$($More Recursions$)(32$ points $=4$ each$)$ Like in many previous

exercises and homework's, find tight asymptotic bounds $($big$-$Theta$)$ for T$($n$)$ is

each of the cases. $\backslash \backslash \backslash \backslash$

$$$ T$($n$)=2$T$($n$/4)+$ n$^2\backslash$sqrt$\{$n$\}$$$\backslash \backslash \backslash \backslash$

$$$ T$($n$)=$ T$($n $-1)+\backslash$frac$\{1\}\{$n$\}$$$\backslash \backslash \backslash \backslash$

$$ $T$($n$)=1600$T$($n$/4)+$n$!$$ $($hint: answering this shouldn't require too many,

if any, difficult calculations$)\backslash \backslash \backslash \backslash$

$$$ T$($n$)=6$T$($n$/3)+\backslash$frac$\{$n$^4\}\{$log$^\{25\}($n$)\}$$ $($hint: answering this shouldn't require too many, if any, difficult calculations$)\backslash \backslash \backslash \backslash$

$$$ T$($n$)=\backslash$sqrt$\{$n$\}$T$(\backslash$sqrt$\{$n$\})+$ n$ $($hint: when everything fails, you guess and check$)\backslash \backslash \backslash \backslash$

$$$ T$($n$)=$ T$($n$/2)+$ n$(5-$ cos$^2($n$)$ sin$^\{20\}($n$))$$ $($hint: answering this shouldn't require too many, if any, difficult calculations, just think the most basic trigonometric inequality$)\backslash \backslash \backslash \backslash$

$$ $T$($n$)=\backslash$alpha T$($n$/4)+\backslash$frac$\{$n$\}\{2\}$$ $($hint: your answer should depend on the $$\backslash$alpha$ parameter$)\backslash \backslash \backslash \backslash$

$$ $T$($n$)=5$T$($n$/5)+\backslash$frac$\{$n$\}\{$log$\_5($n$)\}$$ $($hint: think of n $=5$m$.$ Also the recursion $T$($n$)=$ T$($n $-1)+\backslash$frac$\{1\}\{$n$\}$$ above may come in handy.$)\backslash \backslash \backslash \backslash$