| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2007 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Proving Rational Function Takes All Real Values |
| Difficulty | Standard +0.8 This FP2 question requires polynomial division to find the oblique asymptote, solving a quadratic inequality/discriminant argument to prove the range is all reals, and synthesizing this into a sketch. While systematic, it demands multiple techniques and the discriminant approach in part (ii) requires insight beyond routine manipulation, placing it moderately above average difficulty. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Quote \(x = a\) | B1 | |
| Attempt to divide out | M1 | Allow M1 for \(y = x\) here; allow \((x-a) + k/(x-a)\) seen or implied |
| Get \(y = x - a\) | A1 | Must be equations |
| A1 | ||
| (ii) Attempt at quad. in \(x\) \((=0)\) | M1 | |
| Use \(b^2 - 4ac \geq 0\) for real \(x\) | M1 | Allow \(>\) |
| Get \(y^2 + 4a^2 \geq 0\) | A1 | |
| State/show their quad. is always \(> 0\) | B1 | Allow \(\geq\) |
| (iii) | B1√ | Two asymptotes from (i) (need not be labelled) |
| B1 | Both crossing points | |
| B1√ | Approaches – correct shape | |
| SR Attempt diff. by quotient/product rule | M1 | |
| Get quadratic in \(x\) for \(dy/dx = 0\) and note \(b^2 - 4ac < 0\) | A1 | |
| Consider horizontal asymptotes | B1 | |
| Fully justify answer | B1 |
**(i)** Quote $x = a$ | B1 |
Attempt to divide out | M1 | Allow M1 for $y = x$ here; allow $(x-a) + k/(x-a)$ seen or implied
Get $y = x - a$ | A1 | Must be equations
| A1 |
**(ii)** Attempt at quad. in $x$ $(=0)$ | M1 |
Use $b^2 - 4ac \geq 0$ for real $x$ | M1 | Allow $>$
Get $y^2 + 4a^2 \geq 0$ | A1 |
State/show their quad. is always $> 0$ | B1 | Allow $\geq$
**(iii)** | B1√ | Two asymptotes from (i) (need not be labelled)
| B1 | Both crossing points
| B1√ | Approaches – correct shape
SR Attempt diff. by quotient/product rule | M1 |
Get quadratic in $x$ for $dy/dx = 0$ and note $b^2 - 4ac < 0$ | A1 |
Consider horizontal asymptotes | B1 |
Fully justify answer | B1 |9 It is given that the equation of a curve is
$$y = \frac { x ^ { 2 } - 2 a x } { x - a }$$
where $a$ is a positive constant.\\
(i) Find the equations of the asymptotes of the curve.\\
(ii) Show that $y$ takes all real values.\\
(iii) Sketch the curve $y = \frac { x ^ { 2 } - 2 a x } { x - a }$.
\hfill \mbox{\textit{OCR FP2 2007 Q9 [11]}}