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(B) For a system of homogeneous linear equations to have infinitely many solutions,the determinant of the coefficient matrix must be zero.$\Delta = \b

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is strictly increasing on $R$

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(B) For a system of homogeneous linear equations to have infinitely many solutions,the determinant of the coefficient matrix must be zero.$\Delta = \begin{vmatrix} 1 & 2 & t \ 6 & 1 & 5t \ 3 & t^2 & 1 \end{vmatrix} = 0$.Expanding the determinant along the first row:$1(1 - 5t^3) - 2(6 - 15t) + t(6t^2 - 3) = 0$$1 - 5t^3 - 12 + 30t + 6t^3 - 3t = 0$$t^3 + 27t - 11 = 0$.Since the problem states the system has infinitely many solutions for all $t \in R$,this implies the determinant must be identically zero for all $t$. However,the expression $t^3 + 27t - 11$ is a polynomial that is not identically zero. This suggests an error in the problem statement's premise regarding 'all $t$'. Assuming the question implies the existence of $t$ values,if we treat the expression as a function $f(t) = t^3 + 27t - 11$,its derivative $f'(t) = 3t^2 + 27$ is always positive. Thus,$f(t)$ is strictly increasing.

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