Calculus questions

Question

Question 1

 

Question 2

Answer 1

(2).

Notice that each of the three complex roots of w = 1 has the same magnitude, 1. This means that the points corresponding to each cube root lie the same distance from the origin; that is, the three points lie on a circle with center at the origin and radius 1.

Furthermore, the arguments of these cube roots are 0°, 120°, and 240°, the difference of consecutive pairs being 120° = 360°/3.This means that the three points are equally spaced on the circle, as shown in Figure.These results are not coincidental.

Graph :

(a).

From the graph, the value of (w + w²) is a real.

The first option is correct.

(b).

Let w be one of the complex cube roots of 1.

w = 1^1/3

w³ = (1^1/3)³

w³ = 1

w³ - 1³ = 0

(w - 1)(w² + w + 1) = 0

w - 1 = 0 and w² + w + 1 = 0

w = 1 and w² + w = - 1.

The exact value of w² + w = - 1.