Define the following terms 1. Power Set 2. Coinitial Vectors 3. Proposition 4. Circle 5. Diagonal Matrix 6. Modulus of a vector 7. Equivalence Relation
[7 marks]I) If U = {x/xN, x ≤ 10} A = {x/xN, x is even integer, 1≤ x ≤ 10} B = {2, 3, 5, 7} C = {1, 5, 6, 8, 10} Find (i) A' (ii) A – B (iii) A (B' - C ) (iv) A – (B C') II) Construct a truth table for each of following compound propositions.
[3 marks]p (q r) (ii) (p q) (q r) (iii) (p q) (p r)
[ marks]Verify that A(BC) = (AB)Cfor the following matrices 1 1 1 1 0 −1 −1 −1 A = [2 2 2] B = [ 0 −1 1 ] C = [ 2 2 ] 3 3 3 −1 1 0 1 1
[7 marks]Solve the following equations by using matrix inversion 5x – y + z = 3x + 2y – 5z = x + 3y – 2z =
[5 marks]Solve the following system of linear equations by Gauss elimination method. 5x - y + z = 2x + 4y = x + y + 5z = -1
[12 marks]Give a direct proof of “If n is an odd integer, then n2 is odd.”
[7 marks]Using mathematical induction show that if n is a positive integer, then 1 + 2 + 22 + … + 2n = 2n+1 - 1
[7 marks]Prove that “if n is an integer and n2 is odd, then n is odd” using proof by contraposition.
[7 marks]Answer the following questions. Justify your answer with proper explanation.
[4 marks]How many cards must be selected from a standard deck of 52 cards to guarantee that at least three cards of the same suit are chosen? (ii) How many cards must be selected to guarantee that at least three hearts are selected?
[3 marks]Find the first six terms of the sequence defined by following recurrence relations and initial conditions a a - a n = n-1 n-21 a 0 = a -1 1 =
[2 marks]How many positive integers less than 1000
[7 marks]are divisible by 7? (ii) are divisible by 7 but not by 11? (iii) are divisible by both 7 and 11? (iv) are divisible by either 7 or 11?
[ marks]are divisible by neither 7 nor 11? (vi) have distinct digits? (vii) have distinct digits and are even?
[ marks]Let a = 2n + 5•3n forn = 0, 1, 2, 3,… n
[ marks]Find a ,a a a a 0 1, 2, 3 and (ii) Show that a 5a - 6a 2 = 1 0
[2 marks]How many positive integers between 1000 and 9999 inclusive
[7 marks]are divisible by 9? (ii) are even? (iii) have distinct digits? (iv) are not divisible by 3?
[ marks]are divisible by 5 or 7? (vi) are not divisible by either 5 or 7? (vii) are divisible by 5 and 7?
[ marks]Find the equation of the circle passing through the points (5, -8), (-2, 9) and (2, 1)
[7 marks]Find the angle between the vectors 3i + j + 2k and 2i – 2j + 4k
[7 marks]Find the area of the triangle, the co-ordinates of whose vertices are (1, 3), (1,2), (-1,1)
[7 marks]Show that the points (8,-10), (7,-3) and (0, -4) are the vertices of a right triangle.
[7 marks]