Mathematics - Binomial Theorem Question with Solution | TestHub

$\mathrm{x}-x^2+\mathrm{x}[\mathrm{x}]=\mathrm{x}-\mathrm{x}(\mathrm{x}-[\mathrm{x}])=\mathrm{x}(1-\{\mathrm{x}\})$ Now $\mathrm{x}+x_1=$ even integer where $x_1=(2-\sqrt{3}{)}^n$ clearly $x_1\epsilon (0,1)\forall \mathrm{n}\epsilon N$. $\begin{array}{rlrl}& \therefore \{\mathrm{x}\}+x_1=\text{ integer }& \Rightarrow \{\mathrm{x}\}+x_1=1& \Rightarrow x(1-\{x\})=x\cdot x_1=(2+\sqrt{3}{)}^n(2-\sqrt{3}{)}^n=1\end{array}$