CAIE P3 2021 March — Question 8 9 marks

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2021
SessionMarch
Marks9
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicComplex numbers 2
TypeModulus and argument calculations
DifficultyStandard +0.3 This is a straightforward multi-part complex numbers question requiring standard techniques: division by multiplying by conjugate (routine), conversion to polar form (direct calculation), vector geometry interpretation (recognizing parallel vectors), and angle calculation using arguments (standard formula). All parts are textbook exercises with no novel insight required, making it slightly easier than average.
Spec4.02d Exponential form: re^(i*theta)4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02k Argand diagrams: geometric interpretation
The complex numbers \(u\) and \(v\) are defined by \(u = -4 + 2\text{i}\) and \(v = 3 + \text{i}\).
  1. Find \(\frac{u}{v}\) in the form \(x + \text{i}y\), where \(x\) and \(y\) are real. [3]
  2. Hence express \(\frac{u}{v}\) in the form \(re^{\text{i}\theta}\), where \(r\) and \(\theta\) are exact. [2]
In an Argand diagram, with origin \(O\), the points \(A\), \(B\) and \(C\) represent the complex numbers \(u\), \(v\) and \(2u + v\) respectively.
  1. State fully the geometrical relationship between \(OA\) and \(BC\). [2]
  2. Prove that angle \(AOB = \frac{3}{4}\pi\). [2]
Question 8:
AnswerMarks Guidance
8(a)Multiply numerator and denominator by 3 – i M1
Obtain numerator – 10 + 10i or denominator 10A1
Obtain final answer – 1 + iA1
3
AnswerMarks Guidance
8(b)State or imply r = 2 B1 FT
3
State or imply that θ= π
AnswerMarks
4B1 FT
2
AnswerMarks Guidance
8(c)State that OA and BC are parallel B1
State that BC = 2OAB1
2
AnswerMarks Guidance
QuestionAnswer Marks
AnswerMarks
8(d)u
Use angle AOB = argu−argv=arg
AnswerMarks
vM1
Obtain the given answerA1
Alternative method for question 8(d)
AnswerMarks
Obtain tan AOB from gradients of OA and OB and the tan(A±B)formulaM1
Obtain the given answerA1
Alternative method for question 8(d)
AnswerMarks
Obtain cosAOB by using the cosine rule or a scalar productM1
Obtain the given answerA1
2
AnswerMarks Guidance
QuestionAnswer Marks 
Question 8:
--- 8(a) ---
8(a) | Multiply numerator and denominator by 3 – i | M1 | OE
Obtain numerator – 10 + 10i or denominator 10 | A1
Obtain final answer – 1 + i | A1
3
--- 8(b) ---
8(b) | State or imply r = 2 | B1 FT
3
State or imply that θ= π
4 | B1 FT
2
--- 8(c) ---
8(c) | State that OA and BC are parallel | B1
State that BC = 2OA | B1
2
Question | Answer | Marks | Guidance
--- 8(d) ---
8(d) | u
Use angle AOB = argu−argv=arg
v | M1
Obtain the given answer | A1
Alternative method for question 8(d)
Obtain tan AOB from gradients of OA and OB and the tan(A±B)formula | M1
Obtain the given answer | A1
Alternative method for question 8(d)
Obtain cosAOB by using the cosine rule or a scalar product | M1
Obtain the given answer | A1
2
Question | Answer | Marks | Guidance
The complex numbers $u$ and $v$ are defined by $u = -4 + 2\text{i}$ and $v = 3 + \text{i}$.

\begin{enumerate}[label=(\alph*)]
\item Find $\frac{u}{v}$ in the form $x + \text{i}y$, where $x$ and $y$ are real. [3]
\item Hence express $\frac{u}{v}$ in the form $re^{\text{i}\theta}$, where $r$ and $\theta$ are exact. [2]
\end{enumerate}

In an Argand diagram, with origin $O$, the points $A$, $B$ and $C$ represent the complex numbers $u$, $v$ and $2u + v$ respectively.

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item State fully the geometrical relationship between $OA$ and $BC$. [2]
\item Prove that angle $AOB = \frac{3}{4}\pi$. [2]
\end{enumerate}

\hfill \mbox{\textit{CAIE P3 2021 Q8 [9]}}
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