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: [position$] = [X^α]; [$speed$] = [X^β]; [$acceleration$] = [X^{p}]; [$linear momentum$] = [X^{q}]; [$force$] = [X^{r}].$ Then $(1) α + p = 2β (2) p + q − r = β (3) p − q + r = α (4) p + q + r = β$

Step 1: Define the dimensions of fundamental quantities in terms of $X$.

Let the dimensions of length, time, and mass be represented in terms of the quantity $X$ with exponents $\alpha$, $\tau$, and $\mu$ respectively.

$$[\text{position}] = [X^{\alpha}]$$ $$[\text{time}] = [X^{\tau}]$$ $$[\text{mass}] = [X^{\mu}]$$

Step 2: Express the exponents of the given physical quantities.

Using the definitions of speed, acceleration, linear momentum, and force, we write their exponents in terms of $\alpha$, $\tau$, and $\mu$.

$$\beta = \alpha - \tau$$ $$p = \alpha - 2\tau$$ $$q = \beta + \mu = \alpha - \tau + \mu$$ $$r = p + \mu = \alpha - 2\tau + \mu$$

Step 3: Verify Relation 1.

Check if $\alpha + p = 2\beta$.

$$\alpha + p = \alpha + (\alpha - 2\tau)$$ $$\alpha + p = 2\alpha - 2\tau$$ $$2\beta = 2(\alpha - \tau) = 2\alpha - 2\tau$$

Since both sides are equal, Relation 1 is always true.

Step 4: Verify Relation 2.

Check if $p + q - r = \beta$.

$$p + q - r = p + (\beta + \mu) - (p + \mu)$$ $$p + q - r = \beta$$

Since both sides are equal, Relation 2 is always true.

Step 5: Verify Relation 3.

Check if $p - q + r = \alpha$.

$$p - q + r = (\alpha - 2\tau) - (\alpha - \tau + \mu) + (\alpha - 2\tau + \mu)$$ $$p - q + r = \alpha - 3\tau$$

Since $\alpha - 3\tau \neq \alpha$ in general, Relation 3 is not always true.

Step 6: Verify Relation 4.

Check if $p + q + r = \beta$.

$$p + q + r = (\alpha - 2\tau) + (\alpha - \tau + \mu) + (\alpha - 2\tau + \mu)$$ $$p + q + r = 3\alpha + 2\mu - 5\tau$$

Since $3\alpha + 2\mu - 5\tau \neq \beta$ in general, Relation 4 is not always true.

Answer: The relations that are always true are (1) $\alpha + p = 2\beta$ and (2) $p + q - r = \beta$.