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Question

Let x2(m3)x+m=0, mR be a quadratic equation. The values of m for which both roots lie in between 1 and 2 is given by


A

mϕ

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B

m[1,9]

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C

m(5,7)

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D

m<10

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Solution

The correct option is A

mϕ


For the given equation, a>0, when compared to ax2+bx+c=0.

The graph is upward parabola as shown.

For roots α,β to be in between 1 & 2, f(1)>0,f(2)>0 and 1<b2a<2.

Firstly, for roots to be real

D0

b24ac0

m1 or m9 . . . (1)

f(x)=x2(m3)x+m (given)

Now, f(1)=1(m3)+m

=4>0 [Always]

f(2)=4(m3)2+m

=42m+6+m

=10m

10m>0

m<10 . . . (2)

1<b2a<2

1<m32<2

2<(m3)<4

5<m<7 . . . (3)

From (1),(2) & (3)

No intersection.

Hence, m belongs to null set.


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