Question 1 - 5 marks You should be able to answer this question after studying Unit 9. (a) You are told that the following two statements are true: If Kira is wagging her tail, then Kira is happy. Kira is happy. Can you deduce from the statements above that the following statement is true? Kira is wagging her tail. Justify your answer. [3] (b) Show that the following statement is false by giving a counter-example: For all positive integers n, we have n 2 47-1 [2] Question 2 - 10 marks You should be able to answer this question after studying Unit 9. (a) Prove that the following statement is true for all real numbers I, y # 3, by using a sequence of equivalences: y = 3x+ 1 I - 3 if and only if x = _ 3y + 1 y - 3 [3) (b) Prove that the following statement is true for all integers n: n + 3 is odd if and only if 5n - 1 is odd. [7] Question 3 - 10 marks You should be able to answer this question after studying Unit 9. Use mathematical induction to prove that the following statement is true: 3 x 5 + 5 x 7 + ... + ( 2n + 1) (2n+ 3) _ n(4n-+ 18n+ 23) 3 for all n 2 1. (You may find it helpful to note that ( k + 1 ) (4 (k + 1) 2 + 18 (k + 1 ) + 23 ) = 4k3 + 30k2 + 71k+ 45.) [10] Question 4 10 marks You should be able to answer this question after studying Unit 9. (a) Use proof by contradiction to show that there is no positive real number x such that It -