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The determinant D = |1+sin²(x) cos²(x) sin(2x); sin²(x) 1+cos²(x) sin(2x); sin²(x) cos²(x) 1+sin(2x)| has maximum value alpha and minimum value beta. Which of the following statements is INCORRECT?

1. alpha + beta⁹⁹ = 4
2. alpha³ − beta¹⁷ = 26
3. (alpha^(2n) − beta^(2n)) is always an even integer for n in N
4. A triangle can be constructed having its sides as alpha, beta and 5*alpha − 13*beta

Correct answer: A triangle can be constructed having its sides as alpha, beta and 5*alpha − 13*beta

Solution

After row reduction the determinant simplifies to 2 + sin(2x), so alpha = 3 (max) and beta = 1 (min). The triangle inequality check gives sides 3, 1, and 5(3)−13(1) = 2; since 1+2 = 3 (not strictly greater), no valid triangle exists, making statement D incorrect.

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