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Question

If z4z = 2, then the maximum value of |z| is equal to

A

-1

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B

+1

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C

2

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D

5

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Solution

The correct option is B

+1


We are asked to find the maximum value of |z|. We will apply the triangle inequalities and try to find relations involving |z|.
z4z|z|+4z2|z|+4|z|Let |z|=r2r+4rr22r+40(r1)2+30
which is always true. We can't conclude anything about the maximum value from this relation. So we will consider
|z|=z4z+4zz4z+4z2+4|z|Let |z|=rr2+4rr22r40If α,β are the roots of r22r4=0, then rϵ[α,β].Also r0The roots are24+162=15r>0,rϵ[15,1+5]. Combining both the conditions, we get
rϵ[0,1+5]Maximum value is 1+5.


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