| Exam Board | CAIE |
|---|---|
| Module | Further Paper 1 (Further Paper 1) |
| Year | 2021 |
| Session | November |
| Marks | 14 |
| 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 curve sketching question requiring asymptote finding, range analysis, modulus transformations, and inequality solving. While it involves several steps and modulus functions, each component uses standard Further Maths techniques (rational function analysis, sketching |f(x)|, graphical inequalities). The final part requires visual reasoning from the sketch but no novel insight—slightly 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| |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x = 3\) | B1 | States vertical asymptote |
| \(y = x + 3 + \frac{9}{x-3}\) leading to \(y = x + 3\) | M1 A1 | Finds oblique asymptote |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(yx - 3y = x^2\) leading to \(x^2 - yx + 3y = 0\) | M1 A1 | Forms quadratic in \(x\) |
| \(y^2 - 4(3y) < 0\) leading to \(y^2 - 12y < 0\) | M1 | Uses that discriminant is negative |
| \(0 < y < 12\) | A1 | AG |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Axes and asymptotes shown correctly | B1 | Axes and asymptotes |
| Branches of \(y = \frac{x^2}{x-3}\) correct | B1 | Branches correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Correct graph of \(y = \left | \frac{x^2}{x-3}\right | \) |
| Correct shape at infinity | B1 | Correct shape at infinity |
| Correct shape of \(y = | x | - 3\) |
| Intercepts \((-3, 0)\), \((3, 0)\), \((0, -1)\) shown | B1 | Correct intercepts with axes (may be seen on graph) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(c \leqslant -3\) | B1 |
## Question 6(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x = 3$ | B1 | States vertical asymptote |
| $y = x + 3 + \frac{9}{x-3}$ leading to $y = x + 3$ | M1 A1 | Finds oblique asymptote |
---
## Question 6(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $yx - 3y = x^2$ leading to $x^2 - yx + 3y = 0$ | M1 A1 | Forms quadratic in $x$ |
| $y^2 - 4(3y) < 0$ leading to $y^2 - 12y < 0$ | M1 | Uses that discriminant is negative |
| $0 < y < 12$ | A1 | AG |
---
## Question 6(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Axes and asymptotes shown correctly | B1 | Axes and asymptotes |
| Branches of $y = \frac{x^2}{x-3}$ correct | B1 | Branches correct |
---
## Question 6(d)(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Correct graph of $y = \left|\frac{x^2}{x-3}\right|$ | B1 FT | FT from sketch in (c) |
| Correct shape at infinity | B1 | Correct shape at infinity |
| Correct shape of $y = |x| - 3$ | B1 | Correct shape of $y = |x| - 3$ |
| Intercepts $(-3, 0)$, $(3, 0)$, $(0, -1)$ shown | B1 | Correct intercepts with axes (may be seen on graph) |
---
## Question 6(d)(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $c \leqslant -3$ | B1 | |
---6 The curve $C$ has equation $\mathrm { y } = \frac { \mathrm { x } ^ { 2 } } { \mathrm { x } - 3 }$.
\begin{enumerate}[label=(\alph*)]
\item Find the equations of the asymptotes of $C$.
\item Show that there is no point on $C$ for which $0 < y < 12$.
\item Sketch C.
\item \begin{enumerate}[label=(\roman*)]
\item Sketch the graphs of $y = \left| \frac { x ^ { 2 } } { x - 3 } \right|$ and $y = | x | - 3$ on a single diagram, stating the coordinates of the intersections with the axes.
\item Use your sketch to find the set of values of $c$ for which $\left| \frac { x ^ { 2 } } { x - 3 } \right| \leqslant | x | + c$ has no solution. [1]
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 1 2021 Q6 [14]}}