| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2010 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Straight Lines & Coordinate Geometry |
| Type | Rectangle or parallelogram vertices |
| Difficulty | Standard +0.3 This is a straightforward coordinate geometry question requiring standard techniques: finding a perpendicular bisector (midpoint + negative reciprocal gradient), using the property that diagonals of a rhombus bisect at right angles, and calculating area. All steps are routine applications of formulas with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.03b Straight lines: parallel and perpendicular relationships1.03e Complete the square: find centre and radius of circle |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Mid-point of \(AC = (2, 3)\) | B1 | Co |
| Gradient of \(AC = \frac{1}{3}\) | ||
| Gradient of \(BD = -3\) | M1 | Use of \(m_1m_2 = -1\) |
| Equation \(y - 3 = -3(x - 2)\) | A1 | Co |
| [3] | ||
| (ii) If \(x = 0, y = 9, B (0, 9)\) | B1√ | √ on his equation. |
| Vector move \(D (4, -3)\) | M1 A1 | Valid method. co. |
| [3] | ||
| (iii) \(AC = \sqrt{40}\) | M1 | Correct use on either \(AC\) or \(BD\). |
| \(BD = \sqrt{160}\) | M1 A1 | Full and correct method. co |
| Area \(= 40\) (or by matrix method M2 A1) | [3] |
(i) Mid-point of $AC = (2, 3)$ | B1 | Co
Gradient of $AC = \frac{1}{3}$ | |
Gradient of $BD = -3$ | M1 | Use of $m_1m_2 = -1$
Equation $y - 3 = -3(x - 2)$ | A1 | Co
| [3] |
(ii) If $x = 0, y = 9, B (0, 9)$ | B1√ | √ on his equation.
Vector move $D (4, -3)$ | M1 A1 | Valid method. co.
| [3] |
(iii) $AC = \sqrt{40}$ | M1 | Correct use on either $AC$ or $BD$.
$BD = \sqrt{160}$ | M1 A1 | Full and correct method. co
Area $= 40$ (or by matrix method M2 A1) | [3] |
---8\\
\includegraphics[max width=\textwidth, alt={}, center]{71fe6352-e0dc-4c3a-8b54-99709a1782ca-3_796_695_1539_726}
The diagram shows a rhombus $A B C D$ in which the point $A$ is ( $- 1,2$ ), the point $C$ is ( 5,4 ) and the point $B$ lies on the $y$-axis. Find\\
(i) the equation of the perpendicular bisector of $A C$,\\
(ii) the coordinates of $B$ and $D$,\\
(iii) the area of the rhombus.
\hfill \mbox{\textit{CAIE P1 2010 Q8 [9]}}