| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2021 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Tangent from external point - intersection or geometric properties |
| Difficulty | Standard +0.3 This is a straightforward circle question requiring completion of the square to find the center, solving a quadratic for intersection points, and using perpendicular gradients to find where two tangents meet. All steps are standard techniques with no novel insight required, making it slightly easier than average. |
| 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 |
| When \(y=0\): \(x^2 - 4x - 77 = 0\ [\Rightarrow (x+7)(x-11)=0\) or \((x-2)^2 = 81]\) | M1 | Substituting \(y=0\) |
| So \(x\)-coordinates are \(-7\) and \(11\) | A1 | |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Centre of circle \(C\) is \((2, -3)\) | B1 | |
| Gradient of \(AC\) is \(-\frac{1}{3}\) or Gradient of \(BC\) is \(\frac{1}{3}\) | M1 | For either gradient (M1 sign error, M0 if \(x\)-coordinate(s) in numerator) |
| Gradient of tangent at \(A\) is \(3\) or Gradient of tangent at \(B\) is \(-3\) | M1 | For either perpendicular gradient |
| Equations of tangents are \(y = 3x + 21\), \(y = -3x + 33\) | A1 | For either equation |
| Meet when \(3x + 21 = -3x + 33\) | M1 | OR: centre of circle has \(x\) coordinate 2 so \(x\) coordinate of point of intersection is 2 |
| Coordinates of point of intersection \((2, 27)\) | A1 | |
| Alternative method: | ||
| Implicit differentiation: \(2y\frac{dy}{dx}\) seen | B1 | |
| \(2x - 4 + 2y\frac{dy}{dx} + 6\frac{dy}{dx} = 0\) | M1 | Fully differentiated \(=0\) with at least one term involving \(y\) differentiated correctly |
| Gradient of tangent at \(A\) is \(3\) or Gradient of tangent at \(B\) is \(-3\) | M1 | For either gradient |
| Equations of tangents are \(y = 3x + 21\), \(y = -3x + 33\) | A1 | For either equation |
| Meet when \(3x + 21 = -3x + 33\) | M1 | OR: centre of circle has \(x\) coordinate 2 so \(x\) coordinate of point of intersection is 2 |
| Coordinates of point of intersection \((2, 27)\) | A1 | |
| 6 |
## Question 10(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| When $y=0$: $x^2 - 4x - 77 = 0\ [\Rightarrow (x+7)(x-11)=0$ or $(x-2)^2 = 81]$ | M1 | Substituting $y=0$ |
| So $x$-coordinates are $-7$ and $11$ | A1 | |
| | **2** | |
---
## Question 10(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Centre of circle $C$ is $(2, -3)$ | B1 | |
| Gradient of $AC$ is $-\frac{1}{3}$ or Gradient of $BC$ is $\frac{1}{3}$ | M1 | For either gradient (M1 sign error, M0 if $x$-coordinate(s) in numerator) |
| Gradient of tangent at $A$ is $3$ or Gradient of tangent at $B$ is $-3$ | M1 | For either perpendicular gradient |
| Equations of tangents are $y = 3x + 21$, $y = -3x + 33$ | A1 | For either equation |
| Meet when $3x + 21 = -3x + 33$ | M1 | OR: centre of circle has $x$ coordinate 2 so $x$ coordinate of point of intersection is 2 |
| Coordinates of point of intersection $(2, 27)$ | A1 | |
| **Alternative method:** | | |
| Implicit differentiation: $2y\frac{dy}{dx}$ seen | B1 | |
| $2x - 4 + 2y\frac{dy}{dx} + 6\frac{dy}{dx} = 0$ | M1 | Fully differentiated $=0$ with at least one term involving $y$ differentiated correctly |
| Gradient of tangent at $A$ is $3$ or Gradient of tangent at $B$ is $-3$ | M1 | For either gradient |
| Equations of tangents are $y = 3x + 21$, $y = -3x + 33$ | A1 | For either equation |
| Meet when $3x + 21 = -3x + 33$ | M1 | OR: centre of circle has $x$ coordinate 2 so $x$ coordinate of point of intersection is 2 |
| Coordinates of point of intersection $(2, 27)$ | A1 | |
| | **6** | |
---10 The equation of a circle is $x ^ { 2 } + y ^ { 2 } - 4 x + 6 y - 77 = 0$.
\begin{enumerate}[label=(\alph*)]
\item Find the $x$-coordinates of the points $A$ and $B$ where the circle intersects the $x$-axis.
\item Find the point of intersection of the tangents to the circle at $A$ and $B$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2021 Q10 [8]}}