If $x = 2 a - 1$ , $y = (2 a - 2)$ and $z = 3 - 4 a$ , then the value of $x^{3} + y^{3} + z^{3}$ will be

6 (3 - 13 a + 18 a^{2} - 8 a^{3})

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6 (3 + 13 a - 18 a^{2} + 8 a^{3})

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6 (3 + 13 a + 18 a^{2} - 8 a^{3})

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6 (3 - 13 a - 18 a^{2} - 8 a^{3})

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Solution

The correct option is A $6 (3 - 13 a + 18 a^{2} - 8 a^{3})$
As given $x = (2 a - 1)$ $y = (2 a - 2)$ $z = (3 - 4 a)$
then the value of $x^{3} + y^{3} + z^{3}$
$\Rightarrow {(2 a - 1)}^{3} + {(2 a - 2)}^{3} + {(3 - 4 a)}^{3}$
$∵ {(x - y)}^{3} = x^{3} - y^{3} - 3 x^{2} y + 3 x y^{2}$
$\Rightarrow {(2 a - 1)}^{3} + {(2 a - 2)}^{3} + {(3 - 4 a)}^{3}$
$\Rightarrow (8 a^{3} - 1 - 12 a^{2} + 6 a) + (8 a^{3} - 8 - 24 a^{2} + 24 a) + (27 - 64 a^{3} - 108 a + 144 a^{2})$
$\Rightarrow 48 a^{3} + 18 + 108 a^{2} - 78 a \Rightarrow 18 - 78 a + 108 a^{2} + 48 a^{3}$
$\Rightarrow 6 (3 - 13 a + 18 a^{2} + 8 a^{3})$

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