| Answer | Marks | Guidance |
|---|---|---|
| \(x = -1\) | B1 | States vertical asymptote |
| \(y = \frac{(x+1)(x-2)+2}{x+1}\) | M1 | Finds oblique asymptote |
| \(y = x - 2\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{dy}{dx} = 1 - 2(x+1)^{-2} = 0 \Rightarrow (x+1)^2 = 2\) | M1 A1 | Differentiates and sets derivative equal to 0 |
| \(\left(-1+\sqrt{2},\ -3+2\sqrt{2}\right),\ \left(-1-\sqrt{2},\ -3-2\sqrt{2}\right)\) | A1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Sketch with axes and asymptotes labelled | B1 | Axes and asymptotes labelled |
| Upper branch with \((0,0)\) and \((1,0)\) stated or clear on scale | B1 | |
| Lower branch correct and all asymptotic approaches correct | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Sketch of \(\left | \frac{x^2-x}{x+1}\right | \) |
| \(\frac{x^2-x}{x+1} = 6\) or \(\frac{x^2-x}{x+1} = -6\) | M2 | Finds critical points; award M1 for each case |
| \(x^2 - 7x - 6 = 0\) or \(x^2 + 5x + 6 = 0\) | ||
| \(x = -3,\ -2,\ \frac{7}{2} - \frac{1}{2}\sqrt{73},\ \frac{7}{2} + \frac{1}{2}\sqrt{73}\) | A1 | |
| \(-3 < x < -2\) and \(\frac{7}{2} - \frac{1}{2}\sqrt{73} < x < \frac{7}{2} + \frac{1}{2}\sqrt{73}\) | A1 |
## Question 7(a):
$x = -1$ | B1 | States vertical asymptote
$y = \frac{(x+1)(x-2)+2}{x+1}$ | M1 | Finds oblique asymptote
$y = x - 2$ | A1 |
**Total: 3**
---
## Question 7(b):
$\frac{dy}{dx} = 1 - 2(x+1)^{-2} = 0 \Rightarrow (x+1)^2 = 2$ | M1 A1 | Differentiates and sets derivative equal to 0
$\left(-1+\sqrt{2},\ -3+2\sqrt{2}\right),\ \left(-1-\sqrt{2},\ -3-2\sqrt{2}\right)$ | A1 A1 |
**Total: 4**
---
## Question 7(c):
Sketch with axes and asymptotes labelled | B1 | Axes and asymptotes labelled
Upper branch with $(0,0)$ and $(1,0)$ stated or clear on scale | B1 |
Lower branch correct and all asymptotic approaches correct | B1 |
**Total: 3**
---
## Question 7(d):
Sketch of $\left|\frac{x^2-x}{x+1}\right|$ | B1 FT | FT from sketch in part (c)
$\frac{x^2-x}{x+1} = 6$ or $\frac{x^2-x}{x+1} = -6$ | M2 | Finds critical points; award M1 for each case
$x^2 - 7x - 6 = 0$ or $x^2 + 5x + 6 = 0$ | |
$x = -3,\ -2,\ \frac{7}{2} - \frac{1}{2}\sqrt{73},\ \frac{7}{2} + \frac{1}{2}\sqrt{73}$ | A1 |
$-3 < x < -2$ and $\frac{7}{2} - \frac{1}{2}\sqrt{73} < x < \frac{7}{2} + \frac{1}{2}\sqrt{73}$ | A1 |
**Total: 5**7 The curve $C$ has equation $\mathrm { y } = \frac { \mathrm { x } ^ { 2 } - \mathrm { x } } { \mathrm { x } + 1 }$.
\begin{enumerate}[label=(\alph*)]
\item Find the equations of the asymptotes of $C$.
\item Find the exact coordinates of the stationary points on $C$.
\item Sketch $C$, stating the coordinates of any intersections with the axes.
\item Sketch the curve with equation $y = \left| \frac { x ^ { 2 } - x } { x + 1 } \right|$ and find in exact form the set of values of $x$ for which $\left| \frac { x ^ { 2 } - x } { x + 1 } \right| < 6$.\\
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 2022 Q7 [15]}}