| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2014 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Straight Lines & Coordinate Geometry |
| Type | Rectangle or parallelogram vertices |
| Difficulty | Moderate -0.3 This is a straightforward coordinate geometry question requiring standard techniques: finding perpendicular gradient for part (i), solving simultaneous equations for intersection in part (ii), and using the parallelogram property (diagonals bisect) for part (iii). All methods are routine AS-level procedures with no novel problem-solving required, making it slightly easier than average. |
| 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 |
|---|---|---|
| Eqn \(AD: y - 6 = \frac{1}{3}(x-2)\) or \(3y = x + 16\) | B1, M1, A1 [3] | oe; use of \(m_1m_2 = -1\) with grad \(AB\); co – OK unsimplified |
| Answer | Marks | Guidance |
|---|---|---|
| \(\rightarrow D(6\frac{1}{2}, 7\frac{1}{2})\) | B1, ✓, M1, A1 [3] | OK unsimplified. ✓ on m of AB. Reasonable algebra leading to \(x =\) or \(y =\) with \(AD\) and \(CD\) |
| Answer | Marks | Guidance |
|---|---|---|
| Length of \(BE = 15\) | B1, B1 [2] | May be implied; co |
**(i)** $m_{AB} = -3$ or $\frac{-9}{3}$
$m_{AD} = \frac{1}{3}$
Eqn $AD: y - 6 = \frac{1}{3}(x-2)$ or $3y = x + 16$ | B1, M1, A1 [3] | oe; use of $m_1m_2 = -1$ with grad $AB$; co – OK unsimplified
**(ii)** Eqn $CD: y - 3 = -3(x-8)$ or $y = -3x + 27$
Sim Eqns
$\rightarrow D(6\frac{1}{2}, 7\frac{1}{2})$ | B1, ✓, M1, A1 [3] | OK unsimplified. ✓ on m of AB. Reasonable algebra leading to $x =$ or $y =$ with $AD$ and $CD$
**(iii)** Use of vectors or mid-point
$\rightarrow E(5, 12)$ or mid-point $(5,4.5)$
Length of $BE = 15$ | B1, B1 [2] | May be implied; co\includegraphics{figure_9}
The diagram shows a trapezium $ABCD$ in which $AB$ is parallel to $DC$ and angle $BAD$ is $90°$. The coordinates of $A$, $B$ and $C$ are $(2, 6)$, $(5, -3)$ and $(8, 3)$ respectively.
\begin{enumerate}[label=(\roman*)]
\item Find the equation of $AD$. [3]
\item Find, by calculation, the coordinates of $D$. [3]
\end{enumerate}
The point $E$ is such that $ABCE$ is a parallelogram.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{2}
\item Find the length of $BE$. [2]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2014 Q9 [8]}}