DP Mathematics HL Questionbank
Powers of complex numbers: de Moivre’s theorem.
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[N/A]Directly related questions
- 17N.1.hl.TZ0.8: Determine the roots of the equation \({(z + 2{\text{i}})^3} = 216{\text{i}}\),...
- 17M.1.hl.TZ2.11c.ii: Hence find the cube roots of \(z\) in modulus-argument form.
- 12M.1.hl.TZ2.12A.b: Hence find the two square roots of \( - 5 + 12{\text{i}}\) .
- 12M.1.hl.TZ2.12A.d: Hence write down the two square roots of \( - 5 - 12{\text{i}}\) .
- 12N.1.hl.TZ0.10c: Let \(z = r\,{\text{cis}}\theta \) , where \(r \in {\mathbb{R}^ + }\) and...
- 12N.1.hl.TZ0.10b: (i) Write \({z_2}\) in modulus-argument form. (ii) Hence solve the equation...
- 12N.1.hl.TZ0.10d: Find the smallest positive value of n for which...
- 08M.2.hl.TZ1.14: \({z_1} = {(1 + {\text{i}}\sqrt 3 )^m}{\text{ and }}{z_2} = {(1 - {\text{i}})^n}\) . (a) ...
- 08M.1.hl.TZ2.14: Let \(w = \cos \frac{{2\pi }}{5} + {\text{i}}\sin \frac{{2\pi }}{5}\). (a) Show that w is a...
- 08N.1.hl.TZ0.13Part A: (a) Use de Moivre’s theorem to find the roots of the equation \({z^4} = 1 - {\text{i}}\)...
- 09M.1.hl.TZ2.12: The complex number z is defined as \(z = \cos \theta + {\text{i}}\sin \theta \) . (a) State...
- 09N.1.hl.TZ0.13b: Let \(w = \cos \theta + {\text{i}}\sin \theta \) . (i) Show that...
- SPNone.2.hl.TZ0.4b: Find the cube root of z which lies in the first quadrant of the Argand diagram, giving your...
- SPNone.2.hl.TZ0.4c: Find the smallest positive integer n for which \({z^n}\) is a positive real number.
- 13M.1.hl.TZ1.1b: Given...
- 13M.1.hl.TZ2.13a: (i) Express each of the complex numbers...
- 13M.1.hl.TZ2.13b: (i) State the solutions of the equation \({z^7} = 1\) for \(z \in \mathbb{C}\), giving them...
- 11N.1.hl.TZ0.2: Find the cube roots of i in the form \(a + b{\text{i}}\), where \(a,{\text{ }}b \in \mathbb{R}\).
- 11M.1.hl.TZ1.13b: Hence show that \(\cos 3\theta = 4{\cos ^3}\theta - 3\cos \theta \) .
- 11M.1.hl.TZ1.13c: Similarly show that \(\cos 5\theta = 16{\cos ^5}\theta - 20{\cos ^3}\theta + 5\cos \theta \) .
- 13N.1.hl.TZ0.12a: Use De Moivre’s theorem to show that...
- 18M.1.hl.TZ1.11c: Let...
- 18M.1.hl.TZ1.11a.ii: Sketch on an Argand diagram the points represented by w0 , w1 , w2 and w3.
- 18M.1.hl.TZ1.11a.i: Express w2 and w3 in modulus-argument form.
- 16M.1.hl.TZ2.12c: Write down the roots of the equation \({z^7} - 1 = 0,{\text{ }}z \in \mathbb{C}\) in terms of...
- 16M.1.hl.TZ2.12b: (i) Expand \((w - 1)(1 + w + {w^2} + {w^3} + {w^4} + {w^5} + {w^6})\). (ii) Hence deduce...
- 16M.1.hl.TZ2.12a: Verify that \(w\) is a root of the equation \({z^7} - 1 = 0,{\text{ }}z \in \mathbb{C}\).
- 16M.1.hl.TZ1.12a: Use de Moivre’s theorem to find the value of...
- 16N.1.hl.TZ0.12a: Determine the value of (i) \(1 + \omega + {\omega ^2}\); (ii) ...
- 15N.1.hl.TZ0.11a: Solve the equation \({z^3} = 8{\text{i}},{\text{ }}z \in \mathbb{C}\) giving your answers in the...
- 15N.1.hl.TZ0.11b: Consider the complex numbers \({z_1} = 1 + {\text{i}}\) and...
- 15M.2.hl.TZ1.12a: (i) Use the binomial theorem to expand \({(\cos \theta + {\text{i}}\sin \theta )^5}\). (ii)...
- 14N.1.hl.TZ0.13a: (i) Show that...
