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 (a)

Question 1 (b)

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
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67 Comments

  1. Shirley

    Nice work! Thank you!

    Reply
    • KK LEE

      You are welcome

  2. David

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

    Reply
    • KK LEE

      Geogebra

  3. genevieve

    thank you so much

    Reply
    • KK LEE

      You are welcome

  4. LEE

    how to draw the Argand diagram for Q3C in details?

    Reply
    • KK LEE

      You are welcome

  5. Erica

    it helps me a lot ! thanks!

    Reply
    • KK LEE

      You are welcome

  6. Maths lover

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

    Reply
    • KK LEE

      Geogebra

  7. hugo

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

    Reply
    • KK LEE

      Geogebra

  8. Jason

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

    Reply
    • KK LEE

      Geogebra

  9. yee

    Thanks

    Reply
    • KK LEE

      You are welcome

  10. yamid

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

    Reply
    • KK LEE

      It depends on your teacher.

    • KANG CHUANG HSIEN

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

  11. Nicole

    thanks! definitely a great reference

    Reply
    • KK LEE

      You are welcome.

  12. TM

    Sir,how to do methodology??
    Tq

    Reply
    • KK LEE

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

  13. Freddy

    How to plot the graph in 1B question?

    Reply
    • KK LEE

      Geogebra

  14. CJoanne

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

    Reply
  15. low

    Hi ,can I know how to use geogebra

    Reply
  16. chenli

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

    Reply
    • KK LEE

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

  17. feicheng

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

    Reply
  18. Renuga

    How to write introduction and methodology?

    Reply
    • KK LEE

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

  19. ericalow

    great guidelines for the candidates ! thanks!

    Reply
    • KK LEE

      You are welcome

  20. choo

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

    Reply
    • KK LEE

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

  21. KANG CHUANG HSIEN

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

    Reply
    • KK LEE

      Yes

  22. Megan

    Sir,may I know how to write a methodology?

    Reply
    • KK LEE

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

  23. ong

    how to write methodology?

    Reply
    • KK LEE

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

  24. cheang

    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?

    Reply
    • cheang

      anyway im from penang

    • KK LEE

      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. KJ

    thank you very much!!

    Reply
  26. Naqe'

    Sir, can you teach how to do the viva?

    Reply
  27. Liew Zheng Hao

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

    Reply
    • KK LEE

      Yes

    • KK LEE

      Yes. You can

  28. priya

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

    Reply
    • KK LEE

      Redo the power 2, 3, 4, 5

  29. Soo Seok Huey

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

    Reply
    • KK LEE

      Using Computer? Or manual?

  30. Yew

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

    Reply
    • KK LEE

      The roots are different

  31. Yugesha

    Sir how to plot the graph and draw the diagrams?

    Reply
    • KK LEE

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

  32. cash

    Semester 2 coursework answer when published???

    Reply
    • KK LEE

      It will be published if they are available

  33. Lim

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

    Reply
    • KK LEE

      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. Michael Hii

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

    Reply
    • KK LEE

      Please find a good teacher to guide you

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