| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2014 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Proving Excluded Range of Rational Function |
| Difficulty | Standard +0.8 This FP2 question requires polynomial division to find oblique asymptote, algebraic manipulation to prove a range restriction (solving a quadratic inequality in x and showing no real solutions), and curve sketching. While systematic, it demands multiple techniques and careful algebraic reasoning beyond standard C1-C3 material, placing it moderately above average difficulty. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.02y Partial fractions: decompose rational functions1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| \(\Rightarrow\) Oblique asymptote: \(y = x+3\) | B1, M1, A1 | Allow if fraction missing; Seen by an answer of \(x+a+\left(\frac{b}{x-3}\right)\). Condone incorrect \(b\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\Rightarrow 4 < y < 8\) | M1, M1, M1, A1 | Attempt to get quad; Finding discriminant; Dealing with inequality to find result |
| (iii) | B1, B1 | Asymptotes; Correct shape |
| [2] |
**(i)** $y = \frac{x^3-8}{x-3}$
Vertical asymptote $x = 3$
$y = \frac{x^3-8}{x-3} = \frac{x^3-9+1}{x-3} = \frac{(x-3)(x+3)+1}{x-3} = x+3+\frac{1}{x-3}$
$\Rightarrow$ Oblique asymptote: $y = x+3$ | B1, M1, A1 | Allow if fraction missing; Seen by an answer of $x+a+\left(\frac{b}{x-3}\right)$. Condone incorrect $b$
**(ii)** $xy - 3y = x^2 - 8 \Rightarrow x^2 - xy + 3y - 8 = 0$
Discriminant is $y^2 - 4 \cdot 3y - 8$
$\Rightarrow y^2 - 12y + 32 < 0 \Rightarrow (y-8)(y-4) < 0$
$\Rightarrow 4 < y < 8$ | M1, M1, M1, A1 | Attempt to get quad; Finding discriminant; Dealing with inequality to find result
**(iii)** | B1, B1 | Asymptotes; Correct shape | $x = 3$ is identified and the other line has +ve gradient. Must include a vertical and oblique (with +ve gradient) asymptotes and curve must approach them.
| [2] |
---5 A curve has equation $y = \frac { x ^ { 2 } - 8 } { x - 3 }$.\\
(i) Find the equations of the asymptotes of the curve.\\
(ii) Prove that there are no points on the curve for which $4 < y < 8$.\\
(iii) Sketch the curve. Indicate the asymptotes in your sketch.
\hfill \mbox{\textit{OCR FP2 2014 Q5 [9]}}