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Question

If complex number z satisfies the inequality |z12i|1, then which of the following option(s) is (are) CORRECT ?

A
maximum value of |z| is 5+1
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B
minimum value of |z| is 5
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C
maximum value of arg(z) is π2
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D
minimum value of arg(z) is tan1(34)
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Solution

The correct options are
A maximum value of |z| is 5+1
C maximum value of arg(z) is π2
D minimum value of arg(z) is tan1(34)
|z12i|1 represents the points inside and on the circle with centre C(1,2) and radius r=1


|z|min=OCr=12+221=51
and |z|max=OC+r=12+22+1=5+1

Clearly, arg(z) is maximum for M(z).
max{arg(z)}=π2
arg(z) is minimum for N(z).
min{arg(z)}=θ
Now, tanθ=tan(π22α)
=cot2α
=1tan2α
=1tan2α2tanα
=1(12)22×12=34
θ=tan134




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