KVPY Sample Paper KVPY Stream-SX Model Paper-5

  • question_answer
    Let \[a,b,p,q\in Q\] and suppose that \[f(x)={{x}^{2}}+ax+b=0\] and \[g(x)={{x}^{3}}+px+q=0\] have a common irrational roots. Then,

    A) \[f(x)\text{divides}\,\text{g}\,(x)\]

    B) \[g(x)=xf(x)\]

    C) \[g\,(x)=(x-b-q)f(x)\]       

    D) None of these

    Correct Answer: A

    Solution :

    Let \[\alpha \in R-Q\]be a common root of \[f(x)=0\]and \[g(x)=0,\]then, \[{{\alpha }^{2}}+a\alpha +b=0\]\[\Rightarrow \]\[{{\alpha }^{2}}=-a\alpha -b\]
    On putting this in \[{{\alpha }^{3}}+p\alpha +q=0,\] we get \[({{a}^{2}}-b+p)\alpha +ab+q=0\]
    As \[\alpha \] is irrational and a, b, p, \[q\in Q.\]
    This give, \[g(x)=(x-a)f(x)\]
    \[\therefore \] \[f(x)\]divides \[g(x).\]

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