| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2016 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Modulus and argument with operations |
| Difficulty | Standard +0.3 This is a straightforward multi-part complex numbers question requiring standard techniques: finding modulus/argument, algebraic manipulation with conjugates, and basic Argand diagram geometry. While it covers multiple skills, each part uses routine methods with no novel insight required, making it slightly easier than average. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02c Complex notation: z, z*, Re(z), Im(z), |z|, arg(z)4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State modulus \(2\sqrt{2}\), or equivalent | B1 | |
| State argument \(-\frac{1}{3}\pi\) (or \(-60°\)) | B1 | [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State answer \(3\sqrt{2} + \sqrt{6}\,i\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| EITHER: Substitute for \(z\) and multiply numerator and denominator by conjugate of \(iz\) | M1 | |
| Simplify numerator to \(4\sqrt{3}+4i\) or denominator to \(8\) | A1 | |
| Obtain final answer \(\frac{1}{2}\sqrt{3}+\frac{1}{2}i\) | A1 | |
| OR: Substitute for \(z\), obtain two equations in \(x\) and \(y\) and solve for \(x\) or \(y\) | M1 | |
| Obtain \(x = \frac{1}{2}\sqrt{3}\) or \(y = \frac{1}{2}\) | A1 | |
| Obtain final answer \(\frac{1}{2}\sqrt{3}+\frac{1}{2}i\) | A1 | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Show points \(A\) and \(B\) in relatively correct positions | B1 | |
| Carry out complete method for finding angle \(AOB\), e.g. calculate the argument of \(\frac{z^*}{iz}\) | M1 | |
| Obtain the given answer | A1 | [3] |
## Question 7:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State modulus $2\sqrt{2}$, or equivalent | B1 | |
| State argument $-\frac{1}{3}\pi$ (or $-60°$) | B1 | [2] |
### Part (ii)(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State answer $3\sqrt{2} + \sqrt{6}\,i$ | B1 | |
### Part (ii)(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| **EITHER:** Substitute for $z$ and multiply numerator and denominator by conjugate of $iz$ | M1 | |
| Simplify numerator to $4\sqrt{3}+4i$ or denominator to $8$ | A1 | |
| Obtain final answer $\frac{1}{2}\sqrt{3}+\frac{1}{2}i$ | A1 | |
| **OR:** Substitute for $z$, obtain two equations in $x$ and $y$ and solve for $x$ or $y$ | M1 | |
| Obtain $x = \frac{1}{2}\sqrt{3}$ or $y = \frac{1}{2}$ | A1 | |
| Obtain final answer $\frac{1}{2}\sqrt{3}+\frac{1}{2}i$ | A1 | [4] |
### Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Show points $A$ and $B$ in relatively correct positions | B1 | |
| Carry out complete method for finding angle $AOB$, e.g. calculate the argument of $\frac{z^*}{iz}$ | M1 | |
| Obtain the given answer | A1 | [3] |
---(i) Find the modulus and argument of $z$.\\
(ii) Express each of the following in the form $x + \mathrm { i } y$, where $x$ and $y$ are real and exact:
\begin{enumerate}[label=(\alph*)]
\item $z + 2 z ^ { * }$;
\item $\frac { z ^ { * } } { \mathrm { i } z }$.\\
(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 $A O B$ is equal to $\frac { 1 } { 6 } \pi$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2016 Q7 [9]}}