Which of the following pairs of scales can yield identical...

The relation between the Fahrenheit and Kelvin scales is given by:

$\dfrac{F - 32}{180} = \dfrac{K - 273.15}{100}$

For the two scales to show the same reading, let $F = K = x$. Substituting this into the equation:

$\dfrac{x - 32}{9} = \dfrac{x - 273.15}{5}$

$5x - 160 = 9x - 2458.35$

$4x = 2298.35$

$x = 574.59$

Thus, the Fahrenheit and Kelvin scales can yield identical numerical values at $574.59$.

For Celsius and Kelvin, the relation is $K = C + 273.15$, which means they can never be equal.

For Fahrenheit and Rankine, the relation is $R = F + 459.67$, which means they can never be equal.

The platinum scale measures temperature in degrees Celsius, so it also cannot equal the Kelvin scale.