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Question

If at least one of the roots of the equation x2(k+2)x+7k4=0 is negative, then k lies in the interval

A
(,1][4,)
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B
R{0}
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C
(,0)[4,)
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D
(,0)
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Solution

The correct option is D (,0)
Let f(x)=x2(k+2)x+7k4

Case 1: Both the roots are negative
(i) D0
D=(k+2)27k0
k23k+40(k32)2+740kR

(ii) ca>0k>0

(iii) ba<0k+2<0k<2

kϕ (1)

Case 2: One root is positive and other root is negative
ca<0
k<0 (2)

Checking boundary condition for case :2
For k=0, we get
x22x=0
x=0,2
Here, no root is negative.

k(1)(2)
kϕ(,0)
k(,0)

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