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Question

# If both roots of the equation x2+ax+2=0 lie in the interval (0,3), then the range of values of a is

A
(,22][22,)
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B
(113,22]
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C
(113,)
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D
(6,0)
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Solution

## The correct option is B (−113,−2√2]Given the quadratic equation: x2+ax+2=0 Let α,β be the roots of the equation. Now 0<α,β<3 which can be represented as: Let f(x)=x2+ax+2 Now, for this condition to happen, we have 3 conditions that needs to be followed, that are: A.D≥0⇒a2−8≥0⇒(a−2√2)(a+2√2)≥0⇒a∈(−∞,−2√2]∪[2√2,∞)⋯(1) B.0<Sum of roots2<3⇒0<−a2<3⇒a∈(−6,0)⋯(2) C.f(0)>0 & f(3)>0 ⇒2>0 & 9+3a+2>0⇒3a+11>0⇒a∈(−113,∞)⋯(3) Thus, from (1), (2) & (3), we get: a∈{(−∞,−2√2]∪[2√2,∞)}∩(−6,0)∩(−113,∞) a∈(−113,−2√2]

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