Consider an expression $Q V = k P T L^\alpha$ where $V, P, T, L$ are volume, pressure, time, and length respectively. The quantity $[Q]$ has dimension $M L^{-1} T^{-1}$. $k$ is a dimensionless constant. The value of the integer $\alpha$ is:

Even if a physical quantity depends upon three quantities,out of which two are dimensionally same,then the formula cannot be derived by the method of dimensions. This statement

$A$ length-scale $(l)$ depends on the permittivity $(\varepsilon)$ of a dielectric material,the Boltzmann constant $(k_B)$,the absolute temperature $(T)$,the number density $(n)$ of certain charged particles,and the charge $(q)$ carried by each of the particles. Which of the following expression$(s)$ for $l$ is(are) dimensionally correct?
$(A)$ $l=\sqrt{\left(\frac{n q^2}{\varepsilon k_B T}\right)}$
$(B)$ $l=\sqrt{\left(\frac{\varepsilon k_B T}{n q^2}\right)}$
$(C)$ $l=\sqrt{\left(\frac{q^2}{\varepsilon n^{2 / 3} k_B T}\right)}$
$(D)$ $l=\sqrt{\left(\frac{q^2}{\varepsilon n^{1 / 3} k_B T}\right)}$

Which of the following equations is dimensionally incorrect?
Where $t=$ time,$h=$ height,$s=$ surface tension,$\theta=$ angle,$\rho=$ density,$a, r=$ radius,$g=$ acceleration due to gravity,$v=$ volume,$p=$ pressure,$W=$ work done,$\Gamma=$ torque,$\varepsilon=$ permittivity,$E=$ electric field,$J=$ current density,$L=$ length.

The dimensional formula of $\frac{1}{\mu_{0} \varepsilon_{0}}$ is . . . . . . .

In a particular system,the units of length,mass,and time are chosen to be $10 \; cm$,$10 \; g$,and $0.1 \; s$ respectively. The unit of force in this system will be equal to (in $; N$)?