Given runtime of three different algorithms as below:

$T_1(N)=N^{2}lgN,T_2(N)=N^3$ and $T_3(N)=(1\mathrm{.}2{)}^N$

Construct a table by varying $N$ from $N=10$ to $200$ and find out which algorithm is worst and which is best. Give proper reasoning for that.

$2.$ Using the definition of big$-$O$,$ big$-$omega $()$ and big$-$theta $()$ determine the best and worst algorithms in the following is given:

$T_1(N)=(N^2),T_2(N)=O(N^2)$ and $T_3(N)=(N^2)$

Give proper justification to your answer.

$3.$ Given a recursive function as below, you need to show all the steps $($do not answer this question by running the computer program in a machine, but need to show the steps by hand calculation$).$

int mystery$($int $n)$

$\{$

if $)<mi\; id="EditorElement\_1184305">(</mi><mn\; id="EditorElement\_1184297">1</mn><mn\; id="EditorElement\_1184298">0</mn>$

$\{$

return $n$;

$\}$

else

$\{$

int $a=\frac{n}{10}$

int $b=n\%10$;

return mystery $(a+b)$;

$\}$

$\}$

Determine the value of mystery$(648)$

$4.$ Write a C$++$ program that would use a recursive function IsPalindrome$($S$,$ first, last$)($please see the pseudocode for the algorithm$)$ to check is a string $S$ is Palindrome or not.