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JEE Main & Advanced Sample Paper JEE Main - Mock Test - 26

  • question_answer
    If \[m=\sum\limits_{r=0}^{\infty }{{{a}^{r}},}\] \[n=\sum\limits_{r=0}^{\infty }{{{b}^{r}},}\]where \[0<a,\text{ }b<1,\] then which of the following equations has roots a and b?

    A) \[mn{{x}^{2}}+(m+n-2mn)x+mn-m-n+1=0\]

    B) \[mn{{x}^{2}}-(2mn+m+n)x+mn+m+mn+1=0\]

    C) \[mn{{x}^{2}}+(2mn+m+n)x+mn+m+n+1=0\]

    D) \[mn{{x}^{2}}-(2mn-m-n)x+mn-m-n+1=0\]

    Correct Answer: D

    Solution :

    [d] \[m=\sum\limits_{r=0}^{\infty }{{{a}^{r}}}=\frac{1}{1-a}\Rightarrow a=\frac{m-1}{m}\] Similarly,    \[b=\frac{n-1}{n}\] Equation having roots a and b is: \[{{x}^{2}}-(a+b)x+ab=0\] or \[{{x}^{2}}-\left( \frac{m-1}{m}+\frac{n-1}{n} \right)x+\frac{(m-1)\,(n-1)}{mn}=0\] or \[mn{{x}^{2}}-(2mn-m-n)x+mn-m-n+1=0\]


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