| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2015 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Straight Lines & Coordinate Geometry |
| Type | Parameter from distance condition |
| Difficulty | Moderate -0.3 This is a straightforward coordinate geometry question requiring standard distance formula application and perpendicular bisector calculation. Part (i) involves setting up and solving a quadratic equation from AB=BC, while part (ii) requires finding midpoint, perpendicular gradient, and line intersection—all routine AS-level techniques with no novel problem-solving required. |
| 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/Working | Marks | Guidance |
| \(AB = 10\) | B1 | |
| \(6^2 + (k-1)^2 = 10^2\) | M1 | Use of Pythagoras |
| \(k = -7\) and \(9\) | A1 [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(m\) of \(AB = -\frac{3}{4}\), \(m\) perp \(= \frac{4}{3}\) | B1 M1 | B1 M1 Use of \(m_1 m_2 = -1\) |
| \(M = (1, 4)\) | B1 | |
| Eqn \(y - 4 = \frac{4}{3}(x-1)\) | M1 A1 | |
| Set \(y\) to \(0\), \(\rightarrow x = -2\) | [5] | Complete method leading to \(D\) |
## Question 6:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $AB = 10$ | **B1** | |
| $6^2 + (k-1)^2 = 10^2$ | **M1** | Use of Pythagoras |
| $k = -7$ and $9$ | **A1** [3] | |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $m$ of $AB = -\frac{3}{4}$, $m$ perp $= \frac{4}{3}$ | **B1 M1** | B1 M1 Use of $m_1 m_2 = -1$ |
| $M = (1, 4)$ | **B1** | |
| Eqn $y - 4 = \frac{4}{3}(x-1)$ | **M1 A1** | |
| Set $y$ to $0$, $\rightarrow x = -2$ | [5] | Complete method leading to $D$ |
---6 Points $A , B$ and $C$ have coordinates $A ( - 3,7 ) , B ( 5,1 )$ and $C ( - 1 , k )$, where $k$ is a constant.\\
(i) Given that $A B = B C$, calculate the possible values of $k$.
The perpendicular bisector of $A B$ intersects the $x$-axis at $D$.\\
(ii) Calculate the coordinates of $D$.
\hfill \mbox{\textit{CAIE P1 2015 Q6 [8]}}