| Exam Board | CAIE |
|---|---|
| Module | Further Paper 1 (Further Paper 1) |
| Year | 2021 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Solve |f(x)| > k using sketch |
| Difficulty | Challenging +1.2 This is a substantial Further Maths curve sketching question requiring algebraic division for asymptotes, differentiation using quotient rule for stationary points, reflection for modulus curve, and solving a modulus inequality. While multi-step and requiring several techniques, each component is relatively standard for Further Maths students, making it moderately above average difficulty but not requiring exceptional insight. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02l Modulus function: notation, relations, equations and inequalities1.02n Sketch curves: simple equations including polynomials1.02o Sketch reciprocal curves: y=a/x and y=a/x^21.02s Modulus graphs: sketch graph of |ax+b|1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x = -1\) | B1 | States vertical asymptote |
| \(y = \dfrac{x(x+1)+9}{x+1} = x + \dfrac{9}{x+1}\) | M1 | Finds oblique asymptote |
| \(y = x\) | A1 | |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\dfrac{dy}{dx} = 1 - 9(x+1)^{-2} = 0 \Rightarrow (x+1)^2 = 9\) | M1 A1 | Differentiates and sets derivative equal to 0 |
| \((2, 5)\) | A1 | |
| \((-4, -7)\) | A1 | |
| Total: 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Axes labelled and correct asymptotes drawn | B1 | |
| Upper branch with \((0, 9)\) stated or shown on diagram | B1 | |
| Lower branch correct and good approach to asymptotes throughout, no extra branches | B1 | |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Sketch of \(\ | y\ | \) with correct shape |
| \(x^2 + x + 9 = \dfrac{13}{2}(x+1)\) or \(x^2 + x + 9 = -\dfrac{13}{2}(x+1)\) | M1 M1 | Finds critical points, award M1 for each case. May state \(x^2+x+9=-\frac{13}{2}(x+1)\) has no real solutions since \(7 > \frac{13}{2}\) |
| \(x^2 - \dfrac{11}{2}x + \dfrac{5}{2} = 0\) or \(x^2 + \dfrac{15}{2}x + \dfrac{31}{2} = 0\) | ||
| \(x = \dfrac{1}{2},\ 5\) | A1 | |
| \(x < \dfrac{1}{2}\) and \(x > 5\) | A1 | |
| Total: 5 |
## Question 7(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x = -1$ | B1 | States vertical asymptote |
| $y = \dfrac{x(x+1)+9}{x+1} = x + \dfrac{9}{x+1}$ | M1 | Finds oblique asymptote |
| $y = x$ | A1 | |
| **Total: 3** | | |
---
## Question 7(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\dfrac{dy}{dx} = 1 - 9(x+1)^{-2} = 0 \Rightarrow (x+1)^2 = 9$ | M1 A1 | Differentiates and sets derivative equal to 0 |
| $(2, 5)$ | A1 | |
| $(-4, -7)$ | A1 | |
| **Total: 4** | | |
---
## Question 7(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Axes labelled and correct asymptotes drawn | B1 | |
| Upper branch with $(0, 9)$ stated or shown on diagram | B1 | |
| Lower branch correct and good approach to asymptotes throughout, no extra branches | B1 | |
| **Total: 3** | | |
---
## Question 7(d):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Sketch of $\|y\|$ with correct shape | B1 FT | FT from sketch in (c) with asymptotes shown |
| $x^2 + x + 9 = \dfrac{13}{2}(x+1)$ or $x^2 + x + 9 = -\dfrac{13}{2}(x+1)$ | M1 M1 | Finds critical points, award M1 for each case. May state $x^2+x+9=-\frac{13}{2}(x+1)$ has no real solutions since $7 > \frac{13}{2}$ |
| $x^2 - \dfrac{11}{2}x + \dfrac{5}{2} = 0$ or $x^2 + \dfrac{15}{2}x + \dfrac{31}{2} = 0$ | | |
| $x = \dfrac{1}{2},\ 5$ | A1 | |
| $x < \dfrac{1}{2}$ and $x > 5$ | A1 | |
| **Total: 5** | | |7 The curve $C$ has equation $\mathrm { y } = \frac { \mathrm { x } ^ { 2 } + \mathrm { x } + 9 } { \mathrm { x } + 1 }$.
\begin{enumerate}[label=(\alph*)]
\item Find the equations of the asymptotes of $C$.
\item Find the 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 $\mathrm { y } = \left| \frac { \mathrm { x } ^ { 2 } + \mathrm { x } + 9 } { \mathrm { x } + 1 } \right|$ and find the set of values of $x$ for which $2 \left| x ^ { 2 } + x + 9 \right| > 13 | x + 1 |$.
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 1 2021 Q7 [15]}}