Complex Numbers St Andrew's JC 2019 P1 Q6 | A Level H2 Mathematics

Hi everyone! Welcome back to another practice question on complex numbers, this is taken from the SAJC 2019 Prelims Paper 1, Question #6.
If you’re all ready let’s jump right in!

Timestamps
00:12 Converting a complex number to exponential form
01:27 Finding smallest values of n which expression is purely imaginary
04:30 Simultaneous equations involving complex numbers
06:47 How to compare Real and Imaginary parts of complex number

The question wants us to express p in exponential form. Just a quick recap, you’ll need TWO ingredients to express a complex number in exponential form. Firstly, find the modulus. The next ingredient is the argument. Now if you’re unsure of getting the argument, you can represent it on an Argand diagram, where p is in the 3rd quadrant. The basic angle is tan inverse rt 3 over 1, giving us pi over 3. Now always take the argument from the positive x axis, so the argument is -2pi over 3. Take note that the argument cannot be more than pi, and that’s why we took a clockwise direction from the x axis to give us -2pi over 3. So with the modulus and argument, we can express p in exponential form.

In part ii, we want to find the two smallest positive whole number values of n for which this expression is a PURELY imaginary number. In such a scenario, let’s try to simplify the expression as much as we can, so that we can make use of its argument to illustrate it as a purely imaginary number.

For this expression to be purely imaginary, it means that the complex number lies only on the y-axis, which is pi over 2. But this also means that the possible arguments of this expression could be pi over 2, 3pi over 2, 5 pi over 2, so on and so forth. So what we can do is to illustrate this relationship by equating the argument to (2k-1) multiplied with pi over 2, where k is an integer.

In part (b), use the substitution method to solve the simultaneous equation; and by comparing the real and imaginary parts of a complex number, we can then find z and w in the cartesian form x+yi.

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