| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2011 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Sketch polar curve |
| Difficulty | Standard +0.3 This is a straightforward multi-part question on curve sketching and function analysis. Parts involve identifying an asymptote (y=0, routine), finding range using calculus or completing the square (standard technique), and analyzing y²=f(x) which requires recognizing symmetry and domain restrictions. All parts are direct applications of standard A-level techniques with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.02u Functions: definition and vocabulary (domain, range, mapping)1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Asymptote \(y=0\) | B1 1 | For correct equation (allow \(x\)-axis) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(y=\frac{5ax}{x^2+a^2} \Rightarrow yx^2-5ax+a^2y=0\) | M1 | For expressing as a quadratic in \(x\) |
| M1 | For using \(b^2-4ac\lessgtr 0\) | |
| \(b^2\geqslant 4ac \Rightarrow 25a^2\geqslant 4a^2y^2 \Rightarrow -\frac{5}{2}\leqslant y\leqslant\frac{5}{2}\) | A1 | For \(25a^2-4a^2y^2\) seen or implied |
| A1 4 | For correct range |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(y=\frac{5ax}{x^2+a^2} \Rightarrow \frac{dy}{dx}=\frac{-5a(x^2-a^2)}{(x^2+a^2)^2}\) | M1* | For differentiating \(y\) by quotient OR product rule |
| \(\frac{dy}{dx}=0 \Rightarrow x=\pm a \Rightarrow y=\pm\frac{5}{2}\) | A1 | For correct values of \(x\) |
| Asymptote, sketch etc \(\Rightarrow -\frac{5}{2}\leqslant y\leqslant\frac{5}{2}\) | M1 | For finding \(y\) values and giving argument for range |
| A1(*dep) 4 | For correct range |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(y=0\) | B1 1 | For correct equation (allow \(x\)-axis) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Maximum \(\sqrt{\frac{5}{2}}\), minimum \(-\sqrt{\frac{5}{2}}\) | B1\(\checkmark\) | For correct maximum f.t. from (i)(b) |
| B1\(\checkmark\) 2 | For correct minimum f.t. from (i)(b); allow decimals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(x\geqslant 0\) | B1 1 | For correct set of values (allow in words) |
## Question 3(i)(a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Asymptote $y=0$ | B1 **1** | For correct equation (allow $x$-axis) |
## Question 3(i)(b):
**Method 1**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y=\frac{5ax}{x^2+a^2} \Rightarrow yx^2-5ax+a^2y=0$ | M1 | For expressing as a quadratic in $x$ |
| | M1 | For using $b^2-4ac\lessgtr 0$ |
| $b^2\geqslant 4ac \Rightarrow 25a^2\geqslant 4a^2y^2 \Rightarrow -\frac{5}{2}\leqslant y\leqslant\frac{5}{2}$ | A1 | For $25a^2-4a^2y^2$ seen or implied |
| | A1 **4** | For correct range |
**Method 2**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y=\frac{5ax}{x^2+a^2} \Rightarrow \frac{dy}{dx}=\frac{-5a(x^2-a^2)}{(x^2+a^2)^2}$ | M1* | For differentiating $y$ by quotient OR product rule |
| $\frac{dy}{dx}=0 \Rightarrow x=\pm a \Rightarrow y=\pm\frac{5}{2}$ | A1 | For correct values of $x$ |
| Asymptote, sketch etc $\Rightarrow -\frac{5}{2}\leqslant y\leqslant\frac{5}{2}$ | M1 | For finding $y$ values and giving argument for range |
| | A1(*dep) **4** | For correct range |
## Question 3(ii)(a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y=0$ | B1 **1** | For correct equation (allow $x$-axis) |
## Question 3(ii)(b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Maximum $\sqrt{\frac{5}{2}}$, minimum $-\sqrt{\frac{5}{2}}$ | B1$\checkmark$ | For correct maximum f.t. from **(i)(b)** |
| | B1$\checkmark$ **2** | For correct minimum f.t. from **(i)(b)**; allow decimals |
## Question 3(ii)(c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $x\geqslant 0$ | B1 **1** | For correct set of values (allow in words) |
---3 The function f is defined by $\mathrm { f } ( x ) = \frac { 5 a x } { x ^ { 2 } + a ^ { 2 } }$, for $x \in \mathbb { R }$ and $a > 0$.\\
(i) For the curve with equation $y = \mathrm { f } ( x )$,
\begin{enumerate}[label=(\alph*)]
\item write down the equation of the asymptote,
\item find the range of values that $y$ can take.\\
(ii) For the curve with equation $y ^ { 2 } = \mathrm { f } ( x )$, write down\\
(a) the equation of the line of symmetry,\\
(b) the maximum and minimum values of $y$,
\item the set of values of $x$ for which the curve is defined.
\end{enumerate}
\hfill \mbox{\textit{OCR FP2 2011 Q3 [9]}}