CAIE P3 2018 November — Question 9 10 marks

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2018
SessionNovember
Marks10
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicComplex Numbers Argand & Loci
TypeRegion shading with multiple inequalities
DifficultyStandard +0.3 This is a straightforward multi-part question testing standard complex number techniques: (i) multiplying by conjugate to simplify a fraction (routine 2-mark exercise), (ii) converting to modulus-argument form using exact values (standard procedure), and (iii) sketching a locus region (circle intersected with half-plane) and finding maximum argument. All parts follow textbook methods with no novel insight required, making it slightly easier than average.
Spec4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines
    1. Without using a calculator, express the complex number \(\frac{2 + 6i}{1 - 2i}\) in the form \(x + iy\), where \(x\) and \(y\) are real. [2]
    2. Hence, without using a calculator, express \(\frac{2 + 6i}{1 - 2i}\) in the form \(r(\cos \theta + i \sin \theta)\), where \(r > 0\) and \(-\pi < \theta \leqslant \pi\), giving the exact values of \(r\) and \(\theta\). [3]
  2. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying both the inequalities \(|z - 3i| \leqslant 1\) and \(\text{Re } z \leqslant 0\), where \(\text{Re } z\) denotes the real part of \(z\). Find the greatest value of \(\arg z\) for points in this region, giving your answer in radians correct to 2 decimal places. [5]
Question 9:
AnswerMarks Guidance
9(a)(i)Multiply numerator and denominator by 1 + 2i, or equivalent M1
use of i2 =−1. Can be implied by −10+10i
5
AnswerMarks
Obtain quotient – 2 + 2iA1
Alternative
Equate to x + iy, obtain two equations in x and y and solve for x or
AnswerMarks Guidance
for yM1 x+2y=2, y−2x=6
Obtain quotient – 2 + 2iA1
2
AnswerMarks Guidance
9(a)(ii)Use correct method to find either r or θ M1
quadrant
AnswerMarks Guidance
Obtain r =2 2, or exact equivalentA1ft ft their x + iy
3
Obtain θ = π from exact work
AnswerMarks Guidance
4A1ft ft on k(-1 + i) for k > 0 Do not ISW
3
AnswerMarks Guidance
QuestionAnswer Marks
AnswerMarks Guidance
9(b)Show a circle with centre 3i B1
Show a circle with radius 1B1ft Follow through their centre provided not at the origin
For clearly unequal scales, should be an ellipse
AnswerMarks Guidance
All correct with even scales and shade the correct regionB1 Im z
3i
1
Re z
AnswerMarks Guidance
Carry out a correct method for calculating greatest value of arg zM1 π 1
e.g.argz= +sin−1
2 3
AnswerMarks
Obtain answer 1.91A1
5
AnswerMarks Guidance
QuestionAnswer Marks 
Question 9:
--- 9(a)(i) ---
9(a)(i) | Multiply numerator and denominator by 1 + 2i, or equivalent | M1 | Requires at least one of 2+10i+12i2 and 1−4i2 together with
use of i2 =−1. Can be implied by −10+10i
5
Obtain quotient – 2 + 2i | A1
Alternative
Equate to x + iy, obtain two equations in x and y and solve for x or
for y | M1 | x+2y=2, y−2x=6
Obtain quotient – 2 + 2i | A1
2
--- 9(a)(ii) ---
9(a)(ii) | Use correct method to find either r or θ | M1 | If only finding θ, need to be looking forθin the correct
quadrant
Obtain r =2 2, or exact equivalent | A1ft | ft their x + iy
3
Obtain θ = π from exact work
4 | A1ft | ft on k(-1 + i) for k > 0 Do not ISW
3
Question | Answer | Marks | Guidance
--- 9(b) ---
9(b) | Show a circle with centre 3i | B1
Show a circle with radius 1 | B1ft | Follow through their centre provided not at the origin
For clearly unequal scales, should be an ellipse
All correct with even scales and shade the correct region | B1 | Im z
3i
1
Re z
Carry out a correct method for calculating greatest value of arg z | M1 | π 1
e.g.argz= +sin−1
2 3
Obtain answer 1.91 | A1
5
Question | Answer | Marks | Guidance
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Without using a calculator, express the complex number $\frac{2 + 6i}{1 - 2i}$ in the form $x + iy$, where $x$ and $y$ are real. [2]

\item Hence, without using a calculator, express $\frac{2 + 6i}{1 - 2i}$ in the form $r(\cos \theta + i \sin \theta)$, where $r > 0$ and $-\pi < \theta \leqslant \pi$, giving the exact values of $r$ and $\theta$. [3]
\end{enumerate}

\item On a sketch of an Argand diagram, shade the region whose points represent complex numbers $z$ satisfying both the inequalities $|z - 3i| \leqslant 1$ and $\text{Re } z \leqslant 0$, where $\text{Re } z$ denotes the real part of $z$. Find the greatest value of $\arg z$ for points in this region, giving your answer in radians correct to 2 decimal places. [5]
\end{enumerate}

\hfill \mbox{\textit{CAIE P3 2018 Q9 [10]}}
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