| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2012 |
| 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 testing basic skills: rearranging to find gradient, writing parallel line equations, solving simultaneous equations, and substituting coordinates. All parts are routine textbook exercises requiring only direct application of standard methods with no problem-solving insight needed. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution1.03a 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 |
| \(y=\frac{4}{3}x-\frac{7}{3}\) | M1 | \(y=\pm\frac{4}{3}x+k\) or \(\frac{\Delta y}{\Delta x}\) with 2 correct points |
| \(\Rightarrow\) grad \(AB = \frac{4}{3}\) | A1 | condone slip in rearranging if gradient correct; condone 1.33 or better |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(y=\) 'their grad' \(x+c\) and attempt to use \(x=3\), \(y=-5\) | M1 | or \(y--5=\) 'their grad \(AB\)'\((x-3)\) or \(4x-3y=k\) and attempt to find \(k\) using \(x=3\) and \(y=-5\) |
| \(y+5=\frac{4}{3}(x-3)\) or \(y=\frac{4}{3}x-\frac{27}{3}\) | A1 | correct equation in any form but must simplify \(--\) to \(+\) |
| \(4x-3y=27\) | A1 | integer coefficients in required form e.g. \(-8x+6y=-54\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(4x-3y=7\) and \(3x-2y=4\) \(\Rightarrow 8x-9x=14-12\) etc | M1 | must use correct pair of equations and attempt to eliminate \(x\) or \(y\) (generous) |
| \(x=-2\) | A1 | |
| \(y=-5\) | A1 | or \(D(-2,-5)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(4(k-2)-3(2k-3)=7\) \(4k-8-6k+9=7\) | M1 | sub \(x=k-2\), \(y=2k-3\) into \(4x-3y=7\) and attempt to multiply out with all \(k\) terms on one side (condone one slip) |
| \(\Rightarrow k=-3\) | A1 |
# Question 2:
## Part (a)(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y=\frac{4}{3}x-\frac{7}{3}$ | M1 | $y=\pm\frac{4}{3}x+k$ or $\frac{\Delta y}{\Delta x}$ with 2 correct points |
| $\Rightarrow$ grad $AB = \frac{4}{3}$ | A1 | condone slip in rearranging if gradient correct; condone 1.33 or better |
## Part (a)(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y=$ 'their grad' $x+c$ and attempt to use $x=3$, $y=-5$ | M1 | **or** $y--5=$ 'their grad $AB$'$(x-3)$ **or** $4x-3y=k$ and attempt to find $k$ using $x=3$ and $y=-5$ |
| $y+5=\frac{4}{3}(x-3)$ or $y=\frac{4}{3}x-\frac{27}{3}$ | A1 | correct equation in any form but must simplify $--$ to $+$ |
| $4x-3y=27$ | A1 | integer coefficients in required form e.g. $-8x+6y=-54$ |
## Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $4x-3y=7$ and $3x-2y=4$ $\Rightarrow 8x-9x=14-12$ etc | M1 | must use **correct pair** of equations and **attempt** to eliminate $x$ or $y$ (generous) |
| $x=-2$ | A1 | |
| $y=-5$ | A1 | or $D(-2,-5)$ |
## Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $4(k-2)-3(2k-3)=7$ $4k-8-6k+9=7$ | M1 | sub $x=k-2$, $y=2k-3$ into $4x-3y=7$ and attempt to multiply out with all $k$ terms on one side (condone one slip) |
| $\Rightarrow k=-3$ | A1 | |
---2 The line $A B$ has equation $4 x - 3 y = 7$.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Find the gradient of $A B$.
\item Find an equation of the straight line that is parallel to $A B$ and which passes through the point $C ( 3 , - 5 )$, giving your answer in the form $p x + q y = r$, where $p , q$ and $r$ are integers.
\end{enumerate}\item The line $A B$ intersects the line with equation $3 x - 2 y = 4$ at the point $D$. Find the coordinates of $D$.
\item The point $E$ with coordinates $( k - 2,2 k - 3 )$ lies on the line $A B$. Find the value of the constant $k$.
\end{enumerate}
\hfill \mbox{\textit{AQA C1 2012 Q2 [10]}}