| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2017 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Sketch rational with quadratic numerator |
| Difficulty | Standard +0.8 This is a comprehensive Further Maths curve sketching question requiring polynomial division to find the oblique asymptote, differentiation using the quotient rule, solving a quadratic for turning points, and synthesizing multiple features. While systematic, it demands more technical facility and integration of techniques than standard A-level questions, placing it moderately above average difficulty. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02n Sketch curves: simple equations including polynomials1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x = 1\) | B1 | |
| \(y = 2 - x\) | M1A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(y' = -1 + 4(1-x)^{-2} = 0\) | M1 | |
| \(\Rightarrow x = -1, 3\) | A1 | |
| Turning points are \((-1, 5)\) and \((3, -3)\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((0, 6)\) | B1 | |
| \(y = 0 \Rightarrow x^2 - 3x + 6\); \(\Delta = 9 - 24 \Rightarrow\) No intersection with \(x\)-axis | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| [Graph with correct asymptotes] | B1 FT | Asymptotes correct |
| [Two branches drawn correctly] | B1B1 | Each branch |
## Question 9(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x = 1$ | B1 | |
| $y = 2 - x$ | M1A1 | |
## Question 9(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $y' = -1 + 4(1-x)^{-2} = 0$ | M1 | |
| $\Rightarrow x = -1, 3$ | A1 | |
| Turning points are $(-1, 5)$ and $(3, -3)$ | A1 | |
## Question 9(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(0, 6)$ | B1 | |
| $y = 0 \Rightarrow x^2 - 3x + 6$; $\Delta = 9 - 24 \Rightarrow$ No intersection with $x$-axis | B1 | |
## Question 9(iv):
| Answer | Marks | Guidance |
|--------|-------|----------|
| [Graph with correct asymptotes] | B1 FT | Asymptotes correct |
| [Two branches drawn correctly] | B1B1 | Each branch |9 The curve $C$ has equation $y = \frac { x ^ { 2 } - 3 x + 6 } { 1 - x }$.\\
(i) Find the equations of the asymptotes of $C$.\\
(ii) Find the coordinates of the turning points of $C$.\\
(iii) Find the coordinates of any intersections with the coordinate axes.\\
(iv) Sketch $C$.
\hfill \mbox{\textit{CAIE FP1 2017 Q9 [11]}}