| Exam Board | OCR MEI |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2010 |
| Session | January |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Sketch polar curve |
| Difficulty | Standard +0.8 This is a multi-part Further Maths question combining calculus (differentiation of arctan with chain rule, then working backwards to evaluate a non-standard integral), coordinate conversion to polar form, and analysis/sketching of a polar curve. While each individual technique is standard for FP2, the integration requires insight to connect the derivative result, and the polar curve analysis demands careful consideration of the denominator's range. This is moderately challenging for Further Maths but not exceptionally difficult. |
| Spec | 1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs1.07l Derivative of ln(x): and related functions1.08h Integration by substitution4.09a Polar coordinates: convert to/from cartesian4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve |
1
\begin{enumerate}[label=(\alph*)]
\item Given that $y = \arctan \sqrt { x }$, find $\frac { \mathrm { d } y } { \mathrm {~d} x }$, giving your answer in terms of $x$. Hence show that
$$\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { x } ( x + 1 ) } \mathrm { d } x = \frac { \pi } { 2 }$$
\item A curve has cartesian equation
$$x ^ { 2 } + y ^ { 2 } = x y + 1$$
\begin{enumerate}[label=(\roman*)]
\item Show that the polar equation of the curve is
$$r ^ { 2 } = \frac { 2 } { 2 - \sin 2 \theta }$$
\item Determine the greatest and least positive values of $r$ and the values of $\theta$ between 0 and $2 \pi$ for which they occur.
\item Sketch the curve.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR MEI FP2 2010 Q1 [18]}}