KVPY SX SB Exam 2018 Question Paper

Question : 82 of 120

Marks: +1, -0

The number of different possible values for the sum x + y + z, where x, y, z are real numbers such that

x^4 + 4y^4 + 16z^4 + 64 = 32xyz

Solution:

S = x + y + z = ?

x ^4 + 4y^4 + 16z^4 + 64 = 32 xyz

……….(1)

{x ^4 + 4y^4 + 16z^4 + 64}/4

≥ (x^4 .4y^4 . 16z^4 .64)^{1\/4 }

{32xyz}/4≥8xyz

8xyz ≥ 8\ xyz

∵ AM = GM
∴

x^4 = 4y^4 = 16z^4 = 64

x = ± 2 √2

y = ± 2

z = ±√ 2

but 32xyz is positive. Hence, no. possible sums are 4.

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