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AP EAMCET201824 Apr 2018Morning ShiftMathematicsComplex NumberActual

The maximum value of the modulus of e^ z^2 on the set z C / 0 Re (z) 1,0 Im (z) 1 is

Options

  1. A2 e
  2. Be
  3. Ce+1
  4. De^2

Correct answer

B. e

Step-by-step solution

aligned & Let z=x+i y 0 x 1,0 y 1 & e^ z^2 =e^ x^2-y^2+2 i x y & e^ z^2 =e^ x^2-y^2 e^ i(2 x y) & |e^ z^2 |=e^ x^2-y^2 & [ |e^ i(2 x y) |=1 ] & 0 x^2 1,0 y^2 1 & 0 x^2 1 & -1 -y^2 0 & -1 x^2-y^2 1 & maximum of |e^ z^2 |=e & aligned

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