11th Class Mathematics Other Series Question Bank Critical Thinking

  • question_answer
    \[a,\,g,\,h\] are arithmetic mean, geometric mean and harmonic mean between two positive numbers x and y respectively. Then identify the correct statement among the following [Karnataka CET 2001]

    A) h is the harmonic mean between a  and g

    B) No such relation exists between a, g and h

    C) g  is the geometric mean between a and h

    D) A  is the arithmetic mean between g  and h

    Correct Answer: C

    Solution :

    \[a=\frac{x+y}{2},g=\sqrt{xy}\] and \[h=\frac{2xy}{x+y}\] \[{{g}^{2}}={{\left( \sqrt{xy} \right)}^{2}}=xy\]...(i), ah\[=\frac{x+y}{2}.\frac{2xy}{x+y}=xy\]     ...(ii) From (i) and (ii), we get \[{{g}^{2}}=ah\] or \[g=\sqrt{ah}\], therefore g is the G.M. between a and h.

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