| Exam Board | CAIE |
|---|---|
| Module | Further Paper 1 (Further Paper 1) |
| Year | 2023 |
| Session | June |
| 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 comprehensive curve sketching question requiring multiple techniques (asymptotes, stationary points, modulus transformations, and inequality solving), but each individual step follows standard A-level procedures. The algebraic manipulation is routine, and while part (e) requires careful consideration of the modulus inequality, it's a direct application of the sketch from part (d) with straightforward algebra. More demanding than average due to length and integration of multiple skills, but no novel insights required. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02m Graphs of functions: difference between plotting and sketching1.02n Sketch curves: simple equations including polynomials1.02s Modulus graphs: sketch graph of |ax+b|1.02t Solve modulus equations: graphically with modulus function1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx |
| Answer | Marks | Guidance |
|---|---|---|
| \(x = 2\) | B1 | States vertical asymptote. |
| \(x^2+2x-15 = (x-2)(x+4)-7 \Rightarrow y = x+4\) | M1 A1 | Finds oblique asymptote. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{dy}{dx} = \frac{(x-2)(2x+2)-(x^2+2x-15)}{(x-2)^2}\) | M1 | Differentiates. |
| \(x^2 - 4x + 11 = 0\) \(\left(\text{or } \frac{dy}{dx} = 1 + \frac{7}{(x-2)^2}\right)\) | A1 | Forms quadratic equation or simplifies \(\frac{dy}{dx}\). |
| \(4^2 - 4(1)(11) = -28 < 0\) (or \(y' > 0\)) There are no turning points. | A1 | Correct conclusion. |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Graph with axes and asymptotes shown | B1 | Axes and asymptotes. |
| Correct branches | B1 | Branches correct. |
| \((0, 7.5), (-5, 0), (3, 0)\) | B1 | States coordinates of intersections with axes. |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Correct graph (modulus) following from (c) | B1FT | FT from sketch in (c). |
| Correct shape at infinity and on \(x\) axis | B1 | Correct shape at infinity and on \(x\) axis. |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{x^2+2x-15}{x-2} = \frac{15}{2}\) or \(\frac{x^2+2x-15}{x-2} = -\frac{15}{2}\); \(x^2 - \frac{11}{2}x = 0\) or \(x^2 + \frac{19}{2}x - 30 = 0\) | M2 | Finds critical points, award M1 for each case. |
| \(x = 0, \frac{11}{2}\) or \(x = -12, \frac{5}{2}\) | A1 | |
| \(-12 < x < 0\) or \(\frac{5}{2} < x < \frac{11}{2}\) | A1FT | |
| 4 |
## Question 6:
**Part 6(a):**
$x = 2$ | B1 | States vertical asymptote.
$x^2+2x-15 = (x-2)(x+4)-7 \Rightarrow y = x+4$ | M1 A1 | Finds oblique asymptote.
## Question 6(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{dy}{dx} = \frac{(x-2)(2x+2)-(x^2+2x-15)}{(x-2)^2}$ | M1 | Differentiates. |
| $x^2 - 4x + 11 = 0$ $\left(\text{or } \frac{dy}{dx} = 1 + \frac{7}{(x-2)^2}\right)$ | A1 | Forms quadratic equation or simplifies $\frac{dy}{dx}$. |
| $4^2 - 4(1)(11) = -28 < 0$ (or $y' > 0$) There are no turning points. | A1 | Correct conclusion. |
| | **3** | |
## Question 6(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Graph with axes and asymptotes shown | B1 | Axes and asymptotes. |
| Correct branches | B1 | Branches correct. |
| $(0, 7.5), (-5, 0), (3, 0)$ | B1 | States coordinates of intersections with axes. |
| | **3** | |
## Question 6(d):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Correct graph (modulus) following from (c) | B1FT | FT from sketch in (c). |
| Correct shape at infinity and on $x$ axis | B1 | Correct shape at infinity and on $x$ axis. |
| | **2** | |
## Question 6(e):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{x^2+2x-15}{x-2} = \frac{15}{2}$ or $\frac{x^2+2x-15}{x-2} = -\frac{15}{2}$; $x^2 - \frac{11}{2}x = 0$ or $x^2 + \frac{19}{2}x - 30 = 0$ | M2 | Finds critical points, award M1 for each case. |
| $x = 0, \frac{11}{2}$ or $x = -12, \frac{5}{2}$ | A1 | |
| $-12 < x < 0$ or $\frac{5}{2} < x < \frac{11}{2}$ | A1FT | |
| | **4** | |6 The curve $C$ has equation $\mathrm { y } = \frac { \mathrm { x } ^ { 2 } + 2 \mathrm { x } - 15 } { \mathrm { x } - 2 }$.
\begin{enumerate}[label=(\alph*)]
\item Find the equations of the asymptotes of $C$.
\item Show that $C$ has no stationary points.
\item Sketch $C$, stating the coordinates of the intersections with the axes.
\item Sketch the curve with equation $\mathrm { y } = \left| \frac { \mathrm { x } ^ { 2 } - 2 \mathrm { x } - 15 } { \mathrm { x } - 2 } \right|$.
\item Find the set of values of $x$ for which $\left| \frac { 2 x ^ { 2 } + 4 x - 30 } { x - 2 } \right| < 15$.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 1 2023 Q6 [15]}}