Question

If $2\alpha =-1-i\sqrt{3}$ and $2\beta =-1+i\sqrt{3}$, then $5{\alpha }^4+5{\beta }^4+7{\alpha }^{-1}{\beta }^{-1}$ is equal to

• A. $-1$
• B. $-2$
• C. $2$
• D. $1$
• E. 0

Answer 1

Answer: (C)

Given that, $2\alpha =-1-i\sqrt{3}$
and $2\beta =-1+i\sqrt{3}$
$\therefore \alpha +\beta =-1$ and $\alpha \beta =1$
Now, $5{\alpha }^4+5{\beta }^4+\frac{7}{\alpha \beta }$
$=5\left[{\left\{{\left(\alpha +\beta \right)}^2-2\alpha {\beta }^3-2\left(\alpha \beta \right)\right\}}^2\right]+\frac{7}{\alpha \beta }$
$=5\left[{\left\{{\left(-1\right)}^2-2\times 1\right\}}^2-2{\left(1\right)}^2\right]+\frac{7}{1}$
$=5\left(1-2\right)+7=2$.