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KVPY Sample Paper KVPY Stream-SX Model Paper-8

  • question_answer
    \[f:R\to R\] such that \[f(x+2y)=f(x)+f(2y)+4xy,\]\[\forall \,\,x,\,\,y\,\,\in \,\,R.\] If \[{{I}_{1}}=\int\limits_{0}^{1}{f(x)dx,}\] \[{{I}_{2}}=\int\limits_{-1}^{0}{f(x)dx,}\] \[{{I}_{3}}=\int\limits_{1/2}^{2}{f(x)dx,}\]then-

    A) \[{{I}_{1}}={{I}_{2}}>{{I}_{3}}\]                 

    B) \[{{I}_{1}}>{{I}_{2}}>{{I}_{3}}\]

    C) \[{{I}_{1}}={{I}_{2}}>{{I}_{3}}\]                 

    D) \[{{I}_{1}}<{{I}_{2}}<{{I}_{3}}\]

    Correct Answer: C

    Solution :

    \[f'(x)=\underset{h\to 0}{\mathop{\lim }}\,\,\,\frac{f(x+h)-f(x)}{h}\]
    \[=\underset{h\to 0}{\mathop{\lim }}\,\,\,\frac{f(x+2(h/2))-f(x+2.0)}{h}\]
    \[=f'(x)=\underset{h\to 0}{\mathop{\lim }}\,\,\,\frac{f(x)+f(h)-4x(h/2)-f(x)-f(0)}{h}\]
    \[=\underset{h\to 0}{\mathop{\lim }}\,\,\,\frac{f(h)-f(0)}{h}+2x\]
    (from given f(x), put\[x=y~=0,\text{ }f\left( 0 \right)=0\])
    \[f'(x)=f'(0)+2x\] \[\Rightarrow f'(x)=2x\]
    \[\Rightarrow f(x)={{x}^{2}}+c,\]put \[x=0\Rightarrow c=0\]
    \[f(x)={{x}^{2}},\]\[{{I}_{1}}=1/3,\]\[{{I}_{2}}=1/3\] and \[{{I}_{3}}=21/8\]
    \[\therefore {{I}_{1}}={{I}_{2}}<{{I}_{3}}\]


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