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

  • question_answer If \[\frac{2x}{2{{x}^{2}}+5x+2}\]>\[\frac{1}{x+1}\], then [IIT 1987]

    A) \[-2>x>-1\]

    B) \[-2\ge x\ge -1\]

    C) \[-2<x<-1\]

    D) \[-2<x\le -1\]

    Correct Answer: C

    Solution :

    Given \[\frac{2x}{2{{x}^{2}}+5x+2}>\frac{1}{x+1}\] Þ  \[\frac{2x}{(2x+1)(x+2)}>\frac{1}{(x+1)}\] Þ  \[\frac{2x}{(2x+1)(x+2)}-\frac{1}{(x+1)}>0\] Þ \[\frac{2x(x+1)-(2x+1)(x+2)}{(x+1)(2x+1)(x+2)}>0\] Þ  \[\frac{2{{x}^{2}}+2x-2{{x}^{2}}-4x-x-2}{(x+1)(x+2)(2x+1)}>0\] Þ \[\frac{-3x-2}{(x+1)(x+2)(2x+1)}>0\] Equating each factor equal to 0, we have\[x=-2,-1,-\frac{2}{3},-\frac{1}{2}\]. It is clear that \[-\frac{2}{3}<x<-\frac{1}{2}\]or\[-2<x<-1\].

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