The determinant | array ccc 1 & (x-3) & (x-3)^ 2 1 & (x-4) & (x-4)^...

The given determinant vanishes, i.e., \left|$\begin{array}{ccc}1 & x-3 & (x-3)^{2} \\ 1 & x-4 & (x-4)^{2} \\ 1 & x-5 & (x-5)^{2} \end{array}$\right|=0 Expanding along $\mathrm{C}_{1}$,$ we get\begin{array}{l} (x-4)(x-5)^{2}-(x-5)(x-4)^{2}-\left\{(x-3)(x-5)^{2}\right. \\ \left.-(x-5)(x-3)^{2}\right\}+(x-3)(x-4)^{2} \\ -(x-4)(x-3)^{2}=0 \\ \Rightarrow(x-4)(x-5)(x-5-x+4) \\ -(x-3)(x-5)(x-5-x+3) \\ +(x-3)(x-4)(x-4-x+3)=0 \end{array}$$\Rightarrow-(x-4)(x-5)+2(x-3)(x-5)-(x-3)$(x-4)=0$\Rightarrow-x^{2}+9 x-20+2 x^{2}-16 x+30-x^{2}+7 x-12=0$\Rightarrow-32+30=0 \Rightarrow-2=0Which is not possible, hence no value of $\mathrm{x} satisfies the given condition.$