| Exam Board | CAIE |
|---|---|
| Module | Further Paper 1 (Further Paper 1) |
| Year | 2021 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Solve |f(x)| > k using sketch |
| Difficulty | Challenging +1.3 This is a substantial Further Maths curve sketching question requiring multiple techniques: finding asymptotes (factorizing denominator), stationary points (quotient rule differentiation), sketching rational functions, then applying modulus transformation and solving an inequality graphically. While multi-step and requiring careful work, these are all standard Further Maths techniques without requiring novel insight or particularly complex algebraic manipulation. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.02n Sketch curves: simple equations including polynomials1.07m Tangents and normals: gradient and equations1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x=\dfrac{1-\sqrt{5}}{2},\quad x=\dfrac{1+\sqrt{5}}{2}\) | B1 | Vertical asymptotes. Must be exact |
| \(y=-1\) | B1 | Horizontal asymptote |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\dfrac{dy}{dx}=\dfrac{(1+x-x^2)(2x-1)-(x^2-x-3)(1-2x)}{(1+x-x^2)^2}\) | M1 | Finds \(\dfrac{dy}{dx}\) |
| \((2x-1)(-2)=0\) | M1 | Sets \(\dfrac{dy}{dx}\) equal to \(0\) and forms equation |
| \(\left(\frac{1}{2},-\frac{13}{5}\right)\) | A1 | WWW |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Graph with both axes labelled and correct asymptotes shown | B1 | Both axes labelled and correct asymptotes shown |
| Correct shape and position with asymptotic behaviour clear | B1 | Correct shape and position, with all asymptotic behaviour clear |
| \(\left(\frac{1}{2}+\frac{1}{2}\sqrt{13},0\right),\quad\left(\frac{1}{2}-\frac{1}{2}\sqrt{13},0\right),\quad(0,-3)\) | B1 | States exact coordinates of intersections with axes |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Graph (FT from (c)) | B1 FT | FT from sketch in (c) |
| Correct shape as \(x\) tends to infinity and intersections with \(x\) axis | B1 | Correct shape as \(x\) tends to infinity and intersections with \(x\) axis |
| \(\dfrac{x^2-x-3}{1+x-x^2}=3\quad\) or \(\quad\dfrac{x^2-x-3}{1+x-x^2}=-3\) | M2 | Finds critical points, award M1 for each case |
| \(4x^2-4x-6=0\quad\) or \(\quad -2x^2+2x=0\) | ||
| \(x=\frac{1}{2}+\frac{1}{2}\sqrt{7},\quad x=\frac{1}{2}-\frac{1}{2}\sqrt{7},\quad x=0,\quad x=1\) | A1 | Must be exact |
| \(x<\frac{1}{2}-\frac{1}{2}\sqrt{7},\quad 0 | A1 FT | Must be three distinct regions and strict inequalities |
## Question 7(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x=\dfrac{1-\sqrt{5}}{2},\quad x=\dfrac{1+\sqrt{5}}{2}$ | B1 | Vertical asymptotes. Must be exact |
| $y=-1$ | B1 | Horizontal asymptote |
---
## Question 7(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\dfrac{dy}{dx}=\dfrac{(1+x-x^2)(2x-1)-(x^2-x-3)(1-2x)}{(1+x-x^2)^2}$ | M1 | Finds $\dfrac{dy}{dx}$ |
| $(2x-1)(-2)=0$ | M1 | Sets $\dfrac{dy}{dx}$ equal to $0$ and forms equation |
| $\left(\frac{1}{2},-\frac{13}{5}\right)$ | A1 | WWW |
---
## Question 7(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Graph with both axes labelled and correct asymptotes shown | B1 | Both axes labelled and correct asymptotes shown |
| Correct shape and position with asymptotic behaviour clear | B1 | Correct shape and position, with all asymptotic behaviour clear |
| $\left(\frac{1}{2}+\frac{1}{2}\sqrt{13},0\right),\quad\left(\frac{1}{2}-\frac{1}{2}\sqrt{13},0\right),\quad(0,-3)$ | B1 | States exact coordinates of intersections with axes |
---
## Question 7(d):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Graph (FT from (c)) | B1 FT | FT from sketch in (c) |
| Correct shape as $x$ tends to infinity and intersections with $x$ axis | B1 | Correct shape as $x$ tends to infinity and intersections with $x$ axis |
| $\dfrac{x^2-x-3}{1+x-x^2}=3\quad$ or $\quad\dfrac{x^2-x-3}{1+x-x^2}=-3$ | M2 | Finds critical points, award M1 for each case |
| $4x^2-4x-6=0\quad$ or $\quad -2x^2+2x=0$ | | |
| $x=\frac{1}{2}+\frac{1}{2}\sqrt{7},\quad x=\frac{1}{2}-\frac{1}{2}\sqrt{7},\quad x=0,\quad x=1$ | A1 | Must be exact |
| $x<\frac{1}{2}-\frac{1}{2}\sqrt{7},\quad 0<x<1,\quad x>\frac{1}{2}+\frac{1}{2}\sqrt{7}$ | A1 FT | Must be three distinct regions and strict inequalities |7 The curve $C$ has equation $y = \frac { x ^ { 2 } - x - 3 } { 1 + x - x ^ { 2 } }$.
\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 $y = \left| \frac { x ^ { 2 } - x - 3 } { 1 + x - x ^ { 2 } } \right|$ and find in exact form the set of values of $x$ for which $\left| \frac { x ^ { 2 } - x - 3 } { 1 + x - x ^ { 2 } } \right| < 3$.\\
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 [14]}}