Let \( \alpha_1, \alpha_2, \dots, \alpha_7 \) be the roots of the equation \( x^7 + 3x^5 - 13x^3 - 15x = 0 \) and \( |\alpha_1| \geq |\alpha_2| \geq \dots \geq |\alpha_7| \). Then \( \alpha_1 \alpha_2 - \alpha_3 \alpha_4 + \alpha_5 \alpha_6 \) is equal to:
Correct Answer: 9
Solution and Explanation
Given equation can be rearranged as
x(x⁶ + 3x⁴ - 13x² - 15) = 0
Clearly x = 0 is one of the roots and the other part can be observed by replacing x² = t, from which we get:
t³ + 3t² - 13t - 15 = 0
⇒ (t - 3)(t² + 6t + 5) = 0
So, t = 3, t = -1, t = -5
Now we are getting:
x² = 3, x² = -1, x² = -5
⇒ x = ±√3, x = ±i, x = ±√5i
From the given condition:
|α₁| ≥ |α₂| ≥ .... ≥ |α₆|
We can clearly say that:
|α₁| = 0 and |α₂| = √5 = |α₅| and |α₄| = √3
= |α₃| and |α₂| = 1 = |α₁|
So we can have:
α₁ = √5i, α₂ = -√5i, α₃ = √3i, α₄ = √3, α₅ = i, α₆ = -i
Hence
α₁ - α₂ - α₃ + α₄ + α₅ + α₆ = 1 - (-3) + 5 = 9
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