Question:
The probability that a randomly selected root of the equation \( 1 + x + x^2 + \ldots + x^{118} = 0 \) satisfies the equation \( x^7 = 1 \), is
The probability that a randomly selected root of the equation \( 1 + x + x^2 + \ldots + x^{118} = 0 \) satisfies the equation \( x^7 = 1 \), is
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The roots of the equation are the 119th roots of unity other than 1. Count how many of them are also 7th roots of unity.
Updated On: Nov 12, 2026
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- \( \frac{1}{59} \)
- \( \frac{3}{59} \)
- \( \frac{7}{118} \)
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The Correct Option is C
Solution and Explanation
The equation is \( \frac{x^{119}-1}{x-1}=0 \), so its roots are the 119th roots of unity except \( x=1 \). Number of roots = 118.
Since \( 119 = 7 \times 17 \), every 7th root of unity is also a 119th root of unity.
The roots of \( x^7=1 \) are 7 in number, and excluding \( x=1 \) leaves 6 of them among our 118 roots.
Probability \( = \frac{6}{118} = \frac{3}{59} \).
Answer: (3) \( \frac{3}{59} \)
Since \( 119 = 7 \times 17 \), every 7th root of unity is also a 119th root of unity.
The roots of \( x^7=1 \) are 7 in number, and excluding \( x=1 \) leaves 6 of them among our 118 roots.
Probability \( = \frac{6}{118} = \frac{3}{59} \).
Answer: (3) \( \frac{3}{59} \)
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