If maximum and minimum values of the determinant

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If maximum and minimum values of the determinant $\left|\begin{array}{ccc}1+\sin ^{2} x & \cos ^{2} x & \sin 2 x \\ \sin ^{2} x & 1+\cos ^{2} x & \sin 2 x \\ \sin ^{2} x & \cos ^{2} x & 1+\sin 2 x\end{array}\right|$ are $\alpha$ and $\beta$, then (A) $\alpha+\beta^{99}=4$ (B) $\alpha^{3}-\beta^{17}=26$ (C) $\left(\alpha^{2 n}-\beta^{2 n}\right)$ is always an even integer for $n \in N$ (D) a triangle can be constructed having its sides as $\alpha-\beta, \alpha+\beta$ and $\alpha+3 \beta$

If maximum and minimum values of the determinant $\left|\begin{array}{ccc}1+\sin ^{2} x & \cos ^{2} x & \sin 2 x \\ \sin ^{2} x & 1+\cos ^{2} x & \sin 2 x \\ \sin ^{2} x & \cos ^{2} x & 1+\sin 2 x\end{array}\right|$ are $\alpha$ and $\beta$, then
(A) $\alpha+\beta^{99}=4$
(B) $\alpha^{3}-\beta^{17}=26$
(C) $\left(\alpha^{2 n}-\beta^{2 n}\right)$ is always an even integer for $n \in N$
(D) a triangle can be constructed having its sides as $\alpha-\beta, \alpha+\beta$ and $\alpha+3 \beta$

Key Concepts

Triangle Inequality

The triangle inequality is a fundamental principle in geometry that states the sum of the lengths of any two sides of a triangle must exceed the length of the remaining side. This concept is crucial when determining whether a set of numerical values can represent the sides of a triangle, as it imposes necessary conditions on their relationships.

Optimization of Functions

Optimization involves finding the maximum or minimum values of a function, often by analyzing its critical points or employing known bounds of the trigonometric functions involved. In problems where expressions depend on variable parameters (like x in sine and cosine functions), understanding these extremal values is key to deducing the range and behavior of the overall expression.

Integer Parity and Exponent Properties

Integer parity refers to whether an integer is even or odd, and this concept interacts with exponentiation in problems where powers of numbers are involved. Analyzing the parity of expressions raised to certain powers can reveal insights into the structure of the numeric expression, such as guaranteeing that certain combinations result in even integers.

Matrix Determinants

Matrix determinants are scalar attributes computed from square matrices that capture key properties such as invertibility and volume scaling. Evaluating a determinant, especially when matrix entries involve functions like trigonometric expressions, often involves identifying symmetries or applying row operations to simplify the computation.

Trigonometric Identities

Trigonometric identities are equations involving sine, cosine, and other trigonometric functions that hold true for all values within their domains. They are essential for rewriting and simplifying expressions, such as converting sin² x and cos² x into equivalent forms, or expressing sin 2x in a different form, which facilitates the evaluation of complex algebraic structures like determinants.

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