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JCECE Engineering JCECE Engineering Solved Paper-2005

  • question_answer
    Let\[f(x)\]be twice differentiable such that\[f'\,\,'(x)=-f(x),\,\,f'(x)=g(x)\], where \[f'(x)\] and\[f'\,\,'(x)\] represent the first and second derivatives of \[f(x)\] respectively. Also if \[h(x)={{[f(x)]}^{2}}+{{[g(x)]}^{2}}\]and \[h(5)=5\], then \[h(10)\] is equal to :

    A) \[3\]                                     

    B) \[10\]

    C) \[13\]                                   

    D) \[5\]

    Correct Answer: D

    Solution :

    We have,                 \[h(x)={{[f(x)]}^{2}}+{{[g(x)]}^{2}}\] On differentiating w.r.t.\[x,\] we get \[h'(x)=2f(x)f'(x)+2g(x)g'(x)\]     ? (i) Given that,                 \[f''(x)=-f(x)\] and        \[f'(x)=g(x)\]                                     ? (ii) On differentiating Eq. (ii), we get                 \[f''(x)=g'(x)\] \[\Rightarrow \]               \[g'(x)=-f(x)\] From Eq. (i)                 \[h'(x)=2f(x)-f'(x)-2f'(x)f(x)\]                          \[=0\] \[\Rightarrow \]               \[h(x)=5\]           \[[\because \,\,h(5)=5]\] \[\therefore \]         \[h(10)=5\]


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