| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2016 |
| Session | March |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Straight Lines & Coordinate Geometry |
| Type | Perpendicular bisector of segment |
| Difficulty | Moderate -0.3 Part (i) is a standard perpendicular bisector question requiring midpoint calculation, gradient, and negative reciprocal—routine coordinate geometry. Part (ii) adds a modest step by requiring students to find the intersection point X and then calculate a distance, but the techniques remain straightforward and commonly practiced at A-level. |
| Spec | 1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.03b Straight lines: parallel and perpendicular relationships1.10f Distance between points: using position vectors |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Mid-point of \(AB = (7, 3)\) soi | B1 | |
| Grad. of \(AB = -2 \rightarrow\) grad of perp. bisector \(= 1/2\) soi | M1 | Use of \(m_1 m_2 = -1\) |
| Eqn of perp. bisector is \(y - 3 = \dfrac{1}{2}(x - 7)\) | A1 [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Eqn of \(CX\) is \(y - 2 = -2(x - 1)\) | M1 | Using their original gradient and \((1, 2)\) |
| \(\dfrac{1}{2}x - \dfrac{1}{2} = -2x + 4\) | DM1 | Solve simultaneously dependent on both previous M's |
| \(x = 9/5\), \(y = 2/5\) | A1 | |
| \(BX^2 = 7.2^2 + 1.4^2\) soi | M1 | |
| \(BX = 7.33\) | A1 [5] |
## Question 5:
### Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Mid-point of $AB = (7, 3)$ soi | B1 | |
| Grad. of $AB = -2 \rightarrow$ grad of perp. bisector $= 1/2$ soi | M1 | Use of $m_1 m_2 = -1$ |
| Eqn of perp. bisector is $y - 3 = \dfrac{1}{2}(x - 7)$ | A1 [3] | |
### Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Eqn of $CX$ is $y - 2 = -2(x - 1)$ | M1 | Using their original gradient and $(1, 2)$ |
| $\dfrac{1}{2}x - \dfrac{1}{2} = -2x + 4$ | DM1 | Solve simultaneously dependent on both previous M's |
| $x = 9/5$, $y = 2/5$ | A1 | |
| $BX^2 = 7.2^2 + 1.4^2$ soi | M1 | |
| $BX = 7.33$ | A1 [5] | |
---5 Two points have coordinates $A ( 5,7 )$ and $B ( 9 , - 1 )$.\\
(i) Find the equation of the perpendicular bisector of $A B$.
The line through $C ( 1,2 )$ parallel to $A B$ meets the perpendicular bisector of $A B$ at the point $X$.\\
(ii) Find, by calculation, the distance $B X$.
\hfill \mbox{\textit{CAIE P1 2016 Q5 [8]}}