1.

Find the conjugate of (i - i2)31. -2 - 2i2. -2 + 2i3. i - 14. 2 + 2i

Answer» Correct Answer - Option 1 : -2 - 2i

Concept:

Let z = x + iy be a complex number.

  • Modulus of z = \(\left| {\rm{z}} \right| = {\rm{}}\sqrt {{{\rm{x}}^2} + {{\rm{y}}^2}} = {\rm{}}\sqrt {{\rm{Re}}{{\left( {\rm{z}} \right)}^2} + {\rm{Im\;}}{{\left( {\rm{z}} \right)}^2}}\)
  • arg (z) = arg (x + iy) = \({\tan ^{ - 1}}\left( {\frac{y}{x}} \right)\)
  • Conjugate of z = x – iy

 

Calculation:

Let z = (i - i2)3

= (i - (-1))3 = (1 + i) 3

Using (a + b) 3 = a3 + b3 + 3a2b + 3ab2

⇒ z = 13 + i3 + 3 × 12 × i + 3 × 1 × i2

= 1 – i + 3i – 3

= -2 + 2i

So, conjugate of (1 + i) 3 is -2 - 2i


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