| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2004 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Standard quadratic with real coefficients |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question testing standard complex number techniques: solving a quadratic using the formula, converting to modulus-argument form, and verifying a simple algebraic property. All parts are routine applications of basic complex number theory with no novel insight required, making it slightly easier than average. |
| Spec | 4.02b Express complex numbers: cartesian and modulus-argument forms4.02i Quadratic equations: with complex roots4.02q De Moivre's theorem: multiple angle formulae |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| (i) EITHER: Solve the quadratic and use \(\sqrt{-1} = i\) | M1 | |
| Obtain roots \(\frac{1}{2} + i\frac{\sqrt{3}}{2}\) and \(\frac{1}{2} - i\frac{\sqrt{3}}{2}\), or equivalent | A1 | |
| OR: Substitute \(x + iy\) and solve for \(x\) or \(y\) | M1 | |
| Obtain correct roots | A1 | Total: 2 |
| (ii) State that the modulus of each root is equal to 1 | B1\(\sqrt{}\) | |
| State that the arguments are \(\frac{1}{3}\pi\) and \(-\frac{1}{3}\pi\) respectively | B1\(\sqrt{}\) + B1\(\sqrt{}\) | Total: 3 |
| [Accept degrees and \(\frac{5}{3}\pi\) instead of \(-\frac{1}{3}\pi\). Accept modulus in form \(\sqrt{\frac{p}{q}}\) or \(\sqrt{n}\), where \(p, q, n\) are integers.] | ||
| (iii) EITHER: Verify \(z^3 = -1\) for each root | B1 + B1 | |
| OR: State \(z^3 + 1 = (z+1)(z^2 - z + 1)\) | B1 | |
| Justify the given statement | B1 | |
| OR: Obtain \(z^3 = z^2 - z\) | B1 | |
| Justify the given statement | B1 | Total: 2 |
## Question 8:
| Answer/Working | Mark | Guidance |
|---|---|---|
| **(i) EITHER:** Solve the quadratic and use $\sqrt{-1} = i$ | M1 | |
| Obtain roots $\frac{1}{2} + i\frac{\sqrt{3}}{2}$ and $\frac{1}{2} - i\frac{\sqrt{3}}{2}$, or equivalent | A1 | |
| **OR:** Substitute $x + iy$ and solve for $x$ or $y$ | M1 | |
| Obtain correct roots | A1 | **Total: 2** |
| **(ii)** State that the modulus of each root is equal to 1 | B1$\sqrt{}$ | |
| State that the arguments are $\frac{1}{3}\pi$ and $-\frac{1}{3}\pi$ respectively | B1$\sqrt{}$ + B1$\sqrt{}$ | **Total: 3** |
| [Accept degrees and $\frac{5}{3}\pi$ instead of $-\frac{1}{3}\pi$. Accept modulus in form $\sqrt{\frac{p}{q}}$ or $\sqrt{n}$, where $p, q, n$ are integers.] | | |
| **(iii) EITHER:** Verify $z^3 = -1$ for each root | B1 + B1 | |
| **OR:** State $z^3 + 1 = (z+1)(z^2 - z + 1)$ | B1 | |
| Justify the given statement | B1 | |
| **OR:** Obtain $z^3 = z^2 - z$ | B1 | |
| Justify the given statement | B1 | **Total: 2** |
---8 (i) Find the roots of the equation $z ^ { 2 } - z + 1 = 0$, giving your answers in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.\\
(ii) Obtain the modulus and argument of each root.\\
(iii) Show that each root also satisfies the equation $z ^ { 3 } = - 1$.
\hfill \mbox{\textit{CAIE P3 2004 Q8 [7]}}