Explanation

(a)
Given equation can be reduced to a quadratic equation.
$\therefore 2x^{2} + x - 11 + \frac{1}{x} + \frac{2}{x^{2}} = 0$
$\Rightarrow 2\left( \ x^{2} + \frac{1}{x^{2}} \right) + \left( \ x + \frac{1}{x} \right) - 11 = 0$
Put $x + \frac{1}{x} = y$
$2\left( y^{2} - 2 \right) + y - 11 = 0$
$\Rightarrow 2y^{2} + y - 15 = 0$
$\Rightarrow y = - 3\ and\frac{5}{2}$
$\Rightarrow \ x + \frac{1}{x} = - 3,\ x + \frac{1}{x} = \frac{5}{2}\$
$\Rightarrow x^{2} + 3x + 1 = 0,\ 2x^{2} - 5x + 2 = 0$
Only 2nd equation has rational roots as $D = 9$ and roots are $\frac{1}{2}$ and 2.