Multiple choice

Consider the following statements in respect of the roots of the equation x3 – 8 = 0: 1. The roots are non-collinear. 2. The roots lie on a circle of unit radius. Which of the above statements is/are correct?

  1. Only 1

  2. Only 2

  3. Both 1 and 2

  4. Neither 1 nor 2
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A Correct answer
Explanation

The roots of x^3 - 8 = 0 are 2, 2w, and 2w^2, where w is a complex cube root of unity. These roots are 2, -1 + i*sqrt(3), and -1 - i*sqrt(3), which are non-collinear in the complex plane, and they lie on a circle of radius 2, not unit radius.

AI explanation

The roots of the equation x3 - 8 = 0 are the cube roots of 8, which can be found using the complex roots formula to include 2, -1 + i√3, and -1 - i√3. The distance between the real root (2, 0) and the two complex roots (-1, ±√3) is √(9 + 3) = 2√3, meaning the points do not lie on a straight line and are non-collinear. Furthermore, the distance from the origin to each root is 2, not 1, so they lie on a circle of radius 2, not a unit radius. Therefore, only the first statement is correct.

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