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1.Relation and Function
normal
Let $A$ denote the set of all real numbers $x$ such that $x^3-[x]^3=\left(x-[x]^3\right)$, where $[x]$ is the greatest integer less than or equal to $x$. Then,
A
$A$ is a discrete set of a least two points
B
$A$ contains an interval, but is not an interval
C
$A$ is an interval, but a proper subset of $(-\infty, \infty)$
D
$A=(-\infty, \infty)$
(KVPY-2020)
Solution
(b)
Given, $x^3-[x]^3=(x-[x])^3$
$(x-\mid x])\left(x^2+[x]^2+x[x]\right)=(x-[x])^3$
$x-[x]=0$ or $x[x]=0$
$\{x\}=0$ or $x[x]=0$
$x \in Z$ or $x \in[0,1)$
Hence, solution in $x \in(0, 1 ) \cup Z$.
Standard 12
Mathematics
