Find the roots of \(w^2=-2\mathrm i\) in the form \(p+q\mathrm i\), where \(p,q\in\mathbb R\).
Hence solve \((1+\mathrm iw)^2=-2\mathrm i\).[5]
Suppose \(z_1=-2\mathrm i\) and \(z_2=\mathrm i-\sqrt3\).
Verify that \(z_1\) and \(z_2\) are roots of the equation \(z^3=8\mathrm i\), where \(z\in\mathbb C\).
Find the third root \(z_3\) in the form \(a+b\mathrm i\), where \(a,b\in\mathbb R\).[6]
On an Argand diagram, \(z_1\), \(z_2\) and \(z_3\) are represented by the points \(A\), \(B\) and \(C\) respectively. Find the area of triangle \(ABC\).[2]
Solution:
Solution locked
Sign in to view the step-by-step solution
Similar questions are unavailable for this question.
Answer:(a)(i) \(1-\mathrm i,-1+\mathrm i\), (ii) \(-1,1+2\mathrm i\) (b)(i) Verified, (ii) \(\sqrt3+\mathrm i\) (c) \(3\sqrt3\)