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JAMIA MILLIA ISLAMIA Jamia Millia Islamia Solved Paper-2004

  • question_answer
        If \[-1,2-3\omega ,2-3{{\omega }^{2}}\]and \[\text{2}+\text{7}+\text{14}+\text{23}+\text{34}...\] where \[\left( {{\text{a}}^{\text{2}}}+{{\text{b}}^{\text{2}}}+{{\text{c}}^{\text{2}}} \right){{P}^{\text{2}}}-\text{ 2}\left( \text{ab}+\text{bc}+\text{cd} \right)P\] then\[+({{b}^{2}}+{{c}^{2}}+{{d}^{2}}\le 0\] has at least :

    A)  four real roots

    B)  two real roots

    C)  four imaginary roots

    D)  none of these

    Correct Answer: B

    Solution :

                    Let all four roots are imaginary then roots of both equations\[p(x)=0\]and\[Q(x)=0\] are imaginary thus\[{{b}^{2}}-4ac<0,{{d}^{2}}+4ac<0,\]so \[{{b}^{2}}+{{d}^{2}}<0,\]which is impossible unless \[b=0\]and\[d=0\] so, if\[b\ne 0\]or\[d\ne 0\]at least two roots must be real. If\[b=0,d=0\]we have the equations \[P(x)=a{{x}^{2}}+c=0\] and        \[Q(x)=-a{{x}^{2}}+c=0\] or,\[{{x}^{2}}=\frac{-c}{a},{{x}^{2}}=\frac{c}{a}\]as one of\[\frac{c}{a},\frac{-c}{a}\]must  be positive, so two roots must be real.


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