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(C) The dimension of time period $T$ is $[T^1]$.The given formula is $T = k \sqrt{\frac{\rho r^3}{S}}$.Dimensions of density $\rho = [M L^{-3}]$.Dimen

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$(A)$ is true but $(R)$ is false

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(C) The dimension of time period $T$ is $[T^1]$.The given formula is $T = k \sqrt{\frac{\rho r^3}{S}}$.Dimensions of density $\rho = [M L^{-3}]$.Dimensions of radius $r = [L]$.Dimensions of surface tension $S = [M T^{-2}]$.Substituting these into the formula: $[RHS] = [M L^{-3} L^3]^{1/2} / [M T^{-2}]^{1/2} = [M^{1/2}] / [M^{1/2} T^{-1}] = [T^1]$.Wait,let us re-evaluate the expression provided in the assertion: $T = k \sqrt{\rho r^3 / S}$.$[RHS] = \sqrt{\frac{[M L^{-3}] [L^3]}{[M T^{-2}]}} = \sqrt{\frac{[M]}{[M T^{-2}]}} = \sqrt{[T^2]} = [T^1]$.Since the dimensions of $LHS$ and $RHS$ match,the assertion $(A)$ is actually true. However,based on the provided options and the logic that the assertion claims the formula is dimensionally correct,and the reason claims it is not,we conclude $(A)$ is true and $(R)$ is false.

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