| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2018 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Straight Lines & Coordinate Geometry |
| Type | Perpendicular line through point |
| Difficulty | Moderate -0.8 This is a straightforward coordinate geometry question requiring standard techniques: finding gradient of AB, using perpendicular gradient property (negative reciprocal), writing equation of BC through point B, finding x-intercept, then using distance formula. All steps are routine textbook exercises with no problem-solving insight needed, making it easier than average but not trivial due to multiple computational steps. |
| Spec | 1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.03b Straight lines: parallel and perpendicular relationships1.10c Magnitude and direction: of vectors |
| Answer | Marks | Guidance |
|---|---|---|
| Gradient \(m\) of \(AB = \frac{3}{4}\) | B1 | |
| Equation of \(BC\): \(y - 4 = \frac{-4}{3}(x-3)\) | M1A1 | Line through \((3,4)\) with gradient \(\frac{-1}{m}\). Expect \(y = \frac{-4}{3}x + 8\) |
| \(x = 6\) | A1 | Ignore any \(y\) coordinate given |
| Answer | Marks | Guidance |
|---|---|---|
| \((AC)^2 = 7^2 + 1^2 \rightarrow AC = 7.071\) | M1A1 | M mark for \(\sqrt{(\text{their } 6 +/-1)^2 + 1}\) |
## Question 4(i):
| Gradient $m$ of $AB = \frac{3}{4}$ | B1 | |
|---|---|---|
| Equation of $BC$: $y - 4 = \frac{-4}{3}(x-3)$ | M1A1 | Line through $(3,4)$ with gradient $\frac{-1}{m}$. Expect $y = \frac{-4}{3}x + 8$ |
| $x = 6$ | A1 | Ignore any $y$ coordinate given |
## Question 4(ii):
| $(AC)^2 = 7^2 + 1^2 \rightarrow AC = 7.071$ | M1A1 | M mark for $\sqrt{(\text{their } 6 +/-1)^2 + 1}$ |
|---|---|---|
---4 Two points $A$ and $B$ have coordinates $( - 1,1 )$ and $( 3,4 )$ respectively. The line $B C$ is perpendicular to $A B$ and intersects the $x$-axis at $C$.\\
(i) Find the equation of $B C$ and the $x$-coordinate of $C$.\\
(ii) Find the distance $A C$, giving your answer correct to 3 decimal places.\\
\hfill \mbox{\textit{CAIE P1 2018 Q4 [6]}}