Number and Algebra — IB DP Math AI SL

Question 1 • 2024 Exam Specimen 7 Marks

Question 2 • 2024 Exam Specimen 6 Marks

Question 3 • 2024 Exam Specimen 6 Marks

Question 4 • 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 5 • 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}

Question 6 • Exam Practice 7 Marks

(a) Solve the equation z^5 = 1 , giving all roots in exponential form e^{i\theta} where -\pi < \theta \le \pi . [2 marks] (b) By considering the sum of roots of z^5 - 1 = 0 , show that \cos\left(\frac{2\pi}{5}\right) + \cos\left(\frac{4\pi}{5}\right) = -\frac{1}{2} . [3 marks] (c) Hence, deduce that \cos\left(\frac{2\pi}{5}\right) = \frac{\sqrt{5} - 1}{4} . [2 marks]

Answer: (a) \ z = e^{i 2k\pi/5}, \quad (b) \ \text{Proved by sum of roots}, \quad (c) \ \cos(2\pi/5) = \frac{\sqrt{5} - 1}{4}