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nth Term of A.P

If 3x-4y=5 ...

Question

If $3 x - 4 y = 5$ and $x y = 3$ , then find $27 x^{3} - 64 y^{3}$ .

A

125

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B

665

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C

225

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D

985

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Solution

The correct option is B 665
$3 x - 4 y = 5$
Cubing both sides,
$(3 x - 4 y)^{3} = 5^{3}$

Applying the formula, ${(a - b)}^{3} = a^{3} - b^{3} - 3 a b (a - b)$
$(3 x)^{3} - (4 y)^{3} - 3 \times 3 x \times 4 y \times (3 x - 4 y)) = 125$
$(3 x)^{3} - (4 y)^{3} - 36 \times x y \times (3 x - 4 y) = 125$
Substituting, $3 x - 4 y = 5; x y = 3$

$27 x^{3} - 64 y^{3} - 36 \times 3 \times 5 = 125$
$27 x^{3} - 64 y^{3} = 125 + 540 = 665$

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