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Question

If the values of m for which exactly one root of the equation x22mx+m21=0 lies in the interval (2,4) belong to set A, then the number of integral points common to set A and R are

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Solution

let f(x)= x22mx+m21 as exactly one root of f(x)=0 lies in the interval, we can take D>0 and f(-2) f(4)<0
Consider D>0: (2m)24.1(m21)>0
4>0mϵR ....................(1)
consider f(-2) f(4) <0:
(4+4m+m21)(168m+m21)<0
(m2+4m+3)(m28m+15)<0
(m+1)(m+3)(m3)(m5)<0
(m+3)(m+1)(m3)(m5)<
mϵ(3,1)(3,5) ..........(2)
Hence, the values of m satisfying (1) and (2) at the same time are mϵ(3,1)(3,5).
271661_136657_ans.png

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