| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2010 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Rational functions with parameters: analysis depending on parameter sign/range |
| Difficulty | Standard +0.3 This is a straightforward FP2 curve sketching question involving rational functions. Part (i) requires identifying asymptotes (routine), part (ii) uses calculus to find the minimum value (standard optimization), and part (iii) finds stationary points by differentiation using quotient rule. While it requires multiple techniques and some algebraic manipulation, all steps follow standard procedures without requiring novel insight or particularly complex reasoning. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks |
|---|---|
| Get \(x = 1, y = 0\) | B1, B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Rewrite as quadratic in \(x\) | M1 | \((x^2y - x(2y+k) + y = 0)\) |
| Use \(b^2 - 4ac \geq 0\) for all real \(x\) | M1 | Allow \(>, =\) here |
| Get correct inequality | A1 | \(4ky + k^2 \geq 0\) |
| State use of \(k>0\) to A.G. | A1 | |
| SC Use differentiation (parts (ii) and (iii)) | M1 | |
| Attempt prod/quotient rule | M1 | |
| Solve \(= 0\) for \(x = -1\) | A1 | |
| Use \(x = -1\) only (reject x=1), \(y = -\frac{k}{4}\) | A1 | |
| Fully justify minimum | B1 | |
| Attempt to justify for all \(x\) | M1 | |
| Clearly get A.G. | A1 |
| Answer | Marks |
|---|---|
| Replace \(y = -\frac{k}{4}\) in quadratic in \(x\) | M1 |
| Get \(x = -1\) only | A1 |
| B1 | Through origin with minimum at \((-1, -\frac{k}{4})\) seen or given in the answer |
| B1 | Correct shape (asymptotes and approaches) |
| Answer | Marks |
|---|---|
| Differentiate and solve \(\frac{dy}{dx} = 0\) for at least one x-value, independent of \(k\) | M1 |
| Get \(x = -1\) only | A1 |
## (i)
Get $x = 1, y = 0$ | B1, B1 |
## (ii)
Rewrite as quadratic in $x$ | M1 | $(x^2y - x(2y+k) + y = 0)$
Use $b^2 - 4ac \geq 0$ for all real $x$ | M1 | Allow $>, =$ here
Get correct inequality | A1 | $4ky + k^2 \geq 0$
State use of $k>0$ to A.G. | A1 |
**SC** Use differentiation (parts (ii) and (iii)) | M1 |
Attempt prod/quotient rule | M1 |
Solve $= 0$ for $x = -1$ | A1 |
Use $x = -1$ only (reject x=1), $y = -\frac{k}{4}$ | A1 |
Fully justify minimum | B1 |
Attempt to justify for all $x$ | M1 |
Clearly get A.G. | A1 |
## (iii)
Replace $y = -\frac{k}{4}$ in quadratic in $x$ | M1 |
Get $x = -1$ only | A1 |
| B1 | Through origin with minimum at $(-1, -\frac{k}{4})$ seen or given in the answer
| B1 | Correct shape (asymptotes and approaches)
**SC** (Start again)
Differentiate and solve $\frac{dy}{dx} = 0$ for at least one x-value, independent of $k$ | M1 |
Get $x = -1$ only | A1 |
---The equation of a curve is
$$y = \frac{kx}{(x-1)^2},$$
where $k$ is a positive constant.
\begin{enumerate}[label=(\roman*)]
\item Write down the equations of the asymptotes of the curve. [2]
\item Show that $y \geq -\frac{1}{4}k$. [4]
\item Show that the $x$-coordinate of the stationary point of the curve is independent of $k$, and sketch the curve. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR FP2 2010 Q8 [10]}}