| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2021 |
| Session | March |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Circle through three points using right angle in semicircle |
| Difficulty | Moderate -0.8 This is a straightforward two-part question requiring standard techniques: finding a circle equation from three points (solving simultaneous equations or recognizing a right angle), then finding a tangent using perpendicular gradients. Both are routine P1 procedures with no problem-solving insight needed, making it easier than average but not trivial due to the algebraic manipulation required. |
| 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 |
|---|---|---|
| Answer | Marks | Guidance |
| (a) Centre of circle is \((4, 5)\) | B1 B1 | |
| (a) \(r^2 = (7-4)^2 + (1-5)^2\) | M1 | OE. Either using *their* centre and \(A\) or \(C\), or using \(A\) and \(C\) and dividing by 2 |
| (a) \(r = 5\) | A1 FT | FT on *their* \((4, 5)\) if used |
| (a) Equation is \((x-4)^2 + (y-5)^2 = 25\) | A1 | OE. Allow \(5^2\) for 25 |
| (b) Gradient of radius \(= \frac{9-5}{7-4} = \frac{4}{3}\) | B1 FT | FT for use of *their* centre |
| (b) Equation of tangent is \(y - 9 = -\frac{3}{4}(x - 7)\) | B1 | or \(y = \frac{-3x}{4} + \frac{57}{4}\) |
## Question 8:
| Answer | Marks | Guidance |
|--------|-------|----------|
| (a) Centre of circle is $(4, 5)$ | B1 B1 | |
| (a) $r^2 = (7-4)^2 + (1-5)^2$ | M1 | OE. Either using *their* centre and $A$ or $C$, **or** using $A$ and $C$ and dividing by 2 |
| (a) $r = 5$ | A1 FT | FT on *their* $(4, 5)$ if used |
| (a) Equation is $(x-4)^2 + (y-5)^2 = 25$ | A1 | OE. Allow $5^2$ for 25 |
| (b) Gradient of radius $= \frac{9-5}{7-4} = \frac{4}{3}$ | B1 FT | FT for use of *their* centre |
| (b) Equation of tangent is $y - 9 = -\frac{3}{4}(x - 7)$ | B1 | or $y = \frac{-3x}{4} + \frac{57}{4}$ |
---8 The points $A ( 7,1 ) , B ( 7,9 )$ and $C ( 1,9 )$ are on the circumference of a circle.
\begin{enumerate}[label=(\alph*)]
\item Find an equation of the circle.
\item Find an equation of the tangent to the circle at $B$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2021 Q8 [7]}}