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12th Class Mathematics Definite Integrals Question Bank Critical Thinking

  • question_answer
    The numbers P, Q and \[R\] for which the function \[f(x)=P{{e}^{2x}}+Q{{e}^{x}}+Rx\] satisfies the conditions \[f(0)=-1,\] \[{f}'(\log 2)=31\] and \[\int_{0}^{\log 4}{[f(x)-Rx]\,dx=\frac{39}{2}}\] are given by

    A) \[P=2,\] \[Q=-3,\] \[R=4\]      

    B) \[P=-5,\] \[Q=2,\] \[R=3\]

    C) \[P=5,\] \[Q=-2,\] \[R=3\]      

    D) \[P=5,\] \[Q=-6,\] \[R=3\]

    Correct Answer: D

    Solution :

    • We have \[f'(x)=2P{{e}^{2x}}+Q{{e}^{x}}+R\]                   
    • Þ \[f'(\log 2)=8P+2Q+R\]                   
    • Also,  \[-1=f(0)=P+Q\]                   
    • \[\frac{39}{2}=\int_{0}^{\log 4}{[f(x)-Rx]dx=\int_{0}^{\log 4}{(P{{e}^{2x}}}+Q}{{e}^{x}})dx\]                   
    • \[\Rightarrow \]\[\frac{15P}{2}+3Q=\frac{39}{2}\]           
    • Now on solving the above equations, we get \[P=5,Q=-6\] and \[R=3\].


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