Question

If z = `π/4(1 + i)^4((1 - sqrt(π)i)/(sqrt(π) + i) + (sqrt(π) - i)/(1 + sqrt(π)i))`, then `(|z|/("amp"^((z))))` is equals to ______. 

Options

• 1

• π

• 3π

• 4

Answer 1

If z = `π/4(1 + i)^4((1 - sqrt(π)i)/(sqrt(π) + i) + (sqrt(π) - i)/(1 + sqrt(π)i))`, then `(|z|/("amp"^((z))))` is equals to 4. 

Explanation:

z = `π/4(1 + i)^4((1 - sqrt(π)i)/(sqrt(π) + i) + (sqrt(π) - i)/(1 + sqrt(π)i))`

= `π/4(1 + i)^4[(1 + π + π + 1)/((sqrt(π) + i)(1 + sqrt(π)i))]`

= `π/4(1 + i)^4 2/i`

= `π/4(2i)^2 2/i`

= 2πi

∴ `(|z|/("amp"^((z)))) = (2π)/(π/4)` = 4