| Exam Board | CAIE |
|---|---|
| Module | Further Paper 1 (Further Paper 1) |
| Year | 2021 |
| Session | November |
| 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 multi-part question requiring curve sketching of a rational function, finding asymptotes and stationary points, then sketching the modulus version and solving a modulus inequality. While it involves several techniques (differentiation, asymptote finding, modulus manipulation), each part follows standard procedures taught in Further Maths. The final inequality requires careful case analysis but is methodical rather than requiring novel insight. Slightly above average difficulty due to length and the modulus inequality component. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02l Modulus function: notation, relations, equations and inequalities1.02n Sketch curves: simple equations including polynomials1.02s Modulus graphs: sketch graph of |ax+b|1.07m Tangents and normals: gradient and equations1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x=1,\ x=-1\) | B1 | Vertical asymptotes |
| \(y=0\) | B1 | Horizontal asymptote |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{dy}{dx} = \frac{(4-4x^2)(4)-(4x+5)(-8x)}{(4-4x^2)^2}\) | M1 | Finds \(\frac{dy}{dx}\) |
| \(16x^2 + 40x + 16 = 0\) | M1 | Sets equal to 0 and forms equation |
| \(\left(-2, \frac{1}{4}\right),\ \left(-\frac{1}{2}, 1\right)\) | A1 A1 | |
| 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Sketch with axes and asymptotes shown | B1 | Axes and asymptotes |
| Correct shape and position for \(y=\frac{4x+5}{4-4x^2}\) | B1 | Correct shape and position |
| \(\left(-\frac{5}{4}, 0\right),\ \left(0, \frac{5}{4}\right)\) | B1 | States coordinates of intersections with axes |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Sketch of \(y=\left | \frac{4x+5}{4-4x^2}\right | \) |
| Correct shape at infinity | B1 | Correct shape at infinity |
| \(\frac{4x+5}{4-4x^2}=\frac{5}{4}\) or \(\frac{4x+5}{4-4x^2}=-\frac{5}{4}\), giving \(5x^2+4x=0\) or \(5x^2-4x-10=0\) | M2 | Finds critical points; award M1 for each case |
| \(x=-\frac{4}{5},\ x=0\) or \(x=\frac{2}{5}-\frac{3}{5}\sqrt{6},\ x=\frac{2}{5}+\frac{3}{5}\sqrt{6}\) | A1 | A0 if \(-1.07, 1.87\) |
| \(\frac{2}{5}-\frac{3}{5}\sqrt{6} < x < -\frac{4}{5},\ \ 0 < x < \frac{2}{5}+\frac{3}{5}\sqrt{6},\ \ x\neq\pm 1\) | A1 FT | Condone exclusion of \(x=\pm 1\) from the range |
| 6 |
## Question 7(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x=1,\ x=-1$ | **B1** | Vertical asymptotes |
| $y=0$ | **B1** | Horizontal asymptote |
| | **2** | |
---
## Question 7(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{dy}{dx} = \frac{(4-4x^2)(4)-(4x+5)(-8x)}{(4-4x^2)^2}$ | **M1** | Finds $\frac{dy}{dx}$ |
| $16x^2 + 40x + 16 = 0$ | **M1** | Sets equal to 0 and forms equation |
| $\left(-2, \frac{1}{4}\right),\ \left(-\frac{1}{2}, 1\right)$ | **A1 A1** | |
| | **4** | |
---
## Question 7(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Sketch with axes and asymptotes shown | **B1** | Axes and asymptotes |
| Correct shape and position for $y=\frac{4x+5}{4-4x^2}$ | **B1** | Correct shape and position |
| $\left(-\frac{5}{4}, 0\right),\ \left(0, \frac{5}{4}\right)$ | **B1** | States coordinates of intersections with axes |
| | **3** | |
---
## Question 7(d):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Sketch of $y=\left|\frac{4x+5}{4-4x^2}\right|$ | **B1 FT** | FT from sketch in part (c) |
| Correct shape at infinity | **B1** | Correct shape at infinity |
| $\frac{4x+5}{4-4x^2}=\frac{5}{4}$ or $\frac{4x+5}{4-4x^2}=-\frac{5}{4}$, giving $5x^2+4x=0$ or $5x^2-4x-10=0$ | **M2** | Finds critical points; award M1 for each case |
| $x=-\frac{4}{5},\ x=0$ or $x=\frac{2}{5}-\frac{3}{5}\sqrt{6},\ x=\frac{2}{5}+\frac{3}{5}\sqrt{6}$ | **A1** | A0 if $-1.07, 1.87$ |
| $\frac{2}{5}-\frac{3}{5}\sqrt{6} < x < -\frac{4}{5},\ \ 0 < x < \frac{2}{5}+\frac{3}{5}\sqrt{6},\ \ x\neq\pm 1$ | **A1 FT** | Condone exclusion of $x=\pm 1$ from the range |
| | **6** | |7 The curve $C$ has equation $\mathrm { y } = \frac { 4 \mathrm { x } + 5 } { 4 - 4 \mathrm { x } ^ { 2 } }$.
\begin{enumerate}[label=(\alph*)]
\item Find the equations of the asymptotes of $C$.
\item Find the coordinates of any stationary points on $C$.
\item Sketch $C$, stating the coordinates of the intersections with the axes.
\item Sketch the curve with equation $y = \left| \frac { 4 x + 5 } { 4 - 4 x ^ { 2 } } \right|$ and find in exact form the set of values of $x$ for which $4 | 4 x + 5 | > 5 \left| 4 - 4 x ^ { 2 } \right|$.\\
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]}}