| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2005 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Straight Lines & Coordinate Geometry |
| Type | Perpendicular bisector of segment |
| Difficulty | Moderate -0.8 This is a straightforward coordinate geometry question requiring standard techniques: finding midpoint, gradient of AB, perpendicular gradient, equation of perpendicular bisector, equation of BC, then solving simultaneous equations. All steps are routine textbook procedures with no problem-solving insight needed, making it easier than average but not trivial due to the multi-step nature. |
| Spec | 1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.03b Straight lines: parallel and perpendicular relationships |
| Answer | Marks | Guidance |
|---|---|---|
| gradient of \(AB = \frac{2}{3}\), Perp= \(-\frac{3}{2}\) | B1 M1 | Co. Use of step/x-step + \(m_1m_2=-1\) for AB |
| Equation \(y - 8 = -\frac{3}{2}(x-5)\) | M1 | Use of \(y-k=m(x-h)\) not \((y+k)\) etc |
| \(\rightarrow 2y + 3x = 31\) (or locus method M1A1M1A1) | A1 | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Sim Eqns \(\rightarrow (7,5)\) | M1A1 DM1A1 | Use of \(y-k=m(x-h)\) not \((y+k)\) etc. co Correct attempt at soln of \(BC\) with his answer to (i). |
(i) $M(5,8)$
gradient of $AB = \frac{2}{3}$, Perp= $-\frac{3}{2}$ | B1 M1 | Co. Use of step/x-step + $m_1m_2=-1$ for AB
Equation $y - 8 = -\frac{3}{2}(x-5)$ | M1 | Use of $y-k=m(x-h)$ not $(y+k)$ etc
$\rightarrow 2y + 3x = 31$ (or locus method M1A1M1A1) | A1 | [4]
(ii) $BC. y = 5(x-6) \rightarrow y = 5x - 30$
Sim Eqns $\rightarrow (7,5)$ | M1A1 DM1A1 | Use of $y-k=m(x-h)$ not $(y+k)$ etc. co Correct attempt at soln of $BC$ with his answer to (i). | [4]7 Three points have coordinates $A ( 2,6 ) , B ( 8,10 )$ and $C ( 6,0 )$. The perpendicular bisector of $A B$ meets the line $B C$ at $D$. Find\\
(i) the equation of the perpendicular bisector of $A B$ in the form $a x + b y = c$,\\
(ii) the coordinates of $D$.
\hfill \mbox{\textit{CAIE P1 2005 Q7 [8]}}