Consider the following functions, where $n$ is a positive integer.

$$ n^{1 / 3}, \log (n), \log (n!), 2^{\log (n)} $$

Which one of the following options lists the functions in increasing order of asymptotic growth rate?

Note: Assume the base of log to be 2 .

Which of the following can be recurrence relation(s) corresponding to an algorithm with time complexity $\Theta(n)$ ?

Consider an array $A=[10,7,8,19,41,35,25,31]$. Suppose the merge sort algorithm is executed on array $A$ to sort it in increasing order. The merge sort algorithm will carry out a total of 7 merge operations.

A merge operation on sorted left array $L$ and sorted right array $R$ is said to be void if the output of the merge operation is the elements of array $L$ followed by the elements of array $R$.

The number of void merge operations among these 7 merge operations is $\_\_\_\_$ . (answer in integer)

Consider an array $A$ of integers of size $n$. The indices of $A$ run from 1 to $n$. An algorithm is to be designed to check whether $A$ satisfies the condition given below.

$\forall i, j \in\{1, \ldots, n-1\}$ such that $i>j,(A[i+1]-A[i])>(A[j+1]-A[j])$

Which one of the following gives the worst case time complexity of the fastest algorithm that can be designed for the problem?