Q1. Check whether the following statements are true or not. Justify your answers with a short proof or a counter example.

• i)) If the contrapositive of a statement is true, then the statement itself is also true. (80 words)
• ii)) a + 3an-1 + 2an-2 = 2" is a linear homogeneous recurrence relation. (80 words)
• iii)) A particular solution of the recurrence relation an -2an-1 + an-2 = 1 has the form Cn². (80 words)
• iv)) There exists a boolean expression in variables x1,x2 and x3 with CNF as (X₁ VX2)∧ (X2 V X3) ∧ (X3 V X₁). (80 words)
• v)) If a dice is rolled thrice, then the probability of getting a 6 each time is 1/72 (80 words)
• vi)) Every odd cycle has the same chromatic and edge chromatic numbers. (80 words)
• vii)) Every Eulerian graph is Hamiltonian. (80 words)
• viii)) S gives the number of ways in which any 3 objects can be placed in any 4 boxes. (80 words)
• ix)) There exists a self-complementary planar graph on 5 or more vertices. (80 words)
• X) The number of partitions of 6 is 10. (80 words)