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Question

If |u|<1,|v|<1, and z=uv1+¯uv, then least value of |z| is


A

|u||v|1+|u||v|

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B

|u|+|v|1+|u||v|

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C

||u||v||1|u||v|

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D

None of these

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Solution

The correct option is A

|u||v|1+|u||v|


Let u=r1eiϕ,v=r2eiθ
where r1<1,r2<11+¯uv=1+r1r2ei(θϕ)
|1+¯uv|2=(1+r1r2cos(θϕ))2+r21r22sin2(θϕ)=1+2r1r2cos(θϕ)+r21r22(1+r1r2)2=(1+|u||v|)2|1+¯uv|1+|u||v|11+|u||v|1|1+¯uv|
Also, ||u||v||<|uv|
Thus, ||u||v||1+|u||v|uv1+¯uv


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