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Question

If the equation x39x2+24x+k=0 has exactly one root in (2,4), then k lies in the interval

A
(20,16)
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B
(16, )
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C
(,20)
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D
None of these
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Solution

The correct option is A (20,16)
Let f(x)=x39x2+24x+k=0
If f has exactly one root in (a,b), then f(a)f(b)<0
For (a,b)(2,4), we can say
f(2)f(4)<0
(89(4)+24(2)+k)(649(16)+24(4)+k)<0
(k+20)(k+16)<0
k(20,16)
f(x)=3(x2)(x4)
Hence in the interval (2,4),f(x) is negative.
Hence, it is monotonic decreasing in that interval and can have only one real root.

Hence, option A.

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