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: $$[position] = [𝑋^{\alpha}]; [speed] = [𝑋^{\beta}]; [acceleration] =[𝑋^{p}]; [linear momentum] = [𝑋^{q}]; [force] = [𝑋^{r}]$$. Then

Only one fundamental dimension $$[X]$$ is allowed, so every physical quantity must appear as some power of $$X$$.

Let the dimensions of time and mass be $$[T]=[X^{\tau}]$$ and $$[M]=[X^{\mu}]$$ respectively.
Given data (powers of $$X$$):
  • position : $$\alpha$$  • speed : $$\beta$$  • acceleration : $$p$$
  • linear momentum : $$q$$  • force : $$r$$

Step 1: Relate $$\alpha,\;\beta,\;p$$ using kinematics.
Speed = position / time  ⇒  $$X^{\beta}=X^{\alpha}\,/\,X^{\tau}\;$$ ⇒ $$\tau=\alpha-\beta$$.
Acceleration = speed / time  ⇒  $$X^{p}=X^{\beta}\,/\,X^{\tau}=X^{\beta-(\alpha-\beta)}=X^{2\beta-\alpha}$$.
Hence $$p = 2\beta - \alpha \quad -(1)$$.

Re-arranging $$(1)$$ gives $$\alpha+p-2\beta=0$$, so the quantity given in Option A is identically zero; Option A is therefore correct.

Step 2: Bring in mass via momentum and force.
Linear momentum = mass × velocity  ⇒  $$X^{q}=X^{\mu}\,X^{\beta}=X^{\mu+\beta}$$, thus $$q=\mu+\beta \quad -(2)$$.
Force = mass × acceleration  ⇒  $$X^{r}=X^{\mu}\,X^{p}=X^{\mu+p}$$, thus $$r=\mu+p \quad -(3)$$.

Step 3: Test the remaining options.

Option B claims  $$p+q-r=\beta$$.
Using $$(2),(3):$$ $$p+q-r = p+(\mu+\beta) - (\mu+p) = \beta,$$ which is true. Hence Option B is correct.

Option C claims  $$p-q+r=\alpha$$.
Left-hand side = $$p-(\mu+\beta)+(\mu+p)=2p-\beta$$.
With $$(1):$$ $$2p-\beta = 2(2\beta-\alpha)-\beta = 3\beta-2\alpha,$$ which equals $$\alpha$$ only if $$\alpha=\beta$$, not guaranteed. So Option C is not necessarily true.

Option D claims  $$p+q+r=\beta$$.
Left-hand side = $$p+(\mu+\beta)+(\mu+p)=2p+2\mu+\beta,$$ which cannot reduce to $$\beta$$ for arbitrary $$\mu,p$$. Hence Option D is incorrect.

Therefore the correct relations are given by Option A and Option B.

Final Answer: Option A and Option B.