IB Maths AI HL Ahl 1 12 Hl Complex Numbers Questions

Practise calculating with Cartesian complex numbers, finding complex roots, using modulus and principal argument, and interpreting Argand diagrams or real and imaginary parts.

Exam points

Calculate with Cartesian complex numbers using i²=-1, including sums, products, quotients and powers.
Find complex roots of real-coefficient quadratics with negative discriminant and express them correctly.
Use modulus, principal argument and Argand diagrams to represent or interpret a complex number geometrically.
Separate and compare real and imaginary parts when proving or solving a complex-number relation.

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Study AHL 1.12 (HL)—Complex numbers conceptsOpen the corresponding Concept lesson for explanations, examples and exam connections.

[Maximum number: 4]

Nanthana is completing an exploration on multiplication using complex numbers. She considers a geometric sequence where $u_{1}=9$ and $r=\frac{2}{3}+\frac{2}{3} \mathrm{i}$ .

Show that Nanthana's claim is correct, stating the exact value of the common ratio for this sequence.

attempt to find |r|

\begin{aligned} & |r|=\sqrt{\left(\frac{2}{3}\right)^{2}+\left(\frac{2}{3}\right)^{2}} \\ & |r|=\frac{2 \sqrt{2}}{3} \end{aligned}

statement that multiplying a number by a complex number changes the modulus of the number by a factor of |r| (may be seen at any stage)

Note: Award at most (M1)A1A0R1 if common ratio is given as $0.942809 \ldots \ldots$ .
therefore it is a geometric sequence

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IB Maths AI HL Ahl 1 12 Hl Complex Numbers Questions question 1