GATE 2025 BT Q.42: solution and answer

Official answer: CSource: Official answer key, GATE 2025 BT (IIT Roorkee)

1. A first-order instrument has the transfer function $G(s)=\frac{K}{\tau s+1}$, where $K$ is the steady-state gain and $\tau$ is the time constant. Here $\tau =40\text{s}$.

2. When the thermometer is exposed to a step change in temperature, its response in deviation form is $\frac{y(t)}{y_{\infty }}=1-e^{-t/\tau }$, where $y(t)$ is the change in reading at time $t$ and $y_{\infty }$ is the final (steady-state) change.

3. Set the response to 95% of steady state: $1-e^{-t/\tau }=0.95\Rightarrow e^{-t/\tau }=0.05$.

4. Take the natural logarithm and solve for $t$: $$t=\tau \ln \left(\frac{1}{0.05}\right)=\tau \ln 20=2.9957\tau$$

5. Substitute $\tau =40\text{s}$: $t=2.9957\times 40=119.83\text{s}\approx 120\text{s}$.

6. Check with the usual rule of thumb: a first-order system reaches 63.2% of its final change in $1\tau$, 86.5% in $2\tau$, 95.0% in $3\tau$ and 98.2% in $4\tau$. So 95% takes about $3\tau =120\text{s}$, which is option (C).