JEE Main & Advanced Sample Paper JEE Main Sample Paper-13

  • question_answer
    If nth root of unity be \[1,\,\,{{a}_{1}},\,\,{{a}_{2}},\,....,\,{{a}_{n-1,}}\] then \[\sum\limits_{r\,=\,\,1}^{n\,-\,1}{\,\,\frac{1}{2+{{a}_{r}}}}\] is equal to

    A)  \[\frac{n\cdot {{2}^{n-1}}}{{{2}^{n}}-1}-1\]                        

    B)  \[\frac{n{{(-2)}^{n-1}}}{{{(-2)}^{n}}-1}-1\]

    C)  \[\frac{n{{(-2)}^{n-1}}}{1+{{(-2)}^{n+1}}}-\frac{1}{3}\]  

    D)  None of these

    Correct Answer: C

    Solution :

     \[=1.86\times \frac{68.4}{342}\times \frac{1}{1}\] \[\therefore \] be nth root of unit. \[=0-{{(0.372)}^{o}}C=-{{0.372}^{o}}C\] \[\pi =\frac{n}{V}RT\] On taking logarithm and then differentiation, we get \[=\frac{{{W}_{B}}}{{{M}_{B}}}\times \frac{RT}{V}\]        \[=\frac{68.4}{342}\times \frac{0.0821}{1}\times 298=4.89\,\,\text{atm}\] On putting \[2C{{H}_{3}}CO{{O}^{\odot -}}\xrightarrow{\,}\,2C{{H}_{3}}CO{{O}^{\bullet }}\,+2{{e}^{-}}\] we get \[2C{{H}_{3}}CO{{O}^{\bullet -}}\xrightarrow{\,}\,2\overset{\bullet }{\mathop{C}}\,{{H}_{3}}+2C{{O}_{2}}\] \[CH_{3}^{\bullet }+\overset{\bullet }{\mathop{C}}\,{{H}_{3}}\xrightarrow{\,}\,C{{H}_{3}}-C{{H}_{3}}(g)\,\] \[C{{O}_{2}}\]    \[C{{H}_{3}}-C{{H}_{3}}\] \[2{{H}_{2}}O+2{{e}^{-}}\xrightarrow{\,}\,{{H}_{2}}+2O{{H}^{-}}\]


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