Maths F4....9

19.
(a)
a - (1/a) = 2√6
[a - (1/a)]² = (2√6)²
a² - 2a(1/a) + (1/a)² = 24
a² - 2 + (1/a²) = 24
a² + (1/a²) = 26

(b)
From (a):
a² + (1/a²) = 26
a² + 2 + (1/a²) = 28
a² + 2a(1/a) + (1/a)² = 28
(a + 1/a)² = 28
a + (1/a) = √28 or a + (1/a) = -√28
a + (1/a) = 2√7 or a + (1/a) = -2√7

(c)
(1/a²) - a²
= (1/a)² - a²
= [(1/a) + a] [(1/a) - a]
= -[a + (1/a)] [a - (1/a)]

Hence,
(1/a²) - a² = -(2√7)(2√6) or (1/a²) - a² = -(-2√7)(2√6)
(1/a²) - a² =-4√42 or (1/a²) - a² =4√42


=====
20.
z² - z + 1
= [(1 + √5i)/2]² - (1 + √5i)/2 + 1
= (1 + 2√5i - 5)/4 - 2(1 + √5i)/4 + 4/4
= (-4 + 2√5i)/4 - (2 + 2√5i)/4 + 4/4
= (-4 + 2√5i - 2 - 2√5i + 4)/4
= -1/2

1/(z² - z + 1)
= 1/(-1/2)
= -2