In a system of units, the units of mass, length and time are 1 quintal, 1 km and 1 h respectively. In this system 1 N force will be equal to

Correct Answer: Option 2 (129.6 new units)

To convert a physical quantity from one system of units to another, we use the method of dimensional analysis.

The dimensional formula for force ($F$) is given by:
$\left[F\right]=\left[{\mathrm{MLT}}^{-2}\right]$
Here, the exponents are:
$a=1$ (for mass)
$b=1$ (for length)
$c=-2$ (for time)

Let the SI system be represented as system 1, and the new system of units be represented as system 2.

For the SI system (System 1):
Numerical value, $n_1=1$
Unit of mass, $M_1=1\text{kg}$
Unit of length, $L_1=1\text{m}$
Unit of time, $T_1=1\text{s}$

For the new system (System 2):
Unit of mass, $M_2=1\text{quintal}=100\text{kg}$
Unit of length, $L_2=1\text{km}=1000\text{m}$
Unit of time, $T_2=1\text{h}=3600\text{s}$

The relationship between the numerical values in two systems of units is given by:
$n_2=n_1{\left(\frac{M_1}{M_2}\right)}^a{\left(\frac{L_1}{L_2}\right)}^b{\left(\frac{T_1}{T_2}\right)}^c$

Substituting the values:
$n_2=1\times {\left(\frac{1\text{kg}}{100\text{kg}}\right)}^1\times {\left(\frac{1\text{m}}{1000\text{m}}\right)}^1\times {\left(\frac{1\text{s}}{3600\text{s}}\right)}^{-2}$

Simplifying the expression:
$n_2=1\times \left(\frac{1}{100}\right)\times \left(\frac{1}{1000}\right)\times {\left(3600\right)}^2$
$n_2=\frac{3600\times 3600}{100\times 1000}$
$n_2=\frac{12960000}{100000}$
$n_2=129.6$

Therefore, 1 N force is equal to 129.6 new units in this system of units.