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Question

If the polynomial equation (x2+x+1)2(m3)(x2+x+1)+m=0,mR has two distinct real roots, then m lies in the interval

A
(454,)
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B
(,454)
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C
(92,)
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D
(9,)
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Solution

The correct option is B (,454)
(x2+x+1)2(m3)(x2+x+1)+m=0
Let t=x2+x+1=(x+12)2+34
t[34,)
Given equation becomes t2(m3)t+m=0 (1)
Let its roots be t1 and t2
(i) For every t>34, there exist two distinct real roots for x2+x+1=t
(ii) For every t<34, there exists no real roots for x2+x+1=t

Given equation will have two distinct roots iff for equation (1), roots are of form
t1<34 and t2>34
i.e., 34 lies in between the roots of equation (1).
f(34)<0
916(m3)34+m<0
m<454

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