Consider the following statements in respect of the determinant \Delta=\begin{vmatrix}k(k+2)&2k+1&1\\ 2k+1&k+2&1\\ 3&3&1\end{vmatrix} I. \Delta is positive if k>0. II. \Delta is negative if k<0. III. \Delta is zero if k=0. How many of the statements given above are correct?

Explanation

Applying row operations R_1 \to R_1 - R_2 and R_2 \to R_2 - R_3 simplifies \Delta to (k-1)^3. For k=0, \Delta=-1 \neq 0. For k=2, \Delta=1 \gt 0, but for k=0.5, \Delta=-0.125 \lt 0. So \Delta is always negative for k \lt 0 (II is true), but not always positive for k \gt 0. Only 1 statement is correct.

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