Roots Of Unity
The sum of all the distinct fifth roots of unity in the complex plane, taken together as complex numbers, equals which value?
Select the correct option:
Solution
0
The n distinct nth roots of unity are the solutions of z^n = 1 and are equally spaced on the unit circle, a configuration central to JEE Advanced. They are the roots of the polynomial z^5 - 1 = 0. By Vieta's formulas, the sum of all roots of z^5 - 1 equals the negative of the coefficient of z^4 divided by the leading coefficient. Since z^5 - 1 has no z^4 term, that coefficient is 0, so the sum of the roots is 0. Geometrically, the five roots are vertices of a regular pentagon centred at the origin, and their position vectors cancel by symmetry. Option 1 counts only the root z = 1. Option 5 wrongly adds the count of roots. Option -1 has no symmetric basis. Hence the sum is 0. Plausibility check: for any n ≥ 2 the nth roots of unity sum to zero because they form a symmetric set about the origin, and the absent z^{n-1} term in z^n - 1 confirms this algebraically.
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About This Question
- Subject
- mathematics
- Chapter
- complex numbers and quadratic equations
- Topic
- roots of unity
- Difficulty
- Easy
- Year
- 2025
Solution
Correct Answer:
0
The n distinct nth roots of unity are the solutions of z^n = 1 and are equally spaced on the unit circle, a configuration central to JEE Advanced. They are the roots of the polynomial z^5 - 1 = 0. By Vieta's formulas, the sum of all roots of z^5 - 1 equals the negative of the coefficient of z^4 divided by the leading coefficient. Since z^5 - 1 has no z^4 term, that coefficient is 0, so the sum of the roots is 0. Geometrically, the five roots are vertices of a regular pentagon centred at the origin, and their position vectors cancel by symmetry. Option 1 counts only the root z = 1. Option 5 wrongly adds the count of roots. Option -1 has no symmetric basis. Hence the sum is 0. Plausibility check: for any n ≥ 2 the nth roots of unity sum to zero because they form a symmetric set about the origin, and the absent z^{n-1} term in z^n - 1 confirms this algebraically.
This easy difficulty mathematics question is from the chapter complex numbers and quadratic equations, covering the topic of roots of unity. It appeared in the 2025 exam.
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