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AP EAMCET202317 May 2023Morning ShiftMathematicsComplex NumberActual

If z=(1-i)^3(x+i) is a purely imaginary number for x=x₁ and if z is a purely real number for x = x ₂ , then x ₁ x ₂=

Options

  1. A-1
  2. B0
  3. C1
  4. D2

Correct answer

A. -1

Step-by-step solution

aligned z & =(1-i)^3(x+i) z & =(1+i-3 i(1-i))(x+i) & =(-2 i-2)(x+i) & z=(2-2 x)-i(2+2 x) aligned If z is purely imaginary: 2-2 x=0 x=1 x₁=1 If z is purely real: aligned & 2+2 x=0 x=-1 & x₂=-1 aligned Now, x₁ x₂=1 (-1)=-1

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