Location of Roots when Compared with a constant 'k'
The set of va...
Question
The set of values of k for which roots of the quadratic equation −x2−2(k−1)x−(k+5)=0 are less than \(1)\ is
A
R−1
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B
[4,∞)
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C
[2,3+√272]
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D
R
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Solution
The correct option is B[4,∞) Let f(x)=−x2−2(k−1)x−(k+5)
On comparing with standard quadratic equation y=ax2+bx+c we get a=−1,b=−2(k−1),c=−(k+5).
We can see that a<0, so the given expression will be a downward opening parabola as shown below ∴ Required conditions are (i)D≥0 D=b2−4ac≥0 D=(−2(k−1))2−4.(−1).(−(k+5)≥0 ⇒4k2−12k−16≥0 ⇒k2−3k−4≥0 ⇒(k−4)(k+1)≥0 ⇒k∈(−∞,−1]∪[4,∞)