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MHT CET202521 Apr 2025Morning ShiftMathematicsComplex NumberActual

The complex numbers x+i 2 x and x-i 2 x,(i= -1 ) are conjugate to each other for,

Options

  1. Ax= n , n Z
  2. Bx= ( n + 1 2 ) , n Z
  3. Cx =(3 n -1) , n Z
  4. DNo value of x

Correct answer

D. No value of x

Step-by-step solution

The complex numbers z₁ = x + i 2x and z₂ = x - i 2x being conjugate implies z₁ = z₂ . The conjugate is z₂ = x + i 2x , so we equate real and imaginary parts separately. For the real parts: x = x x = 1 , giving x = n + 4 for integer n . For the imaginary parts: 2x = 2x 2x = 1 , giving x = m 2 + 8 for integer m . Equating these expressions: n + 4 = m 2 + 8 . Multiply by 8: 8n + 2 = 4m + . Simplifying: 4(2n - m) = -1 , which requires 2n - m = - 1 4 . Since 2n - m must be an integer, this equation has no solution. The

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