Exercise miscellaneous

Evaluate: 18 1i i    +       .

If (x + iy)3 = u + iv, then show that 2 24( - )u v x y x y + = .

If α and β are different complex numbers with β 1= , then find β α 1 αβ - - .

Find the number of non-zero integral solutions of the equation 1 2 x x- i = .

If (a + ib) (c + id) (e + if) (g + ih) = A + iB, then show that (a2 + b2) (c2 + d2) (e2 + f 2) (g2 + h2) = A2 + B 2

If / 1 11 / m i - i   + =   , then find the least positive integral value of m.

For any two complex numbers z1 and z2, prove that Re (z1 z 2) = Re z 1 Re z2 - Im z1 Imz2 .

1 4 1 5 / i

−    −   − + +    to the standard form .

If a ibx iy c id −− = − prove that ( ) 2 222 2 2 2 a bx y c d ++ = + .

If z1 = 2 - i, z2 = 1 + i, find 1 2 1 2 - 1 z z z z + + + .

If a + ib = 2/2 ( ) / 2 1 / x i / x + + , prove that a2 + b2 = ( ) 2 2 ( 1) / 2 1 / x / x / + / + .

Let z1 = 2 - i, z2 = -2 + i. Find (i) 1 2

z /       , (ii) 1 1 1Im z z       .

Find the real numbers x and y if (x - iy) (3 + 5i) is the conjugate of -6 - 24i.

1 1 / i i / i i + −− − + .