A particle of mass 10g moves in a straight line with retardation...

Acceleration can be written as, $a = v\dfrac{dv}{dx}$.

Given: $a = - 2x$(-ve sign is due to retardation)

Therefore,

$v\dfrac{dv}{dx} = - 2x ⇒ vdv = \left(- 2x\right)dx$

Integrating both sides, we get

$\dfrac{1}{2}\left({v}_{f}^{2} - {v}_{i}^{2}\right) = - 2\left(\dfrac{{x}^{2}}{2}\right) = {x}^{2} ⇒ \left|\dfrac{1}{2}m\left({v}_{f}^{2} - {v}_{i}^{2}\right)\right| = m{x}^{2}$

$⇒ | \DeltaKE | = 0 . 01{x}^{2} = {\left(\dfrac{10}{x}\right)}^{- 2}$

Hence, $n = 2$.