| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2005 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Straight Lines & Coordinate Geometry |
| Type | Intersection of two lines |
| Difficulty | Moderate -0.8 This is a straightforward C1 coordinate geometry question requiring standard techniques: finding line equations from point-gradient form, solving simultaneous equations for intersection, and calculating triangle area using coordinates. All steps are routine with no problem-solving insight needed, making it easier than average but not trivial due to the multi-part nature. |
| Spec | 1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.03b Straight lines: parallel and perpendicular relationships |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(y-(-4)=\frac{1}{3}(x-9)\) | M1 A1 | Full method to find equation of \(l_1\); any unsimplified form |
| \(3y-x+21=0\) | A1 | o.e., condone 3 terms with integer coefficients e.g. \(3y+21=x\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Equation of \(l_2\) is: \(y=-2x\) | B1 | o.e. |
| Solving \(l_1\) and \(l_2\): \(-6x-x+21=0\) | M1 | Attempt to solve two linear equations leading to linear equation in one variable |
| \(x_p=3\), \(y_p=-6\) | A1, A1f.t. | 2nd A1 f.t.: only f.t. their \(x_p\) or \(y_p\) in \(y=-2x\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(l_1\) is \(y=\frac{1}{3}x-7\); C is \((0,-7)\) or \(OC=7\) | B1f.t. | Either a correct \(OC\) or f.t. from their \(l_1\) |
| Area of \(\triangle OCP = \frac{1}{2}OC \times x_p = \frac{1}{2}\times 7\times 3 = 10.5\) or \(\frac{21}{2}\) | M1 A1 c.a.o. | For correct attempt in letters or symbols for \(\triangle OCP\); \(-\frac{1}{2}\times 7\times 3\) scores M1 A0 |
## Question 8:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y-(-4)=\frac{1}{3}(x-9)$ | M1 A1 | Full method to find equation of $l_1$; any unsimplified form |
| $3y-x+21=0$ | A1 | o.e., condone 3 terms with integer coefficients e.g. $3y+21=x$ |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Equation of $l_2$ is: $y=-2x$ | B1 | o.e. |
| Solving $l_1$ and $l_2$: $-6x-x+21=0$ | M1 | Attempt to solve two linear equations leading to linear equation in one variable |
| $x_p=3$, $y_p=-6$ | A1, A1f.t. | 2nd A1 f.t.: only f.t. their $x_p$ or $y_p$ in $y=-2x$ |
### Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $l_1$ is $y=\frac{1}{3}x-7$; C is $(0,-7)$ or $OC=7$ | B1f.t. | Either a correct $OC$ or f.t. from their $l_1$ |
| Area of $\triangle OCP = \frac{1}{2}OC \times x_p = \frac{1}{2}\times 7\times 3 = 10.5$ or $\frac{21}{2}$ | M1 A1 c.a.o. | For correct attempt in letters or symbols for $\triangle OCP$; $-\frac{1}{2}\times 7\times 3$ scores M1 A0 |
---8. The line $l _ { 1 }$ passes through the point $( 9 , - 4 )$ and has gradient $\frac { 1 } { 3 }$.
\begin{enumerate}[label=(\alph*)]
\item Find an equation for $l _ { 1 }$ in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.
The line $l _ { 2 }$ passes through the origin $O$ and has gradient - 2 . The lines $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $P$.
\item Calculate the coordinates of $P$.
Given that $l _ { 1 }$ crosses the $y$-axis at the point $C$,
\item calculate the exact area of $\triangle O C P$.\\
\includegraphics[max width=\textwidth, alt={}, center]{5a195cf1-37d9-43e9-ab47-c6892a18ba80-10_187_62_2563_1881}
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 2005 Q8 [10]}}