DP Mathematics HL Questionbank

• Syllabus

1.7

Sub sections and their related questions

Powers of complex numbers: de Moivre’s theorem.

• 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.TZ0.10b: (i)     Write \({z_2}\) in modulus-argument form. (ii)     Hence solve the equation...
• 12N.1.hl.TZ0.10c: Let \(z = r\,{\text{cis}}\theta \) , where \(r \in {\mathbb{R}^ + }\)  and...
• 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...
• 14N.1.hl.TZ0.13a: (i)     Show that...
• 15M.2.hl.TZ1.12a: (i)     Use the binomial theorem to expand \({(\cos \theta  + {\text{i}}\sin \theta )^5}\). (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...
• 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.TZ2.12b: (i)     Expand \((w - 1)(1 + w + {w^2} + {w^3} + {w^4} + {w^5} + {w^6})\). (ii)     Hence deduce...
• 16M.1.hl.TZ2.12c: Write down the roots of the equation \({z^7} - 1 = 0,{\text{ }}z \in \mathbb{C}\) in terms of...
• 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)    ...
• 17M.1.hl.TZ2.11c.ii: Hence find the cube roots of \(z\) in modulus-argument form.
• 17N.1.hl.TZ0.8: Determine the roots of the equation \({(z + 2{\text{i}})^3} = 216{\text{i}}\),...
• 18M.1.hl.TZ1.11a.i: Express w2 and w3 in modulus-argument form.
• 18M.1.hl.TZ1.11a.ii: Sketch on an Argand diagram the points represented by w0 , w1 , w2 and w3.
• 18M.1.hl.TZ1.11c: Let...

\(n\)th roots of a complex number.

• 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.TZ0.10b: (i)     Write \({z_2}\) in modulus-argument form. (ii)     Hence solve the equation...
• 12N.1.hl.TZ0.10c: Let \(z = r\,{\text{cis}}\theta \) , where \(r \in {\mathbb{R}^ + }\)  and...
• 12N.1.hl.TZ0.10d: Find the smallest positive value of n for which...
• 08N.1.hl.TZ0.13Part A: (a)     Use de Moivre’s theorem to find the roots of the equation \({z^4} = 1 - {\text{i}}\)...
• 10M.2.hl.TZ1.4: (a)     Solve the equation \({z^3} = - 2 + 2{\text{i}}\), giving your answers in modulus-argument...
• 15M.1.hl.TZ2.7a: Find three distinct roots of the equation \(8{z^3} + 27 = 0,{\text{ }}z \in \mathbb{C}\) giving...
• 15M.2.hl.TZ1.12b: Find the value of \(r\) and the value of \(\alpha \).
• 15N.1.hl.TZ0.11a: Solve the equation \({z^3} = 8{\text{i}},{\text{ }}z \in \mathbb{C}\) giving your answers in the...
• 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.TZ2.12b: (i)     Expand \((w - 1)(1 + w + {w^2} + {w^3} + {w^4} + {w^5} + {w^6})\). (ii)     Hence deduce...
• 16M.1.hl.TZ2.12c: Write down the roots of the equation \({z^7} - 1 = 0,{\text{ }}z \in \mathbb{C}\) in terms of...
• 16N.1.hl.TZ0.12a: Determine the value of (i)     \(1 + \omega  + {\omega ^2}\); (ii)    ...
• 17N.1.hl.TZ0.8: Determine the roots of the equation \({(z + 2{\text{i}})^3} = 216{\text{i}}\),...