Question

Statement 1: The temperature dependence of the resistance is usually given as $R=R_0\left(1+\alpha \mathrm{\Delta }T\right)$. The resistance of a wire changes from $100\mathrm{\Omega }$ to $150\mathrm{\Omega }$ when its temperature is increased from ${27}^{\circ }C$ to ${227}^{\circ }C$. This implies that $\alpha =2.5\times {10}^{-3}{/}^{\circ }C$ .
Statement 2: $R=R_0\left(1+\alpha \mathrm{\Delta }T\right)$ is valid only when the change in the temperature $\mathrm{\Delta }T$ is small and $\mathrm{\Delta }R=\left(R-R_0\right)<<R_0$ .
(A) Statement 1 is True, Statement 2 is False.
(B) Statement 1 is True, Statement 2 is True; Statement 2 is a correct explanation for Statement 1.
(C) Statement 1 is True, Statement 2 is True; Statement 2 is not the correct explanation for Statement 1.
(D) Statement 1 is False, Statement 2 is True.

Answer 1

Hint As temperature increases of a wire, it’s resistance also increases. We are having a relation given for change in resistance with respect to temperature. So we can get a solution using this equation.

Complete step by step answer
The temperature dependence of the resistance in the question is given as
 $\Rightarrow R=R_0\left(1+\alpha \mathrm{\Delta }T\right)$
Now, according to the question, we have $R_0=100\mathrm{\Omega }$, $R=150\mathrm{\Omega }$, $T_0={27}^{\circ }C$, and $T={227}^{\circ }C$ .
So, $\mathrm{\Delta }T=T-T_0$
 $\Rightarrow \mathrm{\Delta }T={227}^{\circ }C-{27}^{\circ }C={200}^{\circ }C$
Putting these values in (1) we get
 $\Rightarrow 150=100\left(1+200\alpha \right)$
 $\Rightarrow 200\alpha =0.5$
On solving we get
 $\Rightarrow \alpha =2.5\times {10}^{-4}{/}^{\circ }C$
This value matches with the value given in the Statement 1.
But while deriving the expression for the variation of the resistance with the temperature, it is assumed that the change in temperature is very small. But in this case, the change in temperature is
 $\Rightarrow \mathrm{\Delta }T={200}^{\circ }C$
This is a quite large value.
Also, while the derivation of the equation (1) is carried out, it is assumed that the change in resistance is very small compared to the original value, that is
 $\Rightarrow \left(R-R_0\right)<<R_0$
But in this case the change in resistance
 $\Rightarrow \left(R-R_0\right)=150-100$
 $\Rightarrow \left(R-R_0\right)=50\mathrm{\Omega }$
Which is comparable to the original value of resistance. So the above equation (1) cannot be applied to this case. Thus the value of $\alpha$ which is obtained above is incorrect.
Thus the Statement 1 is False.
Also the Statement 2 is True at the same time due to the reasons already stated above.
Hence the correct answer is option D.

Note
Do not blindly jump to the conclusion that Statement 1 is correct after getting the value of $\alpha$ same as that given in the Statement. That value is intentionally given to be the same. The question basically wants to judge the knowledge of the concept mentioned in the Statement 2.