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

Choose the correct option:

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. $n\propto \frac{1}{u}$

A dimensionless quantity,

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

Young's modulus of steel is $1.9 \times 10^{11} ~\text{N/m}^2$. When expressed in CGS units of $\text{dyne/cm}^2$, it will be equal to: $(1 \mathrm{~N}=10^5 \text { dyne, } 1~ \text{m}^2=10^4 ~\text{cm}^2)$
1. $1.9 \times 10^{10}$
2. $1.9 \times 10^{11}$
3. $1.9 \times 10^{12}$
4. $1.9 \times 10^9$

On the basis of dimensions, decide which of the following relations for the displacement of a particle undergoing simple harmonic motion is/are not correct.

(Symbols have their usual meanings.)
Choose the correct option: