DP Mathematics HL Questionbank

• Syllabus

1.5

Sub sections and their related questions

Complex numbers: the number \({\text{i}} = \sqrt { - 1} \) ; the terms real part, imaginary part, conjugate, modulus and argument.

• 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.TZ1.7: Given that z is the complex number \(x + {\text{i}}y\) and that...
• 12M.1.hl.TZ2.6a: m and n are real numbers;
• 12M.1.hl.TZ2.6b: m and n are conjugate complex numbers.
• 12M.1.hl.TZ2.12A.a: Given that \({(x + {\text{i}}y)^2} = - 5 + 12{\text{i}},{\text{ }}x,{\text{ }}y \in...
• 12M.1.hl.TZ2.12A.c: For any complex number z , show that \({(z^*)^2} = ({z^2})^*\) .
• 12N.1.hl.TZ0.10a: (i)     Write down \({z_1}\) in Cartesian form. (ii)     Hence determine...
• 12N.2.hl.TZ0.10: Let \(\omega = \cos \theta  + {\text{i}}\sin \theta \) . Find, in terms of \(\theta \) , the...
• 08M.1.hl.TZ1.1: Express...
• 08M.2.hl.TZ1.10: Find, in its simplest form, the argument of...
• 08M.2.hl.TZ1.14: \({z_1} = {(1 + {\text{i}}\sqrt 3 )^m}{\text{ and }}{z_2} = {(1 - {\text{i}})^n}\) . (a)    ...
• 08M.2.hl.TZ2.9: Consider...
• 11M.1.hl.TZ2.4a: Find AB, giving your answer in the form \(a\sqrt {b - \sqrt 3 } \) , where a ,...
• 11M.1.hl.TZ2.4b: Calculate \({\rm{A\hat OB}}\) in terms of \(\pi \).
• 11M.1.hl.TZ2.12b: Let \(\gamma = \frac{{1 + {\text{i}}\sqrt 3 }}{2}\). (i)     Show that \(\gamma \) is one of the...
• 09M.1.hl.TZ1.1: Consider the complex numbers \(z = 1 + 2{\text{i}}\) and \(w = 2 + a{\text{i}}\) , where...
• 09M.1.hl.TZ1.13Part A: If z is a non-zero complex number, we define \(L(z)\) by the...
• 09N.1.hl.TZ0.2: Find the values of n such that \({\left( {1 + \sqrt 3 {\text{i}}} \right)^n}\) is a real number.
• 09N.1.hl.TZ0.13a: Let \(z = x + {\text{i}}y\) be any non-zero complex number. (i)     Express \(\frac{1}{z}\) in...
• SPNone.2.hl.TZ0.4a: Find the modulus and argument of z , giving the argument in degrees.
• 13M.1.hl.TZ1.1a: If w = 2 + 2i , find the modulus and argument of w.
• 10M.2.hl.TZ1.4: (a)     Solve the equation \({z^3} = - 2 + 2{\text{i}}\), giving your answers in modulus-argument...
• 10M.2.hl.TZ2.9: Given that \(z = \cos \theta + {\text{i}}\sin \theta \) show that (a)    ...
• 10N.1.hl.TZ0.11: Consider the complex number \(\omega = \frac{{z + {\text{i}}}}{{z + 2}}\), where...
• 13M.1.hl.TZ2.7a: Write down the exact values of \(\left| {{z_1}} \right|\) and \(\arg ({z_2})\).
• 11N.2.hl.TZ0.6: The complex numbers \({z_1}\) and \({z_2}\) have arguments between 0 and \(\pi \) radians. Given...
• 11N.2.hl.TZ0.10: Given that...
• 11N.2.hl.TZ0.14d: Hence, show that...
• 11M.1.hl.TZ1.2: Given that \(\frac{z}{{z + 2}} = 2 - {\text{i}}\) , \(z \in \mathbb{C}\) , find z in the form...
• 11M.1.hl.TZ1.13a: Write down the expansion of \({\left( {\cos \theta + {\text{i}}\sin \theta } \right)^3}\) in the...
• 14M.1.hl.TZ1.13: A geometric sequence \(\left\{ {{u_n}} \right\}\), with complex terms, is defined by...
• 14M.1.hl.TZ2.7: Consider the complex numbers \(u = 2 + 3{\text{i}}\) and \(v = 3 + 2{\text{i}}\). (a)     Given...
• 13N.2.hl.TZ0.6: A complex number z is given by...
• 15M.2.hl.TZ1.12a: (i)     Use the binomial theorem to expand \({(\cos \theta  + {\text{i}}\sin \theta )^5}\). (ii)...
• 16N.1.hl.TZ0.12b: Show that \((\omega  - 3{\omega ^2})({\omega ^2} - 3\omega ) = 13\).
• 16N.1.hl.TZ0.12c: Find the values of \(x\) that satisfy the equation \(\left| p \right| = \left| q \right|\).
• 16N.1.hl.TZ0.12d: Solve the inequality...
• 18M.1.hl.TZ2.7a: Find the real part of \(\frac{{z + w}}{{z - w}}\).
• 18M.1.hl.TZ2.7b: Find the value of the real part of \(\frac{{z + w}}{{z - w}}\)...

