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JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Question Bank Solution of quadratic equations and Nature of roots

  • question_answer
    Let\[p,q\in \{1,\,2,\,3,\,4\}\]. The number of equations of the form \[p{{x}^{2}}+qx+1=0\] having real roots is   [IIT Screening 1994]

    A) 15

    B) 9

    C) 7

    D) 8

    Correct Answer: C

    Solution :

    For real roots, discriminant  \[\ge 0\] \[\Rightarrow {{q}^{2}}-4p\ge 0\,\,\Rightarrow \,\,{{q}^{2}}\ge 4p\] For \[p=1,{{q}^{2}}\ge 4\,\,\,\Rightarrow q=2,3,4\]              \[p=2,{{q}^{2}}\ge 8\,\,\,\,\Rightarrow q=3,4\]              \[p=3,{{q}^{2}}\ge 12\,\,\,\,\Rightarrow q=4\]               \[p=4,{{q}^{2}}\ge 16\,\,\,\,\Rightarrow q=4\] Total seven solutions are possible.


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