| Exam Board | CAIE |
|---|---|
| Module | Further Paper 1 (Further Paper 1) |
| Year | 2020 |
| Session | Specimen |
| Marks | 17 |
| 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: finding asymptotes, proving an inequality using calculus, finding stationary points, sketching the original curve, then sketching |f(x)| and solving an inequality. While it involves several steps (5 parts), each individual part uses standard Further Maths techniques (asymptotes of rational functions, differentiation, reflection in x-axis for modulus). The inequality solving from the sketch is routine once the graph is drawn. This is moderately above average difficulty due to length and being Further Maths content, but doesn't require exceptional insight. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02l Modulus function: notation, relations, equations and inequalities1.02n Sketch curves: simple equations including polynomials1.02s Modulus graphs: sketch graph of |ax+b|1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| Vertical: \(x=1\); horizontal: \(y=2\) | B1B1 | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| \(yx^2 - 2yx = 2x^2 - 3x - 2\) | M1 | 1 mark |
| \((y-2)x^2 - (2y-3)x + (y+2) = 0\) | A1 | 1 mark |
| For real \(x\): \((2y-3)^2 - 4(y-2)(y+2) \geq 0\) | M1 | 1 mark |
| \(12y \leq 25\), so \(y \leq \frac{25}{12}\) | A1 | 1 mark |
| 4 |
| Answer | Marks | Guidance |
|---|---|---|
| At turning points: \((x^2 - 2x+1)(4x-3) - (2x^2 - 3x - 2)(2x-2) = 0\) | M1 | 1 mark |
| \(x^2 - 8x + 7 = 0\) | M1 | 1 mark |
| \(x = 7\) (since \(x=1\) is an asymptote) | ||
| Stationary point \(\left(7, \frac{75}{12}\right)\) | A1 | 1 mark |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Axes and asymptotes | B1 | 1 mark |
| \((-0.5, 0)\), \((2, 0)\), \((0, -2)\), \((4, 2)\) | B1 | 1 mark |
| Left hand branch | B1 | 1 mark |
| Right hand branch | B1 | 1 mark |
| 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Correct sketch, FT from (d) | B1FT | 1 mark |
| Finds \(0\), \(\frac{7}{4}\) and \(4\) as critical points | M1 | 1 mark |
| \(x < 0\) | B1 | 1 mark |
| \(\frac{7}{4} < x < 4\) | A1 | 1 mark |
| 4 |
## Question 7(a):
| Vertical: $x=1$; horizontal: $y=2$ | B1B1 | **2** |
## Question 7(b):
| $yx^2 - 2yx = 2x^2 - 3x - 2$ | M1 | 1 mark |
| $(y-2)x^2 - (2y-3)x + (y+2) = 0$ | A1 | 1 mark |
| For real $x$: $(2y-3)^2 - 4(y-2)(y+2) \geq 0$ | M1 | 1 mark |
| $12y \leq 25$, so $y \leq \frac{25}{12}$ | A1 | 1 mark | AG |
| | | **4** |
## Question 7(c):
| At turning points: $(x^2 - 2x+1)(4x-3) - (2x^2 - 3x - 2)(2x-2) = 0$ | M1 | 1 mark |
| $x^2 - 8x + 7 = 0$ | M1 | 1 mark |
| $x = 7$ (since $x=1$ is an asymptote) | | |
| Stationary point $\left(7, \frac{75}{12}\right)$ | A1 | 1 mark |
| | | **3** |
## Question 7(d):
| Axes and asymptotes | B1 | 1 mark | Axes and asymptotes |
| $(-0.5, 0)$, $(2, 0)$, $(0, -2)$, $(4, 2)$ | B1 | 1 mark | |
| Left hand branch | B1 | 1 mark | |
| Right hand branch | B1 | 1 mark | |
| | | **4** |
## Question 7(e):
| Correct sketch, FT from (d) | B1FT | 1 mark | FT from sketch in (d) |
| Finds $0$, $\frac{7}{4}$ and $4$ as critical points | M1 | 1 mark | Finds critical points |
| $x < 0$ | B1 | 1 mark | |
| $\frac{7}{4} < x < 4$ | A1 | 1 mark | |
| | | **4** |
The image shows only a **BLANK PAGE** from what appears to be a Cambridge International AS & A Level Mark Scheme specimen paper (from 2020, © UCLES 2017, Page 12 of 12).
There is **no mark scheme content** to extract from this page — it contains no questions, answers, mark allocations, or guidance notes.7 The curve $C$ has equation $y = \frac { 2 x ^ { 2 } - 3 x - 2 } { x ^ { 2 } - 2 x + 1 }$.\\
(a) State the equations of the asymptotes of $C$.\\
(b) Show that $y \leqslant \frac { 25 } { 12 }$ at all points on $C$.\\
(c) Find the coordinates of any stationary points of $C$.\\
(d) Sketch $C$, stating the coordinates of any intersections of $C$ with the coordinate axes and the asymptotes.\\
(e) Sketch the curve with equation $y = \left| \frac { 2 x ^ { 2 } - 3 x - 2 } { x ^ { 2 } - 2 x + 1 } \right|$ and find the set of values of $x$ for which $\left| \frac { 2 x ^ { 2 } - 3 x - 2 } { x ^ { 2 } - 2 x + 1 } \right| < 2$.\\
\hfill \mbox{\textit{CAIE Further Paper 1 2020 Q7 [17]}}