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)

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