$A$ wire of cross-section area $A$, length $L_1$, resistivity $\rho_1$ and temperature coefficient of resistivity $\alpha_1$ is connected to a second wire of length $L_2$, resistivity $\rho_2$ , temperature coefficient of resistivity $\alpha_1$ and the same area $A$, so that wire carries same current. Total resistance $R$ is independent of temperature for small temperature change if (Thermal expansion effect is negligible)
Diffcult
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Let initial resistance of the wires are $R_{1}$ and $R_{2}$ respectively. Then
$R_{1}^{\prime}+R_{2}^{\prime}=R_{1}+R_{2}$
$\Rightarrow R_{1}\left(1+\alpha_{1} \Delta T\right)+R_{2}\left(1+\alpha_{2} \Delta T\right)$
$=R_{1}+R_{2}$
$\Rightarrow R_{1} \alpha_{1}+R_{2} \alpha_{2}=0$
$\frac{\rho_{1} L_{1}}{A}+\frac{\rho_{2} L_{2}}{A} \alpha_{2}=0$
$\Rightarrow \rho_{1} L_{1} \alpha_{1}+\rho_{2} L_{2} \alpha_{2}=0$
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