Let U = {a, b, c, d, e, f, g}, A = {a, c, d}, B = {a, b, e, g}, C = {c, d, f, g}, D = {a, d, e, f, g}.Find: (i)A ∪ C(ii)B ∪ D(iii)B ∪ (C ∩ D)(iv)C′ ∩ A(v)D′ − B.b. If z1 = 4 − 2i and z2 = 3 + i, find z1 + z2, z1 − z2, z1z2 and z1z2 . Evaluate |z1∗z1| and|z2∗z2|.c. The nth term of a sequence is given as 2n3−12 . Find the 14th and 17th terms of thesequence.d. Expand by the binomial formula (2x − 2y )5.2a. The second and fifth terms of an A.P are 12 and 27 respectively.Find the (i) first term and the common difference, (ii) 15th term of the A.P.b. If z1 = 4 + 3i, z2 = i + 5, and z3 = 3 − 2i, then verify that z1(z2 + z3) = z1z2 + z1z3.c. Find the 6th term in the expansion of (x2 + y2 )15.3a. Show by Venn diagrams that (i) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) (ii) (A ∪ B)′ = A′ ∩ B′.b. Let A = {1, 2, 4, 5, 7}, B = {2, 3, 5, 6, 8}, C = {1, 3, 7, 8, 9}.Find (i) A − B (ii) B − C (iii) A − C(iv)A4B.c. Find the coefficient of x5 in the expansion of (2x2 − 3x )10.4a. The first three terms of an arithmetic progression are x, (2x + 2) and (5x − 3). Find the(i) value of x, (ii) 5th term, (iii) 11th term, (iv) nth term.b. Show that : i4 + i3 − i2 − 2 = −i.c. Find the quadratic equation whose roots are α + 2β and β + 2α where α and β are the rootsof the equation 2x2 − 3x + 1 = 0.5a. Use Demoivre’s theorem to show thatcos3 θ = cos(3θ) + 3 cos θ4b. The sum of the first and third terms of a G.P is 213 and the sum of the second and fourthterms is 134 . Find a and r.c. Given that the roots of the quadratic equation 2x2 − 5x − 3 = 0 are α and β.Find (i) α2 + β2 (ii) 1α + 1β .