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. $33$
• B. $57$
• C. $9$
• D. $24$

Answer 1

The correct option is C

$9$

Explanation for the correct option:

Step 1. Find the value of $a+b$:

Given, ${(2+\alpha )}^4=a+b\alpha$

Let $\alpha =\omega$ $\left[\mathbf{\because }{\mathit{\omega }}^{\mathbf{3}}\mathbf{}\mathbf{=}\mathbf{}\mathbf{1}\right]$

Step 2. Put the value of $\alpha$ in the given equation, we get

${(2+\omega )}^4=a+b\omega$

$\Rightarrow$$2^4+4\cdot 2^3\omega +6·2^2{\omega }^2+4·2{\omega }^3+{\omega }^4=a+b\omega$

$\Rightarrow$ $16+32\omega +24{\omega }^2+8+\omega =a+b\omega$

$\Rightarrow$ $24+24{\omega }^2+33\omega =a+b\omega$

$\Rightarrow$ $-24\omega +33\omega =a+b\omega$

$\Rightarrow$ $9\omega =a+b\omega$

$\Rightarrow$ $b=9$ and $a=0$

$\mathbf{\therefore }\mathit{a}\mathbf{}\mathbf{+}\mathbf{}\mathit{b}\mathbf{}\mathbf{=}\mathbf{}\mathbf{9}$

Hence, Option ‘C’ is Correct.