Question

Given that $\alpha$ is root of $x^4+x^2=1$ then the value of ${({\alpha }^6+2{\alpha }^4)}^{1000}$ is

• A. ${\left(\frac{1}{\sqrt{5+1}}\right)}^{500}$
• B. ${\left[\frac{\left(\sqrt{5-1}\right)}{2}\right]}^{1000}$
• C. $1$
• D. $Noneofthese$

Answer 1

The correct option is C $1$

$x^4+x^2=1$
$\sin ce\alpha is\text{the root of give equation}$
$\text{Hence}$
${\alpha }^4+{\alpha }^2-1=0........\left(i\right)$
$multiplyby{\alpha }^{2}bothsides$
$\Rightarrow {\alpha }^2({\alpha }^4+{\alpha }^2-1)=0$
$\Rightarrow {\alpha }^6+{\alpha }^4-{\alpha }^2=0$
$\Rightarrow {\alpha }^6+{\alpha }^4-(-{\alpha }^4+1)=0\left[from\left(i\right)\right]$
$\Rightarrow {\alpha }^6+2{\alpha }^4-1=0$
$\Rightarrow {\alpha }^6+2{\alpha }^4=1$
$Hence$
${({\alpha }^6+2{\alpha }^4)}^{1000}$
$={\left(1\right)}^{1000}=1$