| 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 asymptote finding, differentiation for stationary points, curve sketching, and solving an inequality involving modulus functions. While it involves several techniques and the final part requires careful consideration of the modulus transformation, each individual step follows standard A-level procedures without requiring novel insight. The inequality in part (d) is more challenging than typical but can be solved systematically by considering regions. This is moderately above average difficulty for Further Maths. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.02n Sketch curves: simple equations including polynomials1.02s Modulus graphs: sketch graph of |ax+b|1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x=1,\ x=-1\) | 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 \(x=\pm1\), \(y=0\) | B1 | Axes and asymptotes |
| Correct shape and position of \(y=\frac{4x+5}{4-4x^2}\) | 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 \(-1.07, 1.87\) |
| \(\frac{2}{5}-\frac{3}{5}\sqrt{6} < x < -\frac{4}{5},\ \ 0 < x < \frac{2}{5}+\frac{3}{5}\sqrt{6},\ \ x\neq\pm1\) | A1 FT | Condone exclusion of \(x=\pm1\) 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 $x=\pm1$, $y=0$ | B1 | Axes and asymptotes |
| Correct shape and position of $y=\frac{4x+5}{4-4x^2}$ | 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 $-1.07, 1.87$ |
| $\frac{2}{5}-\frac{3}{5}\sqrt{6} < x < -\frac{4}{5},\ \ 0 < x < \frac{2}{5}+\frac{3}{5}\sqrt{6},\ \ x\neq\pm1$ | A1 FT | Condone exclusion of $x=\pm1$ from the range |
| | **6** | |7 The curve $C$ has equation $y = \frac { 4 x + 5 } { 4 - 4 x ^ { 2 } }$.
\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]}}