Consider the following statements in respect of the determinant Δ = \(\Delta = \begin{vmatrix} k(k + 2) & 2k + 1 & 1 \\ 2k + 1 & k + 2 & 1 \\ 3 & 3 & 1 \end{vmatrix}\) I. Δ is positive if k > 0.
II. Δ is negative if k < 0.
III. Δ is zero if k = 0.
How many of the statements given above are correct?
One
1, it's negative for 0 < k < 1 and zero at k = 1, making the statement only partially correct. Statement II is entirely correct because for all k < 0, (k - 1)³ is always negative. Statement III is incorrect since at k = 0, Δ equals (-1)³ = -1 ≠ 0. Therefore, only Statement II is completely correct out of the three given statements. The answer is that exactly one statement is correct.
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