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The maximum and minimum values of the determinant D = |1+sin²(x) cos²(x) sin(2x)| |sin²(x) 1+cos²(x) sin(2x)| |sin²(x) cos²(x) 1+sin(2x)| are alpha and beta respectively. Which one of the following is INCORRECT?

1. alpha + beta⁹⁹ = 4
2. alpha³ - beta¹⁷ = 26
3. (alpha^(2n) - beta^(2n)) is always an even integer for n belonging to natural numbers
4. alpha * beta = 4

Correct answer: alpha * beta = 4

Solution

Using R1->R1-R2 and R2->R2-R3 the determinant simplifies to 2 + sin(2x), giving alpha = 3 (max) and beta = 1 (min). Then alpha*beta = 3*1 = 3, not 4, making option D incorrect. Options A, B, C all check out correctly.

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