| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2010 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Chord length calculation |
| Difficulty | Standard +0.3 This is a standard C2 circle question with routine steps: (a) finding circle equation using distance formula, (b) finding tangent using perpendicular gradient, (c) chord length using perpendicular distance from center. All techniques are textbook exercises requiring no novel insight, though part (c) involves multiple steps with coordinate geometry and Pythagoras, making it slightly above average difficulty for C2. |
| 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 |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \((10-2)^2+(7-1)^2\) or \(\sqrt{(10-2)^2+(7-1)^2}\) | M1 A1 | |
| \((x\pm2)^2+(y\pm1)^2=k\) (\(k\) a positive value) | M1 | |
| \((x-2)^2+(y-1)^2=100\) (accept \(10^2\) for 100) | A1 | Answer only scores full marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Gradient of radius \(=\frac{7-1}{10-2}=\frac{6}{8}\) (or equiv.) | B1 | Must be seen in part (b) |
| Gradient of tangent \(=\frac{-4}{3}\) (using perpendicular gradient method) | M1 | |
| \(y-7=m(x-10)\) eqn. in any form of line through \((10,7)\) with any numerical gradient (except 0 or \(\infty\)) | M1 | |
| \(y-7=\frac{-4}{3}(x-10)\) or equiv. (ft gradient of radius, dep. on both M marks) | A1ft | |
| \(\{3y=-4x+61\}\) | N.B. A1 only available as ft after B0; unsimplified version scores A mark; equation must at some stage be exact |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\sqrt{r^2-\left(\frac{r}{2}\right)^2}\) | M1 | Condone sign slip if evidence of correct use of Pythagoras |
| \(=\sqrt{10^2-5^2}\) or numerically exact equivalent | A1 | |
| \(PQ\left(=2\sqrt{75}\right)=10\sqrt{3}\) | A1 | Simplest surd form \(10\sqrt{3}\) required for final mark |
## Question 10:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $(10-2)^2+(7-1)^2$ or $\sqrt{(10-2)^2+(7-1)^2}$ | M1 A1 | |
| $(x\pm2)^2+(y\pm1)^2=k$ ($k$ a positive value) | M1 | |
| $(x-2)^2+(y-1)^2=100$ (accept $10^2$ for 100) | A1 | Answer only scores full marks |
**(4 marks)**
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Gradient of radius $=\frac{7-1}{10-2}=\frac{6}{8}$ (or equiv.) | B1 | Must be seen in part (b) |
| Gradient of tangent $=\frac{-4}{3}$ (using perpendicular gradient method) | M1 | |
| $y-7=m(x-10)$ eqn. in any form of line through $(10,7)$ with any numerical gradient (except 0 or $\infty$) | M1 | |
| $y-7=\frac{-4}{3}(x-10)$ or equiv. (ft gradient of radius, dep. on both M marks) | A1ft | |
| $\{3y=-4x+61\}$ | | N.B. A1 only available as ft after B0; unsimplified version scores A mark; equation must at some stage be exact |
**(4 marks)**
### Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\sqrt{r^2-\left(\frac{r}{2}\right)^2}$ | M1 | Condone sign slip if evidence of correct use of Pythagoras |
| $=\sqrt{10^2-5^2}$ or numerically exact equivalent | A1 | |
| $PQ\left(=2\sqrt{75}\right)=10\sqrt{3}$ | A1 | Simplest surd form $10\sqrt{3}$ required for final mark |
**(3 marks, 11 total)**10. The circle $C$ has centre $A ( 2,1 )$ and passes through the point $B ( 10,7 )$.
\begin{enumerate}[label=(\alph*)]
\item Find an equation for $C$.
The line $l _ { 1 }$ is the tangent to $C$ at the point $B$.
\item Find an equation for $l _ { 1 }$.
The line $l _ { 2 }$ is parallel to $l _ { 1 }$ and passes through the mid-point of $A B$.\\
Given that $l _ { 2 }$ intersects $C$ at the points $P$ and $Q$,
\item find the length of $P Q$, giving your answer in its simplest surd form.\\
\includegraphics[max width=\textwidth, alt={}, center]{571780c2-945b-4636-b7c3-0bd558d28710-15_115_127_2461_1814}
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 2010 Q10 [11]}}