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JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Question Bank Condition for common roots, Quadratic expressions and Position of roots

  • question_answer
    If S is a set of \[P(x)\] is polynomial of degree \[\le 2\] such that \[P(0)=0,\]\[P(1)=1\],\[P'(x)>0\text{ }\forall x\in (0,\,1)\], then [IIT Screening 2005]

    A) \[S=0\]

    B) \[S=ax+(1-a){{x}^{2}}\text{ }\forall a\in (0,\infty )\]

    C) \[S=ax+(1-a){{x}^{2}}\text{ }\forall a\in R\]

    D) \[S=ax+(1-a){{x}^{2}}\text{ }\forall a\in (0,2)\]

    Correct Answer: D

    Solution :

    Let   \[P(x)=b{{x}^{2}}+ax+c\] As    \[P(0)=0\Rightarrow c=0\] As    \[P(1)=1\Rightarrow a+b=1\]         \[P(x)=ax+(1-a){{x}^{2}}\] Now \[{P}'(x)=a+2(1-a)x\] as \[{P}'(x)>0\] for \[x\in (0,\,1)\] Only option (d) satisfies above condition


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