| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2005 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Straight Lines & Coordinate Geometry |
| Type | Coordinates from geometric constraints |
| Difficulty | Moderate -0.8 This is a straightforward C1 coordinate geometry question requiring only standard techniques: midpoint formula to find coordinates, perpendicular gradient rule, and solving simultaneous equations. All steps are routine with no problem-solving insight needed, making it easier than average but not trivial due to multiple parts. |
| Spec | 1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.03b Straight lines: parallel and perpendicular relationships |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(p = 15, q = -3\) | B1 B1 | (2 marks) |
| (b) Grad. of line \(ADC\): \(m = -\frac{5}{7}\); Grad. of perp. line \(= -\frac{1}{m}\) (\(= \frac{7}{5}\)) | B1, M1 | |
| Equation of \(l\): \(y - 2 = \frac{7}{5}(x - 8)\) | M1 A1ft | |
| \(7x - 5y - 46 = 0\) (Allow rearrangements, e.g. \(5y = 7x - 46\)) | A1 | (5 marks) |
| (c) Substitute \(y = 7\) into equation of \(l\) and find \(x = \ldots\) | M1 | |
| \(\frac{81}{7}\) or \(11\frac{4}{7}\) (or exact equiv.) | A1 | (2 marks) |
**(a)** $p = 15, q = -3$ | B1 B1 | (2 marks)
**(b)** Grad. of line $ADC$: $m = -\frac{5}{7}$; Grad. of perp. line $= -\frac{1}{m}$ ($= \frac{7}{5}$) | B1, M1 |
Equation of $l$: $y - 2 = \frac{7}{5}(x - 8)$ | M1 A1ft |
$7x - 5y - 46 = 0$ (Allow rearrangements, e.g. $5y = 7x - 46$) | A1 | (5 marks)
**(c)** Substitute $y = 7$ into equation of $l$ and find $x = \ldots$ | M1 |
$\frac{81}{7}$ or $11\frac{4}{7}$ (or exact equiv.) | A1 | (2 marks)
**Total: 9 marks**
**Special case:**
(a) If B0 B0 from main scheme, allow M1 for a correct method, e.g. $8 = \frac{1+p}{2}$.
(b) Finding eqn. of $ADC$ instead of $l$ scores M1 A0 A0.
---8.
\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\includegraphics[alt={},max width=\textwidth]{bace07ee-1eb8-43d6-8229-152d1f74ab59-14_687_1196_280_388}
\end{center}
\end{figure}
The points $A ( 1,7 ) , B ( 20,7 )$ and $C ( p , q )$ form the vertices of a triangle $A B C$, as shown in Figure 2. The point $D ( 8,2 )$ is the mid-point of $A C$.
\begin{enumerate}[label=(\alph*)]
\item Find the value of $p$ and the value of $q$.
The line $l$, which passes through $D$ and is perpendicular to $A C$, intersects $A B$ at $E$.
\item Find an equation for $l$, in the form $a x + b y + c = 0$, where $a$, $b$ and $c$ are integers.
\item Find the exact $x$-coordinate of $E$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 2005 Q8 [9]}}