) For the following differential equation locate and classify its singular points on the x-axis

i) 

ii) 

 Find the series solution about  of the equation
.

Check whether  satisfies Lipschitz condition

i) on any rectangle  and ;

ii) on any strip  and ;

iii) on the entire plane.

  Show the minimum heap after each of the following operations
          i)    Create with values 13, 46, 68, 15, 87, 4, 33, 21, 1, 79.
          ii)   Insertion of value 9.
          iii)  Update of value 1 by 71.

 Construct the Huffman code tree for the set of frequencies in the table below:

 Construct a binary search tree with the following keys, inserting them in the given order:
          110, 8, 19, 6, 9, 12, 32.

Construct a MAXHEAP tree on the following list of numbers

15, 22, 7, 10, 43, 52 13, 16,

 Use Dijkstra's algorithm to find the shortest paths in the graph given below with a as the source vertex.

 Use Kruskal's algorithm to find a minimal spanning tree in the graph given below

Apply DFS algorithm to find a spanning tree of the following graph. Also find all the fundamental cycles of the spanning tree.

 Determine an LCS of (1, 1, 1, 1, 0, 1, 1, 0, 1, 0) and (1, 1, 1, 0, 0, 1, 1, 0, 1, 0).

 Taking  v₁ as the root, find a spanning tree of the following graph using the breadth first search (BFS) algorithm.

  Find an optimal parenthesisation of a matrix chain product whose sequence of dimensions is (4,8,7,2,3).

For the following set of points, describe how the CLOSET-PAIR algorithm finds a closest pair of points:

(1,2),(2,−1),(3,−3),(3,2),(−3,4),(−1,2),(2,3),(1,1),(−5,−2).

Show step by step how the merge procedure of merge sort will merge the arrays ,1 ,3 ,4 ,7 11,9 13, 14, and ,2 ,5 ,6 10,8 12, . 

Using bipartite checking algorithm find out whether the graph given below is bipartite or not. 

 Show the results of inserting the keys
   C, R, A, U, J, S, F, T, O, P, M, L, N, W, Q

in order into an empty B-tree with minimum degree 2. Only draw the configurations of the tree just before some node must split, and also draw the final configuration. 

Show step by step how you will construct a binary search tree by inserting keys in the order 12, 6, 4, 7, 8, 10, 7, 5, 13, 16, 11, starting from an empty binary tree. Explain step by step how you will delete the node with the key 13 from the binary tree. (5)
   b) Illustrate the operations of performing counter sort on the array . 

With the help of an example, explain the following:
    i) Algorithm,
    ii) Input and output for an algorithm
    iii) Running time of an algorithm (5)
   b) Illustrate the operations of partition in quick sort on the array
    . 

Analyse the algorithm to find the upper bound for run time of the above algorith.