Question

If $\alpha$, $\beta$ are the roots of the equation $x^2+x+2=0$, then what is $\frac{{\alpha }^{10}+{\beta }^{10}}{{\alpha }^{-10}+{\beta }^{-10}}$ equal to?

• A. $4096$
• B. $2048$
• C. $1024$
• D. $512$

Answer 1

The correct option is C $1024$

Now $\alpha ,\beta$ are the roots of the equation so

$\alpha +\beta =-1$ $\alpha \beta =2$

$\frac{{\alpha }^{10}+{\beta }^{10}}{\frac{1}{{\alpha }^{10}}+\frac{1}{{\beta }^{10}}}={\alpha }^{10}{\beta }^{10}=(\alpha \beta {)}^{10}$

$=(2{)}^{10}$

$=1024$

Hence, C is correct.