| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2011 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Region shading with multiple inequalities |
| Difficulty | Challenging +1.2 This is a multi-part complex numbers question requiring standard techniques (simplifying complex fractions, finding modulus/argument) followed by geometric interpretation of loci. Part (ii) requires finding the minimum distance from origin to a half-line, and part (iii) finding maximum distance to a circle—both involve geometric insight but are standard A-level Further Maths exercises. The calculations are routine once the geometric setup is understood, placing this moderately above average difficulty. |
| Spec | 4.02b Express complex numbers: cartesian and modulus-argument forms4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Multiply numerator and denominator by \((1-2i)\), or equivalent | M1 | |
| Obtain \(-3i\) | A1 | |
| State modulus is \(3\) | A1 | |
| Refer to \(u\) being on negative imaginary axis or equivalent and confirm argument as \(-\frac{1}{2}\pi\) | A1 | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Using correct processes, divide moduli of numerator and denominator | M1 | |
| Obtain \(3\) | A1 | |
| Subtract argument of denominator from argument of numerator | M1 | |
| Obtain \(-\tan^{-1}\frac{1}{2} - \tan^{-1}2\) or \(-0.464 - 1.107\) and hence \(-\frac{1}{2}\pi\) or \(-1.57\) | A1 | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Show correct half-line from \(u\) at angle \(\frac{1}{4}\pi\) to real direction | B1 | |
| Use correct trigonometry to find required value | M1 | |
| Obtain \(\frac{3}{2}\sqrt{2}\) or equivalent | A1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Show, or imply, locus is a circle with centre \((1+i)u\) and radius \(1\) | M1 | |
| Use correct method to find distance from origin to furthest point of circle | M1 | |
| Obtain \(3\sqrt{2} + 1\) or equivalent | A1 | [3] |
## Question 8:
### Part (i):
**Either method:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| Multiply numerator and denominator by $(1-2i)$, or equivalent | M1 | |
| Obtain $-3i$ | A1 | |
| State modulus is $3$ | A1 | |
| Refer to $u$ being on negative imaginary axis or equivalent and confirm argument as $-\frac{1}{2}\pi$ | A1 | [4] |
**Or method:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| Using correct processes, divide moduli of numerator and denominator | M1 | |
| Obtain $3$ | A1 | |
| Subtract argument of denominator from argument of numerator | M1 | |
| Obtain $-\tan^{-1}\frac{1}{2} - \tan^{-1}2$ or $-0.464 - 1.107$ and hence $-\frac{1}{2}\pi$ or $-1.57$ | A1 | [4] |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Show correct half-line from $u$ at angle $\frac{1}{4}\pi$ to real direction | B1 | |
| Use correct trigonometry to find required value | M1 | |
| Obtain $\frac{3}{2}\sqrt{2}$ or equivalent | A1 | [3] |
### Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Show, or imply, locus is a circle with centre $(1+i)u$ and radius $1$ | M1 | |
| Use correct method to find distance from origin to furthest point of circle | M1 | |
| Obtain $3\sqrt{2} + 1$ or equivalent | A1 | [3] |8 The complex number $u$ is defined by $u = \frac { 6 - 3 \mathrm { i } } { 1 + 2 \mathrm { i } }$.\\
(i) Showing all your working, find the modulus of $u$ and show that the argument of $u$ is $- \frac { 1 } { 2 } \pi$.\\
(ii) For complex numbers $z$ satisfying $\arg ( z - u ) = \frac { 1 } { 4 } \pi$, find the least possible value of $| z |$.\\
(iii) For complex numbers $z$ satisfying $| z - ( 1 + \mathrm { i } ) u | = 1$, find the greatest possible value of $| z |$.
\hfill \mbox{\textit{CAIE P3 2011 Q8 [10]}}