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:

Dimensions of electric current are:

1. $\left[M^0{L}^0{T}^{-1}Q\right]$

2. $\left[M{L}^2{T}^{-1}Q\right]$

3. $\left[M^{2}L{T}^{-1}Q\right]$

4. $\left[M^2{L}^2{T}^{-1}Q\right]$

The position of a body with acceleration \(a\) is given by $x=K{a}^m{t}^{\mathrm{n }}$ (assume \(t\) to be time). The values of m and n will be:

1. m = 1, n = 1

2. m = 1, n = 2

3. m = 2, n = 1

4. m = 2, n = 2

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: