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

  • question_answer The value of 'a' for which one root of the quadratic equation \[({{a}^{2}}-5a+3){{x}^{2}}+(3a-1)x+2=0\] is twice as large as the other, is [AIEEE 2003]

    A) \[\frac{2}{3}\]

    B) \[-\frac{2}{3}\]

    C) \[\frac{1}{3}\]

    D) \[-\frac{1}{3}\]

    Correct Answer: A

    Solution :

    Let the roots are a and 2a Þ  \[\alpha +2\alpha =\frac{1-3a}{{{a}^{2}}-5a+3}\]  and \[\,\alpha .2\alpha =\frac{2}{{{a}^{2}}-5a+3}\] Þ \[2\left[ \frac{1}{9}\frac{{{(1-3a)}^{2}}}{{{({{a}^{2}}-5a+3)}^{2}}} \right]=\frac{2}{{{a}^{2}}-5a+3}\] Þ  \[\frac{{{(1-3a)}^{2}}}{({{a}^{2}}-5a+3)}=9\]\[\Rightarrow 9{{a}^{2}}-6a+1=9{{a}^{2}}-45a+27\] Þ \[39a=26\]\[\Rightarrow a=\frac{2}{3}\].

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