If P, Q and R are physical quantities, having different dimensions, then which mathematical operation given below can never be physically meaningful?

CONCEPT:

Physical Quantities and Dimensions

• Physical quantities are properties or characteristics of a system that can be measured or calculated from other measurements.
• Each physical quantity has a dimension that can be represented by a combination of the basic dimensions (mass, length, time, etc.).
• In order for a mathematical operation involving physical quantities to be physically meaningful, the quantities must have compatible dimensions.

EXPLANATION:

• Let's examine the given options:
  • Option 1: (P-Q)/R
    • For this operation to be meaningful, P and Q must have the same dimensions because only quantities with the same dimensions can be subtracted. If P and Q have the same dimensions, (P-Q) will have the same dimension as P (or Q), but it is given P and Q have a different dimension thus it cannot be meaningful quantity.
  • Option 2: PQ-R
    • For this operation to be meaningful, PQ and R must have the same dimensions. PQ represents the product of P and Q, so the dimensions of PQ are the product of the dimensions of P and Q. Unless R has the same dimensions as PQ.
  • Option 3: PQ/R
    • This operation is meaningful if the dimensions of PQ are compatible with the dimensions of R. Specifically, the dimensions of PQ should be equal to the dimensions of R. This means that the dimensions of P multiplied by the dimensions of Q should result in the dimensions of R.
  • Option 4: (PR - Q²)/R
    • For this operation is meaningful, PR and Q² must have the same dimensions. Since Q² represents Q multiplied by itself, its dimensions are the square of the dimensions of Q. PR represents the product of P and R, so its dimensions are the product of the dimensions of P and R. Additionally, (PR - Q²)/R must be dimensionally consistent.

Therefore, the correct answer is option 1: (P-Q)/R.