| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2014 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Argument relationships and tangent identities |
| Difficulty | Standard +0.3 Part (i) is a routine complex division requiring multiplication by conjugate—standard C3/P3 technique. Part (ii) involves multiplying complex numbers and using the argument addition property (arg(wz) = arg(w) + arg(z)), then recognizing that arg relates to arctan. While it requires connecting multiple concepts, the steps are straightforward once the relationship between argument and arctan is understood. This is slightly easier than average as it's a guided multi-part question with clear signposting. |
| Spec | 4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| State or imply \(iw = -3 + 5i\) | B1 | |
| Carry out multiplication by \(\frac{4-i}{4-i}\) | M1 | |
| Obtain final answer \(-\frac{7}{17} + \frac{23}{17}i\) or equivalent | A1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Multiply \(w\) by \(z\) to obtain \(17 + 17i\) | B1 | |
| State \(\arg w = \tan^{-1}\frac{3}{5}\) or \(\arg z = \tan^{-1}\frac{1}{4}\) | B1 | |
| State \(\arg wz = \arg w + \arg z\) | M1 | |
| Confirm given result \(\tan^{-1}\frac{3}{5} + \tan^{-1}\frac{1}{4} = \frac{1}{4}\pi\) legitimately | A1 | [4] |
**(i)**
State or imply $iw = -3 + 5i$ | B1 |
Carry out multiplication by $\frac{4-i}{4-i}$ | M1 |
Obtain final answer $-\frac{7}{17} + \frac{23}{17}i$ or equivalent | A1 | [3]
**(ii)**
Multiply $w$ by $z$ to obtain $17 + 17i$ | B1 |
State $\arg w = \tan^{-1}\frac{3}{5}$ or $\arg z = \tan^{-1}\frac{1}{4}$ | B1 |
State $\arg wz = \arg w + \arg z$ | M1 |
Confirm given result $\tan^{-1}\frac{3}{5} + \tan^{-1}\frac{1}{4} = \frac{1}{4}\pi$ legitimately | A1 | [4]5 The complex numbers $w$ and $z$ are defined by $w = 5 + 3 \mathrm { i }$ and $z = 4 + \mathrm { i }$.\\
(i) Express $\frac { \mathrm { i } w } { z }$ in the form $x + \mathrm { i } y$, showing all your working and giving the exact values of $x$ and $y$.\\
(ii) Find $w z$ and hence, by considering arguments, show that
$$\tan ^ { - 1 } \left( \frac { 3 } { 5 } \right) + \tan ^ { - 1 } \left( \frac { 1 } { 4 } \right) = \frac { 1 } { 4 } \pi$$
\hfill \mbox{\textit{CAIE P3 2014 Q5 [7]}}