Question 5 • Exam Practice
7 Marks
Let z = \cos\theta + i\sin\theta . (a) Use De Moivre's theorem to express \cos(3\theta) in terms of \cos\theta . [4 marks] (b) Hence, solve the equation 8x^3 - 6x - 1 = 0 for x \in \mathbb{R} . [3 marks]
Answer: (a) \ \cos(3\theta) = 4\cos^3\theta - 3\cos\theta, \quad (b) \ x = \cos(\pi/9), \cos(5\pi/9), \cos(7\pi/9)
Question 6 • Exam Practice
6 Marks
Use the principle of mathematical induction to prove that 5^{2n} - 1 is divisible by 24 for all positive integers n \in \mathbb{Z}^+ .
Answer: \text{True for all } n \in \mathbb{Z}^+ \text{ by mathematical induction}