A) \[\frac{{{c}_{1}}{{T}_{1}}+{{c}_{2}}{{T}_{2}}+{{c}_{3}}{{T}_{3}}}{{{m}_{1}}{{c}_{1}}+{{m}_{2}}{{c}_{2}}+{{m}_{3}}{{c}_{3}}}\]
B) \[\frac{{{m}_{1}}{{c}_{1}}{{T}_{1}}+{{m}_{2}}{{c}_{2}}{{T}_{2}}+{{m}_{3}}{{c}_{3}}{{T}_{3}}}{{{m}_{1}}{{c}_{1}}+{{m}_{2}}{{c}_{2}}+{{m}_{3}}{{c}_{3}}}\]
C) \[\frac{{{m}_{1}}{{c}_{1}}{{T}_{1}}+{{m}_{2}}{{c}_{2}}{{T}_{2}}+{{m}_{3}}{{c}_{3}}{{T}_{3}}}{{{m}_{1}}{{T}_{1}}+{{m}_{2}}{{T}_{2}}+{{m}_{3}}{{T}_{3}}}\]
D) \[\frac{{{m}_{1}}{{T}_{1}}+{{m}_{2}}{{T}_{2}}+{{m}_{3}}{{T}_{3}}}{{{c}_{1}}{{T}_{1}}+{{c}_{2}}{{T}_{2}}+{{c}_{3}}{{T}_{3}}}\]
Correct Answer: B
Solution :
Let the final temperature be T °C. Total heat supplied by the three liquids in coming down to 0°C = \[{{m}_{1}}{{c}_{1}}{{T}_{1}}+{{m}_{2}}{{c}_{2}}{{T}_{2}}+{{m}_{3}}{{c}_{3}}{{T}_{3}}\] ..... (i) Total heat used by three liquids in raising temperature from 0oC to ToC = \[{{m}_{1}}{{c}_{1}}T+{{m}_{2}}{{c}_{2}}T+{{m}_{3}}{{c}_{3}}T\] .....(ii) By equating (i) and (ii) we get \[({{m}_{1}}{{c}_{1}}+{{m}_{2}}{{c}_{2}}+{{m}_{3}}{{c}_{3}})\,T\] = \[{{m}_{1}}{{c}_{1}}{{T}_{1}}+{{m}_{2}}{{c}_{2}}{{T}_{2}}+{{m}_{3}}{{c}_{3}}{{T}_{3}}\] Þ \[T=\frac{{{m}_{1}}{{c}_{1}}{{T}_{1}}+{{m}_{2}}{{c}_{2}}{{T}_{2}}+{{m}_{3}}{{c}_{3}}{{T}_{3}}}{{{m}_{1}}{{c}_{1}}+{{m}_{2}}{{c}_{2}}+{{m}_{3}}{{c}_{3}}}\].
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