1.

The number of solutions of `|[x]-2x|=4, "where" [x]]` is the greatest integer less than or equal to x, isA. 2B. 4C. 1D. infinite

Answer» Correct Answer - B
When `x in Z`
CASE I
In this case, we have [x]=x
`:. |[x]-2x|=4 implies |[x]|=4 implies x=+-4`
CASE II When `x in Z`
In this case, we have
`x=n+lambda"where" n in Z "and "0 lt lambda lt 1`
`implies[x]=n`
`:. |[x]-2x|=4`
`implies|n-2(n+lambda)|=4`
`implies n+2lambda=+-4 implies n=+-4-2lambda " "`...........(i)
This is possible, when `lambda=(1)/(2)` Putting `lambda=(1)/(2)` in (i), we get
`n=+-4-1`
`implies n=3,-5 impliesx=3+(1)/(2),-5+(1)/(2) implies x=(7)/(2),-(9)/(2)`
Hence, `x=+-4,(7)/(2),-(9)/(2)`


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