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Manipal Engineering Manipal Engineering Solved Paper-2010

  • question_answer
    If\[f(x)=(ax+b)\sin x+(cx+d)\cos x\], then the values of\[a,\,\,b,\,\,c\]and\[d\]such that\[f(x)=x\cos x\]for all\[x\], are

    A) \[b=c=0,\,\,a=d=1\]

    B) \[b=d=0,\,\,a=c=1\]

    C) \[c=d=0,\,\,a=b=1\]

    D)  None of the above

    Correct Answer: A

    Solution :

    We have,\[f(x)=(ax+b)\sin x+(cx+d)\cos x\] \[f(x)=a\sin x+(ax+b)\cos x+c\cos x\]                                                           \[-(cx+d)\sin x\] But\[f(x)=x\cos x\]for all\[x\](given) \[\therefore \]\[x\cos x=(a-d)\] \[\sin x+(b+c)\cos x+ax\cos x-cx\sin x\] Equating the coefficients of\[\sin x,\,\,\cos x,\,\,x\cos x\] , and\[x\sin x\], we get                 \[a-d=0,\,\,b+c=0,\,\,a=1,\,\,c=0\] \[\therefore \]  \[b=c=0\]and\[a=d=1\]


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