Q.(MCQ II — one or more options correct) If $P, Q, R$ are physical quantities, having different dimensions, which of the following combinations can never be a meaningful quantity?

The key idea is that only quantities with identical dimensions can be added or subtracted. Multiplication and division are always dimensionally valid.

• In (A), $P - Q$ is impossible because $P$ and $Q$ have different dimensions. Hence the whole expression fails.
• In (B), $PQ$ is a product (dimensionally valid), but subtracting $R$ from it is only possible if $PQ$ and $R$ have the same dimensions — which is not guaranteed, so this can also be meaningless.
• (C) $PQ/R$ involves only multiplication and division, so it is always meaningful. …

The rule is simple but powerful: you can only add or subtract quantities that have the same dimensions. Multiplication and division, however, are always allowed between any dimensions — they just produce a new derived quantity. So the moment you see a plus or minus sign, you must check whether the two terms on either side have identical dimensions. If they don't, that combination is physically meaningless.

Let’s denote the dimensions of $P$, $Q$, and $R$ as $[P]$, $[Q]$, and $[R]$. They are all different from each other. We’ll examine each option.

1. Option (A): $(P - Q)/R$

   The subtraction $P - Q$ requires $[P] = [Q]$. But we are told $P$ and $Q$ have different dimensions. So $P - Q$ is itself meaningless. The whole expression therefore can never be a meaningful quantity.

   This option is invalid.

2. Option (B): $PQ - R$

   Here $PQ$ is a product, so its dimensions are $[P][Q]$. For the subtraction $PQ - R$ to be meaningful, we need $[P][Q] = [R]$. Is it possible? Yes — the problem only says $P, Q, R$ have different dimensions, but it doesn’t forbid that the product of two of them equals the dimension of the third. For example, if $P$ is speed, $Q$ is time, and $R$ is distance, then $[P][Q] = [R]$ holds. So this combination can be meaningful.

   This option is valid.

3. Option (C): $PQ/R$

   This is purely multiplication and division — no addition or subtraction. It is always dimensionally meaningful, regardless of what the dimensions are. The result is just a new quantity with dimensions $[P][Q]/[R]$.

   This option is always valid.

4. Option (D): $(PR - Q^2)/R$ …

Method: Dimensional Analysis of Additive Combinations

Concept: Only quantities with identical dimensions can be added or subtracted. Multiplication and division are always dimensionally valid regardless of the dimensions involved — they simply produce a new derived quantity. So the test for each option is: find every $+$ or $-$ inside the expression, and check whether the dimensions on both sides of that sign can possibly match.

Steps

1. Identify the additive/subtractive terms in each option. $P$, $Q$, $R$ are given to have three different, mutually distinct dimensions $[P] \neq [Q] \neq [R]$.

2. Check each option

   Option (A): $(P - Q)/R$

   • The subtraction $P - Q$ requires $[P] = [Q]$.
   • We are told $[P] \neq [Q]$, so $P - Q$ is dimensionally impossible no matter what $R$ is.
   • Can never be meaningful.

   Option (B): $PQ - R$

   • $PQ$ is a pure product, so $[PQ] = [P][Q]$.
   • The subtraction needs $[P][Q] = [R]$. Nothing in the problem forbids this — e.g. if $P$ is speed and $Q$ is time, $[P][Q] = [L]$, which could equal $[R]$ if $R$ is a length.
   • Can be meaningful for a suitable choice of $P, Q, R$.

   Option (C): $PQ/R$

   • Only multiplication and division appear — no additive term to check.
   • Always meaningful, regardless of the dimensions of $P, Q, R$.

   Option (D): $(PR - Q^2)/R$

   • The subtraction needs $[PR] = [Q^2]$, i.e. $[P][R] = [Q]^2$. This is a solvable dimensional constraint (e.g. $P$ = acceleration, $R$ = time$^2$, $Q$ = displacement gives $[P][R] = [L]$, and choosing $Q$ with $[Q]^2 = [L]$ is a valid — if unusual — dimension). …