| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2010 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Chord length calculation |
| Difficulty | Standard +0.3 This is a structured multi-part circle question with clear scaffolding. Parts (a)-(b) are trivial reading from the equation. Part (c) requires setting up a right triangle with the perpendicular from center to chord (Pythagoras), which is standard C2 content. Parts (d)-(e) involve cosine rule and tangent properties but are heavily guided. Slightly above average due to the multi-step nature and geometric reasoning required, but well within typical C2 expectations. |
| Spec | 1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle1.03f Circle properties: angles, chords, tangents |
| Answer | Marks | Guidance |
|---|---|---|
| \(N(2,-1)\) | B1, B1 (2) | B1 for 2 (\(\alpha\)), B1 for \(-1\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(r = \sqrt{\frac{169}{4}} = \frac{13}{2} = 6.5\) | B1 (1) | B1 for 6.5 o.e. |
| Answer | Marks | Guidance |
|---|---|---|
| Complete method: \(x_2 - x_1 = 12\) and \(\frac{x_1+x_2}{2}=2\), giving \(x_1=-4\), \(x_2=8\) | M1, A1ft A1ft | 1st M1 for finding \(x\) coordinates; A1ft, A1ft for \(\alpha-6\) and \(\alpha+6\) |
| \(d^2 = 6.5^2 - 6^2 \Rightarrow d=2.5 \Rightarrow y_2 = y_1 = -3.5\) | M1, A1 (5) | 2nd M1 for method to find \(y\) coordinates; A marks for \(-3.5\) only |
| Answer | Marks | Guidance |
|---|---|---|
| Let \(A\hat{N}B = 2\theta \Rightarrow \sin\theta = \frac{6}{"6.5"} \Rightarrow \theta = (67.38)\ldots\); angle \(ANB = 134.8°\) | M1, A1 (2) | M1 for full method to find \(\theta\) or angle \(ANB\); ft their 6.5; A1 must be \(134.8°\), do not accept \(134.76°\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(AP\) perpendicular to \(AN\), using triangle \(ANP\): \(\tan\theta = \frac{AP}{"6.5"}\); therefore \(AP = 15.6\) | M1, A1cao (2) | M1 for full method to find \(AP\); alternatives: \(\frac{AP}{\sin 67.4} = \frac{12}{\sin 45.2}\), or \(AP = \frac{6}{\sin 22.6}\), or \(AP = \frac{6}{\cos 67.4}\) each worth M1 |
## Question 8:
### Part (a):
| $N(2,-1)$ | B1, B1 (2) | B1 for 2 ($\alpha$), B1 for $-1$ |
### Part (b):
| $r = \sqrt{\frac{169}{4}} = \frac{13}{2} = 6.5$ | B1 (1) | B1 for 6.5 o.e. |
### Part (c):
| Complete method: $x_2 - x_1 = 12$ and $\frac{x_1+x_2}{2}=2$, giving $x_1=-4$, $x_2=8$ | M1, A1ft A1ft | 1st M1 for finding $x$ coordinates; A1ft, A1ft for $\alpha-6$ and $\alpha+6$ |
| $d^2 = 6.5^2 - 6^2 \Rightarrow d=2.5 \Rightarrow y_2 = y_1 = -3.5$ | M1, A1 (5) | 2nd M1 for method to find $y$ coordinates; A marks for $-3.5$ only |
### Part (d):
| Let $A\hat{N}B = 2\theta \Rightarrow \sin\theta = \frac{6}{"6.5"} \Rightarrow \theta = (67.38)\ldots$; angle $ANB = 134.8°$ | M1, A1 (2) | M1 for full method to find $\theta$ or angle $ANB$; ft their 6.5; A1 must be $134.8°$, do not accept $134.76°$ |
### Part (e):
| $AP$ perpendicular to $AN$, using triangle $ANP$: $\tan\theta = \frac{AP}{"6.5"}$; therefore $AP = 15.6$ | M1, A1cao (2) | M1 for full method to find $AP$; alternatives: $\frac{AP}{\sin 67.4} = \frac{12}{\sin 45.2}$, or $AP = \frac{6}{\sin 22.6}$, or $AP = \frac{6}{\cos 67.4}$ each worth M1 |8.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{e3faf018-37a8-48ef-b100-81402a8ec87f-11_1262_1178_203_386}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}
Figure 3 shows a sketch of the circle $C$ with centre $N$ and equation
$$( x - 2 ) ^ { 2 } + ( y + 1 ) ^ { 2 } = \frac { 169 } { 4 }$$
\begin{enumerate}[label=(\alph*)]
\item Write down the coordinates of $N$.
\item Find the radius of $C$.
The chord $A B$ of $C$ is parallel to the $x$-axis, lies below the $x$-axis and is of length 12 units as shown in Figure 3.
\item Find the coordinates of $A$ and the coordinates of $B$.
\item Show that angle $A N B = 134.8 ^ { \circ }$, to the nearest 0.1 of a degree.
The tangents to $C$ at the points $A$ and $B$ meet at the point $P$.
\item Find the length $A P$, giving your answer to 3 significant figures.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 2010 Q8 [12]}}