This question contains Statement-1 and Statement-2. Of the four choices given after the statements, choose the one that best describes the two statements. Statement-1: The temperature dependence of resistance is usually given as $$R = R_0(1 + \alpha\Delta t)$$. The resistance of a wire changes from $$100\Omega$$ to $$150\Omega$$ when its temperature is increased from $$27°C$$ to $$227°C$$. This implies that $$\alpha = 2.5 \times 10^{-3}/°C$$. Statement 2: $$R = R_t(1 + \alpha\Delta T)$$ is valid only when the change in the temperature $$\Delta T$$ is small and $$\Delta R = (R - R_0) \ll R_0$$.

The given relation is an experimental (empirical) relation for metals

$$R = R_0\bigl(1 + \alpha \Delta T\bigr)$$

where $$R_0$$ is the resistance at some reference temperature and $$\alpha$$ is called the (average) temperature coefficient of resistance between the two temperatures considered. Using this single-step linear relation does not require that the temperature change be infinitesimally small; it works quite accurately over a fairly wide range of temperatures for ordinary metals.

Evaluating Statement-1

The wire changes from
$$R_0 = 100 \,\Omega \quad\text{at}\quad T_0 = 27^{\circ}\text{C}$$
to
$$R = 150 \,\Omega \quad\text{at}\quad T = 227^{\circ}\text{C}$$

Hence $$\Delta T = 227 - 27 = 200^{\circ}\text{C}$$. Putting these numbers in the empirical formula:

$$150 = 100\bigl(1 + \alpha \times 200\bigr)$$

$$\Rightarrow\; 1 + 200\alpha = 1.5$$

$$\Rightarrow\; 200\alpha = 0.5 \;\;\Longrightarrow\;\; \alpha = 2.5\times10^{-3}/^{\circ}\text{C}$$

The value quoted in the statement is exactly this, so Statement-1 is true.

Evaluating Statement-2

Statement-2 claims that the linear formula is valid only when the temperature rise is small and consequently $$\Delta R \ll R_0$$. In practice, for common metals the resistance varies almost linearly with temperature over several hundreds of degrees Celsius. The same linear law is routinely used, for example, from room temperature up to the red-hot range in platinum resistance thermometers. Therefore the restriction “only when $$\Delta T$$ is small and $$\Delta R \ll R_0$$” is not correct; the formula remains a good empirical fit even when the change is large (as in the present problem where $$\Delta R = 0.5 R_0$$).

Hence Statement-2 is false.

Combining the results:

Statement-1 is true and Statement-2 is false.

Option A which is: Statement-1 is true, Statement-2 is false.