In the $CGS$ system, the magnitude of the force is $100 \ dynes$. In another system where the fundamental physical quantities are $kilogram$, $meter$, and $minute$, the magnitude of the force is:

Given that the displacement of an oscillating particle is given by $y = A \sin(Bx + Ct + D)$. The dimensional formula for $ABCD$ is: (Here $x$ is position,$t$ is time)

Which of the following is dimensionally incorrect?

In a typical combustion engine,the work done by a gas molecule is given by $W = \alpha^{2} \beta e^{\frac{-\beta x^{2}}{kT}}$,where $x$ is the displacement,$k$ is the Boltzmann constant,and $T$ is the temperature. If $\alpha$ and $\beta$ are constants,the dimensions of $\alpha$ will be:

$A, B, C$ and $D$ are four different physical quantities having different dimensions. None of them is dimensionless. We know that the equation $AD = C \ln(BD)$ holds true. Which of the following combinations is not a meaningful quantity?

Why does the concept of dimension have basic importance?