Question

The greatest integer less than or equal to $(\sqrt{3}+2{)}^5$ is

• A. 721
• B. 722
• C. 723
• D. 724

Answer 1

Answer: (C)

Let $(2+\sqrt{3}{)}^5=I+F$
$[\because I$ is integer, $F$ is fraction $]$
$\because 2-\sqrt{3}<1$
$\Rightarrow (2-\sqrt{3}{)}^5<1$
Let $(2-\sqrt{3}{)}^5=F^{{}^{\prime }}$
$\therefore I+F+F^{{}^{\prime }}=(2+\sqrt{3}{)}^5+(2-\sqrt{3}{)}^5$
$=2\left(2^5+{}^5{C}_2{2}^3(3)+{}^5{C}_{4}2\cdot 3^2\right)$
$I+F+F^{{}^{\prime }}=2(32+240+90)$
$I+F+F^{{}^{\prime }}=724$
$I=724-\left(F+F^{{}^{\prime }}\right)$
$I=724-1=723\left[\because F+F^{{}^{\prime }}=1\right]$