A) \[{{T}_{2}}=\,\,{{({{T}_{1}}^{3}{{T}_{4}})}^{1/4}};\,\,=\,\,{{T}_{3}}\,\,=\,\,{{({{T}_{1}}{{T}_{4}}^{3})}^{1/4}}\]
B) \[{{T}_{2}}=\,\,{{({{T}_{1}}{{T}_{4}})}^{1/2}};\,\,\,{{T}_{3}}\,\,=\,\,{{({{T}_{1}}^{2}{{T}_{4}})}^{1/3}}\]
C) \[{{T}_{2}}=\,\,{{({{T}_{1}}{{T}_{4}}^{2})}^{1/3}};\,\,{{T}_{3}}\,\,=\,\,{{({{T}_{1}}^{2}{{T}_{4}})}^{1/3}}\]
D) \[{{T}_{2}}=\,\,{{({{T}_{1}}^{2}{{T}_{4}})}^{1/3}};\,\,\,\,{{T}_{3}}\,\,=\,\,{{({{T}_{1}}{{T}_{4}}^{2})}^{1/3}}\]
Correct Answer: D
Solution :
[d] |
\[As\,=\,{{\eta }_{1}}={{\eta }_{2}}\] |
\[1-\frac{{{T}_{2}}}{{{T}_{1}}}\,=\,1-\frac{{{T}_{3}}}{{{T}_{2}}}\] |
\[{{T}^{2}}\,=\,\sqrt{{{T}_{1}}{{T}_{3}}}\] |
\[As\,{{\eta }_{2}}=\,{{\eta }_{3}}\] |
\[1-\frac{{{T}^{3}}}{{{T}^{2}}}\,=\,\,1\,-\,\,\frac{{{T}^{4}}}{{{T}^{3}}}\] |
\[{{T}_{3}}=\sqrt{{{T}_{2}}{{T}_{4}}}\] |
\[As\,\,{{\eta }_{1}}={{\eta }_{3}}\] |
\[1-\frac{{{T}_{2}}}{{{T}_{1}}}\,=\,1\,-\,\frac{{{T}^{4}}}{{{T}^{3}}}\,\,\,Hence\,\,{{T}_{2}}{{T}_{3}}\,=\,{{T}_{1}}{{T}_{4}}\,\] |
\[\frac{{{T}_{1}}{{T}_{4}}}{{{T}_{3}}}\,=\,\sqrt{{{T}_{1}}{{T}_{3}}}\,\,\,\,\,\Rightarrow \,\,\,\frac{{{T}^{2}}_{1}{{T}^{2}}_{4}}{{{T}_{3}}^{2}}\,\,=\,\,{{T}_{1}}{{T}_{3}}\] |
\[{{T}_{3}}=\,\,{{({{T}_{1}}{{T}_{4}}^{2})}^{1/3}}\] |
\[{{({{T}_{1}}{{T}_{4}}^{2})}^{1/3}}\,=\,{{({{T}_{2}}{{T}_{4}})}^{1/2}}\] |
\[{{T}_{1}}^{2/3}{{T}_{4}}^{4/3}\,=\,\,{{T}_{2}}{{T}_{4}}\] |
\[{{T}_{2}}\,=\,{{({{T}_{1}}^{2}{{T}_{4}})}^{1/3}}\] |
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