Question

If f(z) = `(7 - z)/(1 - z^2)`, where z = 1 + 2i, then |f(z)| is ______.

Options

• `|z|/2`

• |z|

• 2|z|

• None of these

Answer 1

If f(z) = `(7 - z)/(1 - z^2)`, where z = 1 + 2i, then |f(z)| is `underlinebb(|z|/2)`.

Explanation:

Given that: z = 1 + 2i

|z| = `sqrt((1)^2 + (2)^2) = sqrt(5)`

Now f(z) = `(7 - z)/(1 - z^2)`

= `(7 - (1 + 2i))/(1 - (1 + 2i)^2`

= `(7 - 1 - 2i)/(1 - 1 - 4i^2 - 4i)`

= `(6 - 2i)/(4 - 4i)`

= `(3 - i)/(2 - 2i)`

= `(3 - i)/(2 - 2i) xx (2 + 2i)/(2 + 2i)`

= `(6 + 6i - 2i - 2i^2)/(4 - 4i^2)`

= `(6 + 4i + 2)/(4 + 4)`

= `(8 + 4i)/8`

= `1 + 1/2 i`

So |f(z)| = `sqrt((1)^2 + (1/2)^2)`

= `sqrt(1 + 1/4)`

= `sqrt(5)/2`

= `|z|/2`