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Question

# If both the roots of the quadratic equation x2−mx+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]x2−mx+4=0 Roots are real and distinct ∴ D>0 ⇒m2−16>0 ⇒(m−4)(m+4)>0 ⇒m∈(−∞,−4)∪(4,∞) ⋯(1) Roots lie in the interval [1,5] ⇒1≤α<β≤5 The possible cases are, So, the required conditions are, f(1)≥0 ⇒ 12−m+4≥0 ⇒ m≤5 ⇒ m∈(−∞,5] ⋯(2) f(5)≥0 ⇒ 25−5m+4≥0 ⇒ m≤295 ⇒ 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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