| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric curves and Cartesian conversion |
| Type | Convert to Cartesian (tan/sec/cot/cosec identities) |
| Difficulty | Standard +0.3 This is a straightforward parametric equations question requiring standard techniques: (i) uses the identity sec²θ = 1 + tan²θ to eliminate the parameter, and (ii) applies a given substitution for integration. Both parts follow well-practiced methods with no novel insight required, making it slightly easier than average. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.08e Area between curve and x-axis: using definite integrals1.08h Integration by substitution |
4.\\
\includegraphics[max width=\textwidth, alt={}, center]{c7b867af-0730-459e-9c76-15eb07b9e476-1_465_976_1539_388}
The diagram shows the curve with parametric equations
$$x = \tan \theta , \quad y = \cos ^ { 2 } \theta , \quad - \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 }$$
(i) Find a cartesian equation for the curve.
The shaded region is bounded by the curve, the $x$-axis and the lines $x = - 1$ and $x = 1$.\\
(ii) Using integration, with the substitution $x = \tan u$, find the area of the shaded region.\\
\hfill \mbox{\textit{OCR C4 Q4 [8]}}