KVPY Sample Paper KVPY Stream-SX Model Paper-4

  • question_answer
    A function \[y\text{ }=\text{ }f(x)\] satisfies the condition \[f'(x)\,\,sin\,x+f\,(x)\,\,\cos \,x=1,\,\,f(x)\] being bounded when \[x\to 0.\] If \[I=\int\limits_{0}^{\pi /2}{f(x)dx,}\] then

    A) \[\frac{\pi }{2}<I<\frac{{{\pi }^{2}}}{4}\]      

    B) \[\frac{\pi }{4}<I<\frac{{{\pi }^{2}}}{2}\]

    C) \[1<I<\frac{\pi }{2}\]               

    D) \[0<I<1\]

    Correct Answer: A

    Solution :

    Since, \[\sin x\frac{dy}{dx}+y\cos x=1\]
    \[\frac{dy}{dx}+y\cot x=\text{cosec}\,x\]
    IF \[(e=Ns\,\,B\,\,\omega \,\,\sin \,\,\omega t),\] \[y\sin x=\int{\text{cosec}\,x.\,\sin x\,\,dx}\]
    \[y\sin x=x+C\]
    If \[x\to 0,\] y is finite \[\therefore C=O\]
    \[\therefore y=\frac{x}{\sin x}\]
    Now, \[I<\frac{{{\pi }^{2}}}{4}\] and \[I>\frac{\pi }{2}\]
    Hence, \[\frac{\pi }{2}<I<\frac{{{\pi }^{2}}}{4}\]

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