Thermoelectric effect

The smallest current the galvanometer can detect is that for one division: $I_{min} = 1\ \mu A$
The thermocouple has negligible resistance, so the circuit resistance is that of the galvanometer, 10 $\Omega$.
Smallest detectable emf: $V_{min} = I_{min}\times R = 1\ \mu A\times10\ \Omega = 10\ \mu V$
The thermocouple gives 40 $\mu$V for each $^\circ C$ of temperature difference.
Smallest detectable temperature difference $= \frac{10\ \mu V}{40\ \mu V/^\circ C}$
$= 0.25^\circ C$

The thermo emf of a thermocouple varies with the hot-junction temperature as $E = \alpha t + \frac{1}{2}\beta t^2$, which is a parabola.
The neutral temperature is the temperature at which the thermo emf is maximum.
At a maximum, the slope of the curve is zero: $\frac{dE}{dt} = 0$
Below the neutral temperature the rate is positive and above it the rate is negative, but exactly at the neutral temperature it is zero for every thermocouple.
Hence the rate of increase of thermo emf at the neutral temperature is zero.

The neutral temperature is the temperature at which the thermo emf is maximum, i.e., $\frac{dE}{d\theta} = 0$
$E = 30\theta - \frac{\theta^2}{15}$
$\frac{dE}{d\theta} = 30 - \frac{2\theta}{15}$
Setting it to zero: $30 - \frac{2\theta}{15} = 0$
$\theta = \frac{30\times15}{2} = 225^\circ C$