11th Class Mathematics Complex Numbers and Quadratic Equations Question Bank Critical Thinking

  • question_answer If both the roots of the quadratic equation\[{{x}^{2}}-2kx+{{k}^{2}}+k-5=0\]are less than 5, then \[k\] lies in the interval [AIEEE 2005]

    A) \[(-\infty ,\,4)\]

    B) [4, 5]

    C) (5, 6]

    D) (6, \[\infty \])

    Correct Answer: A

    Solution :

    \[{{x}^{2}}-2kx+{{k}^{2}}+k-5=0\] Roots are less than 5, \[D\ge 0\] \[4{{k}^{2}}-4\,({{k}^{2}}+k-5)\ge 0\]                          ??(i) Þ \[k\le 5\] Þ \[f(5)>0\]                           .....(ii) Þ \[k\in (-\infty ,\,4)\cup (5,\infty )\]; \[-\left( \frac{2k}{2} \right)<5\Rightarrow k<5\]    ?..(iii) form (i), (ii)  and (iii), \[k\in (-\infty ,\,4)\]

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