Q. The time complexities of some standard graph algorithms are given. Match each algorithm with its time complexity ? (n and m are no. of nodes and edges respectively)

a. Bellman Ford algorithm 1. O (m log n)
b. Kruskals algorithm 2. O (n3)
c. Floyd Warshall algorithm 3. O(mn)
d. Topological sorting 4. O(n + m)

Codes :
a b c d

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Q. Let V1 = 2I – J + K and V2 = I + J – K, then the angle between V1 & V2 and a vector perpendicular to both V1 & V2 shall be :

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Q. Let T(n) be the function defined by T(n) = 1 and T(n) = 2T (n/2) + n, which of the following is TRUE ?

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Q. Which of the following permutations can be obtained in the output using a stack of size 3 elements assuming that input, sequence is 1, 2, 3, 4, 5 ?

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Q. How many solutions do the following equation have x1 + x2 + x3 = 11
where x1 ≥ 1, x2 ≥2, x3 ≥ 3

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Q. The number of distinct bracelets of five beads made up of red, blue, and green beads (two bracelets are indistinguishable if the rotation of one yield another) is,

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Q. 58 lamps are to be connected to a single electric outlet by using an extension board each of which has four outlets. The number of extension boards needed to connect all the light is

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Q. The power set of AUB, where A = {2, 3, 5, 7} and B = {2, 5, 8, 9} is

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Q. Which is not the correct statement ?

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Q. Given the following statements :

(i) Recursive enumerable sets are closed under complementation.
(ii) Recursive sets are closed under complementation.

Which is/are the correct statements ?

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Q. Let θ(x, y, z) be the statement “x + y = z” and let there be two quantifications given as

(i) ∀x ∀y  Z ∃ θ(x, y, z)
(ii) ∃Z ∀x ∀y θ(x, y, z)
 
Where x, y, z are real numbers. Then which one of the following is correct ?

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Q. Let P(rn, n) be the statement if "m divides n" where the universe of discourse for both the variables is the set of positive integers. Determine the truth values of each of the-following propositions:

I. ∀m ∀n P(m,n)
II. ∃m  ∀n P(m,n)

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Q. Big O estimate for
( f(x) = (x + 1) log(x2 + 1) +3x2 is given as

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Q. How many edges are there in a forest of t-trees containing a total of n vertices?

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Q. Let f and g be the functions from the set of integers to the set integers defined by

f(x) = 2x + 3 and g(x) = 3x + 2

Then the composition  of f and g and g and f is given as

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Q. A graph is non-planar if and only if it contains a subgraph homomorphic to

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Q. If the primal Linear Programming problem has unbounded  solution, then it's dual problem will  have

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Q. Given the problem to maximize
            f(x), X =(x1, x2 ,..... .xn)
subject to m number of inequality constraints
  gi(x) ≤ bi  , i = 1, 2 ...... m
including the non-negativity constraints x ≥ 0
Which of the following conditions is a Kuhn-Tucker necessary condition for a local maxima at x  ?

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Q. The following Linear Programming problem has :
Max                                   Z =x1 +x2
Subject to                         = x1-x2 ≥ 0                                  
                                             3x1 -x2 ≤ -3                                
      and                              xI , x2  ≥ 0

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Q. Given a flow graph with 10 nodes,13 edges and one connected components, the number of regions and the number  of predicate (decision) nodes in the flow graph will be

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