If (4x + 2y)3 + (4x - 2y)3 = 16(Ax3 + Bxy2), then what is the value of 1/2 ( A^2 + B^2 )?

Answer: Option C

(4x + 2y)³ + (4x - 2y)³ = 16(Ax³ + Bxy²)
⇒ (4x)³ + (2y)³ + 3 × 4x × 2y(4x + 2y) + (4x)³ - (2y)³ - 3 × 4x × 2y(4x - 2y) = 16(Ax³ + Bxy²)
⇒ 2(4x)³ + 96x²y + 48xy² - 96x²y + 48xy² = 16(Ax³ + Bxy²)
⇒ 128x³ + 96xy² = 16(Ax³ + Bxy²)
⇒ 16(8x³ + 6xy²) = 16(Ax³ + Bxy²)
On comparing on both side-
A = 8 & B = 6
Then,
$$\begin{array}{rl}& \frac{1}{2}\left(\sqrt{A^2+B^2}\right)\\ & =\frac{1}{2}\left(\sqrt{64+36}\right)\\ & =\frac{1}{2}\times 10\\ & =5\text{ Answer}\end{array}$$