| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2014 |
| Session | November |
| Marks | 7 |
| 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 two-part coordinate geometry question requiring standard formulas. Part (i) uses the gradient formula and perpendicular gradient relationship (negative reciprocal). Part (ii) applies the distance formula leading to a quadratic equation. Both parts are routine applications of well-practiced techniques with no conceptual challenges, making it slightly easier than average for A-level. |
| Spec | 1.03b Straight lines: parallel and perpendicular relationships1.10f Distance between points: using position vectors |
| Answer | Marks | Guidance |
|---|---|---|
| \(m = \frac{3a + 9 - (2a - 1)}{2a + 4 - a} = \frac{a + 10}{a + 4}\) oe e.g. \(\frac{-a - 10}{-a - 4}\) | M1A1 | cao Allow omission of brackets for M1 |
| Gradient of perpendicular \(= \frac{-(a + 4)}{a + 10}\) oe but not \(\frac{-1}{\left(\frac{a+10}{a+4}\right)}\) | A1 | Do not ISW. Max penalty for erroneous cancellation 1 mark [3] |
| Answer | Marks | Guidance |
|---|---|---|
| \((\sqrt{[(a+4)^2 + (a+10)^2]}) = (\sqrt{260})\) | M1 | Allow their \((a+4), (a+10)\) from (i). Allow \((-a-4)^2\) etc. Allow omission of brackets |
| \((\sqrt{[(a+4)^2 + (a+10)^2]})\) cao | A1 | |
| \((2)(a^2 + 14a - 72) (= 0)\) | A1 | |
| \(a = 4\) or \(-18\) cao | A1 | [4] |
### (i)
$m = \frac{3a + 9 - (2a - 1)}{2a + 4 - a} = \frac{a + 10}{a + 4}$ oe e.g. $\frac{-a - 10}{-a - 4}$ | **M1A1** | cao Allow omission of brackets for M1
Gradient of perpendicular $= \frac{-(a + 4)}{a + 10}$ oe but not $\frac{-1}{\left(\frac{a+10}{a+4}\right)}$ | **A1** | Do not ISW. Max penalty for erroneous cancellation 1 mark [3]
### (ii)
$(\sqrt{[(a+4)^2 + (a+10)^2]}) = (\sqrt{260})$ | **M1** | Allow their $(a+4), (a+10)$ from (i). Allow $(-a-4)^2$ etc. Allow omission of brackets
$(\sqrt{[(a+4)^2 + (a+10)^2]})$ cao | **A1** |
$(2)(a^2 + 14a - 72) (= 0)$ | **A1** |
$a = 4$ or $-18$ cao | **A1** | [4]
---$A$ is the point $(a, 2a - 1)$ and $B$ is the point $(2a + 4, 3a + 9)$, where $a$ is a constant.
\begin{enumerate}[label=(\roman*)]
\item Find, in terms of $a$, the gradient of a line perpendicular to $AB$. [3]
\item Given that the distance $AB$ is $\sqrt{260}$, find the possible values of $a$. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2014 Q6 [7]}}