| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2002 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas Between Curves |
| Type | Curve-Line Intersection Area |
| Difficulty | Moderate -0.8 This is a straightforward area-between-curves question requiring finding an intersection point by solving 3√x = x (which gives x = 9 quickly), then integrating (3√x - x) from 0 to 9. Both steps are routine applications of standard techniques with no conceptual challenges, making it easier than average but not trivial due to the integration setup required. |
| Spec | 1.02q Use intersection points: of graphs to solve equations1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| (i) P is \((9,9)\) | B1 | Correct only – needs both coordinates |
| Answer | Marks | Guidance |
|---|---|---|
| Area = 54 | M1, A1, DM1, M1, A1 | Used once to find area under a curve or line correct only; Use of his limits correctly; Anywhere – correct attempt at area of triangle Correct only |
| Answer | Marks |
|---|---|
| Subtract the areas \(\Rightarrow 13.5\) |
**(i)** P is $(9,9)$ | B1 | Correct only – needs both coordinates
**(ii)** Area under curve $= \int y \, dx = 3x^{(7/2)} + (3/2)$
Use of limits in either part
Area = 54 | M1, A1, DM1, M1, A1 | Used once to find area under a curve or line correct only; Use of his limits correctly; Anywhere – correct attempt at area of triangle Correct only
Area under line $= \frac{1}{2}x^2$ or uses $\frac{1}{2}bh = 40.5$
Subtract the areas $\Rightarrow 13.5$ | |3\\
\includegraphics[max width=\textwidth, alt={}, center]{b2cefbd6-6e89-495a-9f42-60f76c8c5975-2_629_659_715_740}
The diagram shows the curve $y = 3 \sqrt { } x$ and the line $y = x$ intersecting at $O$ and $P$. Find\\
(i) the coordinates of $P$,\\
(ii) the area of the shaded region.
\hfill \mbox{\textit{CAIE P1 2002 Q3 [6]}}