| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2007 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Region shading with multiple inequalities |
| Difficulty | Standard +0.3 This is a standard two-part complex numbers question requiring routine calculations (simplifying a complex fraction, finding modulus/argument, squaring) followed by sketching a region defined by two inequalities. The perpendicular bisector interpretation of |z-u²|<|z-u| is a common A-level technique, making this slightly easier than average. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks |
|---|---|
| (i) EITHER: Carry out multiplication of numerator and denominator by \(-1 - i\), or solve for \(x\) or \(y\) | M1 |
| Obtain \(u = -1 - i\), or any equivalent of the form \((a + ib)/c\) | A1 |
| State modulus of \(u\) is \(\sqrt{2}\) or 1.41 | A1 |
| State argument of \(u\) is \(-\frac{3}{4}\pi\) (\(-2.36\)) or \(-135°\), or \(\frac{5}{4}\pi\) (3.93) or \(225°\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Divide the modulus of the numerator by that of the denominator | M1 | |
| State modulus of \(u\) is \(\sqrt{2}\) or 1.41 | A1 | |
| Subtract argument of denominator from that of the numerator, or equivalent | M1 | |
| State argument of \(u\) is \(-\frac{3}{4}\pi\) (\(-2.36\)) or \(-135°\), or \(\frac{5}{4}\pi\) (3.93) or \(225°\) | A1 | |
| Carry out method for finding the modulus or the argument of \(u^2\) | M1 | |
| State modulus of \(u\) is 2 and argument of \(u^2\) is \(\frac{1}{2}\pi\) (1.57) or \(90°\) | A1 | Total: 6 marks |
| (ii) Show \(u\) and \(u^2\) in relatively correct positions | B1 \(\checkmark\) | |
| Show a circle with centre at the origin and radius 2 | B1 | |
| Show the line which is the perpendicular bisector of the line joining \(u\) and \(u^2\) correctly | B1 \(\checkmark\) | |
| Shade the correct region, having obtained \(u\) and \(u^2\) correctly | B1 | Total: 4 marks |
(i) EITHER: Carry out multiplication of numerator and denominator by $-1 - i$, or solve for $x$ or $y$ | M1 |
Obtain $u = -1 - i$, or any equivalent of the form $(a + ib)/c$ | A1 |
State modulus of $u$ is $\sqrt{2}$ or 1.41 | A1 |
State argument of $u$ is $-\frac{3}{4}\pi$ ($-2.36$) or $-135°$, or $\frac{5}{4}\pi$ (3.93) or $225°$ | A1 |
OR:
Divide the modulus of the numerator by that of the denominator | M1 |
State modulus of $u$ is $\sqrt{2}$ or 1.41 | A1 |
Subtract argument of denominator from that of the numerator, or equivalent | M1 |
State argument of $u$ is $-\frac{3}{4}\pi$ ($-2.36$) or $-135°$, or $\frac{5}{4}\pi$ (3.93) or $225°$ | A1 |
Carry out method for finding the modulus or the argument of $u^2$ | M1 |
State modulus of $u$ is 2 and argument of $u^2$ is $\frac{1}{2}\pi$ (1.57) or $90°$ | A1 | **Total: 6 marks**
(ii) Show $u$ and $u^2$ in relatively correct positions | B1 $\checkmark$ |
Show a circle with centre at the origin and radius 2 | B1 |
Show the line which is the perpendicular bisector of the line joining $u$ and $u^2$ correctly | B1 $\checkmark$ |
Shade the correct region, having obtained $u$ and $u^2$ correctly | B1 | **Total: 4 marks**
---8 The complex number $\frac { 2 } { - 1 + \mathrm { i } }$ is denoted by $u$.\\
(i) Find the modulus and argument of $u$ and $u ^ { 2 }$.\\
(ii) Sketch an Argand diagram showing the points representing the complex numbers $u$ and $u ^ { 2 }$. Shade the region whose points represent the complex numbers $z$ which satisfy both the inequalities $| z | < 2$ and $\left| z - u ^ { 2 } \right| < | z - u |$.
\hfill \mbox{\textit{CAIE P3 2007 Q8 [10]}}