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STPM 2018 Term 1 Mathematics (T) Coursework PBS Assignment

STPM Coursework Sample Solution

Question

A complex number is an extension of a real number and it can be represented in Cartesian and polar
forms. In this assignment, you are required to explore the powers and roots of complex numbers.

1

(a) Let z =1+i. Find z^n, where n = 2, 3, 4, …, and represent the complex numbers on an Argand diagram.
(b) Rewrite z=1+i in polar form and repeat task (a).
(c) Comment on the methods used in (a) and (b).

2 Let z=r(cos pi/5+isin pi/5). Find z^n, where n = 2, 3, 4, …, with a value of r in each of the following cases:
(a) 0 (b) r>1.
Represent the complex numbers on an Argand diagram and comment on your results.

3. Find the n complex numbers which satisfy the equation z^n=a+bi, for n>3 and represent them on an Argand diagram in each of the following cases:
(a) a<>0, b=0,
(b) a=0, b<>0,
(c) a<>0, b<>0.
State the conjugate pairs of the roots, if any.

Solution

Please check with your teacher for all the requirements. All sample below are for reference only.

Question 1 (c)

Both methods (a) and (b) are suitable for small value of n when finding z^n. For n is large, polar form will be more suitable to explore the power of complex numbers. Actually polar form is more suitable to find powers of complex number because of the de Moivre’s theorem. We can express z^n easily by using de Moivre’s theorem.

Question 2 (a)

Modulus of z^n decrease as n increase. Each higher power is pi/5 rad further along and closer to 0. Five powers are displayed. It forms a spiral and this spiral is called an exponential spiral.

Question 2 (b)

Modulus of z^n increases as n increase. Each higher power is pi/5 rad further along and further to 0. Five powers are displayed. It forms a spiral and this spiral is called an exponential spiral.

Question 3 (a)

You can try to modify the value of b. Example, use b<0, or any other numbers.

Extra sample of Argand digram with n=6 and n=7

Question 3 (b)

You can try to modify the value of a. Example, use a<0, or any other numbers.

Extra sample of Argand digram with n=6 and n=7

Question 3 (c)

You can try to modify the values of a and b. Example, use a<0, b>0 or any other numbers.

Extra sample of Argand digram with n=6 and n=7 KK LEE

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1. Nice work! Thank you!

• You are welcome

2. How to plot the argand diagram using computer? Can you please teach me??

• Geogebra

3. thank you so much

• You are welcome

4. how to draw the Argand diagram for Q3C in details?

• You are welcome

5. it helps me a lot ! thanks!

• You are welcome

6. Sir, I am curious about what program did you used to plot the graph? May you share it with us? Thank in advance Sir.

• Geogebra

7. may i know roughly how to draw the graphs for question 3?

• Geogebra

8. Um sir. May I know which app or software you use to plot those Argand diagrams?

• Geogebra

9. Thanks

• You are welcome

10. Sir,do the points in 2 a and 2b need to connect together?

• It depends on your teacher.

• Sir, can I ask the methodology and results is seperated?

11. thanks! definitely a great reference

• You are welcome.

12. Sir,how to do methodology??
Tq

• Please check with your school teacher for methodology. This page shows the result only.

13. How to plot the graph in 1B question?

• Geogebra

14. Thank you soooo much!
This is the only website that I can refer from and it is very useful 🙂
I’ll search for this website again for the next coming sem math t assignment 🙂
Really appreciate your effort to upload the answer T T

15. Hi ,can I know how to use geogebra

• 16. sorry sir can i know about how to plot question 1 in spiral shape?

• You can use geogebra or use paper. It depends on your school teacher

17. sorry sir can i ask about how to plot question 1 in spiral shape

18. How to write introduction and methodology?

• Please check with your school teacher for methodology and intro. This page shows the result only.

19. great guidelines for the candidates ! thanks!

• You are welcome

20. sir lee
may i ask how to put methodology in our assignment?

• Please check with your school teacher for methodology. This page shows the result only.

21. Sir, can I ask the methodology and results is seperated?

• Yes

22. Sir,may I know how to write a methodology?

• Please check with your school teacher for methodology. This page shows the result only.

23. how to write methodology?

• Please check with your school teacher for methodology. This page shows the result only.

24. z^n=(-1)^k(2)^2k n=4k
(-1)^k(2)^2k(1+i) n=4k+1

please guide me about where does the solution
come from?
(-1)^k(2)^2k?

• anyway im from penang

• From observation.

It is like 2=x, if you call x=2, or 2=2x, if you call x=1. It depends on how you define it. you can write any expression if your teacher wants it.

25. thank you very much!!

26. Sir, can you teach how to do the viva?

27. Sir graph for ques 1a can be in spiral shape?

• Yes

• Yes. You can

28. Sir how to do 1)b) repeat the task?

• Redo the power 2, 3, 4, 5

29. Sir can i know how to connect those points become an exponential spiral?

• Using Computer? Or manual?

30. May i know is the 3c’s argond diagram same as the 3b? *hand drawing

• The roots are different

31. Sir how to plot the graph and draw the diagrams?

• You can use geogebra to draw it. Download it and install in your computer or phone

32. Semester 2 coursework answer when published???

• It will be published if they are available

• When can i get semester 3 coursework results??

33. Hi sir, can I know what is the conjugate pair for 3(a) when n=6

• Conjugate for x+yi is x-yi. If you have the first complex root is 2+3i, then the second root will be 2-3i

34. math is too difficult to me even I do more exercise , how I should do to master every chapter

• Please find a good teacher to guide you

35. When can i get semester 3 coursework results??

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