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Question

If atleast one of the root of the equation x2(a3)x+a=0 is 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) f(1)>0 and f(2)>0
a<10 (1)
(ii) 1<b2a<2
1<a32<2
5<a<7 (2)
(iii) D0 D=(a3)24a0
a210a+90(a1)(a9)0a(,1][9,) (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)

(4)(5)a(10,)

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