The time dependence of a physical quantity $P$ is given by $P = P_0 e^{-\alpha t^2}$,where $\alpha$ is a constant and $t$ is the time. Then the constant $\alpha$ has:

If the velocity of light $C$,the gravitational constant $G$,and Planck's constant $h$ are chosen as the fundamental units,the dimension of density in the new system is

Let force $F = A \sin(Ct) + B \cos(Dx)$,where $x$ and $t$ are displacement and time respectively. The dimensions of $\frac{C}{D}$ are the same as the dimensions of

If $E$ and $E_0$ denote energies at time $t$ and $t_0$ respectively,and $L$ and $L_0$ denote distances from some point at $t$ and $t_0$ respectively,then which of the following equations can be declared to be incorrect on dimensional grounds?
$(A) E = \frac{2 E_0 L}{L_0}$
$(B) E = E_0 e^{-\frac{2 L}{L_0}}$
$(C) E = 2 L e^{-\frac{L}{E_0}}$
$(D) E = 2 \left( \frac{E_0}{L_0} \right) e^{-\frac{L}{L_0}}$

If the unit of force is $100\,N$,unit of length is $10\,m$ and unit of time is $100\,s$,what is the unit of mass in this system of units?

If $\text{force} = \frac{\alpha}{\text{density} + \beta^3}$,then the dimensional formulae of $\alpha$ and $\beta$ are respectively: