Question : If $a+b=3$ and $a b=2$, then what is the value of $2 a^3+2 b^3$?
Option 1: 25
Option 2: 27
Option 3: 18
Option 4: 20
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Correct Answer: 18
Solution :
Given, $a+b=3$ and $a b=2$
Consider, $a+b=3$
Cubing both sides,
$(a+b)^3 = 3^3$
⇒ $a^3 + b^3 + 3ab(a+b) = 27$
⇒ $a^3 + b^3 + 3\times 2\times 3 = 27$ [Putting the given values]
⇒ $a^3 + b^3 + 18 = 27$
⇒ $a^3 + b^3 = 9$
⇒ The value of $2 a^3+2 b^3 = 2(a^3+b^3)$ = 2 × 9 = 18
Hence, the correct answer is 18.
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