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Question

If the equation 22x+a2x+1+a+1=0 has roots of opposite sign, then a lies in the interval

A
(,23)
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B
(1,23)
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C
(,1)
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D
(1,0)
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Solution

The correct option is B (1,23)
22x+a2x+1+a+1=0 (1)
Let the roots be α,β
Assuming 2x=t,
t2+2at+(a+1)=0(2)
Let the roots of equation be t1,t2
As equation (1) have roots of opposite signs, so
<α<0<β<0<2α<1<2β<
0<t1<1<t2<
f(0)>0a+1>0a>1(3)
1 lies in between the roots for equation (2),
f(1)<0
(1+2a+a+1)<0
a<23(4)

From equation (1) and (2),
a(1,23)




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