| Exam Board | OCR |
|---|---|
| Module | FP1 AS (Further Pure 1 AS) |
| Year | 2018 |
| Session | March |
| Marks | 6 |
| Topic | Complex Numbers Argand & Loci |
| Type | Modulus-argument form conversion |
| Difficulty | Moderate -0.8 This is a straightforward application of modulus-argument form conversion and multiplication rules. Part (i) requires calculating |z₁| = 5 and arg(z₁) = arctan(-4/3) ≈ -0.927 radians using standard formulas. Part (ii) simply applies the rule that |z₁z₂| = |z₁||z₂| and arg(z₁z₂) = arg(z₁) + arg(z₂), requiring minimal problem-solving beyond direct recall and calculation. |
| Spec | 4.02b Express complex numbers: cartesian and modulus-argument forms4.02e Arithmetic of complex numbers: add, subtract, multiply, divide |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \( | z_i | = 5\) |
| \(\arg(z_i) = \tan^{-1}\frac{-4}{3}\) | M1 | Or could be implied by eg \(\tan^{-1}\frac{-4}{3}\) |
| "\(5(\cos(-0.927) + i\sin(-0.927))\) or \(5(\cos(5.36) + i\sin(5.36))\)" or "[5.5.36] or Scis(5.36)" | A1 | Or \([5,-0.927]\) or Scis(-0.927) |
| [3] | ||
| (ii) \(30\times...\) | B1ft | 6×their 5 |
| \(...(cos(-2.5 + \theta) + i\sin(-2.5 + \theta))\) | M1 | where \(\theta\) is their argument from (i) seen or implied |
| \(30(\cos(2.86) + i\sin(2.86))\) cao | A1 | |
| [3] |
**(i)** $|z_i| = 5$ | B1 |
$\arg(z_i) = \tan^{-1}\frac{-4}{3}$ | M1 | Or could be implied by eg $\tan^{-1}\frac{-4}{3}$
"$5(\cos(-0.927) + i\sin(-0.927))$ or $5(\cos(5.36) + i\sin(5.36))$" or "[5.5.36] or Scis(5.36)" | A1 | Or $[5,-0.927]$ or Scis(-0.927)
| [3] |
**(ii)** $30\times...$ | B1ft | 6×their 5
$...(cos(-2.5 + \theta) + i\sin(-2.5 + \theta))$ | M1 | where $\theta$ is their argument from (i) seen or implied
$30(\cos(2.86) + i\sin(2.86))$ cao | A1 |
| [3] |
---1 (i) The complex number 3-4i is denoted by $z _ { 1 }$. Write $z _ { 1 }$ in modulus-argument form, giving your angle in radians to 3 significant figures.\\
(ii) The complex number $z _ { 2 }$ has modulus 6 and argument - 2.5 radians.
Express $z _ { 1 } z _ { 2 }$ in modulus-argument form with the angle in radians correct to 3 significant figures.
\hfill \mbox{\textit{OCR FP1 AS 2018 Q1 [6]}}