Question

$\alpha=\sin 36^{\circ}$ is a root of which of the following equation

• A. $10 x^{4}-10 x^{2}-5=0$
• B. $16 x^{4}+20 x^{2}-5=0$
• C. $16 x^{4}-20 x^{2}+5=0$
• D. $16 x^{4}-10 x^{2}+5=0$

Answer 1

Answer: (C)

$\cos 72^{\circ}=\frac{\sqrt{5}-1}{4}$
$\Rightarrow 1-2 \sin ^{2} 36^{\circ}=\frac{\sqrt{5}-1}{4}$
$\Rightarrow 4-8 \alpha^{2}=\sqrt{5}-1$
$\Rightarrow 5-8 \alpha^{2}=\sqrt{5}$
$\Rightarrow \left(5-8 \alpha^{2}\right)^{2}=5$
$\Rightarrow 25+64 \alpha^{4}-80 \alpha^{2}=5$
$\Rightarrow 64 \alpha^{4}-80 \alpha^{2}+20=0$
$\Rightarrow 16 \alpha^{4}-20 \alpha^{2}+5=0$