AP DSC School Assistant Math Model Test 2

Concept:The n^th roots of unity are the complex numbers that satisfy $z^n = 1$. Using De Moivre's theorem, these roots are equally spaced on the unit circle and each root is obtained by multiplying the previous one by a constant complex factor, hence they form a geometric progression (G.P.).Explanation:Step 1: Write 1 in polar form: $1 = \cos 2m\pi + i\sin 2m\pi$, where $m$ is an integer.Step 2: The n^th root is: $1^{1/n} = (\cos 2m\pi + i\sin 2m\pi)^{1/n} = \cos\left(\frac{2m\pi}{n}\right) + i\sin\left(\frac{2m\pi}{n}\right)$.Step 3: For $m = 0, 1, 2, \dots, n-1$, we get the n distinct roots: $1,\ \cos\frac{2\pi}{n}+i\sin\frac{2\pi}{n},\ \cos\frac{4\pi}{n}+i\sin\frac{4\pi}{n},\ \dots,\ \cos\frac{2(n-1)\pi}{n}+i\sin\frac{2(n-1)\pi}{n}$.Step 4: Let $\alpha = \cos\frac{2\pi}{n} + i\sin\frac{2\pi}{n}$. Then the second root is $\alpha$, the third is $\alpha^2$, and so on. Thus the roots are $1, \alpha, \alpha^2, \alpha^3, \dots, \alpha^{n-1}$.Step 5: Here, each term (except the first) is obtained by multiplying the previous term by $\alpha$. This is the definition of a geometric progression (G.P.) with common ratio $\alpha$.Answer:B. G.P.