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Question

If both the roots of the quadratic equation x2mx+4=0 are real and distinct and they lie in the interval [1,5], then m lies in the interval :

A
(4,5)
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B
(3,4)
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C
(4,5]
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D
(5,6)
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Solution

The correct option is C (4,5]
x2mx+4=0
Roots are real and distinct
D>0
m216>0
(m4)(m+4)>0
m(,4)(4,) (1)
Roots lie in the interval [1,5] 1α<β5
The possible cases are,


So, the required conditions are,
f(1)0
12m+40
m5
m(,5] (2)

f(5)0
255m+40
m295
m(,295] (3)

1<b2a<5
1<m2<5
2<m<10
m (2,10) (4)

Thus, from equations (1),(2),(3) and (4), we get
m(4,5]

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