| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2023 |
| Session | November |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Circle from diameter endpoints |
| Difficulty | Moderate -0.8 Part (a) requires substituting x=0 and solving a straightforward quadratic to find two y-coordinates. Part (b) uses the standard result that a circle with diameter endpoints has a specific form, requiring only the midpoint and distance formula. Both parts are routine applications of circle geometry with no problem-solving insight needed, making this easier than average. |
| Spec | 1.02b Surds: manipulation and rationalising denominators1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((0-3)^2 + (y-5)^2 = 40\) | M1 | OE. Substitute \(x = 0\), may use \(y^2 - 10y - 6 = 0\). |
| \(y = 5 \pm \sqrt{31}\) | A1 | OE. Must be surd form. |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\left\{x^2 + (y-5)^2\right\} = \{31\}\), allow \((x-0)^2\) | B1FT B1FT | B1 FT for *their* 5 and B1 FT for *their* 31. Don't allow surd form. |
| 2 |
**Question 2(a):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(0-3)^2 + (y-5)^2 = 40$ | **M1** | OE. Substitute $x = 0$, may use $y^2 - 10y - 6 = 0$. |
| $y = 5 \pm \sqrt{31}$ | **A1** | OE. Must be surd form. |
| | **2** | |
---
**Question 2(b):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\left\{x^2 + (y-5)^2\right\} = \{31\}$, allow $(x-0)^2$ | **B1FT B1FT** | B1 FT for *their* 5 and B1 FT for *their* 31. Don't allow surd form. |
| | **2** | |2 The circle with equation $( x - 3 ) ^ { 2 } + ( y - 5 ) ^ { 2 } = 40$ intersects the $y$-axis at points $A$ and $B$.
\begin{enumerate}[label=(\alph*)]
\item Find the $y$-coordinates of $A$ and $B$, expressing your answers in terms of surds.
\item Find the equation of the circle which has $A B$ as its diameter.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2023 Q2 [4]}}