| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2018 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Straight Lines & Coordinate Geometry |
| Type | Rectangle or parallelogram vertices |
| Difficulty | Moderate -0.5 This is a straightforward coordinate geometry problem requiring finding equations of lines (using two points and perpendicular gradient) and solving simultaneous equations to find an intersection point. While it involves multiple steps, each technique is standard A-level fare with no novel insight required. The kite context provides clear geometric constraints that guide the solution method. |
| Spec | 1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.03b Straight lines: parallel and perpendicular relationships |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply \(dx = -2\cos\theta\sin\theta\, d\theta\), or equivalent | B1 | |
| Substitute for \(x\) and \(dx\), and use Pythagoras | M1 | |
| Obtain integrand \(\pm 2\cos^2\theta\) | A1 | |
| Justify change of limits and obtain given answer correctly | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Obtain indefinite integral of the form \(a\theta + b\sin 2\theta\) | M1* | |
| Obtain \(\theta + \dfrac{1}{2}\sin 2\theta\) | A1 | |
| Use correct limits correctly | M1(dep*) | |
| Obtain answer \(\dfrac{1}{6}\pi\) with no errors seen | A1 |
## Question 5(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply $dx = -2\cos\theta\sin\theta\, d\theta$, or equivalent | B1 | |
| Substitute for $x$ and $dx$, and use Pythagoras | M1 | |
| Obtain integrand $\pm 2\cos^2\theta$ | A1 | |
| Justify change of limits and obtain given answer correctly | A1 | |
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## Question 5(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Obtain indefinite integral of the form $a\theta + b\sin 2\theta$ | M1* | |
| Obtain $\theta + \dfrac{1}{2}\sin 2\theta$ | A1 | |
| Use correct limits correctly | M1(dep*) | |
| Obtain answer $\dfrac{1}{6}\pi$ with no errors seen | A1 | |5\\
\includegraphics[max width=\textwidth, alt={}, center]{e835a60b-fbeb-49fb-ba6b-ac12c702d487-08_558_785_258_680}
The diagram shows a kite $O A B C$ in which $A C$ is the line of symmetry. The coordinates of $A$ and $C$ are $( 0,4 )$ and $( 8,0 )$ respectively and $O$ is the origin.\\
(i) Find the equations of $A C$ and $O B$.\\
(ii) Find, by calculation, the coordinates of $B$.\\
\hfill \mbox{\textit{CAIE P3 2018 Q5 [7]}}