Question

The solution set for `[x]{x}=1` (where {x} and [x] are respectively, fractional part function and greatest integer function) is `R^+-(0,1)` (b) `r^+-{1}` `{m+1/m m in I-{0}}` `{m+1/m m in I-{1}}`
A. `R^(+)-(0,1)`
B. `R^(+)-{1}`
C. `{m+(1)/(m)//m in I-{0}}`
D. `{m+(1)/(m)//m in N-{1}}`

Answer 1

Correct Answer - D
`[x]{x}=1`
Obviousuly, x cannot be negative.
Further, `[x] ne 0`.
For `[x]=1, {x}=1, ` which is not possible.
Hence, ` x notin [0,2).`
For ` x in [2,3),[x] =2 " or " {x} =1//2 " or " x=5//2=2+1//2.`
For ` x in [3,4),[x] =3 " or " {x} =1//3 " or " x=3+1//3.`
For ` x in [4,5),[x] =4 " or " {x} =1//4 " or " x=4+1//4.`
Thus, solution is `x=m+(1)/(m),` where `m in N-{1}.`