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} $
$= 5\left[\left\{\left(-1\right)^{2}-2\times1\right\}^{2}-2\left(1\right)^{2}\right]+\frac{7}{1}$
$=5\left(1-2\right)+7=2$.