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(D) $1$. The loss in gravitational potential energy of the mass $M$ as it moves down by distance $l$ is given by $\Delta PE = Mgl$. Thus,statement $(I

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$I, III$ and $IV$

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(D) $1$. The loss in gravitational potential energy of the mass $M$ as it moves down by distance $l$ is given by $\Delta PE = Mgl$. Thus,statement $(I)$ is correct.$2$. The elastic potential energy (strain energy) stored in a stretched wire is given by $U = \frac{1}{2} 	imes 	ext{force} 	imes 	ext{elongation} = \frac{1}{2} Mgl$. Thus,statement $(III)$ is correct and $(II)$ is incorrect.$3$. According to the work-energy theorem,the work done by the gravitational force is equal to the sum of the elastic potential energy stored in the wire and the heat produced (due to oscillations/damping). Work done by gravity = $Mgl$.Elastic potential energy = $\frac{1}{2} Mgl$.Heat produced = Work done - Elastic potential energy = $Mgl - \frac{1}{2} Mgl = \frac{1}{2} Mgl$. Thus,statement $(IV)$ is correct.Therefore,statements $(I), (III),$ and $(IV)$ are correct.

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