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JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Question Bank Mock Test - Complex Numbers and Quadratic Equations

  • question_answer
    If \[{{b}_{1}}{{b}_{2}}\]=2(\[{{c}_{1}}+{{c}_{2}}\]), then at least one of the equations \[{{x}^{2}}+{{b}_{1}}x+{{c}_{1}}=0\]and \[{{x}^{2}}+{{b}_{2}}x+{{C}_{2}}=0\]has

    A)  imaginary

    B)  real roots

    C) purely imaginary roots

    D) none of these

    Correct Answer: B

    Solution :

    [b]
    \[Let\,\,{{D}_{1}}\,\,and\text{ }{{D}_{2}}\] be desicriminants of \[{{x}^{2}}+{{b}_{1}}x+{{c}_{1}}=0\]
    \[and\text{ }{{x}^{2}}+{{b}_{2}}x+{{c}_{2}}=0,respectively.\,\,Then,\]
    \[{{D}_{1}}+{{D}_{2}}={{b}_{1}}^{2}-4{{c}_{1}}+{{b}_{2}}^{2}-4{{c}_{2}}\]
    \[=({{b}_{1}}^{2}+{{b}_{2}}^{2})-4({{c}_{1}}+{{c}_{2}})\]
    \[={{b}_{1}}^{2}+{{b}_{2}}^{2}-2{{b}_{1}}{{b}_{2}}[\therefore {{b}_{1}}{{b}_{2}}=2({{c}_{1}}+{{c}_{2}})]\]
    \[={{({{b}_{1}}-{{b}_{2}})}^{2}}\ge 0\]
    \[\Rightarrow {{D}_{1}}\ge 0\,\,or\text{ }{{D}_{2}}\ge 0\,\,or\text{ }{{D}_{1}}\,and\,{{D}_{2}}\] both are positive
    Hence, at least one of the equations has real roots.
     


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