If z is a complex number satisfying|z^3 + z^{–3}| \leq 2 , then the maximum possible value of |z + z^{–1}| is

KVPY SX SB Exam 2015 Question Paper

Question : 2 of 120

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If z is a complex number satisfying

|z^3 + z^{–3}| ≤ 2

, then the maximum possible value of

|z + z^{–1}|

Solution:

|z^3 + z^{–3}| ≤ 2

|z^3 +1/ z^{3}| ≤ 2

|(z+1/z)(z^2+1/z^2-1)|≤2

| z+1/z ||((z +1/z) ^2)-3|≤2

| z+1/z |{ \{} |{{| z +1/z|} ^2 -3|\} ≤2

\{∵ |z_1 - z_2| ≥ ||z_1| – |z_2||\}

|z+1/z|_\max=2

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