| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2022 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Line-circle intersection points |
| Difficulty | Moderate -0.3 This is a straightforward two-part question requiring standard techniques: finding the equation of a line through two points (one being the circle's centre), then solving simultaneous equations to find intersection points. While it involves multiple steps and 'exact form' requires careful algebra, these are routine P1 procedures with no conceptual challenges or novel problem-solving required. |
| Spec | 1.02q Use intersection points: of graphs to solve equations1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.03d Circles: equation (x-a)^2+(y-b)^2=r^2 |
| Answer | Marks | Guidance |
|---|---|---|
| Equation of \(BC\) is \(\{y=\}\{2\}\{-3x\}\) | B2, 1, 0 | OE forms \(y+4=-3(x-2)\) or \(y-2=-3(x-0)\) |
| Answer | Marks | Guidance |
|---|---|---|
| \((x-2)^2+(2-3x+4)^2=20\) | \*M1 | OE Sub line equation into equation of circle to eliminate \(y\) |
| \(10(x-2)^2=20\) or \([10](x^2-4x+2)[=0]\) | A1 | OE Accept \((10x^2-40x+20)\) |
| \(x-2=[\pm]\sqrt{2}\) or \(x=\frac{4[\pm]\sqrt{16-8}}{2}\) | DM1 | Correctly solving *their* quadratic. |
| \(x=2-\sqrt{2}\) | A1 | OE only solution. Answer only SC B1 if DM1 not scored. |
| \(y=3\sqrt{2}-4\) | A1 | OE only solution. Answer only SC B1 if DM1 not scored. |
## Question 7(a):
Equation of $BC$ is $\{y=\}\{2\}\{-3x\}$ | **B2, 1, 0** | OE forms $y+4=-3(x-2)$ or $y-2=-3(x-0)$
**2 total**
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## Question 7(b):
$(x-2)^2+(2-3x+4)^2=20$ | **\*M1** | OE Sub line equation into equation of circle to eliminate $y$
$10(x-2)^2=20$ or $[10](x^2-4x+2)[=0]$ | **A1** | OE Accept $(10x^2-40x+20)$
$x-2=[\pm]\sqrt{2}$ or $x=\frac{4[\pm]\sqrt{16-8}}{2}$ | **DM1** | Correctly solving *their* quadratic.
$x=2-\sqrt{2}$ | **A1** | OE only solution. Answer only **SC B1** if DM1 not scored.
$y=3\sqrt{2}-4$ | **A1** | OE only solution. Answer only **SC B1** if DM1 not scored.
**5 total**
---7\\
\includegraphics[max width=\textwidth, alt={}, center]{89a18f20-a4d6-4a42-8b00-849f4fb89692-10_887_1003_258_571}
The diagram shows the circle with equation $( x - 2 ) ^ { 2 } + ( y + 4 ) ^ { 2 } = 20$ and with centre $C$. The point $B$ has coordinates $( 0,2 )$ and the line segment $B C$ intersects the circle at $P$.
\begin{enumerate}[label=(\alph*)]
\item Find the equation of $B C$.
\item Hence find the coordinates of $P$, giving your answer in exact form.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2022 Q7 [7]}}