The dimensions of $\frac{1}{2}{\mathrm{\epsilon }}_0{\mathrm{E}}^2$ where ${\epsilon }_0$ is the permittivity of free space and E is the electric field, are:

1. [ML²T^-2]

2. [ML^-1T^-2]

3.  [ML²T^-1]

4.  [MLT^-1]

Dimensions of resistance in an electrical circuit, in terms of dimension of mass M, length L, time T, and current I, would be:

1. $\left[{\mathrm{ML}}^2{\mathrm{T}}^{-3}{\mathrm{I}}^{-1}\right]$

2. $\left[{\mathrm{ML}}^2{\mathrm{T}}^{-2}\right]$

3. $\left[{\mathrm{ML}}^2{\mathrm{T}}^{-1}{\mathrm{I}}^{-1}\right]$

4. $\left[{\mathrm{ML}}^2{\mathrm{T}}^{-3}{\mathrm{I}}^{-2}\right]$

The velocity \(v\) of a particle at time \(t\) is given by \(v=at+\dfrac{b}{t+c}\), where \(a,\) \(b\) and \(c\) are constants. The dimensions of \(a,\) \(b\) and \(c\) are respectively:
1. \(\left[{LT}^{-2}\right],[{L}] \text { and }[{T}]\)
2. \( {\left[{L}^2\right],[{T}] \text { and }\left[{LT}^2\right]}  \)
3. \( {\left[{LT}^2\right],[{LT}] \text { and }[{L}]}  \)
4. \( {[{L}],[{LT}] \text { and }\left[{T}^2\right]}\)

In an experiment, the percentage errors that occurred in the measurement of physical quantities \(A,\) \(B,\) \(C,\) and \(D\) are \(1\%\), \(2\%\), \(3\%\), and \(4\%\) respectively. Then, the maximum percentage of error in the measurement of \(X,\) where \(X=\frac{A^2 B^{\frac{1}{2}}}{C^{\frac{1}{3}} D^3}\), will be:
1. \(10\%\)
2. \(\frac{3}{13}\%\)
3. \(16\%\)
4. \(-10\%\)

The dimensions of ${\left({\mathrm{\mu }}_0{\mathrm{\epsilon }}_0\right)}^{-1/2}$ are:

1.  $\left[{\mathrm{L}}^{-1}\right]$ $\mathrm{T}$

2.  $\left[{\mathrm{LT}}^{-1}\right]$

3.  $\left[{\mathrm{L}}^{-1/2}\right]$ ${\mathrm{T}}^{1/2}$

4.  $\left[{\mathrm{L}}^{-1/2}\right]$ ${\mathrm{T}}^{-1/2}$

If force (\(F\)), velocity (\(\mathrm{v}\)), and time (\(T\)) are taken as fundamental units, the dimensions of mass will be:
1. \([FvT^{-1}]\)
2. \([FvT^{-2}]\)
3. \([Fv^{-1}T^{-1}]\)
4. \([Fv^{-1}T]\)