{dy}{dx} = -\frac{2a(x-a)}{(x-2a)^3} = 0$ | **M1** | Sets $\frac{dy}{dx} = 0$ |
| Solves for $x$ | **M1** | |
| Since $x \neq 2a$, stationary point is $(a, 0)$ | **A1** | |
| **Total: 3** | | |
---
## Question 6(b)(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(x-a)^2 = (x-a)(x-2a)$ or $(x-a)^2 = -(x-a)(x-2a)$ | **M1** | Finds a critical point |
| $x = a$ and $\frac{3}{2}a$ **or** $(a, 0)$ **or** $(\frac{3}{2}a, 1)$ | **A1** | Both $x$ values or one correct point |
| $(a, 0)$ and $(\frac{3}{2}a, 1)$ | **A1** | |
| **Total: 3** | | |
---
## Question 6(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| [Axes and asymptotes shown] | **B1** | Axes and asymptotes |
| [$C_3$ curve shown] | **B1 FT** | $C_3$ FT from sketch in part (a). Clearly identified |
| [Correct shape of $C_2$] | **B1** | Correct shape of $C_2$. Clearly identified |
| [Relative positions and intersections] | **B1** | Relative positions and intersections correct |
| $x \leqslant \frac{3}{2}a$ | **B1** | Accept algebraic method |
| **Total: 5** | | |
---6 Let $a$ be a positive constant.
\begin{enumerate}[label=(\alph*)]
\item The curve $C _ { 1 }$ has equation $\mathrm { y } = \frac { \mathrm { x } - \mathrm { a } } { \mathrm { x } - 2 \mathrm { a } }$.
Sketch $C _ { 1 }$.
The curve $C _ { 2 }$ has equation $\mathrm { y } = \left( \frac { \mathrm { x } - \mathrm { a } } { \mathrm { x } - 2 \mathrm { a } } \right) ^ { 2 }$. The curve $C _ { 3 }$ has equation $\mathrm { y } = \left| \frac { \mathrm { x } - \mathrm { a } } { \mathrm { x } - 2 \mathrm { a } } \right|$.
\item \begin{enumerate}[label=(\roman*)]
\item Find the coordinates of any stationary points of $C _ { 2 }$.
\item Find also the coordinates of any points of intersection of $C _ { 2 }$ and $C _ { 3 }$.
\end{enumerate}\item Sketch $C _ { 2 }$ and $C _ { 3 }$ on a single diagram, clearly identifying each curve. Hence find the set of values of $x$ for which $\left( \frac { x - a } { x - 2 a } \right) ^ { 2 } \leqslant \left| \frac { x - a } { x - 2 a } \right|$.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 1 2020 Q6 [13]}}