Which of the following sets cannot enter into the list of fundamental quantities in any system of units?

Options

A. length, mass and velocity
B. pressure, density and velocity
C. force, velocity and time
D. force, momentum and time

Answer

B, D

Explanation

For (A) : Length `[L]`, mass `[M]` and velocity `[LT^(-1)]` are independent.
for (B): Pressure `[M'L^(-1)T^(-2)]`, density `[M' L^(-3)T^(0)]` and velocity `[M^(0)LT^(-1)]` are dependent as
`|(1,-1,-2),(1,-3,0),(0,1,-1)|=1(3-0)+1(-1-0)-2(1-0)=0`
Note: Like coplanar vectors, for dependent quantites, determinant of powers of M, L, T's must be zero.
For (C): Force `[MLT^(-2)]`, velocity `[LT^(-1)]` and time `[T]` are independent as
`|(1,1,-2),(0,1,-1),(0,0,1)|=1(1-0)-1(0-0)-2(0-0)=1 ne 0`