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Question

If atleast one of the root of the equation x2(a3)x+a=0 is greater than 2, then a lies in the interval

A
[7,9]
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B
[7,)
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C
[9,)
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D
[7,9)
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Solution

The correct option is C [9,)
Let f(x)=x2(a3)x+a

Case 1: Both roots are greater than 2,


Required conditions are
(i)D0
D=(a3)24a0
a210a+90(a1)(a9)0a(,1][9,) (1)

f(2)>04(a3)2+a>0a<10 (2)
Now, x coordinate of the vertex will be greater than 2.
a32>2a>7 (3)
From equation (1),(2) and (3),
a[9,10) (4)

Case 2: One root is greater than 2 and other root is less than 2

2 lies in between the roots
f(2)<0
4(a3)2+a<0
a>10 (5)

Case 3: One root is greater than 2 and other root (lowest root) =2

Lowest root is 2.
Required conditions are (i) f(2)=0 and (ii) b2a>0
f(2)=0a=10 ...(6)
b2a>0a>7 ...(7)
From (6) and (7)
a=10 ...(8)

From (4),(5) and (8)
a[9,)

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