If 16 x2 + y2 = 48 and xy = 2, x, y > 0, then the value of (64x3 + y3) is :
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Correct answer: A
Explanation
The correct answer is option 1 i.e. 320
(a + b)3 = a3 + b3 + 3(ab)(a + b)
⇒ 16x2 + y2 = 48
⇒ (4x)2 + y2 = 48
⇒ (4x)2 + y2 + 8xy = 48 + 8xy [∵ xy = 2]
⇒ (4x + y)2 = 64
⇒ 4x + y = 8
By applying (a + b)3 = a3 + b3 + 3(ab)(a + b)
⟹ (4x + y)3 = 83
⟹ 64x3 + y3 + 3(4xy)(4x + y) = 512
⟹ 64x3 + y3 + (3 × 8 × 8) = 512
⟹ 64x3 + y3 + 192 = 512
⇒ 64x3 + y3 = 320