| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2011 |
| Session | June |
| Marks | 7 |
| 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 asymptotes, then proving a range result by rearranging to form a quadratic in x and using discriminant analysis. While the techniques are standard for Further Maths, the 'show y can take all values' requires insight into discriminant conditions that goes beyond routine manipulation, making it moderately challenging but within expected FP2 scope. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Asymptote \(x = 2\) | B1 | For correct equation |
| \(y = x - 4 - \frac{13}{x-2}\) \(\Rightarrow\) asymptote \(y = x-4\) | M1 | For dividing out (remainder not required) |
| A1 | For correct equation of asymptote (ignore any extras) | |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(x^2 - (y+6)x + (2y-5) = 0\) | M1 | For forming quadratic in \(x\) — N.B. answer given |
| \(b^2 - 4ac \geq 0 \Rightarrow (y+6)^2 - 4(2y-5) \geq 0\) | M1 | For considering discriminant |
| \(\Rightarrow y^2 + 4y + 56 \geq 0\) | A1 | For correct simplified expression in \(y\) soi |
| \(\Rightarrow (y+2)^2 + 52 \geq 0\): this is true \(\forall y\), so \(y\) takes all values | A1 | For completing square (or equivalent) and correct conclusion www |
| METHOD 2 | ||
| Obtain \(\frac{dy}{dx} = \frac{x^2 - 4x + 17}{(x-2)^2}\) OR \(1 + \frac{13}{(x-2)^2}\) | M1 | For finding \(\frac{dy}{dx}\) either by direct differentiation or dividing out first |
| A1 | For correct expression oe | |
| \(\Rightarrow \frac{dy}{dx} \geq 1\ \forall x\), so \(y\) takes all values | M1 | For drawing a conclusion |
| A1 | For correct conclusion www | |
| Total: 4 | ||
| Alternate scheme (Sketching graph): | ||
| Graph correct approaching asymptotes from both sides | B1 | A graph with no explanation can only score 2 |
| Graph completely correct | B1 | |
| Explanation about no turning values | B1 | |
| Correct conclusion | B1 |
# Question 2(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Asymptote $x = 2$ | B1 | For correct equation |
| $y = x - 4 - \frac{13}{x-2}$ $\Rightarrow$ asymptote $y = x-4$ | M1 | For dividing out (remainder not required) |
| | A1 | For correct equation of asymptote (ignore any extras) |
| **Total: 3** | | |
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# Question 2(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x^2 - (y+6)x + (2y-5) = 0$ | M1 | For forming quadratic in $x$ — **N.B. answer given** |
| $b^2 - 4ac \geq 0 \Rightarrow (y+6)^2 - 4(2y-5) \geq 0$ | M1 | For considering discriminant |
| $\Rightarrow y^2 + 4y + 56 \geq 0$ | A1 | For correct simplified expression in $y$ **soi** |
| $\Rightarrow (y+2)^2 + 52 \geq 0$: this is true $\forall y$, so $y$ takes all values | A1 | For completing square (or equivalent) and correct conclusion **www** |
| **METHOD 2** | | |
| Obtain $\frac{dy}{dx} = \frac{x^2 - 4x + 17}{(x-2)^2}$ OR $1 + \frac{13}{(x-2)^2}$ | M1 | For finding $\frac{dy}{dx}$ either by direct differentiation or dividing out first |
| | A1 | For correct expression **oe** |
| $\Rightarrow \frac{dy}{dx} \geq 1\ \forall x$, so $y$ takes all values | M1 | For drawing a conclusion |
| | A1 | For correct conclusion **www** |
| **Total: 4** | | |
| **Alternate scheme (Sketching graph):** | | |
| Graph correct approaching asymptotes from both sides | B1 | A graph with no explanation can only score 2 |
| Graph completely correct | B1 | |
| Explanation about no turning values | B1 | |
| Correct conclusion | B1 | |
---2 A curve has equation $y = \frac { x ^ { 2 } - 6 x - 5 } { x - 2 }$.\\
(i) Find the equations of the asymptotes.\\
(ii) Show that $y$ can take all real values.
\hfill \mbox{\textit{OCR FP2 2011 Q2 [7]}}