Problem

Problem

If maximum and minimum values of the determinant ∣1+sin⁡2xcos⁡2xsin⁡2xsin⁡2x1+cos⁡2xsin⁡2xsin⁡2xcos⁡2x1+sin⁡2x∣\begin{vmatrix} 1 + \sin^2 x & \cos^2 x & \sin 2x \\ \sin^2 x & 1 + \cos^2 x & \sin 2x \\ \sin^2 x & \cos^2 x & 1 + \sin 2x \end{vmatrix} are α\alpha and β\beta , then a. α+β99=4\alpha + \beta^{99} = 4 b. α3−β17=26\alpha^3 - \beta^{17} = 26 c. (α2n−β2n)(\alpha^{2n} - \beta^{2n}) is always an even integer for n∈Nn \in \mathbb{N} d. a triangle can be drawn having its sides as α,β\alpha, \beta and α−β\alpha - \beta

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