Question bank: Complex Numbers: Identities and Equations

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Question 1:

Medium

In the study of complex numbers, the imaginary unit i plays a significant role. It is defined as the square root of -1 and has interesting properties when raised to different powers. Based on this knowledge, calculate the value of i raised to 2019. To do this, it is important to remember that the powers of i follow a repetitive cycle.
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Question 2:

Medium

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Question 3:

Easy

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Question 4:

Medium

During an experiment in the physics laboratory, the third-year high school students were challenged to analyze the movements of a pendulum. To calculate the spatial variation accurately, the students decided to use complex numbers. One of the students wrote a point in the complex plane as A = a + bi, where a = 2 and b = 2√3. Calculate the distance OA (from point A to the origin) and write the complex number A in trigonometric form (module and argument).
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Question 5:

Medium

One of the practical applications of complex numbers is the study of electrical circuits. Suppose you need to analyze an electrical circuit whose voltage is represented by a complex number in trigonometric form, with a magnitude of 2 and an angle of 30° with respect to the real axis. Consider this voltage as V1. To continue the study of the circuit, you need to determine the value of the square of V1. What is the expression of this complex number in trigonometric form?
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