| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2010 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Region shading with multiple inequalities |
| Difficulty | Standard +0.3 Part (a) requires direct substitution to verify a root and stating the conjugate—routine procedures. Part (b) involves sketching a standard circle and half-line on an Argand diagram, then shading their intersection—straightforward loci work requiring no novel insight, though the visualization and accurate sketching adds minor complexity beyond pure algebraic manipulation. |
| Spec | 4.02i Quadratic equations: with complex roots4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| (a) EITHER: Substitute \(1 + i\sqrt{3}\), attempt complete expansions of the \(x^3\) and \(x^2\) terms | M1 | — |
| Use \(i^2 = -1\) correctly at least once | B1 | — |
| Complete the verification correctly | A1 | — |
| State that the other root is \(1 - i\sqrt{3}\) | B1 | — |
| OR1: State that the other root is \(1 - i\sqrt{3}\) | B1 | — |
| State quadratic factor \(x^2 - 2x + 4\) | B1 | — |
| Divide cubic by 3-term quadratic reaching partial quotient \(2x + k\) | M1 | — |
| Complete the division obtaining zero remainder | A1 | — |
| OR2: State factorisation \((2x + 3)(x^2 - 2x + 4)\), or equivalent | B1 | — |
| Make reasonable solution attempt at a 3-term quadratic and use \(i^2 = -1\) | M1 | — |
| Obtain the root \(1 + i\sqrt{3}\) | A1 | — |
| State that the other root is \(1 - i\sqrt{3}\) | B1 | [4] |
| (b) Show point representing \(1 + i\sqrt{3}\) in relatively correct position on an Argand diagram | B1 | — |
| Show circle with centre at \(1 + i\sqrt{3}\) and radius 1 | B1√ | — |
| Show line for \(\arg z = \frac{1}{3}\pi\) making \(\frac{1}{3}\pi\) with the real axis | B1 | — |
| Show line from origin passing through centre of circle, or the diameter which would contain the origin if produced | B1 | — |
| Shade the relevant region | B1√ | [5] |
**(a)** **EITHER:** Substitute $1 + i\sqrt{3}$, attempt complete expansions of the $x^3$ and $x^2$ terms | M1 | —
Use $i^2 = -1$ correctly at least once | B1 | —
Complete the verification correctly | A1 | —
State that the other root is $1 - i\sqrt{3}$ | B1 | —
**OR1:** State that the other root is $1 - i\sqrt{3}$ | B1 | —
State quadratic factor $x^2 - 2x + 4$ | B1 | —
Divide cubic by 3-term quadratic reaching partial quotient $2x + k$ | M1 | —
Complete the division obtaining zero remainder | A1 | —
**OR2:** State factorisation $(2x + 3)(x^2 - 2x + 4)$, or equivalent | B1 | —
Make reasonable solution attempt at a 3-term quadratic and use $i^2 = -1$ | M1 | —
Obtain the root $1 + i\sqrt{3}$ | A1 | —
State that the other root is $1 - i\sqrt{3}$ | B1 | [4]
**(b)** Show point representing $1 + i\sqrt{3}$ in relatively correct position on an Argand diagram | B1 | —
Show circle with centre at $1 + i\sqrt{3}$ and radius 1 | B1√ | —
Show line for $\arg z = \frac{1}{3}\pi$ making $\frac{1}{3}\pi$ with the real axis | B1 | —
Show line from origin passing through centre of circle, or the diameter which would contain the origin if produced | B1 | —
Shade the relevant region | B1√ | [5]
---\begin{enumerate}[label=(\alph*)]
\item The equation $2x^3 - x^2 + 2x + 12 = 0$ has one real root and two complex roots. Showing your working, verify that $1 + i\sqrt{3}$ is one of the complex roots. State the other complex root. [4]
\item On a sketch of an Argand diagram, show the point representing the complex number $1 + i\sqrt{3}$. On the same diagram, shade the region whose points represent the complex numbers $z$ which satisfy both the inequalities $|z - 1 - i\sqrt{3}| \leq 1$ and $\arg z \leq \frac{1}{4}\pi$. [5]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2010 Q8 [9]}}