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Question

If z be a complex number for which |2zcosθ+z2|=1, then the minimum value of |z|
is ......................

A
31
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B
3+1
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C
21
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D
2+1
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Solution

The correct option is C 21
|z2+2zcosθ|
=|z(z+2cosθ)|
=|z|.|z+2cosθ|
=1
Now
|z|=1 and
|z+2cosθ|=1
Now
|z+2cosθ||z|+|2cosθ|
Considering
|z+2cosθ|=|z|+|2cosθ|=1
Hence
|z|=|2cosθ|±1
Considering z=|2cosθ|1 we get the minimum value at multiples of θ=450
Hence
z=21.

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