Cartesian form \(z = a + {\text{i}}b\) .

• 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.TZ1.7: Given that z is the complex number \(x + {\text{i}}y\) and that...
• 12M.1.hl.TZ2.6a: m and n are real numbers;
• 12M.1.hl.TZ2.6b: m and n are conjugate complex numbers.
• 12M.1.hl.TZ2.12A.a: Given that \({(x + {\text{i}}y)^2} = - 5 + 12{\text{i}},{\text{ }}x,{\text{ }}y \in...
• 12M.1.hl.TZ2.12A.c: For any complex number z , show that \({(z^*)^2} = ({z^2})^*\) .
• 12N.1.hl.TZ0.10a: (i)     Write down \({z_1}\) in Cartesian form. (ii)     Hence determine...
• 11M.1.hl.TZ2.12b: Let \(\gamma = \frac{{1 + {\text{i}}\sqrt 3 }}{2}\). (i)     Show that \(\gamma \) is one of the...
• 10M.2.hl.TZ1.4: (a)     Solve the equation \({z^3} = - 2 + 2{\text{i}}\), giving your answers in modulus-argument...
• 11M.1.hl.TZ1.2: Given that \(\frac{z}{{z + 2}} = 2 - {\text{i}}\) , \(z \in \mathbb{C}\) , find z in the form...
• 11M.1.hl.TZ1.13a: Write down the expansion of \({\left( {\cos \theta + {\text{i}}\sin \theta } \right)^3}\) in the...
• 15N.1.hl.TZ0.11b: Consider the complex numbers \({z_1} = 1 + {\text{i}}\) and...

Sums, products and quotients of complex numbers.

• 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.TZ1.7: Given that z is the complex number \(x + {\text{i}}y\) and that...
• 12M.1.hl.TZ2.6a: m and n are real numbers;
• 12M.1.hl.TZ2.6b: m and n are conjugate complex numbers.
• 12M.1.hl.TZ2.12A.a: Given that \({(x + {\text{i}}y)^2} = - 5 + 12{\text{i}},{\text{ }}x,{\text{ }}y \in...
• 12M.1.hl.TZ2.12A.c: For any complex number z , show that \({(z^*)^2} = ({z^2})^*\) .
• 12N.1.hl.TZ0.10a: (i)     Write down \({z_1}\) in Cartesian form. (ii)     Hence determine...
• 08N.1.hl.TZ0.13Part B: (a)     Expand and simplify \((x - 1)({x^4} + {x^3} + {x^2} + x + 1)\) . (b)     Given that b is...
• 11M.1.hl.TZ2.12b: Let \(\gamma = \frac{{1 + {\text{i}}\sqrt 3 }}{2}\). (i)     Show that \(\gamma \) is one of the...
• 10M.1.hl.TZ2.13: Consider...
• 11M.1.hl.TZ1.2: Given that \(\frac{z}{{z + 2}} = 2 - {\text{i}}\) , \(z \in \mathbb{C}\) , find z in the form...
• 11M.1.hl.TZ1.13a: Write down the expansion of \({\left( {\cos \theta + {\text{i}}\sin \theta } \right)^3}\) in the...
• 15N.1.hl.TZ0.11b: Consider the complex numbers \({z_1} = 1 + {\text{i}}\) and...
• 16N.1.hl.TZ0.12b: Show that \((\omega  - 3{\omega ^2})({\omega ^2} - 3\omega ) = 13\).
• 16N.1.hl.TZ0.12c: Find the values of \(x\) that satisfy the equation \(\left| p \right| = \left| q \right|\).
• 16N.1.hl.TZ0.12d: Solve the inequality...
• 18M.1.hl.TZ2.7a: Find the real part of \(\frac{{z + w}}{{z - w}}\).
• 18M.1.hl.TZ2.7b: Find the value of the real part of \(\frac{{z + w}}{{z - w}}\)...