DP Mathematics HL Questionbank

• Syllabus

1.6

Sub sections and their related questions

Modulus–argument (polar) form $z = r\left( {\cos \theta + {\text{i}}\sin \theta } \right) = r{\text{cis}}\theta = r{e^{{\text{i}}\theta }}$

• 12M.1.hl.TZ1.3: If ${z_1} = a + a\sqrt 3 i$ and ${z_2} = 1 - i$, where a is a real constant, express...
• 12M.1.hl.TZ2.12B.e: Express each of the four roots of the equation in the form $r{{\text{e}}^{{\text{i}}\theta }}$ .
• 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...
• 09M.1.hl.TZ2.7: Given that ${z_1} = 2$ and ${z_2} = 1 + \sqrt 3 {\text{i}}$ are roots of the cubic equation...
• 09N.1.hl.TZ0.2: Find the values of n such that ${\left( {1 + \sqrt 3 {\text{i}}} \right)^n}$ is a real number.
• SPNone.1.hl.TZ0.10a: Calculate $\frac{{{z_1}}}{{{z_2}}}$ giving your answer both in modulus-argument form and...
• SPNone.1.hl.TZ0.10b: Using your results, find the exact value of tan 75° , giving your answer in the form...
• 10M.2.hl.TZ1.4: (a)     Solve the equation ${z^3} = - 2 + 2{\text{i}}$, giving your answers in modulus-argument...
• 10M.1.hl.TZ2.13: Consider...
• 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.2.hl.TZ0.14a: Show that $\left| {{{\text{e}}^{{\text{i}}\theta }}} \right| = 1$.
• 14M.1.hl.TZ2.7: Consider the complex numbers $u = 2 + 3{\text{i}}$ and $v = 3 + 2{\text{i}}$. (a)     Given...
• 14M.2.hl.TZ2.13a: Consider $z = r(\cos \theta  + {\text{i}}\sin \theta ),{\text{ }}z \in \mathbb{C}$. Use...
• 14M.2.hl.TZ2.13b: Given $u = 1 + \sqrt 3 {\text{i}}$ and $v = 1 - {\text{i}}$, (i)     express $u$ and $v$...
• 14M.2.hl.TZ2.13d: Find the area of triangle O${\text{A}}'$${\text{B}}'$.
• 13N.1.hl.TZ0.12d: Show that...
• 14M.2.hl.TZ2.13c: Plot point A and point B on the Argand diagram.
• 13N.1.hl.TZ0.12c: Hence show that ${\cos ^4}\theta  = p\cos 4\theta  + q\cos 2\theta  + r$, where...
• 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...
• 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.2.hl.TZ2.1a: Express $z$ in the form $a + {\text{i}}b$, where $a,\,b \in \mathbb{Q}$.
• 18M.2.hl.TZ2.1b: Find the exact value of the modulus of $z$.
• 18M.2.hl.TZ2.1c: Find the argument of $z$, giving your answer to 4 decimal places.

The complex plane.

• 12M.1.hl.TZ1.3: If ${z_1} = a + a\sqrt 3 i$ and ${z_2} = 1 - i$, where a is a real constant, express...
• 12M.1.hl.TZ2.12B.d: Draw the four roots on the complex plane (the Argand diagram).
• 12M.1.hl.TZ2.12B.e: Express each of the four roots of the equation in the form $r{{\text{e}}^{{\text{i}}\theta }}$ .
• 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...
• 11M.1.hl.TZ2.4a: Find AB, giving your answer in the form $a\sqrt {b - \sqrt 3 }$ , where a ,...
• 14M.1.hl.TZ1.13: A geometric sequence $\left\{ {{u_n}} \right\}$, with complex terms, is defined by...
• 15M.1.hl.TZ2.7b: The roots are represented by the vertices of a triangle in an Argand diagram. Show that the area...
• 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.2.hl.TZ2.1a: Express $z$ in the form $a + {\text{i}}b$, where $a,\,b \in \mathbb{Q}$.
• 18M.2.hl.TZ2.1b: Find the exact value of the modulus of $z$.
• 18M.2.hl.TZ2.1c: Find the argument of $z$, giving your answer to 4 decimal places.