19N.1.AHL.TZ0.H_5

pestleMathematicsAAHLPaper 119N· ahl-1-14-complex-roots-of-polynomials-conjugate-roots-de-moivres-powers-&-roots-of-complex-numbers, ahl-1-12-complex-numbers-cartesian-form-and-argand-diagsource ↗

Consider the equation ${z^4} = - 4$ , where $z \in \mathbb{C}$ .

Solve the equation, giving the solutions in the form $a + {\text{i}}b$ , where $a{\text{, }}b \in \mathbb{R}$ .

[5]

The solutions form the vertices of a polygon in the complex plane. Find the area of the polygon.

[2]

Markscheme / solution

METHOD 1

$\left| z \right| = \sqrt[4]{4}\left( { = \sqrt 2 } \right)$ (A1)

${\text{arg}}\left( {{z_1}} \right) = \frac{\pi }{4}$ (A1)

first solution is $1 + {\text{i}}$ A1

valid attempt to find all roots (De Moivre or +/− their components) (M1)

other solutions are $- 1 + {\text{i}}$ , $- 1 - {\text{i}}$ , $1 - {\text{i}}$ A1

METHOD 2

${z^4} = - 4$

${\left( {a + {\text{i}}b} \right)^4} = - 4$

attempt to expand and equate both reals and imaginaries. (M1)

${a^4} + 4{a^3}b{\text{i}} - 6{a^2}{b^2} - 4a{b^3}{\text{i}} + {b^4} = - 4$

$\left( {{a^4} - 6{a^4} + {a^4} = - 4 \Rightarrow } \right)a = \pm 1$ and $\left( {4{a^3}b - 4a{b^3} = 0 \Rightarrow } \right)a = \pm b$ (A1)

first solution is $1 + {\text{i}}$ A1

valid attempt to find all roots (De Moivre or +/− their components) (M1)

other solutions are $- 1 + {\text{i}}$ , $- 1 - {\text{i}}$ , $1 - {\text{i}}$ A1

[5 marks]

complete method to find area of ‘rectangle' (M1)

$= 4$ A1

[2 marks]

Examiners’ report

[N/A]