| Exam Board | CAIE |
|---|---|
| Module | Further Paper 1 (Further Paper 1) |
| Year | 2024 |
| 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 multi-part Further Maths question involving rational function analysis, asymptotes, stationary points, and modulus sketching. While it requires several techniques (quotient rule differentiation, asymptote finding, reflection of negative portions), each part follows standard procedures. Part (d)(iii) requires solving inequalities using the sketch, which adds modest problem-solving demand, but the given intervals provide strong scaffolding. The algebraic work is routine for Further Maths students, making this moderately above average difficulty. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02m Graphs of functions: difference between plotting and sketching1.02n Sketch curves: simple equations including polynomials1.02s Modulus graphs: sketch graph of |ax+b|1.02t Solve modulus equations: graphically with modulus function1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks |
|---|---|
| \(x = -2\) | B1 |
| \(y = \frac{(x+2)(x+a-2)-2a+5}{x+2} = x+a-2+\frac{5-2a}{x+2}\) | M1 |
| \(y = x+a-2\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{(x+2)(2x+a)-(x^2+ax+1)}{(x+2)^2}\) | M1 | Differentiates |
| \(x^2+4x+2a-1=0\) \(\left(\text{or } \frac{\mathrm{d}y}{\mathrm{d}x} = 1 + \frac{2a-5}{(x+2)^2}\right)\) | A1 | Forms quadratic equation or simplifies \(\frac{\mathrm{d}y}{\mathrm{d}x}\). Not from wrong working |
| \(16-4(2a-1) = 20-8a < 0\) (or \(y' > 0\)) \(\Rightarrow\) No stationary points | M1 A1 | Consideration of discriminant or sign of \(y'\) with correct conclusion |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Axes and asymptotes labelled (vertical asymptote \(x = -2\), oblique asymptote \(y = x + a - 2\)) | B1 | Axes and asymptotes labelled |
| Branches correct | B1 | Asymptotes may cross above, on or below the \(x\)-axis |
| \(\left(0, \frac{1}{2}\right)\) | B1 | May be seen on their diagram |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Graph of \(\left | \frac{x^2+ax+1}{x+2}\right | \) with correct shape |
| Everything correct (approach to vertical asymptote and cusps correct) | B1 | |
| Total | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Horizontal line \(y = a\) drawn on diagram | B1 | |
| Total | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{x^2+ax+1}{x+2} = a\) or \(\frac{x^2+ax+1}{x+2} = -a\) | M1 | Or direct use of \(x = \pm 3\) |
| \(x^2 + 1 - 2a = 0\) or \(x^2 + 2ax + 1 + 2a = 0\) | ||
| \(-a - \sqrt{a^2-2a-1} < x < -\sqrt{2a-1},\ -a+\sqrt{a^2-2a-1} < x < \sqrt{2a-1}\) | ||
| \(a = 5\) | A1 | |
| Total | 2 |
## Question 6(a):
| $x = -2$ | B1 | |
| $y = \frac{(x+2)(x+a-2)-2a+5}{x+2} = x+a-2+\frac{5-2a}{x+2}$ | M1 | |
| $y = x+a-2$ | A1 | |
---
## Question 6(b):
| $\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{(x+2)(2x+a)-(x^2+ax+1)}{(x+2)^2}$ | M1 | Differentiates |
| $x^2+4x+2a-1=0$ $\left(\text{or } \frac{\mathrm{d}y}{\mathrm{d}x} = 1 + \frac{2a-5}{(x+2)^2}\right)$ | A1 | Forms quadratic equation or simplifies $\frac{\mathrm{d}y}{\mathrm{d}x}$. Not from wrong working |
| $16-4(2a-1) = 20-8a < 0$ (or $y' > 0$) $\Rightarrow$ No stationary points | M1 A1 | Consideration of discriminant or sign of $y'$ with correct conclusion |
## Question 6(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Axes and asymptotes labelled (vertical asymptote $x = -2$, oblique asymptote $y = x + a - 2$) | B1 | Axes and asymptotes labelled |
| Branches correct | B1 | Asymptotes may cross above, on or below the $x$-axis |
| $\left(0, \frac{1}{2}\right)$ | B1 | May be seen on their diagram |
| **Total** | **3** | |
---
## Question 6(d)(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Graph of $\left|\frac{x^2+ax+1}{x+2}\right|$ with correct shape | B1 | FT from their attempt in part 6(c) |
| Everything correct (approach to vertical asymptote and cusps correct) | B1 | |
| **Total** | **2** | |
---
## Question 6(d)(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Horizontal line $y = a$ drawn on diagram | B1 | |
| **Total** | **1** | |
---
## Question 6(d)(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{x^2+ax+1}{x+2} = a$ or $\frac{x^2+ax+1}{x+2} = -a$ | M1 | Or direct use of $x = \pm 3$ |
| $x^2 + 1 - 2a = 0$ or $x^2 + 2ax + 1 + 2a = 0$ | | |
| $-a - \sqrt{a^2-2a-1} < x < -\sqrt{2a-1},\ -a+\sqrt{a^2-2a-1} < x < \sqrt{2a-1}$ | | |
| $a = 5$ | A1 | |
| **Total** | **2** | |
---6 The curve $C$ has equation $\mathrm { y } = \frac { \mathrm { x } ^ { 2 } + \mathrm { ax } + 1 } { \mathrm { x } + 2 }$, where $a > \frac { 5 } { 2 }$.
\begin{enumerate}[label=(\alph*)]
\item Find the equations of the asymptotes of $C$.
\item Show that $C$ has no stationary points.
\item Sketch $C$, stating the coordinates of the point of intersection with the $y$-axis and labelling the asymptotes.
\item \begin{enumerate}[label=(\roman*)]
\item Sketch the curve with equation $\mathrm { y } = \left| \frac { \mathrm { x } ^ { 2 } + \mathrm { ax } + 1 } { \mathrm { x } + 2 } \right|$.
\item On your sketch in part (i), draw the line $\mathrm { y } = \mathrm { a }$.
\item It is given that $\left| \frac { \mathrm { x } ^ { 2 } + \mathrm { ax } + 1 } { \mathrm { x } + 2 } \right| < \mathrm { a }$ for $- 5 - \sqrt { 14 } < x < - 3$ and $- 5 + \sqrt { 14 } < x < 3$.
Find the value of $a$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 1 2024 Q6 [15]}}