| Exam Board | CAIE |
|---|---|
| Module | Further Paper 1 (Further Paper 1) |
| Year | 2024 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Simple rational function analysis |
| Difficulty | Moderate -0.3 This is a straightforward curve sketching question requiring standard techniques: identifying asymptotes (denominator never zero, limit as x→∞), finding stationary points via quotient rule, and reflecting for the y² variant. All steps are routine A-level methods with no novel insight required, though the multi-part structure and Further Maths context place it slightly below average difficulty overall. |
| Spec | 1.02m Graphs of functions: difference between plotting and sketching1.02n Sketch curves: simple equations including polynomials1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x^2 + 3 = 0\) has no real roots | B1 | OE |
| \(y = 0\) | B1 | Horizontal asymptote |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{dy}{dx} = \frac{(x^2+3)-(x+1)(2x)}{(x^2+3)^2}\) | M1 | Finds \(\frac{dy}{dx}\) |
| \(x^2 + 2x - 3 = 0\) | M1 | Sets equal to 0 and simplifies to quadratic equation |
| \(\left(-3,\ -\frac{1}{6}\right),\ \left(1,\ \frac{1}{2}\right)\) | A1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Graph with axes and approach to asymptote | B1 | Axes and approach to asymptote |
| Correct smooth shape and position | B1 | Correct smooth shape and position |
| \((-1,\ 0),\ \left(0,\ \frac{1}{3}\right)\) | B1 | States coordinates of intersections with axes, may be shown on their graph |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Sections for \(x \geq 0\) and \(x < -1\) correct | B1 | Sections for \(x \geq 0\) and \(x < -1\) correct |
| \(-1 \leq x < 0\) correct | B1 | \(-1 \leq x < 0\) correct |
| \((-1,\ 0),\ \left(0,\ \pm\frac{1}{\sqrt{3}}\right)\) | B1 | States coordinates of intersections with axes, may be shown on their graph |
| \(\left(1,\ \pm\frac{1}{\sqrt{2}}\right)\) | B1 | May be shown on their graph |
## Question 6(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x^2 + 3 = 0$ has no real roots | B1 | OE |
| $y = 0$ | B1 | Horizontal asymptote |
## Question 6(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{dy}{dx} = \frac{(x^2+3)-(x+1)(2x)}{(x^2+3)^2}$ | M1 | Finds $\frac{dy}{dx}$ |
| $x^2 + 2x - 3 = 0$ | M1 | Sets equal to 0 and simplifies to quadratic equation |
| $\left(-3,\ -\frac{1}{6}\right),\ \left(1,\ \frac{1}{2}\right)$ | A1 A1 | |
## Question 6(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Graph with axes and approach to asymptote | B1 | Axes and approach to asymptote |
| Correct smooth shape and position | B1 | Correct smooth shape and position |
| $(-1,\ 0),\ \left(0,\ \frac{1}{3}\right)$ | B1 | States coordinates of intersections with axes, may be shown on their graph |
## Question 6(d):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Sections for $x \geq 0$ and $x < -1$ correct | B1 | Sections for $x \geq 0$ and $x < -1$ correct |
| $-1 \leq x < 0$ correct | B1 | $-1 \leq x < 0$ correct |
| $(-1,\ 0),\ \left(0,\ \pm\frac{1}{\sqrt{3}}\right)$ | B1 | States coordinates of intersections with axes, may be shown on their graph |
| $\left(1,\ \pm\frac{1}{\sqrt{2}}\right)$ | B1 | May be shown on their graph |
---6 The curve $C$ has equation $y = \frac { x + 1 } { x ^ { 2 } + 3 }$.
\begin{enumerate}[label=(\alph*)]
\item Show that $C$ has no vertical asymptotes and state the equation of the horizontal asymptote. [2]
\item Find the coordinates of any stationary points on $C$.\\
\includegraphics[max width=\textwidth, alt={}, center]{7eb2abb1-68f4-4cc8-8314-f436906d6c4e-12_2715_35_144_2012}\\
\includegraphics[max width=\textwidth, alt={}, center]{7eb2abb1-68f4-4cc8-8314-f436906d6c4e-13_2718_33_141_23}
\item Sketch $C$, stating the coordinates of the intersections with the axes.
\item Sketch $y ^ { 2 } = \frac { x + 1 } { x ^ { 2 } + 3 }$, stating the coordinates of the stationary points and the intersections with the axes.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 1 2024 Q6 [13]}}