If $a + b = 5$ and $a^{2} + b^{2} = 11$ then $a^{3} + b^{3} = 20$

True

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False

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Solution

The correct option is A True
$a + b = 5 \Rightarrow {(a + b)}^{2} = 25$
$\Rightarrow a^{2} + b^{2} + 2 a b = 25$
$\Rightarrow 11 + 2 a b = 25$ since $a^{2} + b^{2} = 11$ (given)
$\Rightarrow 2 a b = 25 - 11 = 14$
$\Rightarrow a b = 7$
$a^{3} + b^{3} = {(a + b)}^{3} - 3 a b (a + b)$
$\Rightarrow a^{3} + b^{3} = {(5)}^{3} - 3 \times 7 (5)$
$\Rightarrow a^{3} + b^{3} = 125 - 105 = 20$
Hence the given ststement is true.

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