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 \(\mathrm{v}=\mathrm{at}+\frac{\mathrm{b}}{\mathrm{t}+\mathrm{c}}\). The dimensions of \(\mathrm{a}\), \(\mathrm{b}\), and \(\mathrm{c}\) are respectively:
1. \( {\left[\mathrm{LT}^{-2}\right],[\mathrm{L}],[\mathrm{T}]} \)
2. \( {\left[\mathrm{L}^2\right],[\mathrm{T}] \text { and }\left[\mathrm{LT}^2\right]} \)
3. \( {\left[\mathrm{LT}^2\right],[\mathrm{LT}] \text { and }[\mathrm{L}]} \)
4. \( {[\mathrm{L}],[\mathrm{LT}], \text { and }\left[\mathrm{T}^2\right]}\)

If the dimensions of a physical quantity are given by ${\mathrm{M}}^{\mathrm{a}}{\mathrm{L}}^{\mathrm{b}}{\mathrm{T}}^{\mathrm{c}},$ then the physical quantity will be:

The dimensional formula of pressure is:

1. $\left[{\mathrm{MLT}}^{-2}\right]$

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

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

4. $\left[{\mathrm{MLT}}^2\right]$

The dimensional formula for impulse is:
1. \([MLT^{-2}]\)
2.  \([MLT^{-1}]\)
3.  \([ML^2T^{-1}]\)
4. \([M^2LT^{-1}]\)

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 kg, 60 m, and 60 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

If the units of force and length, each is increased by four times, then the unit of energy increases by:

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

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

2.  $\left[{\mathrm{LT}}^{-1}\right]$

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

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

For \(10^{(at+3)}\), the dimensions of \(a\) will be:

1. ${\mathrm{M}}^0{\mathrm{L}}^0{\mathrm{T}}^0$

2. ${\mathrm{M}}^0{\mathrm{L}}^0{\mathrm{T}}^1$

3. ${\mathrm{M}}^0{\mathrm{L}}^0{\mathrm{T}}^{-1}$

4. None of these