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JEE Main & Advanced Sample Paper JEE Main - Mock Test - 27

  • question_answer
    If \[{{I}_{1}}=\int\limits_{0}^{1}{{{2}^{{{x}^{2}}}}dx},\,\,{{I}_{2}}=\int\limits_{0}^{1}{{{2}^{{{x}^{3}}}}dx}\], \[\int\limits_{1}^{2}{{{2}^{{{x}^{2}}}}dx}\] and \[{{I}_{4}}=\int\limits_{1}^{2}{{{2}^{{{x}^{3}}}}dx}\] then

    A) \[{{I}_{2}}>{{I}_{1}}\]                 

    B)   \[{{I}_{1}}>{{I}_{2}}\]

    C)   \[{{I}_{3}}={{I}_{4}}\]                 

    D)   \[{{I}_{3}}>{{I}_{4}}\]

    Correct Answer: B

    Solution :

    \[{{I}_{1}}=\,\,\int\limits_{0}^{1}{{{2}^{{{x}^{2}}}}dx,\,\,{{I}_{2}}=\int\limits_{0}^{1}{{{2}^{{{x}^{3}}}}dx,\,{{I}_{3}}=}\int\limits_{1}^{2}{{{2}^{{{x}^{2}}}}dx,\,{{I}_{4}}=}\int\limits_{1}^{2}{{{2}^{{{x}^{3}}}}dx}}\] \[\forall \,0<x<1,\,\,{{x}^{2}}>{{x}^{3}}\] \[\Rightarrow \,\,\,\int\limits_{0}^{1}{{{2}^{{{x}^{2}}}}dx,\,\,>\int\limits_{0}^{1}{{{2}^{{{x}^{3}}}}dx\,\,\Rightarrow \,}\,{{I}_{1}}>{{I}_{2}}}\] Also\[\forall \,1<x<2\,\,\,{{x}^{2}}<{{x}^{3}}\,\,\,\Rightarrow \int\limits_{1}^{2}{{{2}^{{{x}^{2}}}}dx<\int\limits_{1}^{2}{{{2}^{{{x}^{3}}}}dx\Rightarrow }\,\,{{I}_{3}}<{{I}_{4}}}\]


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