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Question

If xR, (log{x})23log[x]+2=0 and 12x3|x1|=1, where {.} is the fractional part function and [.] is the greatest integer function, then the number of non-integral value(s) of x is

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Solution

There are two simultaneous equations. So, we will try to solve the easier one first.
12x3|x1|=1
12x=3|x1|
|x1|=2(1+x)

If x1, we get
x1=2(1+x)x=3
Not possible

If x<1, we get
1x=2(1+x)x=13

Now, putting x=13, we get
(log{x})23log[x]+2=0(log{13})23log[13]+2=0
Which is not defined as [13]=0

Hence, no solution exists.

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