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JEE Main & Advanced Sample Paper JEE Main Sample Paper-7

  • question_answer
    If \[a,\,b,\,c\in R,\] and \[a\ne 0\] and the equation \[a{{x}^{2}}+bx+c=0\] has no real roots, then

    A)  \[{{(a+c)}^{2}}<{{b}^{2}}\]                         

    B)  \[(a+{{c}^{2}})={{b}^{2}}\]

    C)  \[{{(a+c)}^{2}}>{{b}^{2}}\]                         

    D)  \[({{a}^{2}}+c)={{b}^{2}}\]

    Correct Answer: C

    Solution :

    \[\because \]  The equation has no real roots.    \[\therefore \]  \[{{b}^{2}}-4ac<0\Rightarrow \,4ac>{{b}^{2}}\]                  ?(i) Also, \[{{(a-c)}^{2}}\ge 0\Rightarrow \,{{a}^{2}}+{{c}^{2}}\ge 2ac\]           ?(ii) On adding these two inequalities, we get \[{{a}^{2}}+{{c}^{2}}+2ac>{{b}^{2}}\] \[\Rightarrow \,\,{{(a+c)}^{2}}>{{b}^{2}}\]


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