CAT 2017 Quantitative Ability

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Question : 2 of 34

Marks: +1, -0

If x.y.z are real numbers such that

x + y + z = 4

and

x^2 + y^2 + z^2 = 6

, then x, y, z lie in

$[\;\;{3}/{2},2]$
$[\;\;{2}/{3},2]$
$[0,\;\;{2}/{3}]$
None of these

Solution:

x+y+z=4\;\text"and"\;x^{2}+y^{2}+z^{2}=6

∴y + z = 4 - x

yz=\;\;{1}/{2}\{(y+z)^{2}-(y^{2}+z^{2})\}

=\;\;{1}/{2}\{(4-x)^{2}-(6-x^{2})\}

⇒ \;\;yz=x^{2}-4x+5

Hence, y and z are the roots of

t^{2}-(4-x)t+(x^{2}-4x+5)≥0

Since the roots y and z are real

(4-x)^{2}-4(x^{2}-4x+5)≥0

⇒ \; 3x^2 - 8x- 4 ≤ 0

⇒(3x-2)(x-2)≤ 0

⇒x∈[\;\;{2}/{3},2]

\;\text"By symmetry"\;y,z∈\left\[\;\;{2}/{3},2]

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