For a Carnot engine \[{{T}_{1}}>{{T}_{2}}.\] When \[{{T}_{2}}\] is decreased by \[\Delta T\] with \[{{T}_{1}}\] remaining same then efficiency is \[{{\eta }_{1}},\] and when \[{{T}_{1}}\] is increased by \[\Delta T\] with \[{{T}_{2}}\] remaining same, efficiency is \[{{\eta }_{2}}.\] Which one of the following is the correct expression for \[\left( {{\eta }_{1}}-{{\eta }_{2}} \right)?\]

• A. \[\frac{({{T}_{2}}-{{T}_{1}})\,\,\Delta \,T+{{(\Delta \,T)}^{2}}}{{{T}_{2}}({{T}_{2}}+\Delta \,T)}\]
• B. \[\frac{({{T}_{2}}-{{T}_{1}})\,\,\Delta \,T+{{(\Delta \,T)}^{2}}}{{{T}_{1}}({{T}_{1}}+\Delta \,T)}\]
• C. \[\frac{({{T}_{1}}-{{T}_{2}})\,\,\Delta \,T+{{(\Delta \,T)}^{2}}}{{{T}_{1}}({{T}_{1}}+\Delta \,T)}\]
• D. \[\frac{({{T}_{1}}-{{T}_{2}})\,\,\Delta \,T+{{(\Delta T)}^{2}}}{{{T}_{1}}({{T}_{1}}+\Delta T)}\]