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 $(2-\sqrt{3})^{5}=F^{'}$
$\therefore I+F+F^{'}=(2+\sqrt{3})^{5}+(2-\sqrt{3})^{5}$
$=2\left(2^{5}+{ }^{5} C_{2} 2^{3}(3)+{ }^{5} C_{4} 2 \cdot 3^{2}\right)$
$I+F+F^{'}=2(32+240+90)$
$I+F+F^{'}=724$
$I=724-\left(F+F^{'}\right)$
$I=724-1=723 \,\,\,\left[\because F+F^{'}=1\right]$