| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2013 |
| Session | June |
| Marks | 6 |
| Topic | Complex Numbers Argand & Loci |
| Type | Modulus-argument form conversion |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question testing basic complex number operations: calculating modulus using Pythagoras, finding argument using arctan (with attention to quadrant), and finding the reciprocal by multiplying by the conjugate. All three parts are standard textbook exercises requiring direct application of well-practiced techniques with no problem-solving or insight needed. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms |
| Answer | Marks | Guidance |
|---|---|---|
| Obtain \( | z | = 29\) — A1 [2] |
**(i)** Attempt use of Pythagoras — M1
Obtain $|z| = 29$ — A1 **[2]**
**(ii)** Attempt fully correct argument for arg $z$ using tan ratio or equivalent — M1
State $134°$ or $2.33$ rad — A1 **[2]**
**(iii)** Show or imply multiplication by conjugate or equivalent method — M1
Obtain $(-20 - 21\text{i})/841$ — A1 **[2]** **[6]**7 The complex number $z$ is given by $- 20 + 21 \mathrm { i }$. Showing all your working,\\
(i) find the value of $| z |$,\\
(ii) calculate the value of $\arg z$ correct to 3 significant figures,\\
(iii) express $\frac { 1 } { z }$ in the form $x + \mathrm { i } y$, where $x$ and $y$ are real numbers.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2013 Q7 [6]}}