\[(a)\;\mu_k=1-\frac{1}{x^2} \quad (b)\;\mu_k=\sqrt {1-\frac{1}{x^2}} \quad (c)\;\mu_s=1-\frac{1}{x^2} \quad (d)\;\mu_s=\sqrt{1-\frac{1}{x^2}} \]

On a rough surface

$a=(g \sin \theta-\mu g \cos \theta)$

$time =xt$

$S=ut+\large\frac{1}{2} $$ at^2$

$\quad=0+\large\frac{1}{2} $$[g \sin \theta -\mu g \cos \theta](xt)^2$-----(1)

On plane surface

$a=g \sin \theta$

$time=t$

$S=0+\large\frac{1}{2}$$ (g \sin \theta)t^2$-----(2)

Solving (1) and (2)

We get $\mu =1-\large\frac{1}{x^2}\quad $$[\mu=\mu_k$ as the body is moving$]$

Hence a is the correct answer

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