MATH135 Lecture Notes - Lecture 9: Parallelogram
14 May 2018
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MATH 135 Fall 2015: Extra Practice Set 9
These problems are for extra practice and are not to be handed. Solutions will not be posted but, unlike
assignment problems, they may discussed in depth on Piazza.
•The warm-up exercises are intended to be fairly quick and easy to solve. If you are unsure about any
of them, then you should review your notes and possibly speak to an instructor before beginning the
corresponding assignment.
•The recommended problems supplement the practice gained by doing the corresponding assignment.
Some should be done as the material is learned and the rest can be left for exam preparation.
•A few more challenging extra problems are also included for students wishing to push themselves
even harder. Do not worry if you cannot solve these more difficult problems.
Warm-up Exercises
1. Write z=9 + i
5−4iin the form r(cos θ+isin θ) with r≥0 and 0 ≤θ < 2π.
2. Write (√3 + i)4in standard form.
Recommended Problems
1. Find all z∈Cwhich satisfy
(a) z2+ 2z+ 1 = 0,
(b) z2+ 2z+ 1 = 0,
(c) z2=1 + i
1−i.
2. (a) Find all w∈Csatisfying w2=−15 + 8i,
(b) Find all z∈Csatisfying z2−(3 + 2i)z+5+i= 0.
3. Let z, w ∈C. Prove that if zw = 0 then z= 0 or w= 0.
4. Let a, b, c ∈C. Prove: if |a|=|b|=|c|= 1, then a+b+c=1
a+1
b+1
c.
5. Find all z∈Csatisfying z2=|z|2.
6. Find all z∈Csatisfying |z+ 1|2≤3 and shade the corresponding region in the complex plane.
7. Prove that if |z|= 1 or |w|= 1 and zw 6= 1, then
z−w
1−zw
= 1.
8. Show that |Re(z)|+|Im(z)| ≤ √2|z|.
9. Prove that ∀z, w ∈C,|z−w|2+|z+w|2= 2(|z|2+|w|2) (This is the Parallelogram Identity).
10. Use De Moivre’s Theorem (DMT) to prove that sin(4θ) = 4 sin θcos3θ−4 sin3θcos θ.
11. Let n∈Nand a, b ∈R. Show that z= (a+bi)n+ (a−bi)nis real.
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Document Summary
Math 135 fall 2015: extra practice set 9. These problems are for extra practice and are not to be handed. Solutions will not be posted but, unlike assignment problems, they may discussed in depth on piazza: the warm-up exercises are intended to be fairly quick and easy to solve. If you are unsure about any of them, then you should review your notes and possibly speak to an instructor before beginning the corresponding assignment: the recommended problems supplement the practice gained by doing the corresponding assignment. Some should be done as the material is learned and the rest can be left for exam preparation: a few more challenging extra problems are also included for students wishing to push themselves even harder. Do not worry if you cannot solve these more di cult problems. Warm-up exercises: write z , write ( 3 + i)4 in standard form.
