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5. a) Write the expression x1 ∨x2 ∧x3 ∨x4 in conjunction normal form and disjunctive
normal form
4. a) A die is rolled twice and the sum of the numbers that appear is observed. What is
the probability that the sum is either a perfect square or a perfect cube?
3c) If 5 points are chosen in a square of side 2cm, show that there will always be two
points at a distance of at most √
2cm.
3b) A bank pays you 4.5% interest per year. In addition, you receive |100 as bonus at
the end of the year (after the interest is paid). Find a recurrence for the amount of
money after n years if you invest |2000.
3. a) If the solution of the recurrence relation αun−1 +βun−2 = f(n),(n ≥ 2) is
un = 1−2n+3.2
n
, then determine the values of α,β and f(n).
2. a) Consider the propositions ‘2+3 = 5’ and ’The Sun rises in the West’.
i) Write the disjunction of the statements and give its truth value.
ii) Write the conjunction of the statements and give its truth value.
iii) Write the exclusive disjunction of the statements and give its truth value. (3)
b) Prove or disprove the following statement. (2)
“If m divides a
n −b
n
, then m divides abn −ban
also.”
c) Show that any tree with exactly two vertices of degree 1 is a path. (3)
d) Write down the converse of each of the following statements: (2)
i) If p is a prime number and a and b are any two natural numbers and if p
divides a or b, then p divides ab.
ii) In a triangle 4ABC, if AB2 +AC2 = BC2
, then ∠BAC = 90◦
.
2 d) Write down the converse of each of the following statements: (2)
i) If p is a prime number and a and b are any two natural numbers and if p
divides a or b, then p divides ab.
ii) In a triangle 4ABC, if AB2 +AC2 = BC2
, then ∠BAC = 90◦
.
2c) Show that any tree with exactly two vertices of degree 1 is a path.
2b) Prove or disprove the following statement. (2)
“If m divides a
n −b
n
, then m divides abn −ban
also.”
2. a) Consider the propositions ‘2+3 = 5’ and ’The Sun rises in the West’.
i) Write the disjunction of the statements and give its truth value.
ii) Write the conjunction of the statements and give its truth value.
iii) Write the exclusive disjunction of the statements and give its truth value.
