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MHT CET202525 Apr 2025Morning ShiftMathematicsComplex NumberActual

Let z be the complex number with Im ( z )=10 and satisfing 2 z - n 2 z + n =2 i-1 , where i= -1 , for some natural number ' n ' then

Options

  1. An =20 and Re ( z )=10
  2. Bn =20 and Re ( z )=-10
  3. Cn =40 and Re ( z )=10
  4. Dn =40 and Re ( z )=-10

Correct answer

D. n =40 and Re ( z )=-10

Step-by-step solution

The complex number is given by z = x + 10i , since Im (z) = 10 . Substituting z into the equation: 2(x + 10i) - n 2(x + 10i) + n = 2i - 1 Simplify the numerator and denominator: (2x - n) + 20i (2x + n) + 20i = -1 + 2i Multiply both sides by the denominator: (2x - n) + 20i = (-1 + 2i)((2x + n) + 20i) Expand the right-hand side: (2x - n) + 20i = -1(2x + n) - 20i + (4x + 2n)i + 40i^2 Since i^2 = -1 : (2x - n) + 20i = -2x - n - 20i + (4x + 2n)i - 40 Group real and imaginary parts: Real: 2x - n = -2x - n - 40 Imaginary:

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