Complex Numbers

Answer :

        z          = x – iy

                      z^(1/3) =  p + iq

                           z          = (p + iq)³

                             = p³ + (iq)³ + 3p²iq + 3pq²i²

                             = p³ – iq³ – 3pq² + 3p²iq

                   hence

                                      x = p³ – 3pq²

                                      y = q³ – 3p²q

                             ∴ (x/p) = p² – 3q² and (y/q) = q² – 3p²

                             ∴ (x/p) + (y/q) = – 2p² – 2q²

                             ∴ [{(x/p) / (y/a)} / (p² + q²)] = – 2.

Answer :

   z + iw = 0

                             ∴ z = – iw

                             ∴ z = iw

                             ∴ w = – iz

                   given : Arg(zw) = π

                             ∴ Arg(– iz²) = π

                             ∴ Arg( – i) + 2 Arg(z) = π

                             ∴ [(– π) / 2] + 2 Arg(z) = π

                             ∴ 2 Arg(z)  = (3π / 2)

                             ∴ Arg(z) = (3π / 4)

Answer :

       w         = [z / {z – (1/3)i}]

                       |w| = |(x + iy) / {x + iy – (i/3)}| = 1

                   ∴       |(x + iy) / {(x + i{y – (i/3)}}| = 1

                   ∴       x² + y² = x² + {y – (i/3)}²

                   ∴       y² = y² + (1/9) – (2/3)y

                   ∴       (1/9) = (2/3)y

                   ∴       (1/3) = 2y

                   ∴       6y = 1

                   ∴       z lies on a line

Answer :

z² + z + 1 = 0

                   ∴       (z – 1) (z² + z + 1) = 0

                   ∴       z³ – 1 = 0

                   ∴       z³ = 1 ⇒ (1/z) = z²

                   also z² + z = – 1, z⁴ = z

                   ∴       [z + (1/z)]² + [z² + (1 / z²)]² + [z³ + (1 / z³)]² + [z⁴ + (1 / z⁴)]² + [z⁵ + (1 / z⁵)]² + [z⁶ + (1 / z⁶)]²

                             = (z + z²)² + (z² + z)² + (1 + 1)² + (z + z²)² + (z² + z)² + (1 + 1)²

                             = (– 1)² + (– 1)² + 4 + (– 1)² + (– 1)² + 4 ------- z² + z + 1 = 0

                             = 1 + 1 + 4 + 1 + 1 + 4

                             = 12

Answer :

|z – (4/z)| = 2

                             |z| = |z – (4/z) + (4/z)| ≤ |z – (4/z)| + |(4/z)|

                   ∴       |z| ≤ 2 + |(4/z)|

                   ∴       |z| ≤ 2 + (4 / |z|)

                   ∴       |z|² ≤ 2|z| + 4

                   ∴       |z|² – 2|z| + 1 ≤ |1 + 4|

                   ∴       (|z| – 1)² ≤ 5

                   ∴       (|z| – 1) ≤ √5

                   ∴       – √5 ≤ |z| – 1 ≤ √5

                   ∴       – √5 + 1 ≤ |z| ≤ √5 + 1

                   ∴       maximum value of z is √5 + 1