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$ a)\; \large\frac{P}{mg \sin \theta+\mu mg \cos \theta} \\ b)\; \large\frac{P}{mg \sin \theta-\mu mg \cos \theta} \\ c) \;\large\frac{2P}{mg \sin \theta-\mu mg \cos \theta} \\ d)\; \large\frac{3P}{mg \sin \theta-\mu mg \cos \theta}$

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$a=\large\frac{dv}{dt}$

Force due to power P is

$F=\large\frac{p}{v}$$;f=\mu R$

$F-mg \sin \theta-\mu R=ma$

$\large\frac{P}{v}$$-mg \sin \theta - \mu mg \cos \theta=m \large\frac{dv}{dt}$

Velocity is maximum when $\large\frac{dv}{dt}$$=0$

$\large\frac{P}{v_{\Large max}}$$= mg \sin \theta +\mu mg \cos \theta$

$v_{max}= \large\frac{P}{mg \sin \theta+\mu mg \cos \theta}$

Hence a is the correct answer

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