KVPY2020MathematicsFunctions
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
Options
- AA is a discrete set of at least two points
- BA contains an interval, but is not an interval
- CA is an interval, but a proper subset of ( - ∞ , ∞ )
- DA = ( - ∞ , ∞ )
Correct answer
B. A contains an interval, but is not an interval
Step-by-step 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  or  [ x ] = 0  or  x = [ x ] ⇒ x ∈ Z ∪ [ 0 , 1 )