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 ${(z + 2{\text{i}})^3} = 216{\text{i}}$,...