Applying the principle of homogeneity of dimensions,determine which one is correct. Where $T$ is time period,$G$ is gravitational constant,$M$ is mass,and $r$ is the radius of the orbit.

The equation of a circle is given by $x^2+y^2=a^2$,where $a$ is the radius. If the equation is modified to change the origin to a point other than $(0,0)$,find the correct dimensions of $A$ and $B$ in the new equation: $(x-At)^2+(y-\frac{t}{B})^2=a^2$. The dimensions of $t$ are given as $[T^{-1}]$.

If energy $(E)$,velocity $(V)$,and time $(T)$ are chosen as the fundamental quantities,the dimensional formula of surface tension will be

If dimensions of critical velocity $v_c$ of a liquid flowing through a tube are expressed as $[\eta^x \rho^y r^z]$,where $\eta, \rho,$ and $r$ are the coefficient of viscosity of the liquid,density of the liquid,and radius of the tube respectively,then the values of $x, y,$ and $z$ are given by:

$A$ 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?
$(A)$ $l=\sqrt{\left(\frac{n q^2}{\varepsilon k_B T}\right)}$
$(B)$ $l=\sqrt{\left(\frac{\varepsilon k_B T}{n q^2}\right)}$
$(C)$ $l=\sqrt{\left(\frac{q^2}{\varepsilon n^{2 / 3} k_B T}\right)}$
$(D)$ $l=\sqrt{\left(\frac{q^2}{\varepsilon n^{1 / 3} k_B T}\right)}$

If $x = at + bt^2$,where $x$ is the distance travelled by the body in kilometres while $t$ is the time in seconds,then the units of $b$ are