| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2005 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Roots of polynomial equations |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question requiring standard techniques: quadratic formula with complex coefficients (routine), converting to modulus-argument form (standard calculation), and sketching points on an Argand diagram (basic visualization). While it involves multiple steps, each is a textbook exercise with no problem-solving insight required, making it slightly easier than average. |
| Spec | 4.02b Express complex numbers: cartesian and modulus-argument forms4.02i Quadratic equations: with complex roots4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Use quadratic formula, or the method of completing the square, or the substitution \(z = x + iy\) to find a root, using \(i^2 = -1\) | M1 | |
| Obtain a root, e.g. \(2 + i\) | A1 | |
| Obtain the other root \(-2 + i\) | A1 | 3 |
| [Roots given as \(\pm 2 + i\) earn A1 + A1.] | ||
| (ii) Obtain modulus \(\sqrt{5}\) (or \(2.24\)) of both roots | B1√ | |
| Obtain argument of \(2 + i\) as \(26.6°\) or \(0.464\) radians (allow \(\pm1\) in final figure) | B1√ | |
| Obtain argument of \(-2 + i\) as \(153.4°\) or \(2.68\) radians (allow \(\pm1\) in final figure) | B1√ | 3 |
| [SR: in applying the follow through to the roots obtained in (i), if both roots are real or pure imaginary, the mark for the moduli is not available and only B1√ is given if both arguments are correct; also if one of the two roots is real or pure imaginary and the other is neither then B1√ is given if both moduli are correct and B1√ if both arguments are correct.] | ||
| (iii) Show both roots on an Argand diagram in relatively correct positions | B1√ | 1 |
| [This follow through is only available if at least one of the two roots is of the form \(x + iy\) where \(xy \neq 0\).] |
**(i)** Use quadratic formula, or the method of completing the square, or the substitution $z = x + iy$ to find a root, using $i^2 = -1$ | M1 |
Obtain a root, e.g. $2 + i$ | A1 |
Obtain the other root $-2 + i$ | A1 | 3
[Roots given as $\pm 2 + i$ earn A1 + A1.] | |
**(ii)** Obtain modulus $\sqrt{5}$ (or $2.24$) of both roots | B1√ |
Obtain argument of $2 + i$ as $26.6°$ or $0.464$ radians (allow $\pm1$ in final figure) | B1√ |
Obtain argument of $-2 + i$ as $153.4°$ or $2.68$ radians (allow $\pm1$ in final figure) | B1√ | 3
[SR: in applying the follow through to the roots obtained in (i), if both roots are real or pure imaginary, the mark for the moduli is not available and only B1√ is given if both arguments are correct; also if one of the two roots is real or pure imaginary and the other is neither then B1√ is given if both moduli are correct and B1√ if both arguments are correct.] | |
**(iii)** Show both roots on an Argand diagram in relatively correct positions | B1√ | 1
[This follow through is only available if at least one of the two roots is of the form $x + iy$ where $xy \neq 0$.] | |3 (i) Solve the equation $z ^ { 2 } - 2 \mathrm { i } z - 5 = 0$, giving your answers in the form $x + \mathrm { i } y$ where $x$ and $y$ are real.\\
(ii) Find the modulus and argument of each root.\\
(iii) Sketch an Argand diagram showing the points representing the roots.
\hfill \mbox{\textit{CAIE P3 2005 Q3 [7]}}