12th Class Mathematics Sample Paper Mathematics Olympiad - Sample Paper-3

  • question_answer Let \[\mathbf{f}\left( \mathbf{x}+\mathbf{y} \right)=\mathbf{f}\left( \mathbf{x} \right).\mathbf{f}\left( \mathbf{y} \right),\forall \mathbf{x},\mathbf{y}\]; where \[f(0)\ne 0.\,if\,f(5)=2\]and \[\mathbf{f}'\left( \mathbf{0} \right)=\mathbf{3}\], then f'(5) is equal to

    A) 2                     

    B) 4              

    C) 6                                 

    D) 8

    Correct Answer: C

    Solution :

    [c] \[f'(5)=\underset{h\to 0}{\mathop{\lim }}\,\frac{f(5+h)-f(5)}{h}=\underset{h\to 0}{\mathop{\lim }}\,\frac{f(5+h)-f(5+0)}{h}\] \[=\underset{h\to 0}{\mathop{\lim }}\,\frac{f(5).f(h)-.f(0)}{h}[\because \,(x+y)=f(x).f(y)]\,\] \[=\underset{h\to 0}{\mathop{\lim }}\,\frac{f(5)[f(h)-f(0)]}{h}=f(5)\underset{h\to 0}{\mathop{\lim }}\,\frac{f(0+h)-f(0)}{h}\] \[=f\left( 5 \right).f'\left( 0 \right)=2\times 3=6\] Hence, option [c] is correct.


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