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Question

If at least one of the roots of the equation x2(a3)x+a=0 lies in the interval (1,2), then a lies in the interval

A
[9,)
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B
(10,)
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C
[9,10)
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D
(5,7)(10,)
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Solution

The correct option is B (10,)
Let f(x)=x2(a3)x+a

Case 1: Both the roots lie in the interval (1,2)
(i) D0(a3)24a0
a210a+90(a1)(a9)0a(,1][9,) (1)

(ii) 1<b2a<21<a32<25<a<7 (2)

(iii) f(1)>0 f(2)>04>0 10a>0a<10 (3)
(1)(2)(3)aϕ (4)

Case 2: Exactly one root lies in the interval (1,2)
f(1)f(2)<0
4(10a)<0
a(10,) (5)

Checking for boundary condition, when a=10, then
f(x)=x27x+10=(x2)(x5)f(x)=0x=2,5(1,2)

Hence, from equation (4) and (5), we get
x(10,)

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