Location of Roots when Compared with a Constant 'k'
If the roots ...
Question
If the roots of x2−6kx+(2−2k+9k2)=0 are greater than 3, then the range of k is
A
(−∞,1)∪(119,∞)
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B
(1,119)
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C
(1,∞)
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D
(119,∞)
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Solution
The correct option is D(119,∞) Let f(x)=x2−6kx+(2−2k+9k2), whose roots are α,β
Given: α,β>3
Conditions: (i)D≥0⇒36k2−4(2−2k+9k2)≥0 ⇒36k2−8+8k−36k2≥0 ⇒8k≥8 ⇒k≥1…(1)
(ii)−b2a>3Here, a=1,b=−6k⇒6k2>3⇒k>1…(2)
(iii)f(3)>032−18k+2−2k+9k2>0⇒9k2−20k+11>0 ⇒(k−1)(9k−11)>0 ⇒k<1 or k>119…(3)
Taking common region from all the three regions (1),(2)&(3), we get k∈(119,∞)