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The equation a*z² + a*z + 1 = 0 has roots that are purely imaginary, where a = cos(theta) + i*sin(theta) and i = sqrt(-1). Which of the following conclusions are correct?

1. cos(theta) = 2*sin(pi/10)
2. z = +-i*sqrt(2*sin(pi/10))
3. z = +-sqrt(2*sin(pi/10))
4. sin(theta) = 2*sin(pi/10)

Correct answer: cos(theta) = 2*sin(pi/10)

Solution

Substituting z = i*t into az² + az + 1 = 0 and equating real and imaginary parts of the resulting complex equation leads to the condition cos(theta) = 2*sin(pi/10), confirming option A.

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