| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Straight Lines & Coordinate Geometry |
| Type | Intersection of two lines |
| Difficulty | Moderate -0.3 This is a straightforward C1 coordinate geometry question requiring standard techniques: finding gradient and equation of a line through two points, writing equation from gradient, finding intersection point, and calculating distances. Part (c) involves more steps but follows routine procedures with no novel insight required, making it slightly easier than average overall. |
| 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 |
|---|---|
| (a) \(\text{grad} = \frac{7-4}{9-7} = \frac{3}{2}\) | M1 A1 |
| \(\therefore y - 4 = \frac{3}{2}(x - 7)\) | M1 |
| Answer | Marks |
|---|---|
| \(3x - 2y - 13 = 0\) | A1 |
| (b) \(y = 8x\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(x = -1\) \(\therefore R(-1, -8)\) | M1 A1 | |
| \(OP = \sqrt{7^2 + 4^2} = \sqrt{49 + 16} = \sqrt{65}\) | M1 A1 | |
| \(OR = \sqrt{(-1)^2 + (-8)^2} = \sqrt{1 + 64} = \sqrt{65}\) \(\therefore OP = OR\) | A1 | (10 marks) |
(a) $\text{grad} = \frac{7-4}{9-7} = \frac{3}{2}$ | M1 A1 |
$\therefore y - 4 = \frac{3}{2}(x - 7)$ | M1 |
$2y - 8 = 3x - 21$
$3x - 2y - 13 = 0$ | A1 |
(b) $y = 8x$ | B1 |
(c) at $R$, $3x - 2(8x) - 13 = 0$
$x = -1$ $\therefore R(-1, -8)$ | M1 A1 |
$OP = \sqrt{7^2 + 4^2} = \sqrt{49 + 16} = \sqrt{65}$ | M1 A1 |
$OR = \sqrt{(-1)^2 + (-8)^2} = \sqrt{1 + 64} = \sqrt{65}$ $\therefore OP = OR$ | A1 | (10 marks)The points $P$ and $Q$ have coordinates $(7, 4)$ and $(9, 7)$ respectively.
\begin{enumerate}[label=(\alph*)]
\item Find an equation for the straight line $l$ which passes through $P$ and $Q$. Give your answer in the form $ax + by + c = 0$, where $a$, $b$ and $c$ are integers. [4]
\end{enumerate}
The straight line $m$ has gradient $8$ and passes through the origin, $O$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Write down an equation for $m$. [1]
\end{enumerate}
The lines $l$ and $m$ intersect at the point $R$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Show that $OP = OR$. [5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 Q8 [10]}}