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Question

If e|sinx|+e|sinx|+4a=0 will have exactly four different solutions in [0,2π], then

A
aR
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B
a[e4,14]
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C
a[1e24e,]
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D
no real value of a exists
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Solution

The correct option is D no real value of a exists
Given : e|sinx|+e|sinx|+4a=0
Let t=e|sinx|t[1,e]
Now,
t+1t+4a=0t2+4at+1=0

For the given equation to have four different solutions, this quadratic expression should have two distinct roots in [1,e].
So,required conditions are
(1)D>016a24>0|a|>12
(2)f(1)=1+4a+10a12
(3)f(e)=e2+4ae+10a1e24e
(4)1<b2a<e1<2a<ee2<a<12
Clearly , there is no value of a satisfying the above inequalities simultaneously.

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