If \(u_1\) and \(u_2\) are the units selected in two systems of measurement and \(n_1\) and \(n_2\) are their numerical values, then:

The velocity \(v\) of a particle at time \(t\) is given by \({v}={at}+\frac{{b}}{{t}+{c}}.\) The dimensions of \({a}\), \({b}\), and \({c}\) are respectively:
1. \( {\left[{LT}^{-2}\right],[{L}],[{T}]} \)
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 the relation, \(y=a \cos (\omega t-k x)\), the dimensional formula for \(k\) will be:
1. \( {\left[M^0 L^{-1} T^{-1}\right]} \)
2. \({\left[M^0 L T^{-1}\right]} \)
3. \( {\left[M^0 L^{-1} T^0\right]} \)
4. \({\left[M^0 L T\right]}\)

When units of mass, length, and time are taken as \(10~\text{kg}, 60~\text{m}~\text{and}~60~\text{s}\) respectively, the new unit of energy becomes \(x\) times the initial SI unit of energy. The value of \(x\) will be:
1. \(10\)
2. \(20\)
3. \(60\)
4. \(120\)

The dimensions of \((\mu_0\varepsilon_0)^{\frac{-1}{2}}\) are:
1. \(\left[L^{-1}T\right]\)
2. \(\left[LT^{-1}\right]\)
3. \(\left[L^{{-1/2}}T^{{1/2}}\right]\)
4. \(\left[L^{{-1/2}}T^{{-1/2}}\right]\)

For the expression, \(10^{(at+3)}\), the dimensions of \(a\) will be:
1. \(\left[M^0L^0T^{0}\right]\)
2. \(\left[M^0L^0T^{1}\right]\)
3. \(\left[M^0L^0T^{-1}\right]\)
4. None of these