KVPY 2020-21 SB SX Exam Previous Papers

Question : 12 of 120

Marks: +1, -0

Let

A

denote the set of all real numbers

x

such that

x^{3}-[x]^{3}=(x-[x])^{3}

, where

[x]

is the greatest integer less than or equal to

{x}

. Then

Solution:

({x}-[{x}])({x}^{2}+[{x}]^{2}+{x}[{x}])

=({x}-[{x}])({x}^{2}+[{x}]^{2}-2 x[{x}])

\⇒({x}-[{x}])(3 {x}[{x}])=0

\⇒{x}=0

[{x}]=0

{x}=[{x}]

\⇒{x} \∈{Z} \∪[0,1)

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