If the resistance of a conductor is $$5\,\Omega $$  at $${50^ \circ }C$$  and $$7\,\Omega $$  at $${100^ \circ }C,$$  then the mean temperature coefficient of resistance (of the material) is

Solution :
Temperature coefficient of resistance is defined as the increase in resistance per unit original resistance per degree rise of temperature and is given by
$$\alpha = \frac{{{R_t} - {R_o}}}{{{R_o} \times t}}\,\,{\text{or}}\,\,{R_t} = {R_o}\left( {1 + \alpha t} \right)$$
$${R_t} =$$  Resistance at final temperature
$${R_o} =$$  Resistance at initial temperature
$$t =$$  Change in temperature
Case I
$$5 = {R_o}\left[ {1 + \alpha \left( {50} \right)} \right]\,.......\left( {\text{i}} \right)$$
Case II
$$7 = {R_o}\left[ {1 + \alpha \left( {100} \right)} \right]\,.......\left( {{\text{ii}}} \right)$$
Dividing Eq. (i) by Eq. (ii)
$$\eqalign{ & \frac{5}{7} = \frac{{1 + 50\alpha }}{{1 + 100\alpha }} \cr & \therefore 5 + 500\alpha = 7 + 350\alpha \cr & \therefore \alpha = \frac{2}{{150}} = 0.01{/^ \circ }C \cr} $$