UPSC NDA Math Model Paper 6

Let Z=ii
As we know that, any complex number z = x + iy with modulus r and argument θ can also be expressed as z=r⋅ei⋅θ where eiθ=cos‌θ+i‌sin‌θ.
∵ eiπ∕2=cos(

π

2

)+i‌sin(

π

2

)
⇒eiπ∕2=i
For the z = i, we have r = 1 and θ = π/2
Similarly, the given complex number Z can be expressed in the euler form as
ii=(e

iπ

2

)i=ei2

π

2

=e−

π

2

⇒Z=e−π∕2
∴ Imaginary part of Z=ii is 0.