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Sometimes it is convenient to construct a system of units so that all quantities can be expressed in terms of only one physical quantity. In one such system, dimensions of different quantities are given in terms of a quantity X as follows:
$[p o s i t i o n] = [X^{α}] : [s p e e d] = [X^{β}]; [a c c e l e r a t i o n] = [X^{p}];$ $[l i n e a r m o m e n t u m] = [X^{q}]; [f o r c e] = [X^{r}] . T h e n$

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$α + p = 2 β$

$p + q - r = β$

$p - q + r = α$

$p + q + r = β$

answer is A, B.

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Detailed Solution

$\begin{matrix} m = x^{α} \\ m / s = x^{β} \end{matrix}$

$\begin{matrix} (m / s^{2}) = x^{p} \\ m (m / s^{2}) = {(m / s)}^{2} \end{matrix}$

$\Rightarrow α + p = 2 β$

$p + q - r = β$

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