Differences in Mathematica's behaviour with identical functions

Question

I have three functions whose identity was verified, but WM doesn't behave itself equally with them.

The code is:

θ[x_] := Boole[x >= 0]
{R, L, C1, Rl, T, Um} := {500, 0.2, 2*10^(-10), 10^3, 10^(-3), 10};
{t1, t2, t3} := {T/2, T/3, T/6};
u1[t_] := Um/t3*(t*θ[t] - (t - t3) θ[t - t3] - (t - t2) θ[t - t2] + (t - t1) θ[t - t1])
U1[s_] := LaplaceTransform[u1[t], t, s]

A10[ω_] := Abs[U1[s]] /. s -> I*ω
Solve[A10[ω] == 0 && ω ∈ Interval[{10000, 20000}], ω]

A11[ω_]:=(1.2*10^5/ω^2)*Sqrt[2(Cos[10^(-3)ω/6])^3-2(Cos[10^(-3)ω/6])^2-2Cos[10^(-3)ω/6]+2]
Solve[A11[ω] == 0 && ω ∈ Interval[{10000,20000}],ω]

A12[ω_]:=Abs[-(6*10^4/ω^2)*(1-E^(-10^(-3)*I*ω/6)-E^(-10^(-3)*I*ω/3)+E^(-10^(-3)*I*ω/2))]
Solve[A12[ω] == 0 && ω ∈ Interval[{10000,20000}],ω]

Plot[A10[ω], {ω, 0, 80000}, PlotRange -> {{0, 80000}, {0, 0.004}}]
Plot[A11[ω], {ω, 0, 80000}, PlotRange -> {{0, 80000}, {0, 0.004}}]
Plot[A12[ω], {ω, 0, 80000}, PlotRange -> {{0, 80000}, {0, 0.004}}]
(*Plot performance velocity was compared by isolating each two plots with comment syntax*)

First phenomenon is that Solve[] doesn't process A11[ω]==0 equation correctly, returning {{ω -> {18849.6}}} instead of {{ω -> {6000 π}}} and raising such a message:

Solve::ratnz: Solve was unable to solve the system with inexact coefficients. The answer was obtained by solving a corresponding exact system and numericizing the result.

Second phenomenon is that Plot[] for A10[ω] performs unbelievably slow. Thus, I've got two questions...

1. Why I observe problems with getting A11[ω]==0 roots?
2. Why A10[ω] plot performance speed is so different from others?

Thanks in advance.

Answer 1

By typing 1.2*10^5 you are telling Mathematica to do things in floating point. Change this to 120000 and you get the desired exact form. To make it plot faster use Set (shortcut =) rather than SetDelayed (shortcut :=), that is, define

A10[ω_] = Abs[U1[s]] /. s -> I*ω

instead of

A10[ω_] := Abs[U1[s]] /. s -> I*ω

Now it plots almost instantly. (The only difference is the colon before the equals sign).