Question : If $x^{3}-y^{3}=81$ and $x-y=3$, what is the value of $x^{2}+y^{2}$?
Option 1: 18
Option 2: 21
Option 3: 27
Option 4: 36
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Correct Answer: 21
Solution :
Given: $x^{3}-y^{3}=81$ and $x-y=3$
We know, $x^{3}-y^{3}=(x-y)^{3}+3xy(x-y)$
Substituting the values of $x^{3}-y^{3}$ and $x-y$, we get,
$81=3^{3}+3×3xy$
$⇒xy=6$
Now, $x^{2}+y^{2}=(x-y)^{2}+2xy=3^{2}+2\times 6=21$
Hence, the correct answer is 21.
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