Question

If \( \omega \) is a complex cube root of unity, then \( \cos \left( \sum_{k=1}^{2} (k - \omega)(k - \omega^2) \frac{\pi}{175} \right) = \)

• A. \( -1 \)
• B. \( 0 \)
• C. \( 1 \)
• D. \( 5 \)

Hint:

Properties of complex cube roots of unity: \( 1 + \omega + \omega^2 = 0 \) and \( \omega^3 = 1 \). When simplifying expressions involving \( \omega \), these properties are crucial.

Answer 1

The Correct Option is A

We are given the expression \( \cos \left( \sum_{k=1}^{2} (k - \omega)(k - \omega^2) \frac{\pi}{175} \right) \).
Let's first evaluate the term \( (k - \omega)(k - \omega^2) \).
$$ (k - \omega)(k - \omega^2) = k^2 - k\omega^2 - k\omega + \omega^3 $$ Since \( \omega \) is a complex cube root of unity, we know that \( 1 + \omega + \omega^2 = 0 \) and \( \omega^3 = 1 \).
So, \( -\omega^2 - \omega = 1 \).
$$ (k - \omega)(k - \omega^2) = k^2 - k(\omega^2 + \omega) + 1 = k^2 - k(-1) + 1 = k^2 + k + 1 $$ Now let's evaluate the sum for \( k = 1 \) and \( k = 2 \): For \( k = 1 \): \( 1^2 + 1 + 1 = 3 \) For \( k = 2 \): \( 2^2 + 2 + 1 = 4 + 2 + 1 = 7 \) The sum becomes: $$ \sum_{k=1}^{2} (k - \omega)(k - \omega^2) = 3 + 7 = 10 $$ Now we can substitute this back into the cosine expression: $$ \cos \left( \sum_{k=1}^{2} (k - \omega)(k - \omega^2) \frac{\pi}{175} \right) = \cos \left( 10 \times \frac{\pi}{175} \right) = \cos \left( \frac{10\pi}{175} \right) = \cos \left( \frac{2\pi}{35} \right) $$ There seems to be a mistake in my calculation or understanding, as the result does not match any of the options.
Let me recheck the steps.
Re-evaluating \( (k - \omega)(k - \omega^2) \): $$ (k - \omega)(k - \omega^2) = k^2 - k(\omega + \omega^2) + \omega^3 $$ Using \( 1 + \omega + \omega^2 = 0 \), we have \( \omega + \omega^2 = -1 \) and \( \omega^3 = 1 \).
$$ (k - \omega)(k - \omega^2) = k^2 - k(-1) + 1 = k^2 + k + 1 $$ Sum for \( k = 1 \) and \( k = 2 \): For \( k = 1 \): \( 1^2 + 1 + 1 = 3 \) For \( k = 2 \): \( 2^2 + 2 + 1 = 7 \) The sum is \( 3 + 7 = 10 \).
The expression is \( \cos \left( 10 \frac{\pi}{175} \right) = \cos \left( \frac{2\pi}{35} \right) \).
This still doesn't match the options.
Let me check if I misinterpreted the question.
Perhaps the question intended a different range for \( k \).
Let's assume there was a typo and the result should lead to one of the options.
Consider the possibility that the sum was intended to be something else.
If the result of the sum was such that \( \frac{\text{sum} \times \pi}{175} \) is a multiple of \( \pi \), \( \frac{\pi}{2} \), etc.
, we could get the given options.
Let's review the properties of cube roots of unity again.
\( \omega = e^{i 2\pi/3} = \cos(2\pi/3) + i \sin(2\pi/3) = -\frac{1}{2} + i \frac{\sqrt{3}}{2} \) \( \omega^2 = e^{i 4\pi/3} = \cos(4\pi/3) + i \sin(4\pi/3) = -\frac{1}{2} - i \frac{\sqrt{3}}{2} \) If the question was \( \cos(\pi) \), the answer would be \( -1 \).
For this, we would need \( \frac{\text{sum} \times \pi}{175} = (2n + 1)\pi \) for some integer \( n \).
This means \( \text{sum} = 175 (2n + 1) \).
Our sum is 10, which doesn't fit this form.
Let's consider if there's a simplification I missed.
Rethink the problem statement: "If \( \omega \) is a complex cube root of unity.
.
.
" This implies either \( \omega = e^{i 2\pi/3} \) or \( \omega = e^{i 4\pi/3} \).
The result \( k^2 + k + 1 \) is independent of the choice of \( \omega \).
Given the correct answer is \( -1 \), we need \( \cos \left( 10 \frac{\pi}{175} \right) = -1 \), which implies \( \frac{2\pi}{35} = (2n + 1)\pi \), or \( \frac{2}{35} = 2n + 1 \), which is not possible for integer \( n \).
There must be an error in the question or the provided correct answer.
However, following the steps correctly, the argument of the cosine is \( \frac{2\pi}{35} \), and \( \cos \left( \frac{2\pi}{35} \right) \) is not equal to \( -1, 0, 1, \) or \( 5 \).
Assuming the correct answer provided is indeed \( -1 \), there might be a subtlety in the problem statement I am overlooking, or a typo.
If the sum was over a different range of \( k \), or if the fraction involved 10 instead of the sum.
Let's assume the question intended \( \cos(\pi) = -1 \).
Then \( \sum_{k=1}^{2} (k - \omega)(k - \omega^2) \frac{\pi}{175} = \pi \), which means \( 10 \frac{\pi}{175} = \pi \), or \( 10 = 175 \), which is false.
Given the discrepancy, and strictly following the problem as stated, the answer should be \( \cos \left( \frac{2\pi}{35} \right) \), which is not among the options.
However, since I must choose one of the provided correct answers, and there seems to be an issue with the question as transcribed or my understanding, I cannot definitively arrive at \( -1 \).
However, if there was a mistake and the sum resulted in a multiple of 175, e.
g.
, if the sum was 175, then \( \cos(\pi) = -1 \).
But our sum is 10.
Let's proceed with the next question, acknowledging the potential issue with this one.