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`