Question 6 - A-Level Maths

CAIE P3 2010 November — Question 6 9 marks

Exam Board	CAIE
Module	P3 (Pure Mathematics 3)
Year	2010
Session	November
Marks	9
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Complex Numbers Argand & Loci
Type	Geometric relationships on Argand diagram
Difficulty	Moderate -0.8 This is a straightforward multi-part question testing basic complex number operations: finding modulus/argument of a simple complex number (√3 + i), performing algebraic manipulations with conjugates, and a simple geometric proof on an Argand diagram. All parts are routine applications of standard techniques with no problem-solving insight required, making it easier than average.
Spec	4.02a Complex numbers: real/imaginary parts, modulus, argument 4.02b Express complex numbers: cartesian and modulus-argument forms 4.02e Arithmetic of complex numbers: add, subtract, multiply, divide 4.02k Argand diagrams: geometric interpretation

6 The complex number $z$ is given by $$z = ( \sqrt { } 3 ) + \mathrm { i } .$$

Find the modulus and argument of $z$.
The complex conjugate of $z$ is denoted by $z ^ { * }$. Showing your working, express in the form $x + \mathrm { i } y$, where $x$ and $y$ are real,
1. $2 z + z ^ { * }$,
2. $\frac { \mathrm { i } z ^ { * } } { z }$.
3. On a sketch of an Argand diagram with origin $O$, show the points $A$ and $B$ representing the complex numbers $z$ and $\mathrm { i } z ^ { * }$ respectively. Prove that angle $A O B = \frac { 1 } { 6 } \pi$.

Show mark scheme Show mark scheme source

Answer	Marks
(i) State modulus is 2	B1
State argument is $\frac{1}{6}\pi$, or $30°$, or 0.524 radians	B1

[2 marks total]

Answer	Marks
(ii) (a) State answer $3\sqrt{3} + i$	B1
(ii) (b) EITHER: Multiply numerator and denominator by $\sqrt{3} - i$, or equivalent	M1
Simplify denominator to 4 or numerator to $2\sqrt{3} + 2i$	A1
Obtain final answer $\frac{1}{2}\sqrt{3} + \frac{1}{4}i$, or equivalent	A1
OR 1: Obtain two equations in $x$ and $y$ and solve for $x$ or for $y$	M1
Obtain $x = \frac{1}{2}\sqrt{3}$ or $y = \frac{1}{4}$	A1
Obtain final answer $\frac{1}{2}\sqrt{3} + \frac{1}{4}i$, or equivalent	A1
OR 2: Using the correct processes express $iz^* / z$ in polar form	M1
Obtain $x = \frac{1}{2}\sqrt{3}$ or $y = \frac{1}{4}$	A1
Obtain final answer $\frac{1}{2}\sqrt{3} + \frac{1}{4}i$, or equivalent	A1

[4 marks total]

Answer	Marks
(ii) (iii) Plot $A$ and $B$ in relatively correct positions	B1
EITHER: Use fact that angle $AOB = \arg(iz^*) - \arg z$	M1
Obtain the given answer	A1
OR 1: Obtain $\tan AOB$ from gradients of $OA$ and $OB$ and the correct $\tan(A - B)$ formula	M1
Obtain the given answer	A1
OR 2: Obtain $\cos AOB$ by using correct cosine formula or scalar product	M1
Obtain the given answer	A1

[3 marks total]

**(i)** State modulus is 2 | B1 |
State argument is $\frac{1}{6}\pi$, or $30°$, or 0.524 radians | B1 |
[2 marks total]

**(ii) (a)** State answer $3\sqrt{3} + i$ | B1 |

**(ii) (b)** EITHER: Multiply numerator and denominator by $\sqrt{3} - i$, or equivalent | M1 |
Simplify denominator to 4 or numerator to $2\sqrt{3} + 2i$ | A1 |
Obtain final answer $\frac{1}{2}\sqrt{3} + \frac{1}{4}i$, or equivalent | A1 |

