11th Class Mathematics Other Series Question Bank Critical Thinking

  • question_answer If \[a,\ b,\ c\] are in H.P., then the value of \[\left( \frac{1}{b}+\frac{1}{c}-\frac{1}{a} \right)\,\left( \frac{1}{c}+\frac{1}{a}-\frac{1}{b} \right)\], is  [MP PET 1998; Pb. CET 2000]

    A) \[\frac{2}{bc}+\frac{1}{{{b}^{2}}}\]

    B)   \[\frac{3}{{{c}^{2}}}+\frac{2}{ca}\]

    C) \[\frac{3}{{{b}^{2}}}-\frac{2}{ab}\]

    D) None of these

    Correct Answer: C

    Solution :

    \[a,\ b,\ c\]are in H.P., then  \[\frac{1}{a},\ \frac{1}{b},\ \frac{1}{c}\] are in A.P. \[\Rightarrow \]\[\frac{1}{b}-\frac{1}{a}=\frac{1}{c}-\frac{1}{b}\] Now, \[\left( \frac{1}{b}+\frac{1}{c}-\frac{1}{a} \right)\,\left( \frac{1}{c}+\frac{1}{a}-\frac{1}{b} \right)\] \[=\left( \frac{3}{b}-\frac{2}{a} \right)\,\left( \frac{1}{b} \right)=\frac{3}{{{b}^{2}}}-\frac{2}{ab}\].

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