Let` z= a+ ib` (where` a,b,in R` and` i = sqrt(-1)` such that `|2z+3i| = |z^(2)|` identify the correct statement(s)?

Options

A. `|z|_("maximum")` is equal to `3`
B. `|z|_("maximum")` is equal to `1`
C. If `|z|_("maximum")` when `z=alpha + ibeta (alpha,beta in R and I = sqrt (-1) then (alpha^(3)+beta^(3))` is equal to `27`
D. If `|z|_("maximum") "when" z= + iy (x,y in R "and" i = sqrt(-1) "then" (x^(2) +2y^(2))` is equal to 2`

Answer

A, B, C, D

Explanation

Given `|2z+3i| = |z^(2)|`
` because |2z+3i|le2|z|+3`
`implies |z|^(2)le2|z|+3`
`implies 0 le|z|le 3 " "...(1)`
Again `|2z+3i|ge|2|z|-3|`
`implies |z|^(2)ge|2|z|-3|`
`therefore |z|ge1" " …..(2)`
So,(1)`nn`(2)gives
`1le|z|le3`
Also,`|z|_("maximum")impliesz=3i`
So, `alpha=0,beta=3`
`&|z|("maximum") impliesz=-i`
So,`x=0, y=-1`