Multiple choice

If $\alpha$ and $\beta$ are the roots of the quadratic equation, $x^2 + x \sin \theta - 2\sin \theta = 0, \theta \in \left(0, \dfrac{\pi}{2}\right)$, then $\dfrac{\alpha^{12} + \beta^{12}}{(\alpha^{-12} + \beta^{-12}) (\alpha - \beta)^{24}}$ is equal to:

  1. $\dfrac{2^6}{(\sin \theta + 8)^{12}}$
  2. $\dfrac{2^{12}}{(\sin \theta - 8)^6}$
  3. $\dfrac{2^{12}}{(\sin \theta - 4)^{12}}$
  4. $\dfrac{2^{12}}{(\sin \theta + 8)^{12}}$
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D Correct answer
Explanation

The roots satisfy alpha + beta = -sin(theta) and alpha * beta = -2sin(theta). The expression simplifies using the properties of roots and trigonometric identities to the given result.

AI explanation

Expanding the denominator requires multiplying the sum of inverse twelfth powers by the twenty-fourth power of the difference of the roots. Since the roots are a and b, the term (a^-12 + b^-12) is equal to (a^12 + b^12) divided by (ab)^12. The product of this with (a - b)^24 forms a denominator of (a^12 + b^12) multiplied by ((a - b)^2 / (ab))^12. The expression therefore simplifies to 1 / (((a - b)^2 / (ab))^12), and because (a - b)^2 = (a + b)^2 - 4ab, we can substitute the sum of roots -sin(t) and the product -2sin(t) to get ((sin^2(t) + 8sin(t)) / (-2sin(t)))^12. Factoring out sin(t) from the numerator yields ((sin(t) + 8) / (-2))^12, which is equivalent to (sin(t) + 8)^12 / 2^12. Taking the reciprocal of this quantity gives the final result of 2^12 / (sin(t) + 8)^12.

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