OCR MEI
Further Pure Core
2021
November
— Question 10
Exam Board
OCR MEI
Module
Further Pure Core (Further Pure Core)
Year
2021
Session
November
Topic
Complex numbers 2
10
Show on an Argand diagram the points representing the three cube roots of unity.
Find the exact roots of the equation \(z ^ { 3 } - 1 = \sqrt { 3 } \mathrm { i }\), expressing them in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta < \pi\).
The points representing the cube roots of unity form a triangle \(\Delta _ { 1 }\). The points representing the roots of the equation \(z ^ { 3 } - 1 = \sqrt { 3 } \mathrm { i }\) form a triangle \(\Delta _ { 2 }\).
State a sequence of two transformations that maps \(\Delta _ { 1 }\) onto \(\Delta _ { 2 }\).
The three roots in part (b)(i) are \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\).
By simplifying \(z _ { 1 } + z _ { 2 } + z _ { 3 }\), verify that the sum of these roots is zero.