| Exam Board | CAIE |
|---|---|
| Module | Further Paper 1 (Further Paper 1) |
| Year | 2023 |
| Session | November |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Multiple transformations including squared |
| Difficulty | Challenging +1.2 This is a comprehensive curve sketching question requiring asymptote finding, differentiation for stationary points, and transformation analysis. While it involves multiple parts and the reciprocal transformation in parts (d)-(e), each component uses standard A-level techniques (quotient rule, asymptote identification, inequality solving). The rational function is straightforward, and the reciprocal transformation is a familiar concept. More demanding than average due to length and the final inequality, but no novel insights required. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02n Sketch curves: simple equations including polynomials1.02o Sketch reciprocal curves: y=a/x and y=a/x^21.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x = -1\) | B1 | Vertical asymptote. |
| \(y = \frac{(x+1)(x-1)+1}{x+1}\) | M1 | Oblique asymptote. |
| \(y = x - 1\) | A1 | |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{dy}{dx} = \frac{x^2 + 2x}{(x+1)^2} = 0\) | M1 | Sets \(\frac{dy}{dx} = 0\). |
| \((0, 0),\ (-2, -4)\) | A1 | |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Axes and asymptotes shown correctly | B1 | Axes and asymptotes. |
| Left branch correct | B1 | Left branch correct. |
| Right branch correct | B1 | Right branch correct. |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\left(-2, -\frac{1}{4}\right)\) | B1 B1 | B1 for each correct coordinate. SC B1 for \(\left(-2, -\frac{1}{4}\right)\) and \((0,0)\). |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Left branch correct | B1 | Left branch correct. |
| Right branch correct | B1 | Right branch correct. |
| \(\frac{x^2}{x+1} = 1\) or \(\frac{x^2}{x+1} = -1\), leading to \(x^2 - x - 1 = 0\) | M2 | Finds critical points, award M1 for each case. |
| \(x = \frac{1}{2} - \frac{1}{2}\sqrt{5}\) or \(x = \frac{1}{2} + \frac{1}{2}\sqrt{5}\) | A1 | |
| \(x < -1,\ \frac{1}{2} - \frac{1}{2}\sqrt{5} < x < \frac{1}{2} + \frac{1}{2}\sqrt{5},\ x \neq 0\) | B1 | Condone missing \(x \neq 0\). |
| 6 |
## Question 7(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x = -1$ | B1 | Vertical asymptote. |
| $y = \frac{(x+1)(x-1)+1}{x+1}$ | M1 | Oblique asymptote. |
| $y = x - 1$ | A1 | |
| | **3** | |
## Question 7(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{dy}{dx} = \frac{x^2 + 2x}{(x+1)^2} = 0$ | M1 | Sets $\frac{dy}{dx} = 0$. |
| $(0, 0),\ (-2, -4)$ | A1 | |
| | **2** | |
## Question 7(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Axes and asymptotes shown correctly | B1 | Axes and asymptotes. |
| Left branch correct | B1 | Left branch correct. |
| Right branch correct | B1 | Right branch correct. |
| | **3** | |
## Question 7(d):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\left(-2, -\frac{1}{4}\right)$ | B1 B1 | B1 for each correct coordinate. SC B1 for $\left(-2, -\frac{1}{4}\right)$ and $(0,0)$. |
| | **2** | |
## Question 7(e):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Left branch correct | B1 | Left branch correct. |
| Right branch correct | B1 | Right branch correct. |
| $\frac{x^2}{x+1} = 1$ or $\frac{x^2}{x+1} = -1$, leading to $x^2 - x - 1 = 0$ | M2 | Finds critical points, award M1 for each case. |
| $x = \frac{1}{2} - \frac{1}{2}\sqrt{5}$ or $x = \frac{1}{2} + \frac{1}{2}\sqrt{5}$ | A1 | |
| $x < -1,\ \frac{1}{2} - \frac{1}{2}\sqrt{5} < x < \frac{1}{2} + \frac{1}{2}\sqrt{5},\ x \neq 0$ | B1 | Condone missing $x \neq 0$. |
| | **6** | |7 The curve $C$ has equation $y = f ( x )$, where $f ( x ) = \frac { x ^ { 2 } } { x + 1 }$.
\begin{enumerate}[label=(\alph*)]
\item Find the equations of the asymptotes of $C$.
\item Find the coordinates of any stationary points on $C$.
\item Sketch $C$.
\item Find the coordinates of any stationary points on the curve with equation $\mathrm { y } = \frac { 1 } { \mathrm { f } ( \mathrm { x } ) }$.
\item Sketch the curve with equation $y = \frac { 1 } { f ( x ) }$ and find, in exact form, the set of values for which
$$\frac { 1 } { \mathrm { f } ( x ) } > \mathrm { f } ( x ) .$$
If you use the following page to complete the answer to any question, the question number must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 1 2023 Q7 [16]}}