| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2005 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Single locus sketching |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question requiring routine techniques: substitution to verify a root, using conjugate root theorem, and sketching a perpendicular bisector locus. All parts are standard textbook exercises with no novel problem-solving required, making it slightly easier than average. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.02i Quadratic equations: with complex roots4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks |
|---|---|
| Substitute \(x = 1 + 2i\) and attempt expansions | M1 |
| Use \(i^2 = -1\) correctly at least once | M1 |
| Complete the verification correctly | A1 |
| Answer | Marks |
|---|---|
| State that the other complex root is \(1 - 2i\) | B1 |
| Answer | Marks |
|---|---|
| Show \(1 + 2i\) in relatively correct position | B1 |
| Sketch a locus which | B1 |
| (a) is a straight line | B1 |
| (b) relative to the point representing \(1 + 2i\) (call it \(A\)), passes through the mid-point of \(OA\) | B1 |
| (c) intersects \(OA\) at right angles | B1 |
**(i)**
Substitute $x = 1 + 2i$ and attempt expansions | M1 |
Use $i^2 = -1$ correctly at least once | M1 |
Complete the verification correctly | A1 |
**Total for (i): [3]**
**(ii)**
State that the other complex root is $1 - 2i$ | B1 |
**Total for (ii): [1]**
**(iii)**
Show $1 + 2i$ in relatively correct position | B1 |
Sketch a locus which | B1 |
(a) is a straight line | B1 |
(b) relative to the point representing $1 + 2i$ (call it $A$), passes through the mid-point of $OA$ | B1 |
(c) intersects $OA$ at right angles | B1 |
**Total for (iii): [4]**
---7 The equation $2 x ^ { 3 } + x ^ { 2 } + 25 = 0$ has one real root and two complex roots.\\
(i) Verify that $1 + 2 \mathrm { i }$ is one of the complex roots.\\
(ii) Write down the other complex root of the equation.\\
(iii) Sketch an Argand diagram showing the point representing the complex number $1 + 2 \mathrm { i }$. Show on the same diagram the set of points representing the complex numbers $z$ which satisfy
$$| z | = | z - 1 - 2 \mathrm { i } |$$
\hfill \mbox{\textit{CAIE P3 2005 Q7 [8]}}