KVPY SA Exam 2015 Question Paper

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Question : 3 of 80

Marks: +1, -0

Let x ,y, z be positive reals. Which of the following implies x = y = z ?
(I)

x ^3 + y^3 + z^3 = 3 xyz

(II)

x ^3 + y ^2z + yz^2 = 3 xyz

(III)

x ^3 + y ^2z + z^2x = 3 xyz

(IV)

(x + y + z)^3 =27 xyz

I, IV only
I, II, IV only
I, II and III only
All of them

Solution:

(I)

{x ^{3 }+y^{3 }+ z^{3 }} /3 = (x^3 y^ 3 z^ 3 ) ^{1\/3}

Hence x = y = z {AM = GM}
(II)

{x ^{3 }+y^{2}z+ yz^{2 }} /3 ≥ (x^3 y^ 3 z^ 3 ) ^{1\/3}

(AM = GM)
(III)

{x ^{3 }+{y^{2}z}+ z^{2 }x} = 3xyz

{{x ^{3 }+{y^{2}z}/2+ {y^{2}z}/2+z^2x}/4≥ ({x^4 y^ 4z^ 4}/4 ) ^{1\/4}

⇒

{3xyz}/4≥(xyz)/√2

Not possible
(IV)

{x+y+z}/3≥(xyz)^{1\/3}

⇒

(x + y + z)^3 = 27 xyz.

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