Q1. Let (A,*) be a group.
Show that (A, *) is abelian group if and only if a2*b2= (a*b)
2 for all a and b in A
(07)
Q2. Let (A,*) and (B, #) be two algebraic system. The Cartesian
product of (A,*) and (B, #) is an algebraic system (A X B, $), where $ is a
binary operation such that for any (a1, b1) and (a2, b2) in A X B
(a1,
b1) $ (a2, b2) = (a1*a2, b1#b2)
Show that Cartesian product of
two groups is a group. (07)
Q3. Ten (distinct) passengers got into elevator on ground floor of
20 story building. What is the probability that they will all get off at
different floors? (08)
Q4. There are 10 adjacent parking slots in the parking lot. When
you arrive in your new Rolls Royals, there are already seven cars in parking
lot. What is the probability that you can find two adjacent unoccupied spaces
for your Rolls Royals? (08)
Q5. Seven kind of military equipment are to be flown to a
destination by five cargo planes. There are four kinds of units of each kind.
And five planes can carry 8, 8,5,4,4 units respectively. We want to find that,
whether is it possible to load equipment such a way that no two units of same
kind are on one plane. How can transport
network be used to find out the same? Also give the transport network which
will be suitable for given scenario. (10)
Q6. A zoo wants to set up natural habitats to exhibit its animal.
Unfortunately some animals will eat some of the others if given the
opportunity. How can graph model and coloring be used to determine the numbers
of different habitats needed and the placement of animals in theses habitats? (10)
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