Date | November 2019 | Marks available | 2 | Reference code | 19N.1.AHL.TZ0.H_5 |
Level | Additional Higher Level | Paper | Paper 1 (without calculator) | Time zone | Time zone 0 |
Command term | Find | Question number | H_5 | Adapted from | N/A |
Question
Consider the equation z4=−4, where z∈C.
Solve the equation, giving the solutions in the form a+ib, where a, b∈R.
The solutions form the vertices of a polygon in the complex plane. Find the area of the polygon.
Markscheme
METHOD 1
|z|=4√4(=√2) (A1)
arg(z1)=π4 (A1)
first solution is 1+i A1
valid attempt to find all roots (De Moivre or +/− their components) (M1)
other solutions are −1+i, −1−i, 1−i A1
METHOD 2
z4=−4
(a+ib)4=−4
attempt to expand and equate both reals and imaginaries. (M1)
a4+4a3bi−6a2b2−4ab3i+b4=−4
(a4−6a4+a4=−4⇒)a=±1 and (4a3b−4ab3=0⇒)a=±b (A1)
first solution is 1+i A1
valid attempt to find all roots (De Moivre or +/− their components) (M1)
other solutions are −1+i, −1−i, 1−i A1
[5 marks]
complete method to find area of ‘rectangle' (M1)
=4 A1
[2 marks]