If $a - b = 3$ and $a^{3} - b^{3} = 117$ , then $a + b$ is equal to:

5

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7

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9

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11

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Solution

The correct option is C $7$
Given, $a - b = 3$
$\Rightarrow a - 3 = b$ ...(i)
and $a^{3} - b^{3} = 117$
$\Rightarrow (a - b) (a^{2} + a b + b^{2}) = 117$ ...(ii)
Divide (ii) by (i), we get
$∴ a^{2} + a b + b^{2} = \frac{117}{3} = 39$ ...(iii)
Put the value of $b$ in eq. (iii),
$\Rightarrow a^{2} + a (a - 3) + (a - 3)^{2} = 39$
$\Rightarrow 3 a^{2} - 9 a + 9 = 39$
$\Rightarrow a^{2} - 3 a - 10 = 0$
$\Rightarrow (a + 2) (a - 5) = 0$
$\Rightarrow a = - 2 o r 5$
and $b = - 5$ or $2$
$\Rightarrow a + b = 5 + 2 = 7$

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