| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x = -1,\ x = 2\) | B1 | Vertical asymptotes |
| \(y = 1\) | B1 | Horizontal asymptote |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{dy}{dx} = \frac{(x^2 - x - 2)(2x) - (x^2 + 2)(2x - 1)}{(x^2 - x - 2)^2}\) | M1* | Finds \(\frac{dy}{dx}\) |
| \(x^2 + 8x - 2 = 0\) | DM1 | Sets equal to 0 and forms equation |
| \((-8.2, 0.9),\ (0.2, -0.9)\) | A1 A1 | Condone \(\left(-4 - 3\sqrt{2}, \frac{2}{3}\sqrt{2}\right),\ \left(-4 + 3\sqrt{2}, -\frac{2}{3}\sqrt{2}\right)\) |
| 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| [sketch with asymptotes] | B1 | Axes and all three asymptotes |
| [sketch] | B1 | Correct shape and position, crossing horizontal asymptote |
| \((0, -1)\) | B1 | States \((0, -1)\) coordinates of intersection with axes, may be seen on diagram |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| [sketch of \( | y | \)] |
| [sketch] | B1 | All correct |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{x^2+2}{x^2-x-2} = 1\) or \(\frac{x^2+2}{x^2-x-2} = -1\) | M2 | Finds critical points, award M1 for each case |
| \(x + 4 = 0\) or \(2x^2 - x = 0\) | ||
| \(x = -4\) or \(x = 0,\ x = \frac{1}{2}\) | A1 | |
| \(-4 < x < -1,\ 0 < x < \frac{1}{2},\ x > 2\) | B1 | Must have three distinct regions. Condone \(\leqslant -1\) and \(\geqslant 2\) |
| 4 |
## Question 7(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x = -1,\ x = 2$ | B1 | Vertical asymptotes |
| $y = 1$ | B1 | Horizontal asymptote |
| | **2** | |
## Question 7(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{dy}{dx} = \frac{(x^2 - x - 2)(2x) - (x^2 + 2)(2x - 1)}{(x^2 - x - 2)^2}$ | M1* | Finds $\frac{dy}{dx}$ |
| $x^2 + 8x - 2 = 0$ | DM1 | Sets equal to 0 and forms equation |
| $(-8.2, 0.9),\ (0.2, -0.9)$ | A1 A1 | Condone $\left(-4 - 3\sqrt{2}, \frac{2}{3}\sqrt{2}\right),\ \left(-4 + 3\sqrt{2}, -\frac{2}{3}\sqrt{2}\right)$ |
| | **4** | |
## Question 7(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| [sketch with asymptotes] | B1 | Axes and all three asymptotes |
| [sketch] | B1 | Correct shape and position, crossing horizontal asymptote |
| $(0, -1)$ | B1 | States $(0, -1)$ coordinates of intersection with axes, may be seen on diagram |
| | **3** | |
## Question 7(d):
| Answer | Marks | Guidance |
|--------|-------|----------|
| [sketch of $|y|$] | B1 FT | FT from sketch in (c) |
| [sketch] | B1 | All correct |
| | **2** | |
## Question 7(e):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{x^2+2}{x^2-x-2} = 1$ or $\frac{x^2+2}{x^2-x-2} = -1$ | M2 | Finds critical points, award M1 for each case |
| $x + 4 = 0$ or $2x^2 - x = 0$ | | |
| $x = -4$ or $x = 0,\ x = \frac{1}{2}$ | A1 | |
| $-4 < x < -1,\ 0 < x < \frac{1}{2},\ x > 2$ | B1 | Must have three distinct regions. Condone $\leqslant -1$ and $\geqslant 2$ |
| | **4** | |7 The curve $C$ has equation $y = f ( x )$, where $f ( x ) = \frac { x ^ { 2 } + 2 } { x ^ { 2 } - x - 2 }$.
\begin{enumerate}[label=(\alph*)]
\item Find the equations of the asymptotes of $C$.
\item Find the coordinates of any stationary points on $C$, giving your answers correct to 1 decimal place.
\item Sketch $C$, stating the coordinates of any intersections with the axes.
\item Sketch the curve with equation $\mathrm { y } = \frac { 1 } { \mathrm { f } ( \mathrm { x } ) }$.
\item Find 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 [15]}}