Question : If $a^3=117+b^3$ and $a=3+b$, then the value of $(a+b)$ is:
Option 1: $\pm 7$
Option 2: $\pm 49$
Option 3: $\pm 13$
Option 4: $0$
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Correct Answer: $\pm 7$
Solution :
Given: $a^{3} - b^{3} = 117$ and $a - b = 3$
We know, $(a^{3} - b^{3}) = (a - b)(a^{2} + b^{2} + ab)$
$⇒(a^{3} - b^{3}) = (a - b)[(a - b)^{2} + 3ab]$
$⇒117 = 3[3^{2} + 3ab]$
$⇒39 = [3^{2} + 3ab]$
$⇒ab = 10$
Now, $(a+b)^{2} = (a-b)^{2}+4ab$
$⇒(a+b)^{2}=(3)^{2}+4×10$
$⇒(a+b)^{2}=9+40=49$
$\therefore a + b = \pm 7$
Hence, the correct answer is $\pm 7$.
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