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Question

# If the roots of x2 - (a - 3)x + a = 0 are such that at least one of the root is greater than 2, then find the range of a.

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, ∞) x2 - (a - 3)x + a = 0 D= (a−3)2 - 4a = a2 - 10a + 9 = (a - 1)(a - 9) Case 1: Both roots are greater than 2 (i) D≥0⇒ (a−1)(a−9)≥0 ⇒a∈(−∞, 1]∪[9, ∞) (ii) f(2)>0⇒4−(a−3)2+a>0 ⇒ a<10 (iii) −b2a>2⇒a−32>2⇒a>7 ∴ a∈ [9, 10) Case 2 :- One root is greater than 2 and other is less than or equal to 2 (i) D≥0 ⇒(a−1)(a−9)≥0 a∈(−∞,1]∪[9, ∞) (ii) f(2)≤0 ⇒4−(a−3)2+a≤0 →a≥10 ∴ a∈[10, ∞) So from both the cases we get, a∈[9, ∞)

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