Complex Numbers Part 2 - Smarter Maths
Complex Numbers Part 2 - Smarter Maths
Complex Numbers Part 2 - Smarter Maths
Exponential Form
EXT 2: Complex Numbers (Ext2), N2 Using Complex Numbers (Ext2)
Solving Equations with Complex Numbers
Teacher: Sally Gorman
Exam Equivalent Time: 171 minutes (based on HSC allocation of 1.5 minutes approx. per mark)
Questions
1. Complex Numbers, EXT2 N2 2017 HSC 1 MC
The complex number is chosen so that form the vertices of the regular polygon
shown.
Which of the following could be the complex number ?
A.
B.
C.
D.
3. Complex Numbers, EXT2 N1 EQ-Bank 5 MC 7. Complex Numbers, EXT2 N1 EQ-Bank 3 MC
In which quadrant of the complex plane is the complex number found? Which of the following is the complex number ?
A. I A.
B. II
B.
C. III
C.
D. IV
D.
4. Complex Numbers, EXT2 N1 SM-Bank 1 MC
8. Complex Numbers, EXT2 N1 2020 HSC 9 MC
Which of the following is the complex number ?
What is the maximum value of for ?
A.
A.
B.
B.
C.
C.
D. D.
5. Complex Numbers, EXT2 N2 2020 HSC 2 MC 9. Complex Numbers, EXT2 N2 2014 SPEC2 7 MC
Given that is a root of , where and are real, what are the values The sum of the roots of , where , is
of and ?
A.
A.
B.
B.
C.
C.
D.
D.
D.
11. Complex Numbers, EXT2 N2 2016 HSC 10 MC 15. Complex Numbers, EXT2 N2 2009 HSC 2f
i. Find the square roots of (3 marks)
Suppose that
ii. Hence, or otherwise, solve the equation
(2 marks)
What is the value of
A.
16. Complex Numbers, EXT2 N2 2011 HSC 2c
B.
Find, in modulus-argument form, all solutions of (2 marks)
C.
(2 marks)
19. Complex Numbers, EXT2 N1 EQ-Bank 2
ii. Hence, or otherwise, find an expression for involving only powers of (1 mark)
Express the complex number in the exponential form. (2 marks)
. (3 marks)
21. Complex Numbers, EXT2 N2 2020 HSC 11e
14. Complex Numbers, EXT2 N1 SM-Bank 4 22. Complex Numbers, EXT2 N2 2011 HSC 2d
i. Find , given (2 marks) i. Use the binomial theorem to expand (1 mark)
ii. Hence, using the quadratic formula, solve ii. Use de Moivre’s theorem and your result from part (i) to prove that
(1 mark)
(3 marks)
(2 marks)
23. Complex Numbers, EXT2 N2 EQ-Bank 1 26. Complex Numbers, EXT2 N2 2006 HSC 2c
iii. Find the other 4 roots of the equation in exponential form. (3 marks)
i. Find all the 5th roots of in modulus-argument form. (2 marks)
Calculate
giving your answer in the form (3 marks)
Find the cube roots of . Express your answers in modulus-argument form. (3 marks)
25. Complex Numbers, EXT2 N2 2018 HSC 15b
. (3 marks)
34. Complex Numbers, EXT2 N2 2007 HSC 2d 36. Complex Numbers, EXT2 N2 2009 HSC 7b
Let
(3 marks)
The points and on the Argand diagram represent the complex numbers and
respectively.
where is a positive integer. (2 marks)
The triangles and are equilateral with unit sides, so
35. Complex Numbers, EXT2 N2 2014 HSC 15b ii. By substituting where , into part (i), show that
i. Using de Moivre’s theorem, or otherwise, show that for every positive integer ,
. (2 marks)
(3 marks)
ii. Hence, or otherwise, show that for every positive integer divisible by 4,
(2 marks)
38. Complex Numbers, EXT2 N2 2019 HSC 16b Worked Solutions
Let , where and are real numbers.
1. Complex Numbers, EXT2 N2 2017 HSC 1 MC
The roots of are .
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( 3, 3)
ii.
ii.
iii.
14. Complex Numbers, EXT2 N1 SM-Bank 4 15. Complex Numbers, EXT2 N2 2009 HSC 2f
i. i.
ii.
ii.
COMMENT: Since ,
the general formula only produces
2 distinct solutions.
16. Complex Numbers, EXT2 N2 2011 HSC 2c 17. Complex Numbers, EXT2 N1 2013 HSC 11c
i.
ii.
iii.
23. Complex Numbers, EXT2 N2 EQ-Bank 1 24. Complex Numbers, EXT2 N1 2020 HSC 14a
i. i.
ii.
iii.
ii.
ii.
26. Complex Numbers, EXT2 N2 2006 HSC 2c 27. Complex Numbers, EXT2 N2 2009 HSC 2e
i.
ii.
28. Complex Numbers, EXT2 N1 EQ-Bank 6
i.
iii.
35. Complex Numbers, EXT2 N2 2014 HSC 15b
i.
ii.
i.
ii.
37. Complex Numbers, EXT2 N2 2007 HSC 8b
iii.
i.
ii.
iii.
38. Complex Numbers, EXT2 N2 2019 HSC 16b
i.
ii.