| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2016 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Geometric properties using complex numbers |
| Difficulty | Challenging +1.2 This is a multi-part complex numbers geometry question requiring sketch interpretation, division of complex numbers, and angle proof using argument properties. While it involves several steps and the connection between complex arithmetic and geometry, these are standard P3/Further Pure techniques. The angle proof likely uses arg(u/v) = arg(u) - arg(v), which is a well-practiced method. More challenging than routine arithmetic but less demanding than questions requiring novel geometric insight or extended proof chains. |
| 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 |
|---|---|---|
| (i) EITHER: Multiply numerator and denominator of \(\frac{u}{v}\) by \(2 + i\), or equivalent | M1 | |
| Simplify the numerator to \(-5 + 5i\) or denominator to \(5\) | A1 | |
| Obtain final answer \(-1 + i\) | A1 | |
| OR: Obtain two equations in \(x\) and \(y\) and solve for \(x\) or for \(y\) | M1 | |
| Obtain \(x = -1\) or \(y = 1\) | A1 | |
| Obtain final answer \(-1 + i\) | A1 | [3] |
| (ii) Obtain \(u + v = 1 + 2i\) | B1 | |
| In an Argand diagram show points \(A\), \(B\), \(C\) representing \(u\), \(v\) and \(u + v\) respectively | B1 | ✓ |
| State that \(OB\) and \(AC\) are parallel | B1 | |
| State that \(OB = AC\) | B1 | [4] |
| (iii) Carry out an appropriate method for finding angle \(AOB\), e.g. find \(\arg(u/v)\) | M1 | |
| Show sufficient working to justify the given answer \(\frac{7}{4}\pi\) | A1 | [2] |
**(i)** **EITHER:** Multiply numerator and denominator of $\frac{u}{v}$ by $2 + i$, or equivalent | M1 |
Simplify the numerator to $-5 + 5i$ or denominator to $5$ | A1 |
Obtain final answer $-1 + i$ | A1 |
**OR:** Obtain two equations in $x$ and $y$ and solve for $x$ or for $y$ | M1 |
Obtain $x = -1$ or $y = 1$ | A1 |
Obtain final answer $-1 + i$ | A1 | [3]
**(ii)** Obtain $u + v = 1 + 2i$ | B1 |
In an Argand diagram show points $A$, $B$, $C$ representing $u$, $v$ and $u + v$ respectively | B1 | ✓ |
State that $OB$ and $AC$ are parallel | B1 |
State that $OB = AC$ | B1 | [4]
**(iii)** Carry out an appropriate method for finding angle $AOB$, e.g. find $\arg(u/v)$ | M1 |
Show sufficient working to justify the given answer $\frac{7}{4}\pi$ | A1 | [2]
---(i) Sketch this diagram and state fully the geometrical relationship between $O B$ and $A C$.\\
(ii) Find, in the form $x + \mathrm { i } y$, where $x$ and $y$ are real, the complex number $\frac { u } { v }$.\\
(iii) Prove that angle $A O B = \frac { 3 } { 4 } \pi$.
\hfill \mbox{\textit{CAIE P3 2016 Q9 [9]}}