Hello vaibhav101.
I think there is a mistake in the equation. It should be x^2+x-2=0 instead of x^2+x+2=0.
Using the first equation we get:
First, let's find the roots of $$x^2+x-2=0.$$ $$\Rightarrow\ \ x=\frac{-1\pm\sqrt{1+8}}{2}=\frac{-1\pm\sqrt{9}}{2}=\frac{-1\pm 3}{2}$$ This implies that: $$a=\frac{-1+3}{2}=1\ \ \ \ and\ \ \ \ \ \ b=\frac{-1-3}{2}=-2.$$ Hence
$$a^{10}=1\ \ \ and\ \ \ \ \ a^{-10}=1.$$ $$b^{10}=1024\ \ \ and\ \ \ \ \ b^{-10}=\frac{1}{1024}.$$ Now, $$\frac{a^{10}+b^{10}}{a^{-10}+b^{-10}}=\frac{1+1024}{1+\frac{1}{1024}}=\frac{1025}{\frac{1025}{1024}}=1024.$$ Therefore, the correct answer is the option C.
I hope it helps.
