CAT 2023 Slot 1 Solved Paper

Question : 64 of 66

Marks: +1, -0

For some positive and distinct real numbers

x, y

and

z

, if

\frac{1}{\sqrt{y}+\sqrt{z}}

is the arithmetic mean of

\frac{1}{\sqrt{x}+\sqrt{z}}

and

\frac{1}{\sqrt{x}+\sqrt{y}}

, then the relationship which will always hold true, is

Solution:

Given that

\;{1}/{{√{y}}+{√{z}}}

is the arithmetic mean of

\;{1}/{{√{x}}+{√{z}}}

and

\;{1}/{{√{x}}+{√{y}}}

⇒\;{2}/{{√{y}}+{√{z}}}=\;{1}/{{√{x}}+{√{z}}}+\;{1}/{{√{x}}+{√{y}}}

⇒ 2({√{x}}+{√{z}})({√{x}}+{√{y}})=({√{y}}+{√{z}})({√{x}}+{√{y}}+{√{x}}+{√{z}})

⇒ 2(x+{√{x y}}+{√{x z}}+{√{y z}})=2 {√{x y}}+y+{√{y z}}+2 {√{x z}}+{√{y z}}+z

⇒ 2 x=y+z

\Rightarrow y, x, z

are in A.P. as

x

is the arithmetic mean of

y

and

z

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