Question

If a and $b$ are real numbers such that $(2+\alpha {)}^4=a+b\alpha ,$ where $\alpha =\frac{-1+i\sqrt{3}}{2},$ then $a+b$ is equal to

• A. 57
• B. 33
• C. 24
• D. 9

Answer 1

Answer: (D)

$\alpha =\omega \left({\omega }^3=1\right)$
$\Rightarrow (2+\omega {)}^4=a+b\omega$
$\Rightarrow 2^4+4\cdot 2^3\omega +6.2^2{\omega }^3+4.2\cdot {\omega }^3+{\omega }^4=a+b\omega$
$\Rightarrow 16+32\omega +24{\omega }^2+8+\omega =a+b\omega$
$\Rightarrow 24+24{\omega }^2+33\omega =a+b\omega$
$\Rightarrow -24\omega +33\omega =a+b\omega$
$\Rightarrow a=0,b=9$