WJEC Maths for A2 – Pure

15 Test yourself Test yourself ➊ Use proof by contradiction to prove that if a 2 is even, then a must be even. [3] ➋ Prove by contradiction the following proposition: If n is a positive integer and 3 n + 2 n 3 is odd, then n is odd. The ϐirst two lines of the proof are given below. It is given that 3 n + 2 n 3 is odd . Assume that is n is even so that n = 2 k . [4] ➌ Prove by contradiction the following proposition: If a , b are positive real numbers, then a + b ≥ 2 √ ab . The ϐirst line of the proof is given below Assume that positive real numbers a , b exist such that a + b < 2 √ ab . [3] ➍ Prove by contradiction the following proposition: If a and b are odd integers such that 4 is a factor of a − b , then 4 is not a factor of a + b . The ϐirst lines of the proof are given below. Assume that 4 is a factor of a + b. Then there exists an integer, c, such that a + b = 4 c. [3] ➎ Prove by contradiction the following proposition: When x is real, (5 x − 3) 2 + 1 ≥ (3 x − 1) 2 The ϐirst line of the proof is given below. Assume that there is a real value of x such that (5 x − 3) 2 + 1 < (3 x − 1) 2 [3] ➏ Show that √ 5 is irrational. [6] ➐ Prove by contradiction the following proposition: When x is real and 0 ≤ x ≤ π 2 , sin  x + cos  x ≥ 1 [6]