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

Step-by-step solution

Given determinant is = vmatrix k(k+2) & 2k+1 & 1 2k+1 & k+2 & 1 3 & 3 & 1 vmatrix Applying row operations R₁ R₁ - R₂ and R₂ R₂ - R₃ : = vmatrix k^2-1 & k-1 & 0 2k-2 & k-1 & 0 3 & 3 & 1 vmatrix Expanding along the third column: = 1 vmatrix k^2-1 & k-1 2k-2 & k-1 vmatrix = (k^2-1)(k-1) - (2k-2)(k-1) = (k-1) [ k^2 - 1 - 2k + 2 ] = (k-1) (k^2 - 2k + 1) = (k-1)^3 Evaluating the given statements: I. For k = 1 2 > 0 , = ( 1 2 - 1 )^3 = - 1 8 II. For k III. For k = 0 , = (0-1)^3 = -1 0 . Thus, statement III is incorrect. O