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If \(S_k = a^k + b^k + c^k\), then the determinant \[ \Delta = \begin{vmatrix} S_0 & S_1 & S_2 \\ S_1 & S_2 & S_3 \\ S_2 & S_3 & S_4 \end{vmatrix} \] is equal to:

1. \(S_6\)
2. \(S_5 - S_3\)
3. \(S_6 - S_4\)
4. None of these

Correct answer: \(S_6\)

Solution

The determinant is derived from the properties of symmetric sums, where the structure of the matrix reflects the recurrence relations of the sums of powers of the roots. Evaluating the determinant leads to the conclusion that it simplifies to the sum of the sixth powers, hence it equals S_6.

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