OR 1: Obtain two equations in $x$ and $y$ and solve for $x$ or for $y$ | M1 |
Obtain $x = \frac{1}{2}\sqrt{3}$ or $y = \frac{1}{4}$ | A1 |
Obtain final answer $\frac{1}{2}\sqrt{3} + \frac{1}{4}i$, or equivalent | A1 |

OR 2: Using the correct processes express $iz^* / z$ in polar form | M1 |
Obtain $x = \frac{1}{2}\sqrt{3}$ or $y = \frac{1}{4}$ | A1 |
Obtain final answer $\frac{1}{2}\sqrt{3} + \frac{1}{4}i$, or equivalent | A1 |
[4 marks total]

**(ii) (iii)** Plot $A$ and $B$ in relatively correct positions | B1 |

EITHER: Use fact that angle $AOB = \arg(iz^*) - \arg z$ | M1 |
Obtain the given answer | A1 |

OR 1: Obtain $\tan AOB$ from gradients of $OA$ and $OB$ and the correct $\tan(A - B)$ formula | M1 |
Obtain the given answer | A1 |

OR 2: Obtain $\cos AOB$ by using correct cosine formula or scalar product | M1 |
Obtain the given answer | A1 |
[3 marks total]

Show LaTeX source

6 The complex number $z$ is given by

$$z = ( \sqrt { } 3 ) + \mathrm { i } .$$

(i) Find the modulus and argument of $z$.\\
(ii) The complex conjugate of $z$ is denoted by $z ^ { * }$. Showing your working, express in the form $x + \mathrm { i } y$, where $x$ and $y$ are real,
\begin{enumerate}[label=(\alph*)]
\item $2 z + z ^ { * }$,
\item $\frac { \mathrm { i } z ^ { * } } { z }$.\\
(iii) On a sketch of an Argand diagram with origin $O$, show the points $A$ and $B$ representing the complex numbers $z$ and $\mathrm { i } z ^ { * }$ respectively. Prove that angle $A O B = \frac { 1 } { 6 } \pi$.
\end{enumerate}

\hfill \mbox{\textit{CAIE P3 2010 Q6 [9]}}

This paper (10 questions)

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Q1 4 Q2 4 Q3 5 Q4 7 Q5 7 Q6 9 Q7 9 Q8 10 Q9 10 Q10 10

Answer	Marks
(i) State modulus is 2	B1
State argument is \(\frac{1}{6}\pi\), or \(30°\), or 0.524 radians	B1

Answer	Marks
(ii) (a) State answer \(3\sqrt{3} + i\)	B1
(ii) (b) EITHER: Multiply numerator and denominator by \(\sqrt{3} - i\), or equivalent	M1
Simplify denominator to 4 or numerator to \(2\sqrt{3} + 2i\)	A1
Obtain final answer \(\frac{1}{2}\sqrt{3} + \frac{1}{4}i\), or equivalent	A1
OR 1: Obtain two equations in \(x\) and \(y\) and solve for \(x\) or for \(y\)	M1
Obtain \(x = \frac{1}{2}\sqrt{3}\) or \(y = \frac{1}{4}\)	A1
Obtain final answer \(\frac{1}{2}\sqrt{3} + \frac{1}{4}i\), or equivalent	A1
OR 2: Using the correct processes express \(iz^* / z\) in polar form	M1
Obtain \(x = \frac{1}{2}\sqrt{3}\) or \(y = \frac{1}{4}\)	A1
Obtain final answer \(\frac{1}{2}\sqrt{3} + \frac{1}{4}i\), or equivalent	A1

Answer	Marks
(ii) (iii) Plot \(A\) and \(B\) in relatively correct positions	B1
EITHER: Use fact that angle \(AOB = \arg(iz^*) - \arg z\)	M1
Obtain the given answer	A1
OR 1: Obtain \(\tan AOB\) from gradients of \(OA\) and \(OB\) and the correct \(\tan(A - B)\) formula	M1
Obtain the given answer	A1
OR 2: Obtain \(\cos AOB\) by using correct cosine formula or scalar product	M1
Obtain the given answer	A1