Edexcel CP AS 2024 June — Question 5 10 marks

Exam BoardEdexcel
ModuleCP AS (Core Pure AS)
Year2024
SessionJune
Marks10
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicComplex Numbers Argand & Loci
TypeArea calculations in complex plane
DifficultyStandard +0.8 This is a multi-part question requiring understanding of complex loci (circle from modulus condition), geometric interpretation of argument conditions (half-plane), finding their intersection, and calculating the area of a circular segment. Part (d) requires recognizing that the argument condition defines a half-plane, finding where it intersects the circle, and applying the circular segment area formula—this involves multiple conceptual steps beyond routine exercises.
Spec4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines
  1. Given that on an Argand diagram the locus of points defined by \(| z + 5 - 12 i | = 10\) is a circle,
    1. write down,
      1. the coordinates of the centre of this circle,
      2. the radius of this circle.
    2. Show, by shading on an Argand diagram, the set of points defined by
    $$| z + 5 - 12 i | \leqslant 10$$
  2. For the set of points defined in part (b), determine the maximum value of \(| z |\) The set of points \(A\) is defined by $$A = \{ z : 0 \leqslant \arg ( z + 5 - 20 i ) \leqslant \pi \} \cap \{ z : | z + 5 - 12 i | \leqslant 10 \}$$
  3. Determine the area of the region defined by \(A\), giving your answer to 3 significant figures.
Question 5:
Part (a)(i):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\((-5, 12)\) or \(-5 + 12i\)B1 1.1b - Correct centre, condone \((-5, 12i)\)
Part (a)(ii):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(r = 10\)B1 1.1b - Correct radius
Part (b):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
Circle drawn with inside shaded, centre in second quadrant intercepting imaginary axis onlyB1ft 1.1b - Follow through on centre and radius. Centre must be in correct quadrant and intercept axes appropriately
Part (c):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(OC = \sqrt{5^2 + 12^2}\)M1 1.1b
\(z _{\max} = \sqrt{5^2 + 12^2} + 10\)
\(= 23\)A1 1.1b
Part (d):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(\{z: 0, \arg(z+5-20i), \pi\} \Rightarrow y = 20\); \((x+5)^2 + 8^2 = 100 \Rightarrow x = \ldots\) AND finds an angle using e.g. \(\cos\theta = \frac{10^2+10^2-12^2}{2\times10\times10} = 0.28\) or \(a^2 = 10^2 - 8^2 \Rightarrow a = \ldots\{6\}\), \(\sin\left(\frac{1}{2}\theta\right) = \frac{6}{10}\) or \(\cos\left(\frac{1}{2}\theta\right) = \frac{8}{10}\)M1 3.1a
\(\theta = 1.287\ldots\) or \(73.7°\) or \(\frac{1}{2}\theta = 0.6435\ldots\) or \(36.9°\)A1 1.1b
Area \(= \frac{1}{2}\times10^2\times\theta - \frac{1}{2}\times12\times8\) (radians) or Area \(= \pi\times10^2\times\frac{\theta}{360} - \frac{1}{2}\times12\times8\) (degrees) or Area \(= 2\left[\frac{1}{2}\times10^2\times\theta - \frac{1}{2}\times8\times6\right]\)M1 3.1a
\(= \text{awrt } 16.4\)A1 1.1b 
## Question 5:

### Part (a)(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $(-5, 12)$ or $-5 + 12i$ | B1 | 1.1b - Correct centre, condone $(-5, 12i)$ |

### Part (a)(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $r = 10$ | B1 | 1.1b - Correct radius |

### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Circle drawn with inside shaded, centre in second quadrant intercepting imaginary axis only | B1ft | 1.1b - Follow through on centre and radius. Centre must be in correct quadrant and intercept axes appropriately |

### Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $OC = \sqrt{5^2 + 12^2}$ | M1 | 1.1b |
| $|z|_{\max} = \sqrt{5^2 + 12^2} + 10$ | M1 | 3.1a |
| $= 23$ | A1 | 1.1b |

### Part (d):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\{z: 0, \arg(z+5-20i), \pi\} \Rightarrow y = 20$; $(x+5)^2 + 8^2 = 100 \Rightarrow x = \ldots$ AND finds an angle using e.g. $\cos\theta = \frac{10^2+10^2-12^2}{2\times10\times10} = 0.28$ or $a^2 = 10^2 - 8^2 \Rightarrow a = \ldots\{6\}$, $\sin\left(\frac{1}{2}\theta\right) = \frac{6}{10}$ or $\cos\left(\frac{1}{2}\theta\right) = \frac{8}{10}$ | M1 | 3.1a |
| $\theta = 1.287\ldots$ or $73.7°$ or $\frac{1}{2}\theta = 0.6435\ldots$ or $36.9°$ | A1 | 1.1b |
| Area $= \frac{1}{2}\times10^2\times\theta - \frac{1}{2}\times12\times8$ (radians) or Area $= \pi\times10^2\times\frac{\theta}{360} - \frac{1}{2}\times12\times8$ (degrees) or Area $= 2\left[\frac{1}{2}\times10^2\times\theta - \frac{1}{2}\times8\times6\right]$ | M1 | 3.1a |
| $= \text{awrt } 16.4$ | A1 | 1.1b |
\begin{enumerate}
  \item Given that on an Argand diagram the locus of points defined by $| z + 5 - 12 i | = 10$ is a circle,\\
(a) write down,\\
(i) the coordinates of the centre of this circle,\\
(ii) the radius of this circle.\\
(b) Show, by shading on an Argand diagram, the set of points defined by
\end{enumerate}

$$| z + 5 - 12 i | \leqslant 10$$

(c) For the set of points defined in part (b), determine the maximum value of $| z |$

The set of points $A$ is defined by

$$A = \{ z : 0 \leqslant \arg ( z + 5 - 20 i ) \leqslant \pi \} \cap \{ z : | z + 5 - 12 i | \leqslant 10 \}$$

(d) Determine the area of the region defined by $A$, giving your answer to 3 significant figures.

\hfill \mbox{\textit{Edexcel CP AS 2024 Q5 [10]}}
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