If the maximum and minimum values of the determinant vmatrix 1+ ^2x& ^2x& 2x ^2x&1+…

MATHEMATICS · Multi correct · CBSE · Mathematics · Determinants · Determinants

Problem

If the maximum and minimum values of the determinant ∣1+sin⁡2xcos⁡2xsin⁡2xsin⁡2x1+cos⁡2xsin⁡2xsin⁡2xcos⁡2x1+sin⁡2x∣\begin{vmatrix}1+\sin^2x&\cos^2x&\sin2x\\\sin^2x&1+\cos^2x&\sin2x\\\sin^2x&\cos^2x&1+\sin2x\end{vmatrix} are α\alpha and β\beta, respectively, then (1) α+β99=4\alpha+\beta^{99}=4 (2) α3−β17=26\alpha^3-\beta^{17}=26 (3) (α2n−β2n)(\alpha^{2n}-\beta^{2n}) is always an even integer for n∈Nn\in\mathbb{N} (4) A triangle can be drawn having its sides as α\alpha, β\beta, and α−β\alpha-\beta.

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