| Exam Board | OCR MEI |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric curves and Cartesian conversion |
| Type | Convert to Cartesian (polynomial/rational) |
| Difficulty | Moderate -0.8 Part (i) is a straightforward elimination of parameter by substituting t = x/8 into the y equation, yielding a simple quadratic. Part (ii) requires finding the maximum height using standard calculus or completing the square. Both parts are routine applications of well-practiced techniques with no conceptual challenges. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian3.02i Projectile motion: constant acceleration model |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(t = \frac{x}{8}\) | M1 | \(t\) as subject |
| \(y = 10\left(\frac{x}{8}\right) - 5\left(\frac{x}{8}\right)^2\) | M1 A1 | Substitute for \(t\) |
| \(= \frac{5x}{4} - \frac{5x^2}{64}\) | B1 | |
| Total: 4 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(y = \frac{5x}{4} - \frac{5x^2}{64} = \frac{5}{64}\left(64 - (x-8)^2\right)\) | M1 A1 | Or use calculus |
| Maximum height is \(5\) metres (when \(x = 8\)) | B1 | |
| Total: 3 marks |
## Question 5(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $t = \frac{x}{8}$ | M1 | $t$ as subject |
| $y = 10\left(\frac{x}{8}\right) - 5\left(\frac{x}{8}\right)^2$ | M1 A1 | Substitute for $t$ |
| $= \frac{5x}{4} - \frac{5x^2}{64}$ | B1 | |
| **Total: 4 marks** | | |
## Question 5(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $y = \frac{5x}{4} - \frac{5x^2}{64} = \frac{5}{64}\left(64 - (x-8)^2\right)$ | M1 A1 | Or use calculus |
| Maximum height is $5$ metres (when $x = 8$) | B1 | |
| **Total: 3 marks** | | |
---5 A ball is thrown towards a hedge. Its position relative to the point from which it was thrown is given by the parametric equations
$$x = 8 t , y = 10 t - 5 t ^ { 2 }$$
(i) Find the cartesian equation of the trajectory of the ball.\\
(ii) The ball just clears the hedge. What can you say about the height of the hedge?
\hfill \mbox{\textit{OCR MEI C4 Q5 [7]}}