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    Question

    If, 4x + 3y = 18, and 2xy = 12, and 4x > 3y, then find the

    value of 64x 3  - 27y 3 .
    A 1605 Correct Answer Incorrect Answer
    B 1512 Correct Answer Incorrect Answer
    C 1474 Correct Answer Incorrect Answer
    D 1884 Correct Answer Incorrect Answer

    Solution

    2xy = 12

    So, xy = (12/2) = 6 ......(i)

    4x + 3y = 18

    On squaring the above equation, we get,

    (4x + 3y)  2  = 18 2

    Or, 16x 2  + 9y 2  + 2 X 4x X 3y = 324

    Or, 16x 2  + 9y 2  + 24 X 6 = 324 (since, xy = 6)

    Or, 16x 2  + 9y 2  = 324 - 144

    So, 16x 2  + 9y 2  = 180 ....(ii)

    (4x - 3y)  2  = 16x 2  + 9y 2  - 2 X 4x X 3y

    = 180 - 24 X 6 = 180 - 144 = 36 {using values from equations (i) and (ii) }

    Since, 4x > 3y, so, 4x - 3y, cannot be negative.

    So, 4x - 3y = 6 ....(iii)

    We know that, a 3  - b 3  = (a - b) X (a 2  + b 2  + ab) .

    So, (4x)  3  - (3y)  3  = 64x 3  - 27y 3

    = (4x - 3y) X (16x 2  + 9y 2  + 4x X 3y)

    Putting the value of equations (i) , (ii) and (iii) in the above equation, we get,

    = 6 X (180 + 12 X 6)

    = 6 X (180 + 72)

    = 6 X 252 = 1,512

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