Q8 of 202 Page 188

If is purely imaginary and z = –1, show that |z| = 1.

Let z= a + ib

Now,

Given that is purely imaginary ⇒ real part = 0

⇒ a² + b² - 1 = 0

⇒ a² + b² = 1

⇒ |z| = 1

Hence proved.

Find real values of x and y for which

(x⁴ + 2xi) – (3x² + iy) = (3 – 5i) + (1 + 2iy).

If z² + |z|² = 0, show that z is purely imaginary.

If z₁ is a complex number other than –1 such that |z₁| = 1 and z₂ = then show that z2 is purely imaginary.

For all z C, prove that

(i)

(ii) .

(iii) = |z|²

(iv) is real

(v) is 0 or imaginary.