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KVPY Sample Paper KVPY Stream-SX Model Paper-25

  • question_answer
    If the equation \[{{x}^{4}}-4{{x}^{3}}+a{{x}^{2}}+bx+1=\text{ }0\] have four positive roots, then choose the incorrect option

    A) Roots are necessarily integers

    B) \[a+b=2~\]

    C) \[ab=-24~\]

    D) None of these

    Correct Answer: D

    Solution :

    let the roots be \[{{x}_{1}},{{x}_{2}},{{x}_{3}},{{x}_{4}},\]then \[{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+{{x}_{4}}=4\] and \[{{x}_{1}}{{x}_{2}}{{x}_{3}}{{x}_{4}}=1\] \[\Rightarrow \]\[A.M.\,of\,{{x}_{1}},{{x}_{2}},{{x}_{3}},{{x}_{4}}\]
    \[=\operatorname{G}.M.\,of\,{{x}_{1}},{{x}_{2}},{{x}_{3}},{{x}_{4}}\]\[\Leftrightarrow \]\[{{x}_{1}}={{x}_{2}}={{x}_{3}}={{x}_{4}}\]
    \[\therefore \]\[{{x}_{1}}={{x}_{2}}={{x}_{3}}={{x}_{4}}=1\]\[\Rightarrow \]\[{{x}_{1}},{{x}_{2}},{{x}_{3}},{{x}_{4}}\]in A.P. as well as G.P. and in H.P. Also \[{{x}^{4}}-4{{x}^{3}}+a{{x}^{2}}+bx+1={{\left( x-1 \right)}^{4}}\]
    \[\Rightarrow \]\[a=6,b=-4\]


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