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.