| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Straight Lines & Coordinate Geometry |
| Type | Equation of line through two points |
| Difficulty | Standard +0.3 This is a multi-part coordinate geometry question requiring finding line equations (standard C1 skill), finding an intersection point, and verifying a distance equality using the distance formula. While it involves multiple steps, each component is routine and the 'show that' in part (iii) provides the target answer, making it slightly easier than average but still requiring careful algebraic manipulation. |
| Spec | 1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.03b Straight lines: parallel and perpendicular relationships |
| Answer | Marks |
|---|---|
| \(\text{grad} = \frac{7-4}{9-7} = \frac{3}{2}\) | M1 A1 |
| \(\therefore y - 4 = \frac{3}{2}(x-7)\) | M1 |
| \(2y - 8 = 3x - 21\) | |
| \(3x - 2y - 13 = 0\) | A1 |
| Answer | Marks |
|---|---|
| \(y = 8x\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| At \(R\): \(3x - 2(8x) - 13 = 0\) | ||
| \(x = -1\) | M1 | |
| \(\therefore R(-1, -8)\) | 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) |
# Question 9:
## Part (i):
$\text{grad} = \frac{7-4}{9-7} = \frac{3}{2}$ | M1 A1 |
$\therefore y - 4 = \frac{3}{2}(x-7)$ | M1 |
$2y - 8 = 3x - 21$ | |
$3x - 2y - 13 = 0$ | A1 |
## Part (ii):
$y = 8x$ | B1 |
## Part (iii):
At $R$: $3x - 2(8x) - 13 = 0$ | |
$x = -1$ | M1 |
$\therefore R(-1, -8)$ | 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)**
---(i) Find an equation for the straight line $l$ which passes through $P$ and $Q$. Give your answer in the form $a x + b y + c = 0$, where $a$, $b$ and $c$ are integers.
The straight line $m$ has gradient 8 and passes through the origin, $O$.\\
(ii) Write down an equation for $m$.
The lines $l$ and $m$ intersect at the point $R$.\\
(iii) Show that $O P = O R$.\\
\hfill \mbox{\textit{OCR C1 Q9 [10]}}