Q$3$: Prove or disprove

$5n^2+40n+100=0(n^2).$

$\frac{n^3}{1000}=0(n^2).$

$n^k=a(2^n)$

${\displaystyle \underset{i=1}{\overset{n}{}}}logi=(nlogn)$

$7^n=(9^n)$

Q$4$: In each of the following situations, indicate whether $f=O(g),$ or $f=(g),$ or both $f=\frac{O}{g}.$

$f(n)=n-100$ and $g(n)=n-200.$

$f(n)=n^{\frac{1}{2}}$ and $g(n)=n^{\frac{2}{3}}.$

$f(n)=100n+logn$ and $g(n)=n+(logn{)}^2.$

$f(n)=10logn$ and $g(n)=log(n^2).$

$f(n)=\frac{n^2}{logn}$ and $g(n)=n(logn{)}^2.$

$f(n)=n^{\frac{1}{2}}$ and $g(n)=4^{logn}$

$f(n)=n{2}^n$ and $g(n)=3^n.$

$f(n)=2^n$ and $g(n)=2^{n+1}.$

Q$5$: Find the complexity $($Exact and approximate$)$ for the basic operations of the following

algorithms:

sum$=$e; ;

for $(k=1$;$kn$;$k^{**}=2)$

for $(j=1$;$jn$;$++j)$

suint$+$;

sum$2=$;

for $(k=1$;$kn$;$k^{**}=2)$

for $(n)j=0$;$ji$;$j++$Fun$3(\frac{n}{2})nm(n)(n-1,n+1,0)i=0$;$i3(n-1,n,0+1)i=1$;$in$;$i**=2(j=1$;$j$

sum $2+t$;

int Fun$2(intn)$

$\{$

$if(n)$

return $1$;

for $(inti=0$;$i)$

for $(intj=0$;$ji$;$j++)$

constopera tion;

$\}$

return $1*$ Fun$3(\frac{n}{2})$;

void Fun$3(intn,$ int $m,$ int $0)$

$f$

$if(n)$

$f$

cout;

$\}$

Fun $3(n-1,n,0+1)$;

For $(inti=1$;$in$;$i**=2)$

dosonet hing;