| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2012 |
| Session | June |
| Marks | 9 |
| Topic | Straight Lines & Coordinate Geometry |
| Type | Parallel line through point |
| Difficulty | Moderate -0.8 This is a straightforward coordinate geometry question testing basic skills: distance formula, midpoint formula, circle equation from diameter, and parallel line equation. All parts are routine textbook exercises requiring direct application of standard formulas with no problem-solving insight needed. The multi-part structure adds length but not conceptual difficulty. |
| Spec | 1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.03b Straight lines: parallel and perpendicular relationships1.03d Circles: equation (x-a)^2+(y-b)^2=r^2 |
| Answer | Marks | Guidance |
|---|---|---|
| (i)(a) \(AB = \sqrt{6^2 + 8^2} = 10\) | M1, A1 [2] | Use Pythagoras |
| (i)(b) Midpoint of \(AB\) is \((6, -3)\) | B1 [1] | |
| (i)(c) \((x-6)^2 + (y+3)^2 = 25\) AEF | \B1, \B1, \B1 [3] | fit their values from (a) + (b) |
| (ii) Gradient of \(AC\) is \(-0.5\) | M1 | |
| Use of \(y = mx + c\) or equivalent | M1 | \(\Delta y/\Delta x\) |
| Required equation: \(y = -\frac{1}{2}x + \frac{11}{2}\) | A1 [3] | [9] |
**(i)(a)** $AB = \sqrt{6^2 + 8^2} = 10$ | M1, A1 [2] | Use Pythagoras
**(i)(b)** Midpoint of $AB$ is $(6, -3)$ | B1 [1] |
**(i)(c)** $(x-6)^2 + (y+3)^2 = 25$ AEF | \B1, \B1, \B1 [3] | fit their values from (a) + (b)
**(ii)** Gradient of $AC$ is $-0.5$ | M1 |
Use of $y = mx + c$ or equivalent | M1 | $\Delta y/\Delta x$
Required equation: $y = -\frac{1}{2}x + \frac{11}{2}$ | A1 [3] | [9]\includegraphics{figure_2}
The diagram shows a triangle $ABC$. The vertices have coordinates $A(3, -7)$, $B(9, 1)$ and $C(-1, -5)$.
\begin{enumerate}[label=(\roman*)]
\item \begin{enumerate}[label=(\alph*)]
\item Find the length of the side $AB$. [2]
\item Find the coordinates of the mid-point of $AB$. [1]
\item A circle has diameter $AB$. Find the equation of the circle in the form $(x - a)^2 + (y - b)^2 = r^2$, where $a$, $b$ and $r$ are constants to be found. [3]
\end{enumerate}
\item Find the equation of the line $l$ passing through $B$ parallel to $AC$. [3]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2012 Q2 [9]}}