If 4x - 5z = 16 and xz = 12, $64 x^{2} - 125 z^{3} =$

14512

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14576

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25833

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15616

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Solution

The correct option is D $15616$
Given, 4x - 5z = 16 and xz = 12
Since, $(a - b)^{2} = a^{2} - 2 a b + b^{2}$
$\Rightarrow (4 x - 5 z)^{2} = 16 x^{2} - 40 x z + 25 z^{2}$
put 4x - 5z = 16 and xz = 12
$\Rightarrow (16)^{2} = 16 x^{2} - 40 (12) + 25 z^{2}$
$\Rightarrow 16 x^{2} + 25 z^{2} = 256 + 480$
$\Rightarrow 16 x^{2} + 25 z^{2} = 736$ .....(1)
Given, $64 x^{3} - 125 z^{3} = (4 x)^{3} - (5 z)^{3}$
Since, $a^{3} - b^{3} = (a - b) (a^{2} + b^{2} + a b)$
$\Rightarrow (4 x)^{3} - (5 z)^{3} = (4 x - 5 z) (16 x^{2} + 25 z^{2} + 20 x z)$
By putting 4x - 5z = 16 , xz = 12 and using (1). we get,
$\Rightarrow (4 x)^{3} - (5 z)^{3} = (16) (736 + 20 \times 12)$
$\Rightarrow (4 x)^{3} - (5 z)^{3} = (16) (976)$
$\Rightarrow (4 x)^{3} - (5 z)^{3} = 15616$
Option D is correct.

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