| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2015 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Geometric properties using complex numbers |
| Difficulty | Standard +0.8 This is a well-structured multi-part question requiring visualization on an Argand diagram, algebraic manipulation of complex numbers, and a non-trivial proof connecting arguments to inverse tangent identities. Part (iii) requires genuine insight to connect arg(u*/u) to the tangent formula, going beyond routine calculation. The proof element and the need to synthesize geometric and algebraic properties elevates this above standard complex number exercises. |
| Spec | 4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02k Argand diagrams: geometric interpretation |
| Answer | Marks |
|---|---|
| (i) Show \(u\) in a relatively correct position | B1 |
| Show \(u^*\) in a relatively correct position | B1 |
| Show \(u^* - u\) in a relatively correct position | B1 |
| State or imply that \(OABC\) is a parallelogram | B1 |
| [4] | |
| (ii) EITHER: Substitute for \(u\) and multiply numerator and denominator by \(3 + i\), or equivalent | M1 |
| Simplify the numerator to \(8 + 6i\) or the denominator to \(10\) | A1 |
| Obtain final answer \(\frac{4}{5} + \frac{3}{5}i\), or equivalent | A1 |
| OR: Substitute for \(u\), obtain two equations in \(x\) and \(y\) and solve for \(x\) or for \(y\) | M1 |
| Obtain \(x = -\frac{3}{5}\) or \(y = \frac{3}{5}\), or equivalent | A1 |
| Obtain final answer \(\frac{4}{5} + \frac{3}{5}i\), or equivalent | A1 |
| [3] | |
| (iii) State or imply \(\arg(u^*/u) = \tan^{-1}(\frac{3}{4})\) | B1 |
| Substitute exact arguments in \(\arg(u^*/u) = \arg u^* - \arg u\) | M1 |
| Fully justify the given statement using exact values | A1 |
| [3] |
**(i)** Show $u$ in a relatively correct position | B1 |
Show $u^*$ in a relatively correct position | B1 |
Show $u^* - u$ in a relatively correct position | B1 |
State or imply that $OABC$ is a parallelogram | B1 |
| [4] |
**(ii)** EITHER: Substitute for $u$ and multiply numerator and denominator by $3 + i$, or equivalent | M1 |
Simplify the numerator to $8 + 6i$ or the denominator to $10$ | A1 |
Obtain final answer $\frac{4}{5} + \frac{3}{5}i$, or equivalent | A1 |
| |
OR: Substitute for $u$, obtain two equations in $x$ and $y$ and solve for $x$ or for $y$ | M1 |
Obtain $x = -\frac{3}{5}$ or $y = \frac{3}{5}$, or equivalent | A1 |
Obtain final answer $\frac{4}{5} + \frac{3}{5}i$, or equivalent | A1 |
| [3] |
**(iii)** State or imply $\arg(u^*/u) = \tan^{-1}(\frac{3}{4})$ | B1 |
Substitute exact arguments in $\arg(u^*/u) = \arg u^* - \arg u$ | M1 |
Fully justify the given statement using exact values | A1 |
| [3] |9 The complex number 3 - i is denoted by $u$. Its complex conjugate is denoted by $u ^ { * }$.\\
(i) On an Argand diagram with origin $O$, show the points $A , B$ and $C$ representing the complex numbers $u , u ^ { * }$ and $u ^ { * } - u$ respectively. What type of quadrilateral is $O A B C$ ?\\
(ii) Showing your working and without using a calculator, express $\frac { u ^ { * } } { u }$ in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.\\
(iii) By considering the argument of $\frac { u ^ { * } } { u }$, prove that
$$\tan ^ { - 1 } \left( \frac { 3 } { 4 } \right) = 2 \tan ^ { - 1 } \left( \frac { 1 } { 3 } \right)$$
\hfill \mbox{\textit{CAIE P3 2015 Q9 [10]}}