| Exam Board | CAIE |
|---|---|
| Module | Further Paper 1 (Further Paper 1) |
| Year | 2022 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Solve |f(x)| > k using sketch |
| Difficulty | Challenging +1.2 This is a structured multi-part question requiring asymptote analysis, differentiation for stationary points, curve sketching, and modulus transformation. While it involves several techniques and the final part requires understanding how |f(x)| = k relates to the sketch, each step is methodical and follows standard A-level procedures. The rational function is well-behaved (no vertical asymptotes simplifies the work), and the modulus part is a standard reflection technique. More demanding than average due to length and the quotient rule differentiation, but no novel insights required. |
| 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 |
| \(-3 < 0\) | M1 A1 | Finds discriminant or roots of \(x^2 + x + 1\), or completes square |
| \(y = 2\) | B1 | Horizontal asymptote |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\dfrac{dy}{dx} = \dfrac{(x^2+x+1)(4x-1)-(2x^2-x-1)(2x+1)}{(x^2+x+1)^2}\) | M1 | Finds \(\dfrac{dy}{dx}\) |
| \(3x^2 + 6x = 0\) | M1 | Sets equal to 0 and forms equation |
| \((0,-1)\) and \((-2, 3)\) | A1 A1 | One point correct, or both \(x\) values. Other point correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(2x^2 - x - 1 - y(x^2 + x + 1) = 0\) | M1 | Forms quadratic equation |
| Finds discriminant \((y+1)^2 - 4(y+1)(y-2)\) AND states \(y\) exists if discriminant \(\geq 0\) OR does not exist if discriminant \(< 0\) | M1 | |
| Finds \((0,-1)\) and \((-2,3)\) | A1 | |
| Explains why they are stationary values | A1 | Double \(x\) roots for \(y=-1\) and \(y=3\) or no vertical asymptote etc |
| Total: 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Sketch with axes and asymptote | B1 | Axes and asymptote |
| Correct shape and position | B1 | Correct shape and position |
| \((1,0)\), \(\left(-\dfrac{1}{2}, 0\right)\), \((0,-1)\) | B1 | States coordinates of intersections with axes |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Sketch showing cusps on \(x\)-axis | B1 FT | FT from sketch in part (c). There must be cusps on the \(x\) axis |
| \(0 < k < 1\) | B1 | |
| Total: 2 |
## Question 5:
### Part 5(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $-3 < 0$ | M1 A1 | Finds discriminant or roots of $x^2 + x + 1$, or completes square |
| $y = 2$ | B1 | Horizontal asymptote |
| **Total: 3** | | |
---
### Part 5(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\dfrac{dy}{dx} = \dfrac{(x^2+x+1)(4x-1)-(2x^2-x-1)(2x+1)}{(x^2+x+1)^2}$ | M1 | Finds $\dfrac{dy}{dx}$ |
| $3x^2 + 6x = 0$ | M1 | Sets equal to 0 and forms equation |
| $(0,-1)$ and $(-2, 3)$ | A1 A1 | One point correct, or both $x$ values. Other point correct |
**Alternative method 5(b):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $2x^2 - x - 1 - y(x^2 + x + 1) = 0$ | M1 | Forms quadratic equation |
| Finds discriminant $(y+1)^2 - 4(y+1)(y-2)$ AND states $y$ exists if discriminant $\geq 0$ OR does not exist if discriminant $< 0$ | M1 | |
| Finds $(0,-1)$ and $(-2,3)$ | A1 | |
| Explains why they are stationary values | A1 | Double $x$ roots for $y=-1$ and $y=3$ or no vertical asymptote etc |
| **Total: 4** | | |
---
### Part 5(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Sketch with axes and asymptote | B1 | Axes and asymptote |
| Correct shape and position | B1 | Correct shape and position |
| $(1,0)$, $\left(-\dfrac{1}{2}, 0\right)$, $(0,-1)$ | B1 | States coordinates of intersections with axes |
| **Total: 3** | | |
---
### Part 5(d):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Sketch showing cusps on $x$-axis | B1 FT | FT from sketch in part (c). There must be cusps on the $x$ axis |
| $0 < k < 1$ | B1 | |
| **Total: 2** | | |5 The curve $C$ has equation $y = \frac { 2 x ^ { 2 } - x - 1 } { x ^ { 2 } + x + 1 }$.
\begin{enumerate}[label=(\alph*)]
\item Show that $C$ has no vertical asymptotes and state the equation of the horizontal asymptote of $C$.
\item Find the coordinates of the stationary points on $C$.
\item Sketch $C$, stating the coordinates of the intersections with the axes.
\item Sketch the curve with equation $y = \left| \frac { 2 x ^ { 2 } - x - 1 } { x ^ { 2 } + x + 1 } \right|$ and state the set of values of $k$ for which $\left| \frac { 2 x ^ { 2 } - x - 1 } { x ^ { 2 } + x + 1 } \right| = k$ has 4 distinct real solutions.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 1 2022 Q5 [12]}}