A dimensionless quantity,

1.	never has a unit
2.	always has a unit
3.	may have a unit
4.	does not exist

Given: \(\int \frac{{dx}}{\sqrt{2{ax}-{x}^{2}}}={a}^{{n}} \sin ^{-1}\left[\frac{{x}}{{a}}-1\right]\)
The value of \(n\) is:
1. \(0\)
2. \(-1\)
3. \(1\)
4. none of these

The dimensions of \(\left [ML^{-1} T^{-2} \right ] \) may correspond to:

Choose the correct option from the given ones:

The radius of a circle is stated as \(2.12\) cm. Its area should be written as:

A physical quantity is measured and the result is expressed as \(nu\) where \(u\) is the unit used and \(n\) is the numerical value. If the result is expressed in various units then: 
1. \(n\propto \mathrm{size~of}~u\) 
2. \(n\propto u^2\) 
3. \(n\propto \sqrt u\) 
4. \(n\propto \frac{1}{u}\)

Which of the following equations is dimensionally correct?
\((I)~~ v=\sqrt{\dfrac{P}{\rho}}~~~~~~(II)~~v=\sqrt{\dfrac{mgl}{I}}~~~~~~(III)~~v=\dfrac{Pr^2}{2\eta l}\)
(where  \(v=\) speed, \(P=\) pressure; \(r,\) \(l\) are lengths; \(\rho=\)  density, \(m=\) mass, \(g=\) acceleration due to gravity, \(I=\) moment of inertia, and \(\eta=\) coefficient of viscosity)

The acceleration due to gravity on the surface of the Earth is \(g=10~\text{m/s}^2\) . The value in \(\text {km/minute}^2\) is: