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    Subject

    Quantitative Aptitude

    Topic

    Ratio

    Find mean proportional between (15+200) and (27−648)\begin{array}{l} \left( {15 + \sqrt {200} } \right){\rm{\ and\ }}\left( {27 - \sqrt {648} } \right)\\ \end{array}

    ক)
    7√5
    খ)
    3√5
    গ)
    4√10
    ঘ)
    9√7
    ঙ)
    None of these

    Explanation

    The mean proportion between ‘a’ and ‘b’ is √ab, if mean proportion is ‘x’, then, ⇒ a : x = x : b ⇒ a/x = x/b ⇒ x 2 = ab Or, √ab = x Now, let the required mean proportion be ‘m’, then (15+200):m=m:(27−648)\left( {15 + \sqrt {200} } \right){\rm{}}:{\rm{m}} = {\rm{m}}:\left( {27 - \sqrt {648} } \right) According to the rule, The required mean proportional, ⇒m=(15+200)×(27−648)⇒m=(15×27−15×648)+(27×200−200×648)⇒m=405−(15×182)+(27×102)−(102×182)⇒m=405−2702+2702−180×2⇒m=405−360=45⇒m=35\begin{array}{l} \Rightarrow m = \sqrt {\left( {15 + \sqrt {200} } \right) \times \left( {27 - \sqrt {648} } \right)} \\ \Rightarrow m = \sqrt {\left( {15 \times 27 - 15 \times \sqrt {648} } \right) + \left( {27 \times \sqrt {200} - \sqrt {200} \times \sqrt {648} } \right)} \\ \Rightarrow m = \sqrt {405 - \left( {15 \times 18\sqrt 2 } \right) + \left( {27 \times 10\sqrt 2 } \right) - \left( {10\sqrt 2 \times 18\sqrt 2 } \right)} \\ \Rightarrow m = \sqrt {405 - 270\sqrt 2 + 270\sqrt 2 - 180 \times 2} \\ \Rightarrow m = \sqrt {405 - 360} = \sqrt {45} \\ \Rightarrow m = 3\sqrt 5 \end{array}

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